Search Filter Category: af ag artin bio character content crystals data demo dirichlet_character doc dq ec g2c gg group hgm intro lattice lf lfunction local_data mf mot mwf nf nf_weil_height riemann sage specialfunction test uploader users zeros varieties repn field Quality: Beta Ok Reviewed More Recently modified Knowls
 A lfunction.history A history of L-functions 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 ag.variety Algebraic variety 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.arithmetic Arithmetic L-function 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 B ec.bad_reduction Bad reduction of an elliptic curve at a prime ec.base_change Base change mf.bianchi.bianchigroup Bianchi Groups mf.bianchi.bianchimodularforms Bianchi modular forms 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 C ec.q.canonical_height Canonical height lattice.catalogue_of_lattices Catalogue of Lattices lfunction.central_character Central character 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 ag.cm_field CM field mf.elliptic.cm_form CM form riemann.stieltjes.cn Coefficients of power series expansion of the zeta function ec.complex_multiplication Complex multiplication ag.complex_multiplication Complex multiplication ec.conductor_label Conductor labels for elliptic curves character.dirichlet.conductor Conductor of a Dirichlet character hgm.conductor Conductor of a hypergeometric motive 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 artin.permutation_container Containing permutation representation artin.database_comment Content of the Artin representations database riemann.stieltjes.contents Contents of our Stieltjes constants tables 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) mf.sl2z.subgroup.cusps Cusps of a subgroup of the modular group D nf.field.data Data for a global number field lattice.data Data for an integral lattice zeros.zeta.contents David Platt's computation of zeros of the zeta 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 lfunction.degree Degree of an L-function lattice.density Density of an integral lattice doc.knowl.description Description field in knowls lattice.determinant Determinant of an integral lattice doc.development Developers Guide mf.hilbert.dimension Dimension of a Hecke constituent mf.elliptic.dimension_galois_orbit Dimension of a Hecke orbit mf.elliptic.hecke-orbit-dimension Dimension of a Hecke orbit artin.dimension Dimension of an Artin representation mwf.dimension Dimension of an eigenvalue lattice.dimension Dimension of an integral lattice character.dirichlet Dirichlet character character.dirichlet.group Dirichlet character group 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 lattice.dual Dual lattice mf.siegel.lift.ikeda Duke-Imamoglu-Ikeda lift E lattice.8.1.1.1.1.top E_8 Lattice local_data ec.local_data content.edit-board Editorial board and management ec.q.37.a1.bottom Elliptic Curve 37a1 ec.invariants Elliptic curve invariants ec.q.invariants Elliptic curve invariants bio.artin Emil Artin ag.endomorphism_algebra Endomorphism algebra g2c.jac_endomorphisms Endomorphism invariants ec.endomorphism Endomorphism of an elliptic curve ec.q.endomorphism_ring Endomorphism ring ag.endomorphism_ring Endomorphism ring ec.endomorphism_ring Endomorphism ring g2c.equation Equation of a genus 2 curve over $\Q$ mf.4.6.0.a.formulas eta function product formula mf.11.2.0.a.formulas eta-product 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 ec.q.galois_rep_image_exc Exceptional case doc.view_knowl Expand a knowl mf.elliptic.extent Extent of data for classical modular forms dq.ec.extent Extent of data for elliptic curves over $\Q$ dq.gg.extent Extent of Galois group data dq.nf.extent Extent of global number field data dq.lf.extent Extent of local number field data dq.zeros.zeta.extent Extent of Riemann zeta zeros 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 character.dirichlet.field_values Field of values mf.1.12.0.a.formulas Formulas for Delta(z) 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 nf.frobenius_cycle_types Frobenius cycle types artin.frobenius_schur_indicator Frobenius-Schur indicator of a representation lfunction.functional_equation Functional equation of an L-function nf.fundamental_units Fundamental units of a number field G g2c.g2_invariants G2-invariants of a genus 2 curve 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 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 nf.generator Generator of a number field nf.field_generator Generator of a number field character.dirichlet.values_on_gens Generators of the group of Dirichlet characters g2c.g2curve Genus 2 curve ag.curve.genus Genus of a smooth curve mf.sl2z.subgroup.genus 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 bio.riemann Georg Friedrich Bernhard Riemann ec.global_minimal_model Global minimal model ec.good_reduction Good reduction lattice.gram Gram matrix demo.knowl.grants Grants supporting the development of knowls mf.siegel.lift.gritsenko Gritsenko lift mf.sl2.subgroup.gamma0n Group $\Gamma_0(N)$ subgroup of $\textrm{SL}(2,\mathbb{Z})$ mf.siegel.group.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 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 lfunction.history.hecke_characters Hecke L-functions 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.5077a1.top Historical information for 5077a1 ec.q.5077.a1.top Historical information for 5077a1 mot.hodgevector Hodge vector of a hypergeometric motive mf.elliptic Holomorphic modular form (elliptic) 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 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 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 mf.sl2z.subgroup.index Index of a subgroup of the modular group intro.description Index page, first sentences users.jhs.note Information about Joe Silverman ec.q.234446a1.top Information for elliptic curve 234446.a1 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 ec.q.389a1.top Information for elliptic curve 389.a1 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 intro Introduction intro.features Introduction/Features intro.tutorial Introduction/Tutorial g2c.invariants Invariants of a genus 2 curve ec.isogeny Isogeny between elliptic curves g2c.isogeny_class Isogeny class of an abelian variety ec.isogeny_class Isogeny class of an elliptic curve 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 lattice.isometry Isometries between integral lattices ec.weierstrass_isomorphism Isomorphism between Weierstrass models ec.isomorphism Isomorphism 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 af.jacobi Jacobi forms character.dirichlet.jacobi_sum Jacobi sum ag.jacobian Jacobian of a curve g2c.jacobian Jacobian of a genus 2 curve 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 ec.q.kodaira_symbol Kodaira symbol ec.kodaira_symbol Kodaira symbol character.dirichlet.kronecker_symbol Kronecker symbol L lfunction.definition L-function g2c.lfunction L-function of an abelian variety over $\Q$ 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) ec.curve_label Labels of elliptic curves over number fields mf.maass.mwf.laplacian Laplace-Beltrami operator (hyperbolic) mf.maass.picard.laplacian Laplacian in 3d hyperbolic upper half-space doc.macros LaTeX macros in the LMFDB lfunction.degree2holo.key Legend for GL(2) holomorphic graph of dots character.dirichlet.legendre_symbol Legendre symbol bio.euler Leonhard Euler 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 crystals.littelmann_path Littelmann path model for crystals doc.lmfdb.improve LMFDB backend lattice.label LMFDB label of an integral lattice lf.invariants Local field invariants ec.local_minimal_discriminant Local minimal discriminant of an elliptic curve ec.local_minimal_model Local minimal model 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.browsegraph Maass form browsing graph explanation 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 uploader.mapping_rules Mapping rules 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 ec.q.minimal_weierstrass_equation Minimal Weierstrass equation over $\mathbb Q$ lattice.minimal_vector Minimum vector length demo.aim.mission_statement Mission statement of AIM mf.siegel.lift.miyawaki Miyawaki lift ec.q.11.a2.top Modular curve $X_0(11)$ ec.q.11a1.top Modular curve $X_0(11)$ ag.modcurve.x0 Modular curve $X_0(N)$ ec.q.11a3.top Modular curve $X_1(11)$ ec.q.11.a3.top Modular curve $X_1(11)$ 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/$\mathbb Q$ ec.q.modular_parametrization Modular parametrization of an elliptic curve over $\mathbb Q$ character.dirichlet.modulus Modulus of a Dirichlet character 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 ec.mordell-weil Mordell-Weil theorem nf.multiplicative_gal_module Multiplicative Galois Module Structure ec.multiplicative_reduction Multiplicative reduction mf.maass.mwf.dimension Multiplicity of an eigenvalue N lattice.name Name of a lattice doc.knowl.naming_conventions Naming conventions for knowls doc.knowl.naming_suggestions Naming suggestions mf.elliptic.newform Newform ag.singular_point Non-singular point (definition) ec.q.galois_rep_image_nsp_cartan Non-split Cartan case 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 nf.defining_polynomial.normalization Normalization of defining polynomials for number fields 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 character.dirichlet.related_fields Number fields related to a Dirichlet character O demo.grassy_knoll obsolete 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 group.order Order of a group ec.minimal_discriminant.order Order of minimal discriminant of an elliptic curve over a number field at a bad prime gg.other_representations Other representations of a group P mf.siegel.group.paramodular Paramodular group mf.siegel.newform.paramodular Paramodular newform character.dirichlet.parity Parity of a Dirichlet character artin.parity Parity of a representation doc.lmfdb.contextualize Presentation of the data in the LMFDB ag.bad_prime Prime of bad reduction character.dirichlet.primitive Primitive Dirichlet character lattice.primitive Primitive integral lattice lfunction.primitive Primitive L-function 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 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 nf.rank Rank of a number field lfunction.history.rankin_selberg Ranking-Selberg convolution, history 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 $\mathbb Q$ nf.regulator Regulator of a number field ec.q.regulator Regulator of elliptic curve lf.residue_field_degree Residue field degree of an extension of local fields 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 dirichlet_character.roadmap Roadmap for Dirichlet characters 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 ag.sato_tate_group Sato-Tate group (of a motive) g2c.st_group_identity_component Sato-Tate group identity component g2c.st_group Sato-Tate group of a genus 2 curve over Q 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 nf.separable Separable extension mf.elliptic.shimura_correspondence Shimura correspondence mf.half_integral_weight.shimura_decomposition Shimura decomposition 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 lfunction.root_number Sign of an L-function lfunction.sign Sign of the functional equation nf.signature Signature of a number field lfunction.signature Signature of an L-function ag.simple Simple gg.simple_name Simple group name lf.slope_content Slope content of a local number field nf.14.2.35632953415471973197.1.bottom Small discriminant field ag.curve.smooth Smoothness of an algebraic curve dq.ec.source Source of elliptic curve data over $\Q$ 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 ec.split_multiplicative_reduction Split multiplicative reduction riemann.stieltjes_constants Stieltjes constants mf.sturm_bound Sturm bound gg.subfields Subfields ec.q.surjective_prime Surjective prime lfunction.symm Symmetric power of an L-function mf.siegel.group.symplectic 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 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 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.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 lattice.theta Theta series for an integral lattice 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 artin.trace_of_complex_conj Trace of complex conjugation doc.knowl.context_free Transclusion (definition) mf.transformation_property Transformation property for modular forms gg.conway_name Transitive group name conventions U lfunction.underlying_object Underlying object of an L-function nf.unit_group Unit group of a number field nf.unramified_prime Unramified (rational) prime of a number field 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 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.hilbert.weight_vector Weight of a Hilbert modular form mf.elliptic.weight Weight of an elliptic modular form nf_weil_height Weil height mot.width Width of a motive doc.wiki Wiki (definition) bio.william_stein William Stein Y mf.siegel.yoshida_lift Yoshida lift of a Hilbert modular form Z lfunction.zfunction Z-function of an L-function lfunction.zeros Zeros of an L-function ? mf.1-12-0-a.formulas mf.elliptic.coefficient_field nf.dirichlet_group lfunction.riemann.bottom mf.maass.mwf.plot users.d_yasaki.note mf.2.8.0.a.formulas mf.3.6.0.a.formulas mf.4.5.1.a.formulas mf.5.4.0.a.formulas mf.6.4.0.a.formulas mf.7.3.3.a.formulas mf.elliptic.explicit_formula sage.cell users.pascal.note mf.half_integer_weight repn.about group.about field.about doc.related_to "related to" field specialfunction.gammarandc $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ g2c.gl2type $\GL_2$-type 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.2450a1.top 2450a1