Search Filter Category: abvar af ag artin av bio character content crystals curve data dc doc dq ec field g2c gg gl2 group hgm intro lattice lf lfunction mf mot mwf nf repn sage specialfunction st st_group test uploader users varieties zeros Quality: Beta Ok Reviewed More Recently modified Knowls
 A mf.gl2.history A brief history of $\textrm{GL}(2)$ modular forms lfunction.history A history of L-functions intro.tutorial A tour of the Riemann zeta function st_group.abelian Abelian ag.abelian_surface Abelian surface ag.abelian_variety Abelian variety varieties.about About varieties g2c.abs_discriminant Absolute discriminant group.galois.absolute Absolute Galois group ec.additive_reduction Additive reduction nf.algebraic_closure Algebraic closure of a field ag.curve Algebraic curve mf.gl2.history.varieties Algebraic varieties ag.variety Algebraic variety st_group.ambient Ambient group of a Sato-Tate group lfunction Analytic L-function ec.q.analytic_sha_order Analytic order of Ш ec.q.analytic_sha_value Analytic order of Sha (numerical value) lfunction.analytic_rank Analytic rank g2c.analytic_rank Analytic rank bound mf.bianchi.anr Analytic rank of a Bianchi newform av.fq.angle_rank Angle rank of a set lfunction.arithmetic Arithmetic L-function nf.arithmetically_equivalent Arithmetically equivalent fields artin.lfunction Artin L-function lfunction.history.artin Artin L-functions nf.assuming_grh Assuming GRH mf.elliptic.atkinlehner Atkin-Lehner eigenvalue mf.maass.mwf.atkinlehner Atkin-Lehner eigenvalues mf.elliptic.atkin-lehner Atkin-Lehner operator $W_Q$ on $X_0(N)$ bio.selberg Atle Selberg lf.automorphism_group Automorphism group of a field extension g2c.aut_grp Automorphism group of an algebraic curve mf.siegel.extent.kp Available samples for $M_2(K(p))$ mf.siegel.extent.sp4z Available samples for $M_k(\mathrm{Sp}(4,\mathbb{Z}))$ mf.siegel.extent.sp6z Available samples for $M_k(\mathrm{Sp}(6,\mathbb{Z}))$ mf.siegel.extent.sp8z Available samples for $M_k(\mathrm{Sp}(8,\mathbb{Z}))$ mf.siegel.extent.sp4z_2 Available samples for $M_{k,2}(\mathrm{Sp}(4,\mathbb{Z}))$ B ec.bad_reduction Bad reduction of an elliptic curve at a prime ec.base_change Base change ag.base_change Base change mf.bianchi.base_change Base-change Bianchi modular forms af.bernoulli_numbers Bernoulli numbers mf.bianchi.bianchigroup Bianchi Groups mf.bianchi.bianchimodularforms Bianchi modular forms mf.bianchi.newform Bianchi newform ec.q.bsdconjecture Birch and Swinnerton-Dyer conjecture ec.q.bsd_invariants Birch and Swinnerton-Dyer invariants of an elliptic curve/$\mathbb{Q}$ ec.q.galois_rep_image_borel Borel case gl2.borel Borel subgroup curve.highergenus.aut.branchpoints Branch Points C g2c.277.a.277.1.top C277 nf.polredabs Canonical defining polynomial for number fields ec.q.canonical_height Canonical height gl2.cartan Cartan subgroup lattice.catalogue_of_lattices Catalogue of Lattices lfunction.central_character Central character of an L-function lfunction.central_point Central point of an L-function lfunction.central_value Central value of an L-function mf.maass.mwf.character Character of a Maass form mf.elliptic.character Character of a modular form gg.character_table.data Character table data nf.class_number Class number of a number field lattice.class_number Class number of an integral lattice mf.bianchi.cm CM Bianchi newforms ag.cm_field CM field mf.elliptic.cm_form CM form mf.elliptic.coefficient_field Coefficient field for newforms mf.gl2.history.combinatorics Combinatorcs and $q$-series dq.mf.bianchi.extent Completeness of the Bianchi modular form data ec.complex_multiplication Complex multiplication ag.complex_multiplication Complex multiplication st_group.component_group Component group of a Sato-Tate group st_group.components Components of a Sato-Tate group ec.conductor_label Conductor labels for elliptic curves character.dirichlet.conductor Conductor of a Dirichlet character character.dirichlet.conrey.conductor Conductor of a Dirichlet character from its Conrey label hgm.conductor Conductor of a hypergeometric motive nf.conductor Conductor of an abelian number field ag.conductor Conductor of an abelian variety artin.conductor Conductor of an Artin representation ec.conductor Conductor of an elliptic curve ec.q.conductor Conductor of an elliptic curve over $\Q$ lfunction.conductor Conductor of an L-function ec.conductor_valuation Conductor valuation data.congruent_numbers Congruent numbers data gg.conjugacy_classes.data Conjugacy class representatives gg.conjugacy_classes Conjugacy classes st_group.connected Connected Sato-Tate group character.dirichlet.conrey.index Conrey index of a Dirichlet character character.dirichlet.conrey Conrey label of Dirichlet characters artin.permutation_container Containing permutation representation artin.database_comment Content of the Artin representations database doc.create_knowl Create a knowl ec.q.cremona_label Cremona label lfunction.critical_line Critical line of an L-function lfunction.critical_strip Critical strip crystals.crystal Crystal mf.elliptic.cusp_form Cusp form (elliptic) group.fuchsian.cusps Cusps of a subgroup of the modular group ag.cyclic_trigonal Cyclic Trigonal Curve D nf.field.data Data for a global number field lf.field.data Data for a local field lattice.data Data for an integral lattice group.small.data Data on small group zeros.zeta.contents David Platt's computation of zeros of the zeta function av.decomposition Decomposition into isogeny factors av.decomposition.data Decomposition into isogeny factors data g2c.decomposition Decomposition of the Jacobian of a genus 2 curve mf.half_integral_weight.dedekind_eta Dedekind eta function lfunction.history.dedekind Dedekind zeta-functions, history hgm.defining_parameters Defining parameters of a hypergeometric motive lf.defining_polynomial Defining polynomial of a local field nf.defining_polynomial Defining Polynomial of a Number Field lf.qp Definition of $\Q_p$ gg.degree Degree lfunction.known_degree1 Degree 1 L-functions, all lfunction.known_degree2 Degree 2 L-functions, examples lfunction.known_degree3 Degree 3 L-functions, examples lfunction.known_degree4 Degree 4 L-functions, examples lf.degree Degree of a local field extension nf.degree Degree of a number field st_group.degree Degree of a Sato-Tate group mf.siegel.degree Degree of a Siegel modular form lfunction.degree Degree of an L-function lattice.density Density of an integral lattice doc.knowl.description Description field in knowls artin.determinant Determinant of an Artin representation lattice.determinant Determinant of an integral lattice doc.development Developers Guide curve.highergenus.aut.dimension Dimension of a family of curves mf.hilbert.dimension Dimension of a Hecke constituent mf.elliptic.dimension_galois_orbit Dimension of a Hecke orbit ag.dimension Dimension of an algebraic variety artin.dimension Dimension of an Artin representation mwf.dimension Dimension of an eigenvalue lattice.dimension Dimension of an integral lattice character.dirichlet Dirichlet character artin.dimensionone Dirichlet character associated with a 1-dimensional Artin representation dq.character.dirichlet.extent Dirichlet character data extent character.dirichlet.group Dirichlet character group nf.dirichlet_group Dirichlet group of an Abelian number field lfunction.dirichlet Dirichlet L-function lfunction.dirichlet_series Dirichlet series lfunction.history.dirichlet Dirichlet's work on L-functions lf.discriminant_exponent Discriminant exponent of a local field mf.1.12.1.a.top Discriminant modular form g2c.discriminant Discriminant of a genus 2 curve nf.discriminant Discriminant of a number field ec.discriminant Discriminant of a Weierstrass equation ec.q.discriminant Discriminant of an elliptic curve over $\mathbb Q$ lf.discriminant_root_field Discriminant root field of a local field ec.discriminant_valuation Discriminant valuation af.divisor_function Divisor function bio.zagier Don Bernard Zagier lattice.dual Dual lattice mf.siegel.lift.ikeda Duke-Imamoglu-Ikeda lift E lattice.8.1.1.1.1.top E_8 Lattice ec.q.37.a1.bottom Elliptic Curve 37a1 ec.q.66.c4.bottom Elliptic curve 66.c4 ec.q.858.k1.bottom Elliptic curve 858.k1 ec.invariants Elliptic curve invariants ec.q.invariants Elliptic curve invariants ec.q.189.b1.top Elliptic curve with label 189.b1 mf.gl2.history.elliptic Elliptic functions bio.artin Emil Artin ag.endomorphism_algebra Endomorphism algebra g2c.jac_endomorphisms Endomorphism invariants g2c.jac_end_lattice Endomorphism lattice ec.endomorphism Endomorphism of an elliptic curve ag.endomorphism_ring Endomorphism ring ec.q.endomorphism_ring Endomorphism ring of an elliptic curve ec.endomorphism_ring Endomorphism ring of an elliptic curve g2c.equation Equation of a genus 2 curve over $\Q$ mf.11.2.1.a.formulas Eta product for mf.11.2.1.a mf.12.3.5.a.formulas Eta product for mf.12.3.5.a mf.14.2.1.a.formulas Eta product for mf.14.2.1.a mf.15.2.1.a.formulas Eta product for mf.15.2.1.a mf.16.3.15.a.formulas Eta product for mf.16.3.15.a mf.2.8.1.a.formulas Eta product for mf.2.8.1.a mf.20.2.1.a.formulas Eta product for mf.20.2.1.a mf.24.2.1.a.formulas Eta product for mf.24.2.1.a mf.27.2.1.a.formulas Eta product for mf.27.2.1.a mf.3.6.1.a.formulas Eta product for mf.3.6.1.a mf.32.2.1.a.formulas Eta product for mf.32.2.1.a mf.36.2.1.a.formulas Eta product for mf.36.2.1.a mf.4.5.3.a.formulas Eta product for mf.4.5.3.a mf.4.6.1.a.formulas Eta product for mf.4.6.1.a mf.5.4.1.a.formulas Eta product for mf.5.4.1.a mf.6.4.1.a.formulas Eta product for mf.6.4.1.a mf.7.3.6.a.formulas Eta product for mf.7.3.6.a mf.8.3.3.a.formulas Eta product for mf.8.3.3.a mf.8.4.1.a.formulas Eta product for mf.8.4.1.a mf.9.4.1.a.formulas Eta product for mf.9.4.1.a mf.144.2.1.a.formulas Eta quotient for mf.144.2.1.a mf.48.2.1.a.formulas Eta quotient for mf.48.2.1.a mf.64.2.1.a.formulas Eta quotient for mf.64.2.1.a mf.80.2.1.b.formulas Eta quotient for mf.80.2.1.b mf.half_integral_weight.eta_quotients Eta quotients g2c.bad_lfactors Euler factors at bad primes specialfunction.gamma Euler gamma function lfunction.riemann.euler_product Euler product for the Riemann zeta function lfunction.euler_product Euler product of an L-function lfunction.history.euler Euler's work on the zeta function st_group.probabilities Event probabilities ec.q.galois_rep_image_exc Exceptional case gl2.exceptional Exceptional subgroup doc.view_knowl Expand a knowl dq.artin.extent Extent of Artin representation data mf.elliptic.extent Extent of data for classical modular forms dq.ec.extent Extent of data for elliptic curves over $\Q$ dq.curve.highergenus.aut.extent Extent of data for families of higher genus curves with automorphisms dq.hecke_algebras.extent Extent of data for Hecke Algebras dq.lattice.extent Extent of data for Integral Lattices dq.rep_galois_modl.extent Extent of data for mod ℓ Galois representations dq.modlmf.extent Extent of data for mod ℓ Modular Forms dq.ecnf.extent Extent of elliptic curve data over number fields dq.gg.extent Extent of Galois group data dq.g2c.extent Extent of genus 2 curve data dq.nf.extent Extent of global number field data dq.mf.hilbert.extent Extent of Hilbert modular form data dq.lf.extent Extent of local number field data dq.zeros.zeta.extent Extent of Riemann zeta zeros data dq.st.extent Extent of Sato-Tate group data F nf.20.2.206007596521214410095208558252435839890349094339.1.bottom Field associated with 19-torsion on an elliptic curve character.dirichlet.field_cut_out Field cut out by a Dirichlet character ag.base_field Field of definition character.dirichlet.value_field Field of values of a Dirichlet character field.about Fields ec.q.52.a1.top First example of a congruence modulo 13 ec.q.988.c1.top First example of a congruence modulo 13 ec.q.47775.be1.top First example of a congruence modulo 17 mf.1.12.1.a.formulas Formula for mf.1.12.1a mf.elliptic.fouriercoefficients Fourier coefficients of a modular form mf.maass.picard.fourierexpansion Fourier expansion of a Maass form on PSL(2,Z[i]) mf.maass.mwf.fourierexpansion Fourier expansion of Maass form mf.maass.mwf.fricke Fricke eigenvalue mf.elliptic.fricke Fricke involution for Gamma_0 av.fq.frobenius_angles Frobenius angles nf.frobenius_cycle_types Frobenius cycle types artin.frobenius_schur_indicator Frobenius-Schur indicator of a representation group.fuchsian Fuchsian group lfunction.functional_equation Functional equation of an L-function group.fuchsian.fundamental_domain Fundamental domain group.sl2z.fundamental_domain Fundamental domain of SL$_2\mathbb{Z}$ nf.fundamental_units Fundamental units of a number field dq.av.fq.further_extent Further extent of the data on isogeny classes of abelian varieties over finite fields G g2c.g2_invariants G2-invariants of a genus 2 curve nf.galois_closure Galois closure of an extension gg.galois_group Galois group nf.galois_group.data Galois group information gg.label Galois group label nf.galois_group Galois group of a number field nf.galois_group.name Galois group of a number field -- naming convention av.fq.galois_group Galois group of an isogeny class of abelian varieties lf.galois_invariants Galois invariants of a local field nf.galois_group.gmodule Galois module information nf.galois_group.gmodule_v4_type_i_i Galois module structure of Type I (i) for units in biquadratic number fields nf.galois_group.gmodule_v4_type_i_ii Galois module structure of Type I (ii) for units in biquadratic number fields nf.galois_group.gmodule_v4_type_i_iii Galois module structure of Type I (iii) for units in biquadratic number fields nf.galois_group.gmodule_v4_type_ii_i Galois module structure of Type II (i) for units in biquadratic number fields nf.galois_group.gmodule_v4_type_ii_ii Galois module structure of Type II (ii) for units in biquadratic number fields nf.galois_group.gmodule_v4_type_iii Galois module structure of Type III for units in biquadratic number fields nf.galois_group.gmodule_v4_type_iv Galois module structure of Type IV for units in biquadratic number fields mf.elliptic.galois-orbits Galois orbit mf.siegel.galois_orbit Galois orbit character.dirichlet.galois_orbit Galois orbit of a Dirichlet character artin.galois_orbit Galois orbit of an Artin representation ec.galois_rep Galois representations attached to an elliptic curve character.dirichlet.gauss_sum Gauss sum lfunction.rh Generalized Riemann hypothesis curve.highergenus.aut.generatingvector Generating Vector nf.generator Generator of a number field nf.field_generator Generator of a number field character.dirichlet.group.generators Generators of a Dirichlet group st_group.generators Generators of a Sato-Tate group g2c.g2curve Genus 2 curve g2c.249.a.249.1.top Genus 2 curve 249.a.249.1 g2c.997.b.997.1.top Genus 2 curve 997.b.997.1 g2c.1116.a.214272.1.top Genus 2 curve of conductor 1116 g2c.976.a.999424.1.top Genus 2 curve of conductor 976 ag.curve.genus Genus of a smooth curve group.sl2z.genus_subgroup Genus of a subgroup of the modular group lattice.genus Genus of an integral lattice g2c.geom_aut_grp Geometric automorphism group g2c.geom_iso_class Geometric isomorphism class of a curve ag.geom_simple Geometrically simple av.geom_simple Geometrically simple bio.riemann Georg Friedrich Bernhard Riemann ec.global_minimal_model Global minimal model ec.good_ordinary_reduction Good ordinary reduction ec.good_reduction Good reduction ec.good_supersingular_reduction Good supersingular reduction lattice.gram Gram matrix mf.siegel.lift.gritsenko Gritsenko lift group.gammann Group $\Gamma(N)$ mf.sl2.subgroup.gamma0n Group $\Gamma_0(N)$ subgroup of $\textrm{SL}(2,\mathbb{Z})$ group.sp2gr.gamma0 Group $\Gamma_0(N)$, subgroup of $\mathrm{Sp}(2g,\mathbb Z)$ mf.sl2.subgroup.gamma1n Group $\Gamma_1(N)$ subgroup of $\textrm{SL}(2,\mathbb{Z})$ gg.group_action_invariants Group action Invariants gg.invariants Group invariants lattice.group_order Group order of an integral lattice group.about Groups mf.growth_condition Growth condition for a modular form H mf.half_integral_weight Half-integral weight modular forms bio.harald_schilly Harald Schilly nf.17.1.604462909807314587353088.1.bottom Harbater number field g2c.has_square_sha Has square Ш lfunction.history.hasse_weil Hasse-Weil L-function mf.bianchi.hecke_algebra Hecke algebra character.hecke Hecke character mf.gl2.history.hecke Hecke Characters lfunction.history.hecke_characters Hecke L-functions mf.elliptic.hecke-operator Hecke operator of index $n$ lfunction.history.hecke_operators Hecke operators, history mf.elliptic.hecke-orbit Hecke orbit lattice.hermite_number Hermite number hgm.field.label HGM Labels crystals.highest_weight Highest weight crystals mf.hilbert Hilbert modular form nf.ideal.label.hmf Hilbert modular form convention for the label of a number field ideal ec.q.5077.a1.top Historical information for 5077a1 (the Gauss elliptic curve) lattice.history History of Lattices mot.hodgevector Hodge vector of a hypergeometric motive mf.elliptic.eisenstein Holomorphic Eisenstein series mf.elliptic Holomorphic modular form (elliptic) abvar.fq.honda_tate Honda-Tate Theorem av.fq.honda_tate Honda-Tate Theorem dq.nf.howcomputed How global number field data was computed content.how-to-cite How to cite the LMFDB dq.zeros.zeta.howcomputed How zeros of the Riemann zeta function were computed mf.upper-half-plane Hyperbolic 2-space mf.bianchi.hyperbolic3space Hyperbolic 3-space ag.hyperelliptic_curve Hyperelliptic curve hgm.familes Hypergeometric family over $\Q$ I nf.ideal_class_group Ideal class group of a number field ec.conductor_norm Ideal norm of conductor doc.knowl.identifier Identifier of a knowl st_group.identity_component Identity component of a Sato-Tate group g2c.igusa_invariants Igusa invariants of a genus 2 curve g2c.igusa_clebsch_invariants Igusa-Clebsch invariants of a genus 2 curve ec.q.galois_rep_image Image of mod p Galois Representation artin.gg_quotient Image of the representation doc.knowl.implicit Implicit creation of knowls gg.int_modules Indecomposable integral representation group.subgroup.index Index of a normal subgroup st_group.index Index of the identity component intro.description Index page, first sentences character.dirichlet.induce Induced Dirichlet character mf.gl2.history.infinite Infinite products users.jhs.note Information about Joe Silverman ec.q.234446.a1.top Information for elliptic curve 234446.a1 ec.q.2450.i1.top Information for elliptic curve 2450.i1 ec.q.389.a1.top Information for elliptic curve 389.a1 av.fq.initial_coefficients Initial coefficients of the Weil q-polynomial nf.integral_basis Integral basis of a number field nf.integral Integral elements lattice.definition Integral Lattice ec.integral_model Integral model ec.q.integral_points Integral points nf.intermediate_fields Intermediate fields lf.intermediate_fields Intermediate fields intro Introduction intro.features Introduction/Features g2c.invariants Invariants of a genus 2 curve lfunction.invariants Invariants of an L-function ag.irreducible Irreducible variety (definition) g2c.529.a.529.1.top Isogenous to modular curve $X_0(23)$ g2c.841.a.841.1.top Isogenous to modular curve $X_0(29)$ g2c.450.a.2700.1.top Isogenous to modular curve $X_0(30)/W_2$ g2c.363.a.11979.1.top Isogenous to modular curve $X_0(33)/W_3$ g2c.1225.a.6125.1.top Isogenous to modular curve $X_0(35)/W_7$ ec.isogeny Isogeny between elliptic curves av.fq.abvar.data Isogeny class data g2c.isogeny_class Isogeny class of an abelian variety av.isogeny_class Isogeny class of an abelian variety ec.isogeny_class Isogeny class of an elliptic curve abvar.decomposition.data Isogeny factor decomposition data ec.isogeny_graph Isogeny graph of an isogeny class of elliptic curves ec.isogeny_matrix Isogeny matrix of an isogeny class of elliptic curves ag.isogeny Isogeny of abelian varieties av.isogeny Isogeny of abelian varieties lattice.isometry Isometries between integral lattices ec.weierstrass_isomorphism Isomorphism between Weierstrass models ec.isomorphism Isomorphism of elliptic curves ec.iwasawa_invariants Iwasawa invariants of elliptic curves J ec.j_invariant_denominator_valuation j-invariant denominator valuation ec.q.j_invariant j-invariant of a rational elliptic curve ec.j_invariant j-invariant of an elliptic curve mf.jacobi Jacobi forms character.dirichlet.jacobi_sum Jacobi sum ag.jacobian Jacobian of a curve g2c.jacobian Jacobian of a genus 2 curve bio.serre Jean-Pierre Serre bio.dirichlet Johann Peter Gustav Lejeune Dirichlet K lattice.kissing Kissing Number character.dirichlet.kloosterman_sum Kloosterman sum doc.knowl Knowl doc.knowl.guidelines Knowl Edit Guidelines test.knowlparam Knowl with parameters g2c.known_rational_points Known rational points ec.q.kodaira_symbol Kodaira symbol ec.kodaira_symbol Kodaira symbol character.dirichlet.kronecker_symbol Kronecker symbol L g2c.good_lfactors L-factors at good primes lfunction.definition L-function g2c.lfunction L-function of an abelian variety over $\mathbb{Q}$ lfunction.spectral_parameters L-function spectral parameters lfunction.history.maass L-functions of Maass forms mf.elliptic.character.label Label convention for Dirichlet characters lf.field.label Label for a local field artin.label Label for an Artin representation ec.q.lmfdb_label Label for an elliptic curve over $\mathbb Q$ g2c.label Label of a genus 2 curve over $\Q$ mf.elliptic.label Label of a holomorphic cusp form on GL(2) mf.hilbert.label Label of an Hilbert Modular Form dq.curve.highergenus.aut.label Labeling convention for families of higher genus curves av.fq.lmfdb_label Labels for isogeny classes of abelian varieties over finite fields nf.ideal_labels Labels for number field ideals curve.highergenus.aut.groupnotation Labels for the small group database in GAP and Magma mf.bianchi.labels Labels of Bianchi modular forms ec.curve_label Labels of elliptic curves over number fields ec.lambda_invariant lambda-invariant of an elliptic curve lfunction.history.langlands_program Langlands program mf.maass.mwf.laplacian Laplace-Beltrami operator (hyperbolic) mf.maass.picard.laplacian Laplacian in 3d hyperbolic upper half-space lfunction.degree4.1.34260324.bottom Large gap between nontrivial zeros doc.macros LaTeX macros in the LMFDB character.dirichlet.conrey.primitive_generator least primitive root modulo p^e lfunction.degree2holo.key Legend for GL(2) holomorphic graph of dots character.dirichlet.legendre_symbol Legendre symbol bio.euler Leonhard Euler mf.bianchi.level Level of a Bianchi modular form mf.hilbert.level_norm Level of a Hilbert modular form mf.elliptic.level Level of a modular form lattice.level Level of an integral lattice lfunction.level Level of an L-function mf.siegel.lift Lift Siegel modular form crystals.littelmann_path Littelmann path model for crystals doc.lmfdb.improve LMFDB backend doc.news.in_the_news LMFDB in the news lattice.label LMFDB label of an integral lattice lf.algebra.data Local algebra nf.local_algebra Local algebra ec.local_data Local data of an elliptic curve lf.invariants Local field invariants ec.local_minimal_discriminant Local minimal discriminant of an elliptic curve ec.local_minimal_model Local minimal model ec.local_root_number Local root number g2c.locally_solvable Locally solvable nf.14.2.79396891942519570432.1.bottom Low discriminant field nf.14.2.536561726709678316933.1.bottom Low discriminant field nf.17.1.239072435685151324847153.1.bottom Low discriminant field nf.19.1.75613185918270483380568064.1.bottom Low discriminant field M mf.maass.mwf Maass form on GL2(R) mf.maass.mwf.weight0 Maass forms of weight 0 mf.maass Maass forms on GL(n) mf.maass.picard Maass forms on PSL(2,Z[i]) mf.siegel.lift.maass Maass lift from Jacobi form to Siegel modular form content.edit-board Management board uploader.mapping_rules Mapping rules ec.maximal_galois_rep Maximal Galois representation st_group.subgroups Maximal subgroups ec.q.3675.g1.top Minimal 17-congruence ec.q.47775.be1 Minimal 17-congruence ec.minimal_discriminant Minimal discriminant ec.minimal_discriminant_norm Minimal discriminant norm g2c.minimal_equation Minimal equation of a genus 2 curve mf.elliptic.minimal Minimal modular form nf.minimal_polynomial Minimal polynomial st_group.supgroups Minimal supergroups ec.q.minimal_weierstrass_equation Minimal Weierstrass equation over $\mathbb Q$ lattice.minimal_vector Minimum vector length mf.siegel.lift.miyawaki Miyawaki lift mf.siegel.lift.miyawaki1 Miyawaki lift of type I, Ikeda-Miyawaki lift mf.siegel.lift.miyawaki2 Miyawaki lift of type II ec.q.11.a2.top Modular curve $X_0(11)$ ec.q.14.a6.top Modular curve $X_0(14)$ ec.q.15.a5.top Modular curve $X_0(15)$ ec.q.17.a3.top Modular curve $X_0(17)$ ec.q.19.a2.top Modular curve $X_0(19)$ ec.q.20.a4.top Modular curve $X_0(20)$ ec.q.21.a5.top Modular curve $X_0(21)$ ec.q.24.a4.top Modular curve $X_0(24)$ g2c.676.a.562432.1.top Modular curve $X_0(26)$ ec.q.27.a3.top Modular curve $X_0(27)$ g2c.196.a.21952.1.top Modular curve $X_0(28)$ g2c.450.a.36450.1.top Modular curve $X_0(3)/W_{10}$ g2c.961.a.923521.1.top Modular curve $X_0(31)$ ec.q.32.a4.top Modular curve $X_0(32)$ ec.q.36.a4.top Modular curve $X_0(36)$ g2c.1369.a.50653.1.top Modular curve $X_0(37)$ ec.q.49.a4.top Modular curve $X_0(49)$ g2c.2500.a.50000.1.top Modular curve $X_0(50)$ ag.modcurve.x0 Modular curve $X_0(N)$ ec.q.11.a3.top Modular curve $X_1(11)$ g2c.169.a.169.1.top Modular curve $X_1(13)$ ec.q.14.a5.top Modular curve $X_1(14)$ ec.q.15.a7.top Modular curve $X_1(15)$ g2c.256.a.512.1.top Modular curve $X_1(16)$ g2c.324.a.648.1.top Modular curve $X_1(18)$ ag.modcurve.x1 Modular curve $X_1(N)$ ec.q.modular_degree Modular degree of an elliptic curve over $\mathbb Q$ ec.q.modular_form Modular form associated to an elliptic curve/$\mathbb Q$ mf.elliptic.particular Modular forms with common names ec.q.modular_parametrization Modular parametrization of an elliptic curve over $\mathbb Q$ character.dirichlet.modulus Modulus of a Dirichlet character st_group.moments Moment statistics curve.highergenus.aut.monodromy Monodromy Group ec.mordell_weil Mordell-Weil group ag.mordell_weil Mordell-Weil group of an abelian variety ec.q.mordell-weil Mordell-Weil group of an elliptic curve over a number field g2c.mordell_weil_rank Mordell-Weil rank ec.mordell-weil Mordell-Weil theorem lfunction.motivic_weight Motivic weight ec.mu_invariant mu-invariant of an elliptic curve nf.multiplicative_gal_module Multiplicative Galois Module Structure ec.multiplicative_reduction Multiplicative reduction mf.maass.mwf.dimension Multiplicity of an eigenvalue mf.multipliersystem Multiplier system N lattice.name Name of a lattice st_group.name Name of a Sato-Tate group doc.knowl.naming_conventions Naming conventions for knowls doc.knowl.naming_suggestions Naming suggestions nf.narrow_class_group Narrow class group nf.narrow_class_number Narrow class number mf.gl2.history.new New modular forms from given ones mf.elliptic.newform Newform lf.newton_polygon Newton polygon of a polynomial ag.singular_point Non-singular point (definition) ec.q.galois_rep_image_nsp_cartan Non-split Cartan case gl2.nonsplit_cartan Non-split Cartan subgroup ec.nonsplit_multiplicative_reduction Non-split multiplicative reduction ec.q.non-surjective_prime Non-surjective prime nf.14.2.20325604337285010030592.1.bottom Nonsolvable field ramified at 2 and 13 nf.18.2.40254497110927943179349807054456171205137.1.bottom Nonsolvable field ramified at only 17 ec.discriminant_norm Norm of discriminant lfunction.normalization Normalization of an L-function nf.defining_polynomial.normalization Normalization of defining polynomials for number fields gl2.normalizer_cartan Normalizer of a Cartan subgroup gl2.normalizer_nonsplit_cartan Normalizer of a non-split Cartan subgroup gl2.normalizer_split_cartan Normalizer of a split Cartan subgroup ec.q.galois_rep_image_norm_nsp_cartan Normalizer of the nonsplit Cartan case ec.q.galois_rep_image_norm_sp_cartan Normalizer of the split Cartan case artin.number_field Number field associated to an Artin representation nf.invariants Number field invariants nf.label Number field label nf.nickname Number field nicknames nf.field.missing Number field not in the database av.fq.number_field Number field of an isogeny class of abelian varieties character.dirichlet.related_fields Number fields related to a Dirichlet character g2c.num_rat_pts Number of rational points O ec.obstruction_class Obstruction class of an elliptic curve ec.q.optimal Optimal elliptic curve over $\mathbb Q$ character.dirichlet.order Order of a Dirichlet character character.dirichlet.conrey.order Order of a Dirichlet character from its label character.dirichlet.group.order Order of a Dirichlet group group.order Order of a group st_group.order Order of the component group av.fq.ordinary Ordinary P abvar.fq.p_rank p-rank av.fq.p_rank p-rank group.paramodular Paramodular group mf.siegel.newform.paramodular Paramodular newform character.dirichlet.parity Parity of a Dirichlet character character.dirichlet.conrey.parity Parity of a Dirichlet character from its Conrey label gg.parity Parity of a Galois group artin.parity Parity of a representation mf.elliptic.petersson-scalar-product Petersson scalar product mf.gl2.history.poincare Poincaré series specialfunction.poincare Poincaré series ag.fq.point_counts Point counts of a variety av.fq.curve_point_counts Point counts of the curve av.polarization Polarization doc.lmfdb.contextualize Presentation of the data in the LMFDB ag.bad_prime Prime of bad reduction ag.good_prime Prime of good reduction g2c.good_reduction Primes of good reduction ag.primitive Primitive algebraic variety character.dirichlet.primitive Primitive Dirichlet character lattice.primitive Primitive integral lattice lfunction.primitive Primitive L-function ag.torsor Principal homogeneous space av.princ_polarizable Principally polarizable gg.4t3.bottom Properties of D_4 character.dirichlet.basic_properties Properties of Dirichlet characters Q mf.hilbert.q_expansion q-expansion of a Hilbert modular form mf.elliptic.q-expansion q-expansion of an elliptic modular form mf.quasi_modular Quasi-modular form curve.highergenus.aut.quotientgenus Quotient Genus R lfunction.history.ramanujan_tau Ramanujan tau L-function lf.ramification_index Ramification index of a prime in a local field nf.ramified_primes Ramified (rational) prime of a number field nf.ramified_prime Ramified prime of a number field artin.ramified_primes Ramified prime of an Artin representation nf.rank Rank of a number field lfunction.history.rankin_selberg Ranking-Selberg convolution, history g2c.all_rational_points Rational points st_group.rational Rational Sato-Tate group st_group.real_dimension Real dimension of a Sato-Tate group character.dirichlet.real Real Dirichlet character ag.real_endomorphism_algebra Real endomorphism algebra ag.real_multiplication Real multiplication ec.q.real_period Real period ec.reduction_type Reduction type ec.q.reduction_type Reduction type of an elliptic curve over $\mathbb Q$ curve.highergenus.aut.refinedpassport Refined Passport nf.regulator Regulator of a number field ec.q.regulator Regulator of elliptic curve mf.gl2.history.remainder Remainder terms repn.about Representation lf.residue_field_degree Residue field degree of an extension of local fields bio.borcherds Richard Ewen Borcherds lfunction.riemann Riemann zeta-function lfunction.history.riemanns_memoir Riemann's memoir nf.ring_of_integers Ring of integers of a number field artin.roadmap Roadmap for Artin representations character.dirichlet.roadmap Roadmap for Dirichlet characters ec.roadmap Roadmap for elliptic curves mf.elliptic.roadmap Roadmap for elliptic modular forms content.roadmap Roadmap for future areas of LMFBD gg.roadmap Roadmap for Galois groups g2c.roadmap Roadmap for genus 2 curves mf.hilbert.roadmap Roadmap for Hilbert modular forms lfunction.roadmap Roadmap for L-functions lf.roadmap Roadmap for local number fields mf.maass.roadmap Roadmap for Maass forms nf.roadmap Roadmap for number fields mf.siegel.roadmap Roadmap for Siegel modular forms nf.root_discriminant Root discriminant of a number field lf.root_number Root number of a local field artin.root_number Root number of an Artin representation ec.q.rouse-classification Rouse classification ec.rouse_label Rouse label S sage.cellexample Sage cell example sage.test Sage cell knowl mf.siegel.lift.saito_kurokawa Saito-Kurokawa lift mf.elliptic.satake_parameters Satake parameters st_group.definition Sato-Tate group ag.sato_tate_group Sato-Tate group (of a motive) dq.st.source Sato-Tate group data source g2c.st_group_identity_component Sato-Tate group identity component st_group.invariants Sato-Tate group invariants st.label Sato-Tate group label st_group.label Sato-Tate group labels g2c.st_group Sato-Tate group of a genus 2 curve over Q st_group.roadmap Sato-Tate group roadmap st_group.subsupgroups Sato-Tate subgroups and supergroups ec.q.21168.ce2.top secp256k1 elliptic curve lfunction.selberg_class.axioms Selberg class axioms lfunction.selberg_class Selberg class of L-functions lfunction.selbergdata Selberg data for the functional equation of an L-function lfunction.self-dual Self-dual L-function ag.selmer_group Selmer group of an abelian variety ec.semi_global_minimal_model Semi-global minimal model nf.separable_algebra Separable algebra nf.separable Separable extension mf.elliptic.shimura_correspondence Shimura correspondence mf.half_integral_weight.shimura_decomposition Shimura decomposition ag.shimura_variety Shimura Variety nf.sibling Sibling fields gg.other_representations Siblings for a Galois group mf.siegel.phi Siegel $\Phi$ operator mf.siegel.cusp_form Siegel cusp form mf.siegel.eisenstein_series Siegel Eisenstein series mf.siegel.klingen_eisenstein_series Siegel Klingen-Eisenstein series mf.siegel Siegel modular form mf.siegel.hecke_operator Siegel modular form Hecke operators mf.siegel.character Siegel modular form with character lfunction.history.siegel Siegel modular forms, history mf.siegel.theta_series Siegel theta series and theta series with pluriharmonics mf.siegel.halfspace Siegel upper half space mf.bianchi.sign Sign of a Bianchi newform lfunction.root_number Sign of an L-function lfunction.sign Sign of the functional equation curve.highergenus.aut.signature Signature of a group action on a curve nf.signature Signature of a number field lfunction.signature Signature of an L-function ag.simple Simple abvar.simple Simple abelian variety av.simple Simple abelian variety gg.simple_name Simple group name bio.wiles Sir Andrew John Wiles lf.slope_content Slope content of a local number field lf.newton_slopes Slopes of the Newton polygon of a polynomial nf.14.2.35632953415471973197.1.bottom Small discriminant field g2c.277.a.top Smallest prime conductor of an abelian surface ag.curve.smooth Smoothness of an algebraic curve gg.solvable Solvable group dq.artin.source Source of Artin representation data dq.mf.bianchi.source Source of Bianchi modular form data dq.character.dirichlet.source Source of Dirichlet character computations dq.ec.source Source of elliptic curve data over $\Q$ dc.ecnf.source Source of elliptic curve data over number fields dq.ecnf.source Source of elliptic curve data over number fields dq.gg.source Source of Galois group data dq.g2c.source Source of genus 2 curve data dq.mf.hilbert.source Source of Hilbert modular form data dq.lattice.source Source of lattice data dq.lf.source Source of Local Field Data dq.curve.highergenus.aut.source Source of the higher genus curves with automorphisms data mf.bianchi.spaces Spaces of Bianchi modular forms ec.q.special_value Special value of an elliptic curve L-function lfunction.special_values Special values of an L-function mf.maass.picard.eigenvalue Spectral parameter of Maass forms on PSL(2,Z[i]) ec.q.galois_rep_image_sp_cartan Split Cartan case gl2.split_cartan Split Cartan subgroup ec.split_multiplicative_reduction Split multiplicative reduction character.dirichlet.group.structure Structure of a Dirichlet group mf.sturm_bound Sturm bound gg.subfields Subfields group.sl2z.subgroup.gamman Subgroup $\Gamma(N)$ of $\textrm{SL}(2,\mathbb{Z})$ group.sl2z.subgroup.gamma0n Subgroup $\Gamma_0(N)$ of $\textrm{SL}(2,\mathbb{Z})$ group.sl2z.subgroup.gamma1n Subgroup $\Gamma_1(N)$ of $\textrm{SL}(2,\mathbb{Z})$ gg.subnormal_series Subnormal series av.fq.supersingular Supersingular ec.q.surjective_prime Surjective prime lfunction.symm Symmetric power of an L-function group.sp2gr Symplectic group T gg.tnumber T-number of a transitive group zeros.first.contents Table of first zeros of L-function ec.tamagawa_number Tamagawa number g2c.tamagawa Tamagawa numbers ec.q.tamagawa_numbers Tamagawa numbers for an elliptic curve ec.padic_tate_module Tate module of an elliptic curve lfunction.history.tates_thesis Tate's thesis lfunction.taxonomy Taxonomy of L-functions test.text Test → !!! mf.maass.mwf.eigenvalue The eigenvalue of a Maass form lattice.24.1.1.0.1.top The Leech lattice lattice.24.1.1.24.1.top The Leech lattice mf.maass.mwf.level The level of a Maass form mf.maass.mwf.ncoefficients The number of coefficients mf.maass.mwf.precision The precision of an eigenvalue mf.siegel.family.kp The ring of paramodular cusp forms $S_k(K(p))$ mf.siegel.family.gamma_2 The ring of Siegel modular forms of degree 2 for $\Gamma(2)$ mf.siegel.family.gamma0_2 The ring of Siegel modular forms of degree 2 for $\Gamma_0(2)$ mf.siegel.family.gamma0_3 The ring of Siegel modular forms of degree 2 with respect to $\Gamma_0(3)$ mf.siegel.family.gamma0_3_psi_3 The ring of Siegel modular forms of degree 2 with respect to $\Gamma_0(3)$ with character $\psi_3$ mf.siegel.family.gamma0_4 The ring of Siegel modular forms of degree 2 with respect to $\Gamma_0(4)$ mf.siegel.family.gamma0_4_psi_4 The ring of Siegel modular forms of degree 2 with respect to $\Gamma_0(4)$ with character mf.siegel.family.sp4z The ring of Siegel modular forms of degree 2 with respect to the full modular group mf.siegel.family.sp6z The ring of Siegel modular forms of degree 3 with respect to the full modular group mf.siegel.family.sp8z The ring of Siegel modular forms of degree 4 with respect to the full modular group mf.maass.mwf.spectralparameter The spectral parameter of a Maass waveform mf.maass.mwf.symmetry The symmetry of a Maass form lf.unramified_totally_ramified_tower The unramified/totally ramified tower of a local field mf.maass.mwf.weight The weight of a Maass form mf.siegel.theta_constant Theta constant mf.half_integral_weight.theta Theta function mf.gl2.history.theta Theta functions lattice.theta Theta series for an integral lattice dq.zeros.zeta.toomany Too many zeros nf.18.2.4768875488962616064464333777.1.bottom Torsion field ec.q.torsion_generators Torsion generators of an elliptic curve over $\mathbb Q$ g2c.torsion_order Torsion order ec.field.torsion_order Torsion order of an elliptic curve ec.torsion_order Torsion order of an elliptic curve ec.q.torsion_order Torsion order of an elliptic curve over $\mathbb{Q}$ g2c.torsion Torsion subgroup ec.torsion_subgroup Torsion subgroup of an elliptic curve ec.q.torsion_subgroup Torsion subgroup of an elliptic curve over $\mathbb Q$ nf.totally_imaginary Totally imaginary nf.totally_positive Totally positive nf.totally_real Totally real st_group.trace_moments Trace moments of a Sato-Tate group artin.trace_of_complex_conj Trace of complex conjugation st_group.trace_zero_density Trace zero density of a Sato-Tate group doc.knowl.context_free Transclusion (definition) mf.transformation_property Transformation property for modular forms gg.conway_name Transitive group name conventions nf.sextic_twin Twin sextic algebra U lfunction.underlying_object Underlying object of an L-function character.unit_group Underlying unit group for a Dirichlet character nf.unit_group Unit group of a number field nf.unramified_prime Unramified (rational) prime of a number field artin.unramified_primes Unramified prime of an Artin representation doc.upload_data Upload data mf.upper_half_plane Upper half-pane V lfunction.value Value of an L-function character.dirichlet.values Values of a Dirichlet character character.dirichlet.values_on_gens Values of a Dirichlet character on generators mf.siegel.family.sp4z_2 Vector valued Siegel modular forms of degree 2 taking values in three-dimensional space mf.siegel.vector_valued Vector-valued Siegel modular forms W ec.weierstrass_coeffs Weierstrass equation or model g2c.num_rat_wpts Weierstrass points of a hyperelliptic curve mf.slash_action Weight $k$ slash-action mf.bianchi.weight Weight of a Bianchi modular form mf.hilbert.weight_vector Weight of a Hilbert modular form st_group.weight Weight of a Sato-Tate group mf.elliptic.weight Weight of an elliptic modular form nf.weil_height Weil height ec.q.weil_height Weil height of a point on an elliptic curve over $\Q$ av.fq.weil_polynomial Weil q-polynomial abvar.fq.weil-polynomial Weil q-Polynomials group.fuchsian.cusps.width Width of a cusp mot.width Width of a motive doc.wiki Wiki (definition) bio.william_stein William Stein Z lfunction.zfunction Z-function of an L-function lfunction.zeros Zeros of an L-function ? lfunction.riemann.bottom mf.maass.mwf.plot sage.cell users.pascal.note content.news ec.galois_rep_image nf.3.1.503.1.top nf.order nf.index nf.monomial_order nf.primitive_element doc.related_to "related to" field specialfunction.gammarandc $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ g2c.gl2type $\GL_2$-type ec.q_curve $\mathbb{Q}$-curves specialfunction.k_bessel $K$-Bessel functions nf.20.2.558854584859141340505879659895771.1.bottom 19-division field for CM elliptic curve for $\Q(\sqrt{-7})$ g2c.two_selmer_rank 2-Selmer rank ec.q.2450.a1.top 2450.a1