af: * arithmetic function af.bernoulli_numbers: * Bernoulli numbers af.divisor_function: * divisor function af.multiplicative: * multiplicative ag.abelian_surface: * abelian surface ag.abelian_variety: * abelian variety ag.affine_space: * Affine space ag.arithmetic_genus: * arithmetic genus ag.aut_group: * automorphism * automorphism group ag.bad_prime: * prime of bad reduction ag.base_change: * base change ag.base_field: * base field ag.bielliptic_curve: * bielliptic curve * bielliptic involution ag.canonical_height: * canonical height * height pairing ag.canonical_model: * canonical map * canonical model ag.cluster_picture: * cluster picture * depth * relative depth * size ag.cohen_macaulay_type: * Cohen-Macaulay ag.complex_multiplication: * complex multiplication ag.conductor: * conductor ag.curve: * algebraic curve ag.curve.genus: * algebraic genus * arithmetic genus * genus * geometric genus ag.curve.smooth: * smooth ag.cyclic_trigonal: * cyclic trigonal curve * trigonal automorphism ag.dimension: * dimension ag.endomorphism_algebra: * endomorphism algebra ag.endomorphism_ring: * endomorphism * endomorphism ring ag.geom_component: * geometric component ag.geom_endomorphism_algebra: * geometric endomorphism algebra ag.geom_endomorphism_ring: * geometric endomorphism ring ag.geom_simple: * geometrically (or absolutely) simple ag.gonality: * $k$-gonality * gonality ag.gonality_geom: * $\overline{k}$-gonality * geometric gonality ag.good_reduction: * bad reduction * good reduction ag.holomorphic_euler_char: * holomorphic Euler characteristic ag.hyperelliptic_curve: * Weierstrass model * conic * hyperelliptic curve ag.irreducible: * geometrically irreducible * irreducible ag.jacobian: * Jacobian ag.minimal_field: * minimal field of definition ag.mordell_weil: * Mordell-Weil group * Mordell-Weil rank * torsion subgroup ag.plane_model: * plane model * singular * smooth ag.potential_good_reduction: * potential good reduction ag.primitive: * primitive ag.projective_space: * Projective space * homogeneous coordinates * projectivization ag.quotient_curve: * quotient curve ag.real_endomorphism_algebra: * real endomorphism algebra ag.real_geom_endomorphism_algebra: * real geometric endomorphism algebra ag.real_multiplication: * real multiplication ag.regulator: * height pairing * regulator ag.riemann_surface: * Riemann surface ag.selmer_group: * $n$-Selmer group ag.shimura_variety: * (connected) Shimura varieties ag.simple: * simple ag.singular_point: * non-singular ag.splitting_field: * splitting field ag.tate_shafarevich: * Tate-Shafarevich group ag.torsor: * principal homogeneous space * torsor ag.variety: * affine algebraic set * affine variety * algebraic varieties * defining ideal * projective algebraic set * projective variety alg.bezout_matrix: * degree $n$ Bézout matrix alg.binary_operation: * binary operation alg.binary_operation.associative: * associative alg.binary_operation.commutative: * commutative alg.binary_operation.identity: * identity * the identity element alg.binary_operation.inverse: * inverse alg.symmetric_bilinear_form: * symmetric bilinear form alg.symplectic_isomorphism: * symplectic isomorphism artin: * Artin representation artin.conductor: * conductor artin.determinant: * determinant artin.dimension: * dimension artin.frobenius_schur_indicator: * Frobenius-Schur indicator artin.galois_orbit: * Galois orbit artin.gg_quotient: * image of the Artin representation artin.number_field: * artin field artin.parity: * even * odd artin.permutation_container: * contains artin.projective_field: * projective Artin field artin.projective_image: * projective image artin.projective_stem_field: * projective stem field artin.ramified_primes: * ramified artin.root_number: * root number * sign artin.stem_field: * Artin stem field artin.unramified_primes: * unramified av.aut_group: * automorphism * automorphism group av.decomposition: * isogeny factors av.deligne_module: * Deligne module av.endomorphism_field: * endomorphism field av.fq.all_polarizations_product: * decomposable * indecomposable av.fq.angle_rank: * angle corank * angle rank av.fq.bass_zfv: * Bass av.fq.conductor: * (relative) conductor * conductor av.fq.conjugate_stable: * CM-involution * conjugate stable av.fq.cyclic_group_points: * cyclic av.fq.endomorphism_ring_notation: * $\#\mathrm{Pic}$ * $\#\mathrm{weak} = 1 $ * $\#\mathrm{weak}$ * $\mathrm{index}$ * $i$ * notation av.fq.frobenius_angles: * Frobenius angles av.fq.group_structure: * rational $k$-points av.fq.index_of_order: * index av.fq.initial_coefficients: * do not include av.fq.l-polynomial: * L-polynomial av.fq.lmfdb_label: * $N :=[\mathcal{O}_{\mathbb{Q}[F]}:R]$ * $d$ * $g$ * $i$ * $j$ * $k$ * $q$ * $w$ * g.q.isog * g.q.isog.N.i * g.q.isog.N.i.w * g.q.isog.N.i.w.j * g.q.isog.N.i.w.j.d.k * isog * label av.fq.maximal_zfv: * maximal av.fq.newton_elevation: * almost ordinary * elevation * eligible av.fq.noncyclic_primes: * non-cyclic at $\ell$ av.fq.number_field: * number field av.fq.order: * maximal order * order * overorder * étale algebra av.fq.ordinary: * ordinary av.fq.p_rank: * $p$-corank * $p$-rank av.fq.picard_group: * Picard group av.fq.singular_dimensions: * invertible * locally principal av.fq.singular_primes: * singular primes * singular support av.fq.supersingular: * supersingular av.fq.weak_equivalence_class: * weak equivalence class av.fq.weil_polynomial: * Weil polynomial * characteristic polynomial av.fq.zfv_index: * index av.fq.zfv_real_index: * real index av.fq.zfv_rel_disc_norm: * relative discriminant av.galois_rep: * $\ell$-adic Galois representation * adelic Galois representation * mod-$\ell$ Galois representation * mod-$m$ Galois representation av.geometrically_simple: * geometrically simple av.geometrically_squarefree: * geometrically squarefree av.is_product: * product abelian variety av.isogeny: * isogenous * isogeny av.isogeny_class: * isogeny class av.polarization: * degree * kernel * polarization av.potential_toric_rank: * potential toric rank av.princ_polarizable: * principally polarizable av.semiabelian_variety: * semiabelian variety av.simple: * simple av.squarefree: * squarefree av.tate_module: * $p$-adic Tate module av.theta_divisor: * symmetric * theta divisor av.twist: * twist av.weak_equivalence_class: * weakly equivalent belyi.abc: * $(a,b,c)$ triple belyi.base_field: * base field belyi.degree: * degree belyi.galmap: * belyi map belyi.genus: * genus belyi.geometry_type: * geometry type belyi.group: * monodromy group belyi.label: * label belyi.num_orbits: * number of Galois orbits belyi.orbit_size: * orbit size belyi.passport: * passport belyi.permutation_triple: * permutation triple * transitive belyi.ramification_type: * ramification type character.dirichlet: * Dirichlet character * modulus character.dirichlet.conductor: * conductor character.dirichlet.conrey.index: * conrey index * index character.dirichlet.conrey.primitive_generator: * conrey (primitive) generator character.dirichlet.degree: * degree character.dirichlet.field_cut_out: * fixed field character.dirichlet.galois_orbit: * galois orbit character.dirichlet.galois_orbit_index: * orbit index character.dirichlet.galois_orbit_label: * label character.dirichlet.gauss_sum: * gauss sum character.dirichlet.group.generators: * generators character.dirichlet.group.order: * order character.dirichlet.group.structure: * structure character.dirichlet.induce: * induced character.dirichlet.jacobi_sum: * jacobi sum character.dirichlet.jacobi_symbol: * jacobi symbol * law of quadratic reciprocity states character.dirichlet.kloosterman_sum: * kloosterman sum character.dirichlet.kronecker_symbol: * Kronecker symbol character.dirichlet.legendre_symbol: * legendre symbol character.dirichlet.minimal: * minimal character.dirichlet.modulus: * dirichlet character * modulus character.dirichlet.order: * order character.dirichlet.parity: * parity character.dirichlet.primitive: * primitive character.dirichlet.principal: * principal * trivial character.dirichlet.real: * real character.dirichlet.value_field: * field of values character.hecke: * Hecke character character.unit_group: * underlying unit group cmf: * modular form cmf.analytic_conductor: * analytic conductor cmf.analytic_rank: * analytic rank cmf.artin_field: * artin field cmf.artin_image: * Artin image cmf.atkin-lehner: * Atkin-Lehner involution cmf.bad_prime: * bad prime * good prime cmf.character: * character cmf.cm_form: * complex multiplication cmf.coefficient_field: * coefficient field * hecke field cmf.coefficient_ring: * Hecke ring * coefficient ring cmf.congruence_subgroup: * congruence subgroup * level * principal congruence subgroup cmf.cusp_form: * cusp form cmf.cusp_form_mathlib_def: * CuspForm $\Gamma$ $k$ cmf.cusp_form_subspace_mathlib_def: * cuspFormSubmodule $\Gamma$ $k$ cmf.decomposition.new.gamma0chi: * {{ knowl(mf.elliptic.newform, title=newform) }} cmf.defining_polynomial: * defining polynomial cmf.dimension: * dimension * relative dimension cmf.dimension_mathlib_def: * module rank cmf.distinguishing_primes: * distinguishes cmf.dualform: * dual cmf.eisenstein: * Eisenstein series * isotropy group cmf.eisenstein_form: * Eisenstein form * Eisenstein subspace cmf.eisenstein_label: * label cmf.eisenstein_newform: * Eisenstein newform cmf.eisenstein_newspace: * new subspace * old subspace cmf.eisenstein_series: * Eisenstein series cmf.embedding: * embedding cmf.embedding_label: * label cmf.eta_quotient: * eta product * eta quotient cmf.fouriercoefficients: * Fourier coefficients * Fourier expansion cmf.fouriercoefficients_mathlib_def: * $n$-th coefficient cmf.fricke: * Fricke involution cmf.galois_conjugate: * galois conjugate cmf.galois_orbit: * Galois orbit * newform orbits cmf.galois_representation: * galois representation * projective galois representation cmf.hecke_operator: * Hecke operator cmf.hecke_orbit: * Hecke orbit cmf.hecke_ring_generators: * coefficient ring generator bound cmf.heckecharpolys: * Hecke characteristic polynomial cmf.inner_twist: * inner twists cmf.inner_twist_count: * inner twist count cmf.inner_twist_multiplicity: * multiplicity cmf.label: * label cmf.labels: * label cmf.level: * level cmf.maximal: * largest * maximal cmf.minimal: * minimal cmf.minimal_twist: * minimal twist cmf.minus_space: * minus subspace cmf.newform: * newform cmf.newform_subspace: * newform subspace cmf.newspace: * new subspace * old subspace cmf.nontrivial_twist: * nontrivial cmf.oldspace: * old cmf.petersson_scalar_product: * Petersson scalar product cmf.petersson_scalar_product_mathlib_def: * petersson cmf.plus_space: * plus subspace cmf.projective_field: * projective field cmf.projective_image: * projective image * types cmf.q-expansion: * $q$-expansion cmf.q-expansion_mathlib_def: * qExpansion cmf.relative_dimension: * relative dimension cmf.rm_form: * real multiplication cmf.satake_angles: * Satake angles cmf.satake_parameters: * Satake parameters cmf.sato_tate: * sato-tate group cmf.self_twist: * self-twist cmf.selfdual: * self dual cmf.shimura_correspondence: * shimura correspondence cmf.space: * space cmf.space_mathlib_def: * ModularForm $\Gamma$ $k$ cmf.space_trace_form: * trace form cmf.stark_unit: * Stark unit cmf.sturm_bound: * Sturm bound cmf.sturm_bound_gamma1: * S cmf.subspaces: * cuspidal subspace * new subspace * old subspace cmf.trace_bound: * trace bound cmf.trace_form: * trace form cmf.twist: * twist * twist class * twist equivalent cmf.twist_minimal: * minimal twist * twist minimal cmf.twist_multiplicity: * multiplicity cmf.weight: * weight combin.hasse_diagram: * Hasse diagram content.announcements: * Beta section * Major revision * New section content.ongoing: * Warning crystals.crystal: * ($u_q(\mathfrak{g})$-)crystal curve.highergenus.aut.braid_equivalence: * braid equivalent * braid operations curve.highergenus.aut.branchpoints: * branch points curve.highergenus.aut.dimension: * dimension curve.highergenus.aut.full: * full group of automorphisms curve.highergenus.aut.generatingvector: * generating vector curve.highergenus.aut.generators: * standard generators curve.highergenus.aut.group_action: * group $G$ acts on a compact Riemann surface $X$ curve.highergenus.aut.groupalgebradecomp: * group algebra decomposition curve.highergenus.aut.monodromy: * (geometric) monodromy group curve.highergenus.aut.quotientgenus: * quotient genus curve.highergenus.aut.refinedpassport: * refined passport curve.highergenus.aut.signature: * signature curve.highergenus.aut.topological_equivalence: * topologically equivalent doc.related_to: * related objects dq.curve.highergenus.aut.source: * corresponding character(s) * dimension of the corresponding {{knowl(ag.shimura_variety, title=shimura variety)}} * equations * hyperelliptic or cyclic trigonal curves * {{knowl(ag.jacobian, title=jacobian variety)}} decomposition dq.mf.hilbert.source: * M ec: * elliptic curve ec.additive_reduction: * additive reduction ec.analytic_sha_order: * Tate-Shafarevich group * analytic order of Sha ec.bad_reduction: * bad reduction ec.base_change: * base change ec.bsdconjecture: * BSD * BSD invariants * Birch and Swinnerton-Dyer ec.canonical_height: * canonical height * height pairing ec.complex_multiplication: * CM discriminant * complex multiplication * potential complex multiplication ec.conductor: * conductor * conductor norm ec.discriminant: * discriminant ec.endomorphism: * endomorphism ec.endomorphism_ring: * endomorphism ring ec.galois_rep: * $\ell$-adic Galois representation * adelic Galois representation * mod-$\ell$ Galois representation * mod-$m$ Galois representation ec.galois_rep_adelic_image: * genus * index * level ec.galois_rep_modell_image: * A4 * A5 * B * Cn * Cs * G * Nn * Ns * S4 ec.geom_endomorphism_ring: * geometric endomorphism ring ec.global_minimal_model: * global minimal model ec.good_ordinary_reduction: * ordinary * ordinary reduction ec.good_reduction: * everywhere good reduction * good reduction ec.good_supersingular_reduction: * supersingular * supersingular reduction ec.integral_model: * integral model ec.intrinsic_torsion: * intrinsic subgroup ec.invariants: * conductor norm * minimal discriminant norm ec.isogeny: * cyclic * degree * endomorphisms * isogenous * isogeny * isogeny classes ec.isogeny_class: * isogeny class (over a field $K$) ec.isogeny_class_degree: * isogeny class degree ec.isogeny_graph: * isogeny graph ec.isogeny_matrix: * isogeny matrix ec.isomorphism: * isomorphism * isomorphism classes ec.j_invariant: * $j$-invariant ec.kodaira_symbol: * kodaira symbol ec.local_data: * local data ec.local_minimal_discriminant: * local minimal discriminant ec.local_minimal_model: * local minimal model ec.maximal_elladic_galois_rep: * maximal * maximal prime ec.maximal_galois_rep: * maximal ec.minimal_discriminant: * minimal discriminant ec.mordell_weil_group: * Mordell-Weil group * Mordell-Weil rank ec.mordell_weil_theorem: * mordell-weil theorem ec.multiplicative_reduction: * multiplicative reduction ec.mw_generators: * Mordell-Weil generators * non-torsion generators * torsion generators * {{KNOWL('ec.mordell_weil_group', 'Mordell-Weil group')}} ec.nonsplit_multiplicative_reduction: * non-split multiplicative reduction ec.obstruction_class: * obstruction class ec.padic_tate_module: * $p$-adic Tate module ec.period: * global period ec.potential_good_reduction: * potential good reduction ec.q: * elliptic curve $E$ over $\mathbb{Q}$ ec.q.abc_quality: * $abc$ quality * quality ec.q.analytic_rank: * analytic rank ec.q.analytic_sha_order: * Analytic Order of Sha * Tate-Shafarevic group ec.q.bsdconjecture: * Birch and Swinnerton-Dyer ec.q.canonical_height: * canonical height * height pairing ec.q.conductor: * conductor ec.q.cremona_label: * Cremona label ec.q.discriminant: * discriminant ec.q.endomorphism_ring: * endomorphism ring ec.q.faltings_height: * Faltings height * stable Faltings height ec.q.faltings_ratio: * Faltings ratio ec.q.frey: * Frey curve * Frey-Hellegouarch curve ec.q.integral_points: * integral points ec.q.j_invariant: * $j$-invariant ec.q.kodaira_symbol: * kodaira symbol ec.q.lmfdb_label: * LMFDB label * conductor * isogeny class index * isogeny class label * isomorphism class index ec.q.manin_constant: * 154 * Manin constant ec.q.minimal_twist: * minimal quadratic twist * minimal quartic twist * minimal sextic twist * minimal twist ec.q.minimal_weierstrass_equation: * minimal weierstrass equation * reduced minimal model ec.q.modular_degree: * modular degree ec.q.modular_parametrization: * modular parametrization ec.q.naive_height: * naive height ec.q.optimal: * optimal ec.q.period_lattice: * Néron lattice * period lattice * real period ec.q.real_period: * homothetic * real period * the ec.q.reduction_type: * additive reduction * bad reduction * cuspidal * good reduction * multiplicative reduction * nodal * non-split * ordinary * reduction type * split * supersingular ec.q.regulator: * height pairing * regulator ec.q.semistable: * semistable ec.q.serre_invariants: * Serre conductor * Serre invariants * Serre weight * exceptional ec.q.special_value: * special value ec.q.szpiro_ratio: * Szpiro ratio ec.q.torsion_growth: * primitive * torsion growth ec.q.torsion_subgroup: * torsion subgroup ec.q_curve: * -curve ec.rank: * rank ec.reduction: * {{KNOWL('ec.potential_good_reduction','potential good reduction')}} * {{KNOWL('ec.semistable','semistable')}} ec.reduction_type: * additive reduction * multiplicative reduction * non-split multiplicative reduction * ordinary * reduction type * split multiplicative reduction * supersingular ec.regulator: * height pairing * regulator ec.ring: * elliptic curve ec.scheme: * elliptic scheme ec.semi_global_minimal_model: * semi-global minimal models ec.semistable: * semistable ec.simple_equation: * short Weierstrass model * simplified equation ec.special_value: * leading coefficient * special value ec.split_multiplicative_reduction: * split multiplicative reduction ec.tamagawa_number: * tamagawa number * tamagawa product ec.torsion_order: * torsion order ec.torsion_subgroup: * torsion structure * torsion subgroup ec.twists: * quadratic twists * quartic twists * sextic twists * twist ec.weierstrass_coeffs: * Weierstrass coefficients * Weierstrass equation * Weierstrass model ec.weierstrass_isomorphism: * Weierstrass isomorphisms * isomorphic over $K$ field.finite.conway_polynomial: * Conway polynomials g2c.abs_discriminant: * absolute discriminant g2c.all_rational_points: * rational points g2c.analytic_rank: * analytic rank g2c.analytic_sha: * analytic order of Sha g2c.aut_grp: * automorphism * automorphism group * hyperelliptic involution g2c.bsd_invariants: * BSD formula * BSD invariants g2c.conductor: * conductor g2c.decomposition: * decompose * split g2c.discriminant: * discriminant g2c.end_alg: * complex multiplication * endomorphism algebra * real multiplication g2c.g2_invariants: * G2-invariants g2c.g2curve: * Weierstrass equation g2c.galois_rep.non_maximal_primes: * nonmaximal primes g2c.galois_rep_image: * 1p3 * 2p2 * ? * cusp * irred * nss g2c.geom_aut_grp: * geometric automorphism group g2c.geom_end_alg: * complex multiplication * geometric endomorphism algebra * quaternionic multiplication * real multiplication g2c.geom_iso_class: * geometric isomorphism class g2c.gl2type: * $\textbf{GL}_2$-type g2c.good_reduction: * bad reduction * good reduction g2c.hasse_weil_conjecture: * Hasse-Weil conjecture g2c.igusa_clebsch_invariants: * Igusa-Clebsch invariants g2c.igusa_invariants: * Igusa invariants g2c.isogeny_class: * isogeny class g2c.jac_end_lattice: * endomorphism lattice g2c.known_rational_points: * known rational points g2c.lfunction: * l-function * l-polynomial g2c.local_invariants: * local invariants g2c.locally_solvable: * locally solvable g2c.maximal_galois_rep: * generic * maximal g2c.minimal_equation: * minimal equation g2c.mordell_weil_rank: * mordell-weil rank g2c.mw_generator: * Mordell-Weil generators * Mumford representation g2c.mw_generator_order: * order g2c.num_rat_wpts: * Weierstrass points g2c.paramodular_conjecture: * paramodular conjecture g2c.real_period: * real period g2c.regulator: * regulator g2c.simple_equation: * simplified equation g2c.st_group: * Sato-Tate group g2c.st_group_identity_component: * component group * identity component g2c.tamagawa: * Tamagawa number * Tamagawa product g2c.torsion: * torsion subgroup g2c.torsion_order: * torsion order * torsion primes g2c.two_selmer_rank: * 2-Selmer rank g2c.two_torsion_field: * 2-torsion field gg.concentrated: * concentrated gg.degree: * degree gg.elusive_group: * elusive group gg.galois_group: * Galois group gg.generic_polynomial: * generic gg.group_action_invariants: * degree * parity * primitive * resolvent fields gg.index: * index gg.int_modules: * indecomposable * integral representation gg.invariants: * abelian * character table * cyclic * order * solvable gg.malle_a: * Malle gg.malle_b: * Malle gg.other_representations: * siblings gg.parity: * parity gg.primitive: * primitive gg.regular_extension: * regular gg.resolvents: * resolvents gg.semiconcentrated: * semiconcentrated gg.subfields: * subfields gg.subnormal_series: * subnormal series gg.transitivity: * $k$-transitive * transitivity number gl2.borel: * B * B.a.b * Borel subgroup gl2.cartan: * Cartan subgroup * extended Cartan subgroup gl2.exceptional: * A4 * A5 * S4 * exceptional subgroup gl2.genus: * genus gl2.index: * index gl2.label: * RSZB label gl2.level: * level gl2.nonsplit_cartan: * Cn * Cn.a.b * non-split Cartan subgroup gl2.normalizer_cartan: * normalizer of a cartan subgroup gl2.normalizer_nonsplit_cartan: * Cn * Nn * Nn.a.b * normalizer of a non-split Cartan subgroup gl2.normalizer_split_cartan: * Cs * Ns * Ns.a.b * Ns.a.b.c * normalizer of a split Cartan subgroup gl2.open: * open subgroup gl2.profinite: * profinite group gl2.split_cartan: * Cs * Cs.a.b * split Cartan subgroup group: * group group.a_group: * A-group group.abelian: * abelian group.abelian_invariants: * Smith abelian invariants * primary abelian invariants group.abelianization: * abelianization group.action: * group action * transitive group.almost_simple: * almost simple group group.alternating: * alternating group group.ambient: * ambient group group.autjugacy_class: * autjugacy class group.autjugate_subgroup: * autjugate group.automorphism: * automorphism group.center: * center group.central: * central group.centralizer: * centralizer group.characteristic_subgroup: * characteristic subgroup group.chevalley: * Chevalley groups group.chief_series: * chief series * composition factors group.commutator_length: * commutator length group.commutator_subgroup: * commutator subgroup * commutators * derived subgroup group.complete: * complete group.complex_character_table: * complex character table group.conjugacy_class: * conjugacy class group.core: * c group.coset: * coset group.coxeter: * Coxeter graph * Coxeter group * irreducible Coxeter group group.cyclic: * cyclic group.derived_series: * derived length * derived series group.dicyclic: * dicyclic group group.dihedral: * dihedral group group.direct_product: * direct factors * direct product group.division: * division * rational conjugacy classes group.division_maximal: * maximal group.division_small_large: * large * small group.element_order: * order group.elementary: * elementary group.exponent: * exponent of a {{KNOWL('group', 'group')}} group.fitting_subgroup: * Fitting subgroup group.frattini_subgroup: * Frattini subgroup group.frobenius: * Frobenius group group.fuchsian: * Fuchsian group * first kind group.fuchsian.cusps: * cusps group.fuchsian.cusps.width: * generators * width group.fuchsian.fundamental_domain: * fundamental domain group.galois.absolute: * absolute Galois group group.gassmann_equivalence: * Gassmann equivalent group.general_linear: * general linear group * projective general linear group group.generalized_quaternion: * (generalized) quaternion group group.generators: * generators group.group_mathlib_defn: * monoid group.group_ring: * group ring group.haar_measure: * Haar measure * normalized Haar measure group.heisenberg: * Heisenberg group group.homomorphism: * group homomorphism group.hyperelementary: * hyperelementary group.inner_automorphism: * inner automorphism group.isoclinism: * isoclinism group.isomorphism: * group isomorphism group.label: * label group.label_complex_group_char: * label group.label_conjugacy_class: * label group.label_rational_group_char: * label of a rational character group.lower_central_series: * lower central series group.maximal_quotient: * maximal quotient group.maximal_subgroup: * maximal subgroup group.metabelian: * metabelian group.min_faithful_linear: * irreducible degree * linear degree group.minimal_normal: * minimal normal subgroup group.mobius_function: * möbius quotient function * möbius subgroup function group.monomial: * M-group * monomial group.nilpotent: * nilpotency class * nilpotent group.nonsplit_product: * extension * non-split extension * split extension group.normal_series: * normal series * subnormal series group.order: * order group.order_stats: * order statistics group.orthogonal: * orthogonal group * projective orthogonal groups * special orthogonal groups group.other_dihedral: * other-dihedral group group.outer_aut: * outer automorphism * outer automorphism group group.over_subgroup: * minimal over-subgroup * over-subgroup group.perfect: * perfect group.permutation_degree: * minimal permutation degree group.permutation_module: * permutation module * permutation representation group.permutation_representation: * degree * faithful * permutation representation * transitive group.pgroup: * $p$-group group.presentation: * polycyclic presentation * polycyclic sequence * polycyclic series * presentation * relative orders group.primary_decomposition: * primary decomposition group.proper_subgroup: * proper group.psl2r: * projective special linear group group.quotient: * quotient group group.radical: * radical group.rank: * rank group.rational_character_table: * rational character table group.rational_group: * rational group.representation.center: * center group.representation.character: * character group.representation.complex_char_deg: * degree group.representation.cyclotomic_n: * conductor group.representation.faithful: * faithful group.representation.image: * image group.representation.irrep: * irreducible * irreps * reducible * representation group.representation.kernel: * kernel group.representation.rational_character: * irreducible rational characters * rational character group.representation.schur_index: * Schur index group.representation.type: * Complex * Quaternionic (or Symplectic) * Real * types group.schur_multiplier: * Schur multiplier group.semi_dihedral: * semi-dihedral group group.semidirect_product: * non-split * semidirect factor * semidirect product * split * trivial semidirect product group.simple: * simple group.size_conjugacy_class: * size group.sl2z: * modular group group.sl2z.genus_subgroup: * genus group.socle: * socle group.solvable: * solvable group.sp2gr: * integral symplectic group group.special_linear: * projective special linear group * special linear group group.spinor_norm: * spinor norm group.sporadic: * sporadic groups group.stem_extension: * stem extension * stem subgroup group.stem_group: * stem group group.subgroup: * subgroup group.subgroup.centralizer: * centralizer group.subgroup.complement: * complement group.subgroup.diagram: * subgroup diagram group.subgroup.hall: * Hall subgroup group.subgroup.index: * index group.subgroup.normal: * normal group.subgroup.normal_closure: * normal closure group.subgroup.normalizer: * normalizer group.subgroup.projective_image: * projective image group.supersolvable: * supersolvable group.sylow_subgroup: * $p$-Sylow subgroup group.symmetric: * symmetric group group.symplectic: * projective symplectic group * symplectic group group.torsion: * torsion subgroup * torsion {{KNOWL('group', 'group')}} group.transitive_degree: * minimal transitive degree group.triangle_group: * Von Dyck group * triangle group group.trivial_subgroup: * non-trivial * trivial subgroup group.type: * Type - length group.under_subgroup: * maximal under-subgroup * under-subgroup group.unitary: * (general) unitary group * projective (general) unitary group * projective special unitary group * special unitary group group.upper_central_series: * upper central series group.weight_function: * weight function group.weyl_group: * Weyl group group.wreath_product: * transitive * wreath product group.z_group: * Z-group hgm.bezout_determinant: * Bézout determinant hgm.bezout_matrix: * Bézout matrix hgm.bezout_module: * Bezout module hgm.conductor: * conductor hgm.cusp_index: * cusp-index hgm.defining_parameters: * defining parameters hgm.degree: * degree hgm.familes: * hypergeometric family hgm.families: * hypergeometric family hgm.good_prime: * bad prime * good prime * prime of bad reduction * prime of good reduction hgm.hodge_vector: * Hodge vector hgm.imprimitivity_index: * rotation number hgm.local_discriminant: * local discriminant hgm.motive: * Hypergeometric motives hgm.rotation_number: * rotation number hgm.signature: * signature hgm.tame: * tame hgm.type: * orthogonal * symplectic * type hgm.weight: * (motivic) weight hgm.wild: * wild hgm.zigzagplot: * zigzag plot * zigzag procedure hmsurface.arithmetic_genus: * arithmetic genus hmsurface.congruence_subgroup: * congruence subgroup * level * principal congruence subgroup hmsurface.elliptic_point: * elliptic * elliptic point hmsurface.euler_number: * (topological) Euler number hmsurface.hmsurface: * Hilbert modular surface hmsurface.hodge_numbers: * Hodge numbers hmsurface.kodaira_dimension: * Kodaira dimension hmsurface.level: * level hmsurface.polarization_module: * polarization cone * polarization module intro: * Breadcrumbs: * Browsing: * Downloading: * Feedback: * Interesting: * Knowls: * Learn more: * Navigating: * Properties: * Random: * Related Objects: * Searching: intro.features: * Breadcrumbs: * Browsing: * Downloading: * Feedback: * Knowls: * Navigating: * Navigation Bars: * Properties: * Related Objects: * Searching: * Zeros of L-functions Search: lattice.additively_indecomposable: * additively indecomposable lattice.algebraic: * algebraic lattice.arithmetic: * arithmetic lattice.automorphism_group: * automorphism group lattice.class_number: * class number lattice.conway_genus_symbol: * Conway genus symbol lattice.covering_radius: * Covering radius * inhomogeneous minimum lattice.definition: * integral * lattice * quadratic space lattice.density: * center density * density * lattice packing problem * sphere packing problem lattice.determinant: * determinant lattice.dimension: * rank lattice.discriminant: * 2-adic unit discriminant * Witt discriminant * absolute discriminant * discriminant * geometric discriminant * half discriminant * quadratic discriminant lattice.discriminant_group: * Gram matrix * discriminant group * torsion bilinear form * torsion quadratic form lattice.dual: * dual * rescaled dual lattice.euclidean_basis: * Euclidean basis lattice.eutactic: * eutactic * strongly eutactic lattice.even_sublattice: * even sublattice lattice.extreme: * extreme lattice.festi_veniani_index: * Festi--Veniani index lattice.genus: * genus lattice.gram: * gram matrix * minkowski reduced form lattice.group_order: * group order lattice.hermite_number: * Hermite number lattice.hyperbolic_plane: * hyperbolic plane lattice.isometry: * class * isometric * isometry class lattice.kissing: * general kissing number problem * kissing number * lattice kissing number problem lattice.level: * level lattice.mass: * mass lattice.minimal_vector: * minimal vector * minimum lattice.minimum: * minimal vector * minimum lattice.nondegenerate: * nondegenerate * positive semidefinite * signature lattice.norm: * norm lattice.norm1_sublattice: * norm 1 sublattice lattice.normalized_minimal_vector: * normalized lattice.orthogonal_decomposition: * indecomposable * orthogonal decomposition lattice.parity: * even * odd lattice.perfect: * perfect lattice.pneighbor: * $p$-neighbor lattice.pneighbor_adjacency: * $p$-neighbor adjacency matrix * p-neighbor adjacency graph lattice.positive_definite: * positive definite lattice.postive_definite: * positive definite lattice.primitive: * primitive lattice.quadratic_form: * even universal * norm * quadratic form * universal lattice.root_lattice: * root lattice * root system * roots * weyl group lattice.root_sublattice: * root lattice * root sublattice lattice.signature: * indefinite * negative definite * signature lattice.similarity: * similar lattice.spherical_design: * spherical $t$-design * spherical $t$-design lattice * strongly perfect lattice.sublattice: * Even sublattice * Max diag sublattice * Min vector sublattice * Norm 1 sublattice * Root sublattice * sublattice lattice.successive_minima: * Successive minima lattice.tensor_decomposition: * tensor decomposable * tensor decomposition * tensor indecomposable * tensor product lattice.theta: * theta series lattice.unimodular: * unimodular lattice.well_rounded: * generated by minimal vectors * strongly well-rounded * well-rounded lf.associated_inertia: * associated inertia lf.automorphism_group: * automorphism group lf.defining_polynomial: * defining polynomial lf.degree: * absolute degree * base degree * degree lf.discriminant_exponent: * absolute discriminant exponent * base discriminant exponent * discriminant * discriminant exponent lf.eisenstein_diagram: * $A$-points * $B$-points * $C$-points * $D$-point * $Z$-points * Eisenstein diagram of $I/K$ * band * critical * index * negligible lf.eisenstein_form: * Eisenstein form lf.eisenstein_polynomial: * eisenstein lf.family_ambiguity: * ambiguity lf.family_base: * base field lf.family_field_count: * field count lf.family_mass: * absolute mass * absolute mass of the family * mass * mass of the family lf.family_polynomial: * absolute family * family * generic polynomial * relative family * semicanonical polynomials lf.galois_invariants: * Galois Mean Slope * Galois Splitting Model * Swan slope * inertia subgroup * tame degree * unramified degree * wild Artin slopes lf.galois_mean_slope: * Galois mean slope lf.galois_splitting_model: * Galois splitting model lf.heights: * height * set of heights lf.herbrand_function: * Herbrand function lf.herbrand_invariant: * Herbrand invariant * arithmetic ram * compatible * geometric ram lf.hidden_slopes: * Galois Artin slopes * Galois Swan slopes * hidden Artin slopes * hidden Swan slopes lf.indices_of_inseparability: * index of inseparability lf.inertia_group: * inertia group lf.intermediate_fields: * intermediate fields lf.jump_set: * jump set lf.local_field: * archimedean local field * local field * nonarchimedean local field lf.local_height_function: * local height * local height function lf.log: * $p$-adic logarithm lf.maximal_ideal: * maximal ideal lf.means: * mean * means * slope polygon lf.newton_polygon: * Newton polygon lf.newton_slopes: * slopes lf.packet: * packet lf.padic_field: * $p$-adic field * local number field lf.qp: * p-adic numbers lf.ramification_index: * absolute ramification index * base ramification index * ramification index * tame ramification index * tamely ramified * totally ramified * uniformizer * unramified * wild ramification exponent lf.ramification_polygon: * ramification polygon * ramification polynomial lf.rams: * rams * small rams * tiny rams lf.residual_polynomials: * equivalence class of residual polynomials * residual polynomial * residual polynomials lf.residue_field: * residue field * residue field characteristic lf.residue_field_degree: * absolute residue field degree * base residue field degree * residue field degree lf.ring_of_integers: * ring of integers lf.root_number: * local root number lf.slope_content: * Artin slope content * Swan slopes * wild Artin slopes lf.slopes: * Artin slope content * Artin slopes * Swan slope content * Swan slopes * visible Artin slopes * visible Swan slopes * wild Artin slopes * wild Swan slopes lf.subfamily: * subfamily lf.swan_slopes: * Swan slopes lf.tame_degree: * Galois tame degree * tame degree lf.top_slope: * top Artin slope lf.unramified_degree: * Galois unramified degree * unramified degree lf.unramified_subfield: * maximal unramified subextension * unramified subfield lf.unramified_totally_ramified_tower: * canonical tower lf.visible_slopes: * visible Artin slopes * visible Swan slopes lf.wild_inertia_group: * wild inertia group lf.wild_segments: * number of wild segments lf.wild_slopes: * wild Artin slopes lfunction: * L-function lfunction.analytic_conductor: * analytic conductor lfunction.analytic_rank: * analytic rank lfunction.arithmetic: * arithmetic lfunction.bad_prime: * bad primes lfunction.central_character: * central character lfunction.central_point: * central point lfunction.central_value: * central value lfunction.coefficient_field: * coefficient field lfunction.completed: * completed L-function lfunction.conductor: * conductor lfunction.critical_line: * critical line lfunction.critical_strip: * critical strip lfunction.degree: * degree lfunction.dirichlet: * Dirichlet L-function lfunction.dirichlet_series: * dirichlet series lfunction.dual: * dual lfunction.euler_product: * Euler factors * Euler product * local factors lfunction.functional_equation: * functional equation lfunction.gamma_factor: * gamma factors lfunction.infinity_factor: * infinity factor lfunction.leading_coeff: * leading coefficient lfunction.motivic_weight: * arithmetic weight * motivic weight lfunction.normalization: * analytic normalization * arithmetic normalization lfunction.rational: * rational lfunction.rh: * riemann hypothesis lfunction.riemann: * Riemann zeta-function lfunction.riemann.euler_product: * euler product lfunction.root_analytic_conductor: * root analytic conductor lfunction.root_angle: * root angle lfunction.root_number: * sign lfunction.selberg_class: * Selberg class lfunction.selberg_class.axioms: * selberg axioms lfunction.selbergdata: * Selberg data lfunction.self-dual: * self-dual lfunction.sign: * root number * sign lfunction.signature: * signature lfunction.spectral_parameters: * spectral parameters lfunction.trivial_zero: * trivial zeros lfunction.zeros: * lowest zero * zeros lfunction.zfunction: * Riemann-Siegel theta function * Z-function mf: * modular form mf.base_change: * b mf.bianchi: * Bianchi modular form mf.bianchi.anr: * analytic rank mf.bianchi.base_change: * base-change mf.bianchi.bianchigroup: * Bianchi group associated to mf.bianchi.cm: * cm * complex multiplication mf.bianchi.fourierbessell: * Fourier-Bessel expansion * K-Bessel functions mf.bianchi.hecke_algebra: * Hecke algebra * Hecke operators mf.bianchi.hyperbolic3space: * Hyperbolic 3-space mf.bianchi.level: * $\mathbf{GL}_2$-levels * $\mathbf{SL}_2$-levels * level label * level norm mf.bianchi.newform: * Bianchi newform * Hecke field * dimension mf.bianchi.sign: * Fricke involution * sign mf.bianchi.spaces: * Hecke field * cuspidal subspace * dimension * newform * newspace * oldforms mf.bianchi.weight: * weight mf.bianchi.weight2: * cuspidal mf.cm: * c mf.ellitpic.self_twist: * self twist mf.half_integral_weight: * modular form of half-integral {{ knowl(mf.elliptic.weight,title=weight)}} mf.half_integral_weight.dedekind_eta: * Dedekind eta function mf.half_integral_weight.theta: * theta function mf.hilbert: * Hilbert cusp form * Hilbert modular form mf.hilbert.dimension: * dimension mf.hilbert.level_norm: * level * level norm mf.hilbert.weight_vector: * parallel weight * weight mf.maass: * maass form mf.maass.exceptional_eigenvalue: * exceptional mf.maass.label: * label mf.maass.mwf: * Maass form mf.maass.mwf.coefficients: * coefficients mf.maass.mwf.dimension: * multiplicity mf.maass.mwf.eigenvalue: * eigenvalue mf.maass.mwf.fourierexpansion: * Fourier expansion mf.maass.mwf.laplacian: * laplace-beltrami operator mf.maass.mwf.level: * level mf.maass.mwf.spectralparameter: * spectral parameter mf.maass.mwf.symmetry: * even * odd mf.maass.mwf.weight: * weight mf.maass.mwf.weight0: * maass form mf.maass.picard: * Maass form * Picard group mf.maass.picard.eigenvalue: * eigenvalue * spectral parameter * true eigenvalue mf.maass.picard.fourierexpansion: * Fourier expansion mf.maass.spectral_index: * spectral index mf.multipliersystem: * multiplier system mf.quasi_modular: * quasi-modular form mf.siegel: * Siegel modular form * scalar-valued Siegel modular form * vector-valued Siegel modular forms mf.siegel.automorphic_type: * automorphic type mf.siegel.character: * Siegel modular form with character $\psi$ mf.siegel.coefficient_field: * Hecke field * coefficient field mf.siegel.coefficient_ring: * Hecke ring * coefficient ring mf.siegel.cusp_form_degree2: * cusp form mf.siegel.defining_polynomial: * defining polynomial mf.siegel.degree: * degree mf.siegel.dimension: * dimension * relative dimension mf.siegel.family: * Paramodular subgroup * Principal congruence subgroup * Siegel parabolic subgroup mf.siegel.hecke_operator: * Hecke algebra * Hecke eigenform * local Hecke algebra mf.siegel.koecher.principle: * Koecher principle mf.siegel.label: * label mf.siegel.level: * level mf.siegel.newform: * new subspace * newform * old * old subspace mf.siegel.newspace: * new subspace * old subspace mf.siegel.phi: * Siegel $\Phi$ operator mf.siegel.slash.op: * slash operator mf.siegel.weight: * weight mf.siegel.weight_k_j: * weight mf.slash_action: * weight $k$ slash-action mf.upper_half_plane: * completed upper * upper half-plane modcurve: * Complex points * Rational points * modular curve modcurve.agreeable: * agreeable * agreeable closure * agreeable quotient modcurve.canonical_field: * canonical field of definition modcurve.cm_discriminants: * CM discriminants * CM points modcurve.cusp_orbits: * cusp orbits * cusps modcurve.cusp_widths: * cusps * width modcurve.cusps: * $\Q$-cusps * cusps * rational cusps modcurve.decomposition: * isogeny decomposition modcurve.elliptic_points: * elliptic point modcurve.embedded_model: * embedded model modcurve.entanglement_index: * entanglement index * entanglement quotient modcurve.fiber_product: * fiber product modcurve.gassmann_class: * Gassmann class modcurve.genus: * genus modcurve.genus_minus_rank: * genus-rank difference modcurve.gonality: * $K$-gonality modcurve.index: * index modcurve.j_invariant_map: * $\boldsymbol{j}$-invariant map * $\boldsymbol{j}$-map modcurve.label: * label modcurve.level: * level modcurve.level_structure: * $H$-level structure * Cyclic $\boldsymbol{N}$-isogeny field degree * Cyclic $\boldsymbol{N}$-torsion field degree * N-torsion field degree * rational modcurve.local_obstruction: * local obstruction modcurve.minimal_twist: * minimal twist * twists modcurve.modular_cover: * degree * kernel * minimal * modular cover modcurve.name: * nonsplit Cartan * split Cartan modcurve.newform_level: * newform level modcurve.other_labels: * (CP) * (RSZB) * (RZB) * (S) * (SZ) modcurve.point_residue_field: * residue field modcurve.psl2index: * $\PSL_2$-index modcurve.quadratic_refinements: * index 2 refinement modcurve.rank: * analytic rank modcurve.relative_index: * relative index modcurve.simple: * simple modcurve.sl2level: * $\SL_2$-level modcurve.xn: * classical modular curve of level $N$ * full modular curve of level $N$ modcurve.xns.remarks: * oriented necklace modcurve.xns_plus.remarks: * necklace modlgal: * mod-$\ell$ Galois representation modlgal.abelian: * abelian * nonabelian modlgal.absolutely_irreducible: * absolutely irreducible modlgal.base_ring_characteristic: * base ring characteristic modlgal.characteristic: * characteristic modlgal.codomain: * codomain modlgal.conductor: * conductor modlgal.determinant: * determinant character modlgal.determinant_index: * determinant index modlgal.dimension: * Galois characters * dimension modlgal.frobenius_charpoly: * characteristic polynomial of Frobenius modlgal.frobenius_determinant: * Frobenius determinant modlgal.frobenius_matrix: * Frobenius matrix modlgal.frobenius_order: * Frobenius order modlgal.frobenius_prime: * Frobenius prime * good prime modlgal.frobenius_trace: * Frobenius trace modlgal.image: * image modlgal.image_abstract_group: * abstract image modlgal.image_index: * index modlgal.image_order: * image order modlgal.min_sib_splitting_field: * minimal sibling of the splitting field modlgal.projective_kernel_polynomial: * projective kernel polynomial modlgal.projective_representation: * projective representation modlgal.ramified: * ramified * unramified modlgal.solvable: * nonsolvable * solvable modlgal.splitting_field: * splitting field modlgal.surjective: * surjective modlgal.top_slope: * top Artin slope mot.hodgevector: * Hodge vector nf: * number field nf.abelian: * abelian nf.abs_discriminant: * absolute discriminant nf.absolute_value: * absolute value * archimedean * equivalent * nonarchimedean * trivial absolute value nf.arithmetically_equivalent: * arithmetically equivalent nf.class_number: * class number nf.class_number_formula: * analytic class number formula nf.cm_field: * CM field nf.complex_embedding: * complex embedding nf.conductor: * conductor nf.defining_polynomial: * defining polynomial nf.degree: * degree nf.degree_mathlib_def: * module rank nf.dirichlet_group: * dirichlet character group nf.discriminant: * discriminant nf.discriminant_root_field: * discriminant root field * polynomial discriminant nf.embedding: * embedding nf.frobenius_cycle_types: * frobenius cycle type * frobenius element * residue degree * unramified nf.fundamental_units: * fundamental units nf.galois_closure: * galois closure nf.galois_group: * Galois group nf.galois_root_discriminant: * Galois root discriminant nf.generator: * generator nf.ideal_class_group: * ideal class group nf.ideal_labels: * ideal label nf.inertial_height_function: * inertial height function * local height function * radical discriminant * weight function nf.inessential_prime: * inessential prime nf.integral: * integral nf.integral_basis: * integral basis nf.intermediate_fields: * intermediate fields nf.is_galois: * galois nf.local_algebra: * local algebra nf.max_p: * maximum ramified prime nf.maximal_cm_subfield: * maximal CM subfield nf.minimal_polynomial: * minimal polynomial nf.minimal_sibling: * minimal sibling nf.monogenic: * monogenic nf.monomial_order: * monomial order nf.narrow_class_group: * narrow class group * strict class group nf.narrow_class_number: * narrow class number * strict class number nf.nickname: * nicknames nf.order: * order nf.padic_completion: * $p$-adic completion of $K$ at $\mathfrak{P}$ nf.place: * completion * complex place * infinite places * place * real place nf.polredabs: * defining polynomial * reduced defining polynomial nf.poly_discriminant: * discriminant nf.prime: * prime * prime above nf.radical_discriminant: * radical discriminant nf.ramified_primes: * ramified prime nf.rank: * rank nf.real_embedding: * real embedding nf.reflex_field: * CM type * reflex field nf.reflex_reflex_field: * reflex field of the reflex field nf.regulator: * regulator nf.relative_class_number: * relative class number nf.ring_of_integers: * ring of integers nf.root_discriminant: * root discriminant nf.separable: * separable nf.separable_algebra: * \'etale $F$-algebra * separable algebra nf.serre_odlyzko_bound: * Serre Odlyzko bound nf.sibling: * siblings nf.signature: * signature nf.stem_field: * stem field * stem fields of the polynomial nf.torsion: * torsion generator nf.totally_imaginary: * totally complex * totally imaginary nf.totally_positive: * totally positive nf.totally_real: * totally real nf.unit_group: * unit group nf.unit_signature_rank: * signature * unit signature rank nf.unramified_prime: * unramified (rational) prime nf.weil_height: * Weil height * absolute logarithmic * logarithmic nf.weil_polynomial: * Weil $q$-polynomial nf.zk_index: * index of the number field rcs: * completeness * reliability * source rcs.cande.lfunction: * Degrees 5 to 40 * {{KNOWL('lfunction.known_degree1','Degree 1')}} * {{KNOWL('lfunction.known_degree2','Degree 2')}} * {{KNOWL('lfunction.known_degree3','Degree 3')}} * {{KNOWL('lfunction.known_degree4','Degree 4')}} rcs.cande.modcurve: * note that this excludes many known rational points rcs.cite.curve.highergenus.aut: * A database of group actions on Riemann surfaces rcs.rigor.modcurve: * known issues with the rational points data * the absence of rational points should not be taken as an indication that none are known * use with caution rcs.rigor.shimcurve: * use with caution rcs.source.curve.highergenus.aut: * Corresponding Character(s) * Dimension of the corresponding {{KNOWL('ag.shimura_variety', "Shimura variety")}} * Equations * Hyperelliptic or Cyclic Trigonal Curves * {{KNOWL('ag.jacobian', "Jacobian variety")}} decomposition repn.highest.weight: * the highest weight ring: * commutative ring * non-unital ring * ring * ring without identity * rng ring.a-field: * $A$-field ring.associate: * associates ring.characteristic: * characteristic ring.dedekind_domain: * Dedekind domain ring.euclidean_domain: * E ring.field: * field ring.field_of_fractions: * field of fractions ring.field_of_fractions_mathlib: * Fraction Ring ring.fractional_ideal: * fractional ideal ring.ideal: * homogeneous * ideal ring.integral: * integral * integral closure ring.integral_domain: * integral domain ring.integrally_closed: * integrally closed ring.irreducible: * irreducible ring.maximal_ideal: * maximal ring.noetherian: * Noetherian ring.prime_ideal: * prime ring.principal_fractional_ideal: * principal fractional ideal ring.principal_ideal_domain: * principal ideal domain ring.unique_factorization_domain: * unique factorization domain ring.unit: * unit ring.zero_divisor: * zero divisor sage.cell: * Sage Cell Server shimcurve: * (quaternionic) Shimura curve of level $H$ * full level $N$ Shimura curve associated to $O$ * level $1$ Shimura curve associated to $O$ shimcurve.aut_pm_mu_o: * automorphisms of $O$ preserving the polarization $\mu$ (up to sign) shimcurve.decomposition: * isogeny decomposition shimcurve.discb: * discriminant * ramified * split shimcurve.disco: * discriminant * reduced discriminant shimcurve.eichler: * Eichler of level $M$ * Eichler {{KNOWL('shimcurve.order','order')}} shimcurve.elliptic_points: * elliptic point shimcurve.enhanced_group: * enhanced group (associated to $(O,\mu)$) shimcurve.genus: * genus shimcurve.genus_minus_rank: * genus-rank difference shimcurve.gonality: * $K$-gonality shimcurve.index: * index shimcurve.is_coarse: * coarse shimcurve.label: * label shimcurve.level: * level shimcurve.local_obstruction: * local obstruction shimcurve.nrdmu: * polarization degree shimcurve.order: * maximal * order shimcurve.polarized_order: * polarization * polarized order shimcurve.quaternion_algebra: * definite * indefinite * nonsplit * quaternion algebra * split shimcurve.rank: * analytic rank shimcurve.simple: * simple specialfunction.gamma: * (Euler) gamma function $\Gamma(z)$ specialfunction.k_bessel: * $k$-bessel functions specialfunction.poincare: * $n$th poincaré series st_group.ambient: * ambient group st_group.component_group: * component group st_group.connected: * connected st_group.definition: * Sato-Tate group st_group.degree: * degree st_group.first_a2_moment: * first $a_2$ moment st_group.fourth_trace_moment: * fourth trace moment st_group.hodge_circle: * Hodge circle st_group.identity_component: * identity component st_group.moment_matrix: * diagonal * moment matrix st_group.moment_simplex: * moment simplex st_group.moments: * Moment statistics * moment sequence st_group.probabilities: * probability st_group.rational: * rational st_group.real_dimension: * real dimension st_group.second_trace_moment: * second trace moment st_group.subgroups: * maximal subgroups st_group.supgroups: * maximal * minimal supergroups st_group.symplectic_form: * symplectic form st_group.trace_moments: * moment st_group.trace_zero_density: * trace zero density st_group.usp: * unitary symplectic group st_group.weight: * weight stats.proportions: * by column * by row * none * vs unconstrained $(a,b,c)$ triple: * belyi.abc $\#\mathrm{Pic}$: * av.fq.endomorphism_ring_notation $\#\mathrm{weak} = 1 $: * av.fq.endomorphism_ring_notation $\#\mathrm{weak}$: * av.fq.endomorphism_ring_notation $\boldsymbol{j}$-invariant map: * modcurve.j_invariant_map $\boldsymbol{j}$-map: * modcurve.j_invariant_map $\ell$-adic Galois representation: * av.galois_rep * ec.galois_rep $\mathbf{GL}_2$-levels: * mf.bianchi.level $\mathbf{SL}_2$-levels: * mf.bianchi.level $\mathrm{index}$: * av.fq.endomorphism_ring_notation $\overline{k}$-gonality: * ag.gonality_geom $\PSL_2$-index: * modcurve.psl2index $\Q$-cusps: * modcurve.cusps $\SL_2$-level: * modcurve.sl2level $\textbf{GL}_2$-type: * g2c.gl2type $A$-field: * ring.a-field $A$-points: * lf.eisenstein_diagram $abc$ quality: * ec.q.abc_quality $B$-points: * lf.eisenstein_diagram $C$-points: * lf.eisenstein_diagram $d$: * av.fq.lmfdb_label $D$-point: * lf.eisenstein_diagram $g$: * av.fq.lmfdb_label $H$-level structure: * modcurve.level_structure $i$: * av.fq.endomorphism_ring_notation * av.fq.lmfdb_label $j$: * av.fq.lmfdb_label $j$-invariant: * ec.j_invariant * ec.q.j_invariant $k$: * av.fq.lmfdb_label $k$-bessel functions: * specialfunction.k_bessel $k$-gonality: * ag.gonality $K$-gonality: * modcurve.gonality * shimcurve.gonality $k$-transitive: * gg.transitivity $N :=[\mathcal{O}_{\mathbb{Q}[F]}:R]$: * av.fq.lmfdb_label $n$-Selmer group: * ag.selmer_group $n$-th coefficient: * cmf.fouriercoefficients_mathlib_def $n$th poincaré series: * specialfunction.poincare $p$-adic completion of $K$ at $\mathfrak{P}$: * nf.padic_completion $p$-adic field: * lf.padic_field $p$-adic logarithm: * lf.log $p$-adic Tate module: * av.tate_module * ec.padic_tate_module $p$-corank: * av.fq.p_rank $p$-group: * group.pgroup $p$-neighbor: * lattice.pneighbor $p$-neighbor adjacency matrix: * lattice.pneighbor_adjacency $p$-rank: * av.fq.p_rank $p$-Sylow subgroup: * group.sylow_subgroup $q$: * av.fq.lmfdb_label $q$-expansion: * cmf.q-expansion $w$: * av.fq.lmfdb_label $Z$-points: * lf.eisenstein_diagram ($u_q(\mathfrak{g})$-)crystal: * crystals.crystal (connected) Shimura varieties: * ag.shimura_variety (CP): * modcurve.other_labels (Euler) gamma function $\Gamma(z)$: * specialfunction.gamma (general) unitary group: * group.unitary (generalized) quaternion group: * group.generalized_quaternion (geometric) monodromy group: * curve.highergenus.aut.monodromy (motivic) weight: * hgm.weight (quaternionic) Shimura curve of level $H$: * shimcurve (relative) conductor: * av.fq.conductor (RSZB): * modcurve.other_labels (RZB): * modcurve.other_labels (S): * modcurve.other_labels (SZ): * modcurve.other_labels (topological) Euler number: * hmsurface.euler_number -curve: * ec.q_curve 154: * ec.q.manin_constant 1p3: * g2c.galois_rep_image 2-adic unit discriminant: * lattice.discriminant 2-Selmer rank: * g2c.two_selmer_rank 2-torsion field: * g2c.two_torsion_field 2p2: * g2c.galois_rep_image ?: * g2c.galois_rep_image \'etale $F$-algebra: * nf.separable_algebra A database of group actions on Riemann surfaces: * rcs.cite.curve.highergenus.aut A-group: * group.a_group A4: * ec.galois_rep_modell_image * gl2.exceptional A5: * ec.galois_rep_modell_image * gl2.exceptional abelian: * gg.invariants * group.abelian * modlgal.abelian * nf.abelian abelian surface: * ag.abelian_surface abelian variety: * ag.abelian_variety abelianization: * group.abelianization absolute degree: * lf.degree absolute discriminant: * g2c.abs_discriminant * lattice.discriminant * nf.abs_discriminant absolute discriminant exponent: * lf.discriminant_exponent absolute family: * lf.family_polynomial absolute Galois group: * group.galois.absolute absolute logarithmic: * nf.weil_height absolute mass: * lf.family_mass absolute mass of the family: * lf.family_mass absolute ramification index: * lf.ramification_index absolute residue field degree: * lf.residue_field_degree absolute value: * nf.absolute_value absolutely irreducible: * modlgal.absolutely_irreducible abstract image: * modlgal.image_abstract_group additive reduction: * ec.additive_reduction * ec.q.reduction_type * ec.reduction_type additively indecomposable: * lattice.additively_indecomposable adelic Galois representation: * av.galois_rep * ec.galois_rep affine algebraic set: * ag.variety Affine space: * ag.affine_space affine variety: * ag.variety agreeable: * modcurve.agreeable agreeable closure: * modcurve.agreeable agreeable quotient: * modcurve.agreeable algebraic: * lattice.algebraic algebraic curve: * ag.curve algebraic genus: * ag.curve.genus algebraic varieties: * ag.variety almost ordinary: * av.fq.newton_elevation almost simple group: * group.almost_simple alternating group: * group.alternating ambient group: * group.ambient * st_group.ambient ambiguity: * lf.family_ambiguity analytic class number formula: * nf.class_number_formula analytic conductor: * cmf.analytic_conductor * lfunction.analytic_conductor analytic normalization: * lfunction.normalization analytic order of Sha: * ec.analytic_sha_order * g2c.analytic_sha Analytic Order of Sha: * ec.q.analytic_sha_order analytic rank: * cmf.analytic_rank * ec.q.analytic_rank * g2c.analytic_rank * lfunction.analytic_rank * mf.bianchi.anr * modcurve.rank * shimcurve.rank angle corank: * av.fq.angle_rank angle rank: * av.fq.angle_rank archimedean: * nf.absolute_value archimedean local field: * lf.local_field arithmetic: * lattice.arithmetic * lfunction.arithmetic arithmetic function: * af arithmetic genus: * ag.arithmetic_genus * ag.curve.genus * hmsurface.arithmetic_genus arithmetic normalization: * lfunction.normalization arithmetic ram: * lf.herbrand_invariant arithmetic weight: * lfunction.motivic_weight arithmetically equivalent: * nf.arithmetically_equivalent artin field: * artin.number_field * cmf.artin_field Artin image: * cmf.artin_image Artin representation: * artin Artin slope content: * lf.slope_content * lf.slopes Artin slopes: * lf.slopes Artin stem field: * artin.stem_field associated inertia: * lf.associated_inertia associates: * ring.associate associative: * alg.binary_operation.associative Atkin-Lehner involution: * cmf.atkin-lehner autjugacy class: * group.autjugacy_class autjugate: * group.autjugate_subgroup automorphic type: * mf.siegel.automorphic_type automorphism: * ag.aut_group * av.aut_group * g2c.aut_grp * group.automorphism automorphism group: * ag.aut_group * av.aut_group * g2c.aut_grp * lattice.automorphism_group * lf.automorphism_group automorphisms of $O$ preserving the polarization $\mu$ (up to sign): * shimcurve.aut_pm_mu_o B: * ec.galois_rep_modell_image * gl2.borel b: * mf.base_change B.a.b: * gl2.borel bad prime: * cmf.bad_prime * hgm.good_prime bad primes: * lfunction.bad_prime bad reduction: * ag.good_reduction * ec.bad_reduction * ec.q.reduction_type * g2c.good_reduction band: * lf.eisenstein_diagram base change: * ag.base_change * ec.base_change base degree: * lf.degree base discriminant exponent: * lf.discriminant_exponent base field: * ag.base_field * belyi.base_field * lf.family_base base ramification index: * lf.ramification_index base residue field degree: * lf.residue_field_degree base ring characteristic: * modlgal.base_ring_characteristic base-change: * mf.bianchi.base_change Bass: * av.fq.bass_zfv belyi map: * belyi.galmap Bernoulli numbers: * af.bernoulli_numbers Beta section: * content.announcements Bezout module: * hgm.bezout_module Bianchi group associated to: * mf.bianchi.bianchigroup Bianchi modular form: * mf.bianchi Bianchi newform: * mf.bianchi.newform bielliptic curve: * ag.bielliptic_curve bielliptic involution: * ag.bielliptic_curve binary operation: * alg.binary_operation Birch and Swinnerton-Dyer: * ec.bsdconjecture * ec.q.bsdconjecture Borel subgroup: * gl2.borel braid equivalent: * curve.highergenus.aut.braid_equivalence braid operations: * curve.highergenus.aut.braid_equivalence branch points: * curve.highergenus.aut.branchpoints Breadcrumbs:: * intro * intro.features Browsing:: * intro * intro.features BSD: * ec.bsdconjecture BSD formula: * g2c.bsd_invariants BSD invariants: * ec.bsdconjecture * g2c.bsd_invariants by column: * stats.proportions by row: * stats.proportions Bézout determinant: * hgm.bezout_determinant Bézout matrix: * hgm.bezout_matrix c: * group.core * mf.cm canonical field of definition: * modcurve.canonical_field canonical height: * ag.canonical_height * ec.canonical_height * ec.q.canonical_height canonical map: * ag.canonical_model canonical model: * ag.canonical_model canonical tower: * lf.unramified_totally_ramified_tower Cartan subgroup: * gl2.cartan center: * group.center * group.representation.center center density: * lattice.density central: * group.central central character: * lfunction.central_character central point: * lfunction.central_point central value: * lfunction.central_value centralizer: * group.centralizer * group.subgroup.centralizer character: * cmf.character * group.representation.character character table: * gg.invariants characteristic: * modlgal.characteristic * ring.characteristic characteristic polynomial: * av.fq.weil_polynomial characteristic polynomial of Frobenius: * modlgal.frobenius_charpoly characteristic subgroup: * group.characteristic_subgroup Chevalley groups: * group.chevalley chief series: * group.chief_series class: * lattice.isometry class number: * lattice.class_number * nf.class_number classical modular curve of level $N$: * modcurve.xn cluster picture: * ag.cluster_picture cm: * mf.bianchi.cm CM discriminant: * ec.complex_multiplication CM discriminants: * modcurve.cm_discriminants CM field: * nf.cm_field CM points: * modcurve.cm_discriminants CM type: * nf.reflex_field CM-involution: * av.fq.conjugate_stable Cn: * ec.galois_rep_modell_image * gl2.nonsplit_cartan * gl2.normalizer_nonsplit_cartan Cn.a.b: * gl2.nonsplit_cartan coarse: * shimcurve.is_coarse codomain: * modlgal.codomain coefficient field: * cmf.coefficient_field * lfunction.coefficient_field * mf.siegel.coefficient_field coefficient ring: * cmf.coefficient_ring * mf.siegel.coefficient_ring coefficient ring generator bound: * cmf.hecke_ring_generators coefficients: * mf.maass.mwf.coefficients Cohen-Macaulay: * ag.cohen_macaulay_type commutative: * alg.binary_operation.commutative commutative ring: * ring commutator length: * group.commutator_length commutator subgroup: * group.commutator_subgroup commutators: * group.commutator_subgroup compatible: * lf.herbrand_invariant complement: * group.subgroup.complement complete: * group.complete completed L-function: * lfunction.completed completed upper: * mf.upper_half_plane completeness: * rcs completion: * nf.place Complex: * group.representation.type complex character table: * group.complex_character_table complex embedding: * nf.complex_embedding complex multiplication: * ag.complex_multiplication * cmf.cm_form * ec.complex_multiplication * g2c.end_alg * g2c.geom_end_alg * mf.bianchi.cm complex place: * nf.place Complex points: * modcurve component group: * g2c.st_group_identity_component * st_group.component_group composition factors: * group.chief_series concentrated: * gg.concentrated conductor: * ag.conductor * artin.conductor * av.fq.conductor * character.dirichlet.conductor * ec.conductor * ec.q.conductor * ec.q.lmfdb_label * g2c.conductor * group.representation.cyclotomic_n * hgm.conductor * lfunction.conductor * modlgal.conductor * nf.conductor conductor norm: * ec.conductor * ec.invariants congruence subgroup: * cmf.congruence_subgroup * hmsurface.congruence_subgroup conic: * ag.hyperelliptic_curve conjugacy class: * group.conjugacy_class conjugate stable: * av.fq.conjugate_stable connected: * st_group.connected conrey (primitive) generator: * character.dirichlet.conrey.primitive_generator conrey index: * character.dirichlet.conrey.index contains: * artin.permutation_container Conway genus symbol: * lattice.conway_genus_symbol Conway polynomials: * field.finite.conway_polynomial corresponding character(s): * dq.curve.highergenus.aut.source Corresponding Character(s): * rcs.source.curve.highergenus.aut coset: * group.coset Covering radius: * lattice.covering_radius Coxeter graph: * group.coxeter Coxeter group: * group.coxeter Cremona label: * ec.q.cremona_label critical: * lf.eisenstein_diagram critical line: * lfunction.critical_line critical strip: * lfunction.critical_strip Cs: * ec.galois_rep_modell_image * gl2.normalizer_split_cartan * gl2.split_cartan Cs.a.b: * gl2.split_cartan cusp: * g2c.galois_rep_image cusp form: * cmf.cusp_form * mf.siegel.cusp_form_degree2 cusp orbits: * modcurve.cusp_orbits cusp-index: * hgm.cusp_index CuspForm $\Gamma$ $k$: * cmf.cusp_form_mathlib_def cuspFormSubmodule $\Gamma$ $k$: * cmf.cusp_form_subspace_mathlib_def cuspidal: * ec.q.reduction_type * mf.bianchi.weight2 cuspidal subspace: * cmf.subspaces * mf.bianchi.spaces cusps: * group.fuchsian.cusps * modcurve.cusp_orbits * modcurve.cusp_widths * modcurve.cusps cyclic: * av.fq.cyclic_group_points * ec.isogeny * gg.invariants * group.cyclic Cyclic $\boldsymbol{N}$-isogeny field degree: * modcurve.level_structure Cyclic $\boldsymbol{N}$-torsion field degree: * modcurve.level_structure cyclic trigonal curve: * ag.cyclic_trigonal decomposable: * av.fq.all_polarizations_product decompose: * g2c.decomposition Dedekind domain: * ring.dedekind_domain Dedekind eta function: * mf.half_integral_weight.dedekind_eta defining ideal: * ag.variety defining parameters: * hgm.defining_parameters defining polynomial: * cmf.defining_polynomial * lf.defining_polynomial * mf.siegel.defining_polynomial * nf.defining_polynomial * nf.polredabs definite: * shimcurve.quaternion_algebra degree: * av.polarization * belyi.degree * character.dirichlet.degree * ec.isogeny * gg.degree * gg.group_action_invariants * group.permutation_representation * group.representation.complex_char_deg * hgm.degree * lf.degree * lfunction.degree * mf.siegel.degree * modcurve.modular_cover * nf.degree * st_group.degree degree $n$ Bézout matrix: * alg.bezout_matrix Degrees 5 to 40: * rcs.cande.lfunction Deligne module: * av.deligne_module density: * lattice.density depth: * ag.cluster_picture derived length: * group.derived_series derived series: * group.derived_series derived subgroup: * group.commutator_subgroup determinant: * artin.determinant * lattice.determinant determinant character: * modlgal.determinant determinant index: * modlgal.determinant_index diagonal: * st_group.moment_matrix dicyclic group: * group.dicyclic dihedral group: * group.dihedral dimension: * ag.dimension * artin.dimension * cmf.dimension * curve.highergenus.aut.dimension * mf.bianchi.newform * mf.bianchi.spaces * mf.hilbert.dimension * mf.siegel.dimension * modlgal.dimension Dimension of the corresponding {{KNOWL('ag.shimura_variety', "Shimura variety")}}: * rcs.source.curve.highergenus.aut dimension of the corresponding {{knowl(ag.shimura_variety, title=shimura variety)}}: * dq.curve.highergenus.aut.source direct factors: * group.direct_product direct product: * group.direct_product Dirichlet character: * character.dirichlet dirichlet character: * character.dirichlet.modulus dirichlet character group: * nf.dirichlet_group Dirichlet L-function: * lfunction.dirichlet dirichlet series: * lfunction.dirichlet_series discriminant: * ec.discriminant * ec.q.discriminant * g2c.discriminant * lattice.discriminant * lf.discriminant_exponent * nf.discriminant * nf.poly_discriminant * shimcurve.discb * shimcurve.disco discriminant exponent: * lf.discriminant_exponent discriminant group: * lattice.discriminant_group discriminant root field: * nf.discriminant_root_field distinguishes: * cmf.distinguishing_primes division: * group.division divisor function: * af.divisor_function do not include: * av.fq.initial_coefficients Downloading:: * intro * intro.features dual: * cmf.dualform * lattice.dual * lfunction.dual E: * ring.euclidean_domain Eichler of level $M$: * shimcurve.eichler Eichler {{KNOWL('shimcurve.order','order')}}: * shimcurve.eichler eigenvalue: * mf.maass.mwf.eigenvalue * mf.maass.picard.eigenvalue eisenstein: * lf.eisenstein_polynomial Eisenstein diagram of $I/K$: * lf.eisenstein_diagram Eisenstein form: * cmf.eisenstein_form * lf.eisenstein_form Eisenstein newform: * cmf.eisenstein_newform Eisenstein series: * cmf.eisenstein * cmf.eisenstein_series Eisenstein subspace: * cmf.eisenstein_form elementary: * group.elementary elevation: * av.fq.newton_elevation eligible: * av.fq.newton_elevation elliptic: * hmsurface.elliptic_point elliptic curve: * ec * ec.ring elliptic curve $E$ over $\mathbb{Q}$: * ec.q elliptic point: * hmsurface.elliptic_point * modcurve.elliptic_points * shimcurve.elliptic_points elliptic scheme: * ec.scheme elusive group: * gg.elusive_group embedded model: * modcurve.embedded_model embedding: * cmf.embedding * nf.embedding endomorphism: * ag.endomorphism_ring * ec.endomorphism endomorphism algebra: * ag.endomorphism_algebra * g2c.end_alg endomorphism field: * av.endomorphism_field endomorphism lattice: * g2c.jac_end_lattice endomorphism ring: * ag.endomorphism_ring * ec.endomorphism_ring * ec.q.endomorphism_ring endomorphisms: * ec.isogeny enhanced group (associated to $(O,\mu)$): * shimcurve.enhanced_group entanglement index: * modcurve.entanglement_index entanglement quotient: * modcurve.entanglement_index equations: * dq.curve.highergenus.aut.source Equations: * rcs.source.curve.highergenus.aut equivalence class of residual polynomials: * lf.residual_polynomials equivalent: * nf.absolute_value eta product: * cmf.eta_quotient eta quotient: * cmf.eta_quotient Euclidean basis: * lattice.euclidean_basis Euler factors: * lfunction.euler_product Euler product: * lfunction.euler_product euler product: * lfunction.riemann.euler_product eutactic: * lattice.eutactic even: * artin.parity * lattice.parity * mf.maass.mwf.symmetry even sublattice: * lattice.even_sublattice Even sublattice: * lattice.sublattice even universal: * lattice.quadratic_form everywhere good reduction: * ec.good_reduction exceptional: * ec.q.serre_invariants * mf.maass.exceptional_eigenvalue exceptional subgroup: * gl2.exceptional exponent of a {{KNOWL('group', 'group')}}: * group.exponent extended Cartan subgroup: * gl2.cartan extension: * group.nonsplit_product extreme: * lattice.extreme faithful: * group.permutation_representation * group.representation.faithful Faltings height: * ec.q.faltings_height Faltings ratio: * ec.q.faltings_ratio family: * lf.family_polynomial Feedback:: * intro * intro.features Festi--Veniani index: * lattice.festi_veniani_index fiber product: * modcurve.fiber_product field: * ring.field field count: * lf.family_field_count field of fractions: * ring.field_of_fractions field of values: * character.dirichlet.value_field first $a_2$ moment: * st_group.first_a2_moment first kind: * group.fuchsian Fitting subgroup: * group.fitting_subgroup fixed field: * character.dirichlet.field_cut_out Fourier coefficients: * cmf.fouriercoefficients Fourier expansion: * cmf.fouriercoefficients * mf.maass.mwf.fourierexpansion * mf.maass.picard.fourierexpansion Fourier-Bessel expansion: * mf.bianchi.fourierbessell fourth trace moment: * st_group.fourth_trace_moment Fraction Ring: * ring.field_of_fractions_mathlib fractional ideal: * ring.fractional_ideal Frattini subgroup: * group.frattini_subgroup Frey curve: * ec.q.frey Frey-Hellegouarch curve: * ec.q.frey Fricke involution: * cmf.fricke * mf.bianchi.sign Frobenius angles: * av.fq.frobenius_angles frobenius cycle type: * nf.frobenius_cycle_types Frobenius determinant: * modlgal.frobenius_determinant frobenius element: * nf.frobenius_cycle_types Frobenius group: * group.frobenius Frobenius matrix: * modlgal.frobenius_matrix Frobenius order: * modlgal.frobenius_order Frobenius prime: * modlgal.frobenius_prime Frobenius trace: * modlgal.frobenius_trace Frobenius-Schur indicator: * artin.frobenius_schur_indicator Fuchsian group: * group.fuchsian full group of automorphisms: * curve.highergenus.aut.full full level $N$ Shimura curve associated to $O$: * shimcurve full modular curve of level $N$: * modcurve.xn functional equation: * lfunction.functional_equation fundamental domain: * group.fuchsian.fundamental_domain fundamental units: * nf.fundamental_units G: * ec.galois_rep_modell_image g.q.isog: * av.fq.lmfdb_label g.q.isog.N.i: * av.fq.lmfdb_label g.q.isog.N.i.w: * av.fq.lmfdb_label g.q.isog.N.i.w.j: * av.fq.lmfdb_label g.q.isog.N.i.w.j.d.k: * av.fq.lmfdb_label G2-invariants: * g2c.g2_invariants galois: * nf.is_galois Galois Artin slopes: * lf.hidden_slopes Galois characters: * modlgal.dimension galois closure: * nf.galois_closure galois conjugate: * cmf.galois_conjugate Galois group: * gg.galois_group * nf.galois_group Galois Mean Slope: * lf.galois_invariants Galois mean slope: * lf.galois_mean_slope Galois orbit: * artin.galois_orbit * cmf.galois_orbit galois orbit: * character.dirichlet.galois_orbit galois representation: * cmf.galois_representation Galois root discriminant: * nf.galois_root_discriminant Galois Splitting Model: * lf.galois_invariants Galois splitting model: * lf.galois_splitting_model Galois Swan slopes: * lf.hidden_slopes Galois tame degree: * lf.tame_degree Galois unramified degree: * lf.unramified_degree gamma factors: * lfunction.gamma_factor Gassmann class: * modcurve.gassmann_class Gassmann equivalent: * group.gassmann_equivalence gauss sum: * character.dirichlet.gauss_sum general kissing number problem: * lattice.kissing general linear group: * group.general_linear generated by minimal vectors: * lattice.well_rounded generating vector: * curve.highergenus.aut.generatingvector generator: * nf.generator generators: * character.dirichlet.group.generators * group.fuchsian.cusps.width * group.generators generic: * g2c.maximal_galois_rep * gg.generic_polynomial generic polynomial: * lf.family_polynomial genus: * ag.curve.genus * belyi.genus * ec.galois_rep_adelic_image * gl2.genus * group.sl2z.genus_subgroup * lattice.genus * modcurve.genus * shimcurve.genus genus-rank difference: * modcurve.genus_minus_rank * shimcurve.genus_minus_rank geometric automorphism group: * g2c.geom_aut_grp geometric component: * ag.geom_component geometric discriminant: * lattice.discriminant geometric endomorphism algebra: * ag.geom_endomorphism_algebra * g2c.geom_end_alg geometric endomorphism ring: * ag.geom_endomorphism_ring * ec.geom_endomorphism_ring geometric genus: * ag.curve.genus geometric gonality: * ag.gonality_geom geometric isomorphism class: * g2c.geom_iso_class geometric ram: * lf.herbrand_invariant geometrically (or absolutely) simple: * ag.geom_simple geometrically irreducible: * ag.irreducible geometrically simple: * av.geometrically_simple geometrically squarefree: * av.geometrically_squarefree geometry type: * belyi.geometry_type global minimal model: * ec.global_minimal_model global period: * ec.period gonality: * ag.gonality good prime: * cmf.bad_prime * hgm.good_prime * modlgal.frobenius_prime good reduction: * ag.good_reduction * ec.good_reduction * ec.q.reduction_type * g2c.good_reduction Gram matrix: * lattice.discriminant_group gram matrix: * lattice.gram group: * group group $G$ acts on a compact Riemann surface $X$: * curve.highergenus.aut.group_action group action: * group.action group algebra decomposition: * curve.highergenus.aut.groupalgebradecomp group homomorphism: * group.homomorphism group isomorphism: * group.isomorphism group order: * lattice.group_order group ring: * group.group_ring Haar measure: * group.haar_measure half discriminant: * lattice.discriminant Hall subgroup: * group.subgroup.hall Hasse diagram: * combin.hasse_diagram Hasse-Weil conjecture: * g2c.hasse_weil_conjecture Hecke algebra: * mf.bianchi.hecke_algebra * mf.siegel.hecke_operator Hecke character: * character.hecke Hecke characteristic polynomial: * cmf.heckecharpolys Hecke eigenform: * mf.siegel.hecke_operator hecke field: * cmf.coefficient_field Hecke field: * mf.bianchi.newform * mf.bianchi.spaces * mf.siegel.coefficient_field Hecke operator: * cmf.hecke_operator Hecke operators: * mf.bianchi.hecke_algebra Hecke orbit: * cmf.hecke_orbit Hecke ring: * cmf.coefficient_ring * mf.siegel.coefficient_ring height: * lf.heights height pairing: * ag.canonical_height * ag.regulator * ec.canonical_height * ec.q.canonical_height * ec.q.regulator * ec.regulator Heisenberg group: * group.heisenberg Herbrand function: * lf.herbrand_function Herbrand invariant: * lf.herbrand_invariant Hermite number: * lattice.hermite_number hidden Artin slopes: * lf.hidden_slopes hidden Swan slopes: * lf.hidden_slopes Hilbert cusp form: * mf.hilbert Hilbert modular form: * mf.hilbert Hilbert modular surface: * hmsurface.hmsurface Hodge circle: * st_group.hodge_circle Hodge numbers: * hmsurface.hodge_numbers Hodge vector: * hgm.hodge_vector * mot.hodgevector holomorphic Euler characteristic: * ag.holomorphic_euler_char homogeneous: * ring.ideal homogeneous coordinates: * ag.projective_space homothetic: * ec.q.real_period Hyperbolic 3-space: * mf.bianchi.hyperbolic3space hyperbolic plane: * lattice.hyperbolic_plane hyperelementary: * group.hyperelementary hyperelliptic curve: * ag.hyperelliptic_curve hyperelliptic involution: * g2c.aut_grp hyperelliptic or cyclic trigonal curves: * dq.curve.highergenus.aut.source Hyperelliptic or Cyclic Trigonal Curves: * rcs.source.curve.highergenus.aut hypergeometric family: * hgm.familes * hgm.families Hypergeometric motives: * hgm.motive ideal: * ring.ideal ideal class group: * nf.ideal_class_group ideal label: * nf.ideal_labels identity: * alg.binary_operation.identity identity component: * g2c.st_group_identity_component * st_group.identity_component Igusa invariants: * g2c.igusa_invariants Igusa-Clebsch invariants: * g2c.igusa_clebsch_invariants image: * group.representation.image * modlgal.image image of the Artin representation: * artin.gg_quotient image order: * modlgal.image_order indecomposable: * av.fq.all_polarizations_product * gg.int_modules * lattice.orthogonal_decomposition indefinite: * lattice.signature * shimcurve.quaternion_algebra index: * av.fq.index_of_order * av.fq.zfv_index * character.dirichlet.conrey.index * ec.galois_rep_adelic_image * gg.index * gl2.index * group.subgroup.index * lf.eisenstein_diagram * modcurve.index * modlgal.image_index * shimcurve.index index 2 refinement: * modcurve.quadratic_refinements index of inseparability: * lf.indices_of_inseparability index of the number field: * nf.zk_index induced: * character.dirichlet.induce inertia group: * lf.inertia_group inertia subgroup: * lf.galois_invariants inertial height function: * nf.inertial_height_function inessential prime: * nf.inessential_prime infinite places: * nf.place infinity factor: * lfunction.infinity_factor inhomogeneous minimum: * lattice.covering_radius inner automorphism: * group.inner_automorphism inner twist count: * cmf.inner_twist_count inner twists: * cmf.inner_twist integral: * lattice.definition * nf.integral * ring.integral integral basis: * nf.integral_basis integral closure: * ring.integral integral domain: * ring.integral_domain integral model: * ec.integral_model integral points: * ec.q.integral_points integral representation: * gg.int_modules integral symplectic group: * group.sp2gr integrally closed: * ring.integrally_closed Interesting:: * intro intermediate fields: * lf.intermediate_fields * nf.intermediate_fields intrinsic subgroup: * ec.intrinsic_torsion inverse: * alg.binary_operation.inverse invertible: * av.fq.singular_dimensions irred: * g2c.galois_rep_image irreducible: * ag.irreducible * group.representation.irrep * ring.irreducible irreducible Coxeter group: * group.coxeter irreducible degree: * group.min_faithful_linear irreducible rational characters: * group.representation.rational_character irreps: * group.representation.irrep isoclinism: * group.isoclinism isog: * av.fq.lmfdb_label isogenous: * av.isogeny * ec.isogeny isogeny: * av.isogeny * ec.isogeny isogeny class: * av.isogeny_class * g2c.isogeny_class isogeny class (over a field $K$): * ec.isogeny_class isogeny class degree: * ec.isogeny_class_degree isogeny class index: * ec.q.lmfdb_label isogeny class label: * ec.q.lmfdb_label isogeny classes: * ec.isogeny isogeny decomposition: * modcurve.decomposition * shimcurve.decomposition isogeny factors: * av.decomposition isogeny graph: * ec.isogeny_graph isogeny matrix: * ec.isogeny_matrix isometric: * lattice.isometry isometry class: * lattice.isometry isomorphic over $K$: * ec.weierstrass_isomorphism isomorphism: * ec.isomorphism isomorphism class index: * ec.q.lmfdb_label isomorphism classes: * ec.isomorphism isotropy group: * cmf.eisenstein jacobi sum: * character.dirichlet.jacobi_sum jacobi symbol: * character.dirichlet.jacobi_symbol Jacobian: * ag.jacobian jump set: * lf.jump_set K-Bessel functions: * mf.bianchi.fourierbessell kernel: * av.polarization * group.representation.kernel * modcurve.modular_cover kissing number: * lattice.kissing kloosterman sum: * character.dirichlet.kloosterman_sum Knowls:: * intro * intro.features known issues with the rational points data: * rcs.rigor.modcurve known rational points: * g2c.known_rational_points Kodaira dimension: * hmsurface.kodaira_dimension kodaira symbol: * ec.kodaira_symbol * ec.q.kodaira_symbol Koecher principle: * mf.siegel.koecher.principle Kronecker symbol: * character.dirichlet.kronecker_symbol l-function: * g2c.lfunction L-function: * lfunction L-polynomial: * av.fq.l-polynomial l-polynomial: * g2c.lfunction label: * av.fq.lmfdb_label * belyi.label * character.dirichlet.galois_orbit_label * cmf.eisenstein_label * cmf.embedding_label * cmf.label * cmf.labels * group.label * group.label_complex_group_char * group.label_conjugacy_class * mf.maass.label * mf.siegel.label * modcurve.label * shimcurve.label label of a rational character: * group.label_rational_group_char laplace-beltrami operator: * mf.maass.mwf.laplacian large: * group.division_small_large largest: * cmf.maximal lattice: * lattice.definition lattice kissing number problem: * lattice.kissing lattice packing problem: * lattice.density law of quadratic reciprocity states: * character.dirichlet.jacobi_symbol leading coefficient: * ec.special_value * lfunction.leading_coeff Learn more:: * intro legendre symbol: * character.dirichlet.legendre_symbol level: * cmf.congruence_subgroup * cmf.level * ec.galois_rep_adelic_image * gl2.level * hmsurface.congruence_subgroup * hmsurface.level * lattice.level * mf.hilbert.level_norm * mf.maass.mwf.level * mf.siegel.level * modcurve.level * shimcurve.level level $1$ Shimura curve associated to $O$: * shimcurve level label: * mf.bianchi.level level norm: * mf.bianchi.level * mf.hilbert.level_norm linear degree: * group.min_faithful_linear LMFDB label: * ec.q.lmfdb_label local algebra: * nf.local_algebra local data: * ec.local_data local discriminant: * hgm.local_discriminant local factors: * lfunction.euler_product local field: * lf.local_field local Hecke algebra: * mf.siegel.hecke_operator local height: * lf.local_height_function local height function: * lf.local_height_function * nf.inertial_height_function local invariants: * g2c.local_invariants local minimal discriminant: * ec.local_minimal_discriminant local minimal model: * ec.local_minimal_model local number field: * lf.padic_field local obstruction: * modcurve.local_obstruction * shimcurve.local_obstruction local root number: * lf.root_number locally principal: * av.fq.singular_dimensions locally solvable: * g2c.locally_solvable logarithmic: * nf.weil_height lower central series: * group.lower_central_series lowest zero: * lfunction.zeros M: * dq.mf.hilbert.source M-group: * group.monomial maass form: * mf.maass * mf.maass.mwf.weight0 Maass form: * mf.maass.mwf * mf.maass.picard Major revision: * content.announcements Malle: * gg.malle_a * gg.malle_b Manin constant: * ec.q.manin_constant mass: * lattice.mass * lf.family_mass mass of the family: * lf.family_mass Max diag sublattice: * lattice.sublattice maximal: * av.fq.maximal_zfv * cmf.maximal * ec.maximal_elladic_galois_rep * ec.maximal_galois_rep * g2c.maximal_galois_rep * group.division_maximal * ring.maximal_ideal * shimcurve.order * st_group.supgroups maximal CM subfield: * nf.maximal_cm_subfield maximal ideal: * lf.maximal_ideal maximal order: * av.fq.order maximal prime: * ec.maximal_elladic_galois_rep maximal quotient: * group.maximal_quotient maximal subgroup: * group.maximal_subgroup maximal subgroups: * st_group.subgroups maximal under-subgroup: * group.under_subgroup maximal unramified subextension: * lf.unramified_subfield maximum ramified prime: * nf.max_p mean: * lf.means means: * lf.means metabelian: * group.metabelian Min vector sublattice: * lattice.sublattice minimal: * character.dirichlet.minimal * cmf.minimal * modcurve.modular_cover minimal discriminant: * ec.minimal_discriminant minimal discriminant norm: * ec.invariants minimal equation: * g2c.minimal_equation minimal field of definition: * ag.minimal_field minimal normal subgroup: * group.minimal_normal minimal over-subgroup: * group.over_subgroup minimal permutation degree: * group.permutation_degree minimal polynomial: * nf.minimal_polynomial minimal quadratic twist: * ec.q.minimal_twist minimal quartic twist: * ec.q.minimal_twist minimal sextic twist: * ec.q.minimal_twist minimal sibling: * nf.minimal_sibling minimal sibling of the splitting field: * modlgal.min_sib_splitting_field minimal supergroups: * st_group.supgroups minimal transitive degree: * group.transitive_degree minimal twist: * cmf.minimal_twist * cmf.twist_minimal * ec.q.minimal_twist * modcurve.minimal_twist minimal vector: * lattice.minimal_vector * lattice.minimum minimal weierstrass equation: * ec.q.minimal_weierstrass_equation minimum: * lattice.minimal_vector * lattice.minimum minkowski reduced form: * lattice.gram minus subspace: * cmf.minus_space mod-$\ell$ Galois representation: * av.galois_rep * ec.galois_rep * modlgal mod-$m$ Galois representation: * av.galois_rep * ec.galois_rep modular cover: * modcurve.modular_cover modular curve: * modcurve modular degree: * ec.q.modular_degree modular form: * cmf * mf modular form of half-integral {{ knowl(mf.elliptic.weight,title=weight)}}: * mf.half_integral_weight modular group: * group.sl2z modular parametrization: * ec.q.modular_parametrization ModularForm $\Gamma$ $k$: * cmf.space_mathlib_def module rank: * cmf.dimension_mathlib_def * nf.degree_mathlib_def modulus: * character.dirichlet * character.dirichlet.modulus moment: * st_group.trace_moments moment matrix: * st_group.moment_matrix moment sequence: * st_group.moments moment simplex: * st_group.moment_simplex Moment statistics: * st_group.moments monodromy group: * belyi.group monogenic: * nf.monogenic monoid: * group.group_mathlib_defn monomial: * group.monomial monomial order: * nf.monomial_order Mordell-Weil generators: * ec.mw_generators * g2c.mw_generator Mordell-Weil group: * ag.mordell_weil * ec.mordell_weil_group Mordell-Weil rank: * ag.mordell_weil * ec.mordell_weil_group mordell-weil rank: * g2c.mordell_weil_rank mordell-weil theorem: * ec.mordell_weil_theorem motivic weight: * lfunction.motivic_weight multiplicative: * af.multiplicative multiplicative reduction: * ec.multiplicative_reduction * ec.q.reduction_type * ec.reduction_type multiplicity: * cmf.inner_twist_multiplicity * cmf.twist_multiplicity * mf.maass.mwf.dimension multiplier system: * mf.multipliersystem Mumford representation: * g2c.mw_generator möbius quotient function: * group.mobius_function möbius subgroup function: * group.mobius_function Néron lattice: * ec.q.period_lattice N-torsion field degree: * modcurve.level_structure naive height: * ec.q.naive_height narrow class group: * nf.narrow_class_group narrow class number: * nf.narrow_class_number Navigating:: * intro * intro.features Navigation Bars:: * intro.features necklace: * modcurve.xns_plus.remarks negative definite: * lattice.signature negligible: * lf.eisenstein_diagram New section: * content.announcements new subspace: * cmf.eisenstein_newspace * cmf.newspace * cmf.subspaces * mf.siegel.newform * mf.siegel.newspace newform: * cmf.newform * mf.bianchi.spaces * mf.siegel.newform newform level: * modcurve.newform_level newform orbits: * cmf.galois_orbit newform subspace: * cmf.newform_subspace newspace: * mf.bianchi.spaces Newton polygon: * lf.newton_polygon nicknames: * nf.nickname nilpotency class: * group.nilpotent nilpotent: * group.nilpotent Nn: * ec.galois_rep_modell_image * gl2.normalizer_nonsplit_cartan Nn.a.b: * gl2.normalizer_nonsplit_cartan nodal: * ec.q.reduction_type Noetherian: * ring.noetherian non-cyclic at $\ell$: * av.fq.noncyclic_primes non-singular: * ag.singular_point non-split: * ec.q.reduction_type * group.semidirect_product non-split Cartan subgroup: * gl2.nonsplit_cartan non-split extension: * group.nonsplit_product non-split multiplicative reduction: * ec.nonsplit_multiplicative_reduction * ec.reduction_type non-torsion generators: * ec.mw_generators non-trivial: * group.trivial_subgroup non-unital ring: * ring nonabelian: * modlgal.abelian nonarchimedean: * nf.absolute_value nonarchimedean local field: * lf.local_field nondegenerate: * lattice.nondegenerate none: * stats.proportions nonmaximal primes: * g2c.galois_rep.non_maximal_primes nonsolvable: * modlgal.solvable nonsplit: * shimcurve.quaternion_algebra nonsplit Cartan: * modcurve.name nontrivial: * cmf.nontrivial_twist norm: * lattice.norm * lattice.quadratic_form norm 1 sublattice: * lattice.norm1_sublattice Norm 1 sublattice: * lattice.sublattice normal: * group.subgroup.normal normal closure: * group.subgroup.normal_closure normal series: * group.normal_series normalized: * lattice.normalized_minimal_vector normalized Haar measure: * group.haar_measure normalizer: * group.subgroup.normalizer normalizer of a cartan subgroup: * gl2.normalizer_cartan normalizer of a non-split Cartan subgroup: * gl2.normalizer_nonsplit_cartan normalizer of a split Cartan subgroup: * gl2.normalizer_split_cartan notation: * av.fq.endomorphism_ring_notation note that this excludes many known rational points: * rcs.cande.modcurve Ns: * ec.galois_rep_modell_image * gl2.normalizer_split_cartan Ns.a.b: * gl2.normalizer_split_cartan Ns.a.b.c: * gl2.normalizer_split_cartan nss: * g2c.galois_rep_image number field: * av.fq.number_field * nf number of Galois orbits: * belyi.num_orbits number of wild segments: * lf.wild_segments obstruction class: * ec.obstruction_class odd: * artin.parity * lattice.parity * mf.maass.mwf.symmetry old: * cmf.oldspace * mf.siegel.newform old subspace: * cmf.eisenstein_newspace * cmf.newspace * cmf.subspaces * mf.siegel.newform * mf.siegel.newspace oldforms: * mf.bianchi.spaces open subgroup: * gl2.open optimal: * ec.q.optimal orbit index: * character.dirichlet.galois_orbit_index orbit size: * belyi.orbit_size order: * av.fq.order * character.dirichlet.group.order * character.dirichlet.order * g2c.mw_generator_order * gg.invariants * group.element_order * group.order * nf.order * shimcurve.order order statistics: * group.order_stats ordinary: * av.fq.ordinary * ec.good_ordinary_reduction * ec.q.reduction_type * ec.reduction_type ordinary reduction: * ec.good_ordinary_reduction oriented necklace: * modcurve.xns.remarks orthogonal: * hgm.type orthogonal decomposition: * lattice.orthogonal_decomposition orthogonal group: * group.orthogonal other-dihedral group: * group.other_dihedral outer automorphism: * group.outer_aut outer automorphism group: * group.outer_aut over-subgroup: * group.over_subgroup overorder: * av.fq.order p-adic numbers: * lf.qp p-neighbor adjacency graph: * lattice.pneighbor_adjacency packet: * lf.packet parallel weight: * mf.hilbert.weight_vector paramodular conjecture: * g2c.paramodular_conjecture Paramodular subgroup: * mf.siegel.family parity: * character.dirichlet.parity * gg.group_action_invariants * gg.parity passport: * belyi.passport perfect: * group.perfect * lattice.perfect period lattice: * ec.q.period_lattice permutation module: * group.permutation_module permutation representation: * group.permutation_module * group.permutation_representation permutation triple: * belyi.permutation_triple petersson: * cmf.petersson_scalar_product_mathlib_def Petersson scalar product: * cmf.petersson_scalar_product Picard group: * av.fq.picard_group * mf.maass.picard place: * nf.place plane model: * ag.plane_model plus subspace: * cmf.plus_space polarization: * av.polarization * shimcurve.polarized_order polarization cone: * hmsurface.polarization_module polarization degree: * shimcurve.nrdmu polarization module: * hmsurface.polarization_module polarized order: * shimcurve.polarized_order polycyclic presentation: * group.presentation polycyclic sequence: * group.presentation polycyclic series: * group.presentation polynomial discriminant: * nf.discriminant_root_field positive definite: * lattice.positive_definite * lattice.postive_definite positive semidefinite: * lattice.nondegenerate potential complex multiplication: * ec.complex_multiplication potential good reduction: * ag.potential_good_reduction * ec.potential_good_reduction potential toric rank: * av.potential_toric_rank presentation: * group.presentation primary abelian invariants: * group.abelian_invariants primary decomposition: * group.primary_decomposition prime: * nf.prime * ring.prime_ideal prime above: * nf.prime prime of bad reduction: * ag.bad_prime * hgm.good_prime prime of good reduction: * hgm.good_prime primitive: * ag.primitive * character.dirichlet.primitive * ec.q.torsion_growth * gg.group_action_invariants * gg.primitive * lattice.primitive principal: * character.dirichlet.principal principal congruence subgroup: * cmf.congruence_subgroup * hmsurface.congruence_subgroup Principal congruence subgroup: * mf.siegel.family principal fractional ideal: * ring.principal_fractional_ideal principal homogeneous space: * ag.torsor principal ideal domain: * ring.principal_ideal_domain principally polarizable: * av.princ_polarizable probability: * st_group.probabilities product abelian variety: * av.is_product profinite group: * gl2.profinite projective (general) unitary group: * group.unitary projective algebraic set: * ag.variety projective Artin field: * artin.projective_field projective field: * cmf.projective_field projective galois representation: * cmf.galois_representation projective general linear group: * group.general_linear projective image: * artin.projective_image * cmf.projective_image * group.subgroup.projective_image projective kernel polynomial: * modlgal.projective_kernel_polynomial projective orthogonal groups: * group.orthogonal projective representation: * modlgal.projective_representation Projective space: * ag.projective_space projective special linear group: * group.psl2r * group.special_linear projective special unitary group: * group.unitary projective stem field: * artin.projective_stem_field projective symplectic group: * group.symplectic projective variety: * ag.variety projectivization: * ag.projective_space proper: * group.proper_subgroup Properties:: * intro * intro.features qExpansion: * cmf.q-expansion_mathlib_def quadratic discriminant: * lattice.discriminant quadratic form: * lattice.quadratic_form quadratic space: * lattice.definition quadratic twists: * ec.twists quality: * ec.q.abc_quality quartic twists: * ec.twists quasi-modular form: * mf.quasi_modular quaternion algebra: * shimcurve.quaternion_algebra Quaternionic (or Symplectic): * group.representation.type quaternionic multiplication: * g2c.geom_end_alg quotient curve: * ag.quotient_curve quotient genus: * curve.highergenus.aut.quotientgenus quotient group: * group.quotient radical: * group.radical radical discriminant: * nf.inertial_height_function * nf.radical_discriminant ramification index: * lf.ramification_index ramification polygon: * lf.ramification_polygon ramification polynomial: * lf.ramification_polygon ramification type: * belyi.ramification_type ramified: * artin.ramified_primes * modlgal.ramified * shimcurve.discb ramified prime: * nf.ramified_primes rams: * lf.rams Random:: * intro rank: * ec.rank * group.rank * lattice.dimension * nf.rank rational: * group.rational_group * lfunction.rational * modcurve.level_structure * st_group.rational rational $k$-points: * av.fq.group_structure rational character: * group.representation.rational_character rational character table: * group.rational_character_table rational conjugacy classes: * group.division rational cusps: * modcurve.cusps rational points: * g2c.all_rational_points Rational points: * modcurve real: * character.dirichlet.real Real: * group.representation.type real dimension: * st_group.real_dimension real embedding: * nf.real_embedding real endomorphism algebra: * ag.real_endomorphism_algebra real geometric endomorphism algebra: * ag.real_geom_endomorphism_algebra real index: * av.fq.zfv_real_index real multiplication: * ag.real_multiplication * cmf.rm_form * g2c.end_alg * g2c.geom_end_alg real period: * ec.q.period_lattice * ec.q.real_period * g2c.real_period real place: * nf.place reduced defining polynomial: * nf.polredabs reduced discriminant: * shimcurve.disco reduced minimal model: * ec.q.minimal_weierstrass_equation reducible: * group.representation.irrep reduction type: * ec.q.reduction_type * ec.reduction_type refined passport: * curve.highergenus.aut.refinedpassport reflex field: * nf.reflex_field reflex field of the reflex field: * nf.reflex_reflex_field regular: * gg.regular_extension regulator: * ag.regulator * ec.q.regulator * ec.regulator * g2c.regulator * nf.regulator related objects: * doc.related_to Related Objects:: * intro * intro.features relative class number: * nf.relative_class_number relative depth: * ag.cluster_picture relative dimension: * cmf.dimension * cmf.relative_dimension * mf.siegel.dimension relative discriminant: * av.fq.zfv_rel_disc_norm relative family: * lf.family_polynomial relative index: * modcurve.relative_index relative orders: * group.presentation reliability: * rcs representation: * group.representation.irrep rescaled dual: * lattice.dual residual polynomial: * lf.residual_polynomials residual polynomials: * lf.residual_polynomials residue degree: * nf.frobenius_cycle_types residue field: * lf.residue_field * modcurve.point_residue_field residue field characteristic: * lf.residue_field residue field degree: * lf.residue_field_degree resolvent fields: * gg.group_action_invariants resolvents: * gg.resolvents riemann hypothesis: * lfunction.rh Riemann surface: * ag.riemann_surface Riemann zeta-function: * lfunction.riemann Riemann-Siegel theta function: * lfunction.zfunction ring: * ring ring of integers: * lf.ring_of_integers * nf.ring_of_integers ring without identity: * ring rng: * ring root analytic conductor: * lfunction.root_analytic_conductor root angle: * lfunction.root_angle root discriminant: * nf.root_discriminant root lattice: * lattice.root_lattice * lattice.root_sublattice root number: * artin.root_number * lfunction.sign root sublattice: * lattice.root_sublattice Root sublattice: * lattice.sublattice root system: * lattice.root_lattice roots: * lattice.root_lattice rotation number: * hgm.imprimitivity_index * hgm.rotation_number RSZB label: * gl2.label S: * cmf.sturm_bound_gamma1 S4: * ec.galois_rep_modell_image * gl2.exceptional Sage Cell Server: * sage.cell Satake angles: * cmf.satake_angles Satake parameters: * cmf.satake_parameters sato-tate group: * cmf.sato_tate Sato-Tate group: * g2c.st_group * st_group.definition scalar-valued Siegel modular form: * mf.siegel Schur index: * group.representation.schur_index Schur multiplier: * group.schur_multiplier Searching:: * intro * intro.features second trace moment: * st_group.second_trace_moment selberg axioms: * lfunction.selberg_class.axioms Selberg class: * lfunction.selberg_class Selberg data: * lfunction.selbergdata self dual: * cmf.selfdual self twist: * mf.ellitpic.self_twist self-dual: * lfunction.self-dual self-twist: * cmf.self_twist semi-dihedral group: * group.semi_dihedral semi-global minimal models: * ec.semi_global_minimal_model semiabelian variety: * av.semiabelian_variety semicanonical polynomials: * lf.family_polynomial semiconcentrated: * gg.semiconcentrated semidirect factor: * group.semidirect_product semidirect product: * group.semidirect_product semistable: * ec.q.semistable * ec.semistable separable: * nf.separable separable algebra: * nf.separable_algebra Serre conductor: * ec.q.serre_invariants Serre invariants: * ec.q.serre_invariants Serre Odlyzko bound: * nf.serre_odlyzko_bound Serre weight: * ec.q.serre_invariants set of heights: * lf.heights sextic twists: * ec.twists shimura correspondence: * cmf.shimura_correspondence short Weierstrass model: * ec.simple_equation siblings: * gg.other_representations * nf.sibling Siegel $\Phi$ operator: * mf.siegel.phi Siegel modular form: * mf.siegel Siegel modular form with character $\psi$: * mf.siegel.character Siegel parabolic subgroup: * mf.siegel.family sign: * artin.root_number * lfunction.root_number * lfunction.sign * mf.bianchi.sign signature: * curve.highergenus.aut.signature * hgm.signature * lattice.nondegenerate * lattice.signature * lfunction.signature * nf.signature * nf.unit_signature_rank similar: * lattice.similarity simple: * ag.simple * av.simple * group.simple * modcurve.simple * shimcurve.simple simplified equation: * ec.simple_equation * g2c.simple_equation singular: * ag.plane_model singular primes: * av.fq.singular_primes singular support: * av.fq.singular_primes size: * ag.cluster_picture * group.size_conjugacy_class slash operator: * mf.siegel.slash.op slope polygon: * lf.means slopes: * lf.newton_slopes small: * group.division_small_large small rams: * lf.rams Smith abelian invariants: * group.abelian_invariants smooth: * ag.curve.smooth * ag.plane_model socle: * group.socle solvable: * gg.invariants * group.solvable * modlgal.solvable source: * rcs space: * cmf.space special linear group: * group.special_linear special orthogonal groups: * group.orthogonal special unitary group: * group.unitary special value: * ec.q.special_value * ec.special_value spectral index: * mf.maass.spectral_index spectral parameter: * mf.maass.mwf.spectralparameter * mf.maass.picard.eigenvalue spectral parameters: * lfunction.spectral_parameters sphere packing problem: * lattice.density spherical $t$-design: * lattice.spherical_design spherical $t$-design lattice: * lattice.spherical_design spinor norm: * group.spinor_norm split: * ec.q.reduction_type * g2c.decomposition * group.semidirect_product * shimcurve.discb * shimcurve.quaternion_algebra split Cartan: * modcurve.name split Cartan subgroup: * gl2.split_cartan split extension: * group.nonsplit_product split multiplicative reduction: * ec.reduction_type * ec.split_multiplicative_reduction splitting field: * ag.splitting_field * modlgal.splitting_field sporadic groups: * group.sporadic squarefree: * av.squarefree stable Faltings height: * ec.q.faltings_height standard generators: * curve.highergenus.aut.generators Stark unit: * cmf.stark_unit stem extension: * group.stem_extension stem field: * nf.stem_field stem fields of the polynomial: * nf.stem_field stem group: * group.stem_group stem subgroup: * group.stem_extension strict class group: * nf.narrow_class_group strict class number: * nf.narrow_class_number strongly eutactic: * lattice.eutactic strongly perfect: * lattice.spherical_design strongly well-rounded: * lattice.well_rounded structure: * character.dirichlet.group.structure Sturm bound: * cmf.sturm_bound subfamily: * lf.subfamily subfields: * gg.subfields subgroup: * group.subgroup subgroup diagram: * group.subgroup.diagram sublattice: * lattice.sublattice subnormal series: * gg.subnormal_series * group.normal_series Successive minima: * lattice.successive_minima supersingular: * av.fq.supersingular * ec.good_supersingular_reduction * ec.q.reduction_type * ec.reduction_type supersingular reduction: * ec.good_supersingular_reduction supersolvable: * group.supersolvable surjective: * modlgal.surjective Swan slope: * lf.galois_invariants Swan slope content: * lf.slopes Swan slopes: * lf.slope_content * lf.slopes * lf.swan_slopes symmetric: * av.theta_divisor symmetric bilinear form: * alg.symmetric_bilinear_form symmetric group: * group.symmetric symplectic: * hgm.type symplectic form: * st_group.symplectic_form symplectic group: * group.symplectic symplectic isomorphism: * alg.symplectic_isomorphism Szpiro ratio: * ec.q.szpiro_ratio tamagawa number: * ec.tamagawa_number Tamagawa number: * g2c.tamagawa tamagawa product: * ec.tamagawa_number Tamagawa product: * g2c.tamagawa tame: * hgm.tame tame degree: * lf.galois_invariants * lf.tame_degree tame ramification index: * lf.ramification_index tamely ramified: * lf.ramification_index Tate-Shafarevic group: * ec.q.analytic_sha_order Tate-Shafarevich group: * ag.tate_shafarevich * ec.analytic_sha_order tensor decomposable: * lattice.tensor_decomposition tensor decomposition: * lattice.tensor_decomposition tensor indecomposable: * lattice.tensor_decomposition tensor product: * lattice.tensor_decomposition the: * ec.q.real_period the absence of rational points should not be taken as an indication that none are known: * rcs.rigor.modcurve the highest weight: * repn.highest.weight the identity element: * alg.binary_operation.identity theta divisor: * av.theta_divisor theta function: * mf.half_integral_weight.theta theta series: * lattice.theta tiny rams: * lf.rams top Artin slope: * lf.top_slope * modlgal.top_slope topologically equivalent: * curve.highergenus.aut.topological_equivalence torsion bilinear form: * lattice.discriminant_group torsion generator: * nf.torsion torsion generators: * ec.mw_generators torsion growth: * ec.q.torsion_growth torsion order: * ec.torsion_order * g2c.torsion_order torsion primes: * g2c.torsion_order torsion quadratic form: * lattice.discriminant_group torsion structure: * ec.torsion_subgroup torsion subgroup: * ag.mordell_weil * ec.q.torsion_subgroup * ec.torsion_subgroup * g2c.torsion * group.torsion torsion {{KNOWL('group', 'group')}}: * group.torsion torsor: * ag.torsor totally complex: * nf.totally_imaginary totally imaginary: * nf.totally_imaginary totally positive: * nf.totally_positive totally ramified: * lf.ramification_index totally real: * nf.totally_real trace bound: * cmf.trace_bound trace form: * cmf.space_trace_form * cmf.trace_form trace zero density: * st_group.trace_zero_density transitive: * belyi.permutation_triple * group.action * group.permutation_representation * group.wreath_product transitivity number: * gg.transitivity triangle group: * group.triangle_group trigonal automorphism: * ag.cyclic_trigonal trivial: * character.dirichlet.principal trivial absolute value: * nf.absolute_value trivial semidirect product: * group.semidirect_product trivial subgroup: * group.trivial_subgroup trivial zeros: * lfunction.trivial_zero true eigenvalue: * mf.maass.picard.eigenvalue twist: * av.twist * cmf.twist * ec.twists twist class: * cmf.twist twist equivalent: * cmf.twist twist minimal: * cmf.twist_minimal twists: * modcurve.minimal_twist type: * hgm.type Type - length: * group.type types: * cmf.projective_image * group.representation.type under-subgroup: * group.under_subgroup underlying unit group: * character.unit_group uniformizer: * lf.ramification_index unimodular: * lattice.unimodular unique factorization domain: * ring.unique_factorization_domain unit: * ring.unit unit group: * nf.unit_group unit signature rank: * nf.unit_signature_rank unitary symplectic group: * st_group.usp universal: * lattice.quadratic_form unramified: * artin.unramified_primes * lf.ramification_index * modlgal.ramified * nf.frobenius_cycle_types unramified (rational) prime: * nf.unramified_prime unramified degree: * lf.galois_invariants * lf.unramified_degree unramified subfield: * lf.unramified_subfield upper central series: * group.upper_central_series upper half-plane: * mf.upper_half_plane use with caution: * rcs.rigor.modcurve * rcs.rigor.shimcurve vector-valued Siegel modular forms: * mf.siegel visible Artin slopes: * lf.slopes * lf.visible_slopes visible Swan slopes: * lf.slopes * lf.visible_slopes Von Dyck group: * group.triangle_group vs unconstrained: * stats.proportions Warning: * content.ongoing weak equivalence class: * av.fq.weak_equivalence_class weakly equivalent: * av.weak_equivalence_class Weierstrass coefficients: * ec.weierstrass_coeffs Weierstrass equation: * ec.weierstrass_coeffs * g2c.g2curve Weierstrass isomorphisms: * ec.weierstrass_isomorphism Weierstrass model: * ag.hyperelliptic_curve * ec.weierstrass_coeffs Weierstrass points: * g2c.num_rat_wpts weight: * cmf.weight * mf.bianchi.weight * mf.hilbert.weight_vector * mf.maass.mwf.weight * mf.siegel.weight * mf.siegel.weight_k_j * st_group.weight weight $k$ slash-action: * mf.slash_action weight function: * group.weight_function * nf.inertial_height_function Weil $q$-polynomial: * nf.weil_polynomial Weil height: * nf.weil_height Weil polynomial: * av.fq.weil_polynomial well-rounded: * lattice.well_rounded Weyl group: * group.weyl_group weyl group: * lattice.root_lattice width: * group.fuchsian.cusps.width * modcurve.cusp_widths wild: * hgm.wild wild Artin slopes: * lf.galois_invariants * lf.slope_content * lf.slopes * lf.wild_slopes wild inertia group: * lf.wild_inertia_group wild ramification exponent: * lf.ramification_index wild Swan slopes: * lf.slopes Witt discriminant: * lattice.discriminant wreath product: * group.wreath_product Z-function: * lfunction.zfunction Z-group: * group.z_group zero divisor: * ring.zero_divisor zeros: * lfunction.zeros Zeros of L-functions Search:: * intro.features zigzag plot: * hgm.zigzagplot zigzag procedure: * hgm.zigzagplot {{ knowl(mf.elliptic.newform, title=newform) }}: * cmf.decomposition.new.gamma0chi {{KNOWL('ag.jacobian', "Jacobian variety")}} decomposition: * rcs.source.curve.highergenus.aut {{KNOWL('ec.mordell_weil_group', 'Mordell-Weil group')}}: * ec.mw_generators {{KNOWL('ec.potential_good_reduction','potential good reduction')}}: * ec.reduction {{KNOWL('ec.semistable','semistable')}}: * ec.reduction {{KNOWL('lfunction.known_degree1','Degree 1')}}: * rcs.cande.lfunction {{KNOWL('lfunction.known_degree2','Degree 2')}}: * rcs.cande.lfunction {{KNOWL('lfunction.known_degree3','Degree 3')}}: * rcs.cande.lfunction {{KNOWL('lfunction.known_degree4','Degree 4')}}: * rcs.cande.lfunction {{knowl(ag.jacobian, title=jacobian variety)}} decomposition: * dq.curve.highergenus.aut.source étale algebra: * av.fq.order