Search Filter Category: af ag artin bio character content crystals data demo dirichlet_character doc ec g2c gg group hgm intro lattice lf lfunction mf mot mwf nf riemann sage specialfunction test uploader users zeros Quality: Beta Ok Reviewed More Recently modified Knowls
 A ag.abelian_surface Abelian surface ag.abelian_variety Abelian variety g2c.abs_discriminant Absolute discriminant group.galois.absolute Absolute Galois group ag.curve Algebraic curve ag.curve.genus Algebraic genus of a smooth curve ag.variety Algebraic variety lfunction Analytic L-function ec.q.analytic_sha_order Analytic order of Ш lfunction.arithmetic Arithmetic L-function artin.lfunction Artin L-function nf.assuming_grh Assuming GRH mf.elliptic.atkinlehner Atkin-Lehner eigenvalue mf.maass.mwf.atkinlehner Atkin-Lehner eigenvalues bio.selberg Atle Selberg lf.automorphism_group Automorphism group of a field extension g2c.aut_grp Automorphism group of an algebraic curve B 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 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 riemann.stieltjes.cn Coefficients of power series expansion of the zeta function ec.complex_multiplication Complex multiplication g2c.conductor Conductor ec.conductor_label Conductor labels for elliptic curves character.dirichlet.conductor Conductor of a Dirichlet character hgm.conductor Conductor of a Hypergeometric Motive artin.conductor Conductor of an Artin representation ec.conductor Conductor of an elliptic curve ec.q.conductor Conductor of an elliptic curve over $\Q$ data.congruent_numbers Congruent numbers data gg.conjugacy_classes.data Conjugacy class representatives gg.conjugacy_classes Conjugacy classes artin.database_comment Content of the Artin representations database riemann.stieltjes.contents Contents of our Stieltjes constants tables doc.create_knowl Create a knowl doc.create_account Create an account ec.q.cremona_label Cremona label lfunction.critical_line Critical line of an L-function lfunction.critical_strip Critical strip crystals.crystal Crystals g2c.g2curve Curve of genus 2 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 zeros.zeta.contents David Platt's computation of zeros of the zeta function 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 doc.knowl.description Description field in knowls 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 character.dirichlet Dirichlet character character.dirichlet.group Dirichlet character group lfunction.dirichlet Dirichlet L-function lfunction.dirichlet_series Dirichlet series lf.discriminant_exponent Discriminant exponent of a local field 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 mf.siegel.lift.ikeda Duke-Imamoglu-Ikeda lift E content.edit-board Editorial board and management ec.q.37.a1.bottom Elliptic Curve 37a1 ec.q.invariants Elliptic curve invariants bio.artin Emil Artin g2c.end_alg Endomorphism algebra ec.endomorphism_ring Endomorphism ring g2c.endomorphism_ring Endomorphism ring ec.q.endomorphism_ring Endomorphism ring g2c.jac_endomorphisms Endomorphisms g2c.q.equation Equation of a genus 2 curve over $\Q$ mf.11.2.0.a.formulas eta-product 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 ec.q.galois_rep_image_exc Exceptional case doc.view_knowl Expand a knowl F 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 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 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 ec.galois_rep Galois representations attached to an elliptic curve character.dirichlet.gauss_sum Gauss sum lfunction.rh Generalized Riemann hypothesis nf.field_generator Generator of a number field character.dirichlet.values_on_gens Generators of the group of Dirichlet characters mf.sl2z.subgroup.genus Genus of a subgroup of the modular group bio.riemann Georg Friedrich Bernhard Riemann 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 $\mathrm{SL}(2,\mathbb Z)$ 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 mf.growth_condition Growth condition for a modular form H mf.half_integral_weight Half-integral weight modular forms bio.harald_schilly Harald Schilly mf.elliptic.hecke-orbit Hecke orbit 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 mot.hodgevector Hodge vector of a hypergeometric motive mf.elliptic Holomorphic modular form (elliptic) content.how-to-cite How to cite the LMFDB 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 ec.discriminant_norm Ideal norm of discriminant doc.knowl.identifier Identifier of a knowl 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 ec.q.234446a1.top Information for elliptic curve 234446.a1 ec.q.389a1.top Information for elliptic curve 389.a1 nf.integral_basis Integral basis of a number field ec.q.integral_points Integral points nf.intermediate_fields Intermediate fields intro Introduction intro.features Introduction/Features intro.tutorial Introduction/Tutorial ec.isogeny Isogenies between elliptic curves 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 J ec.q.j_invariant j-invariant ec.j_invariant j-invariant of an elliptic curve af.jacobi Jacobi forms character.dirichlet.jacobi_sum Jacobi sum bio.dirichlet Johann Peter Gustav Lejeune Dirichlet K lattice.kissing Kissing Number character.dirichlet.kloosterman_sum Kloosterman sum doc.knowl Knowl test.knowlparam Knowl with parameters ec.q.kodaira_symbol Kodaira symbol character.dirichlet.kronecker_symbol Kronecker symbol L lfunction.definition L-function lfunction.spectral_parameters L-function spectral parameters mf.elliptic.character.label Label convention for Dirichlet characters lf.field.label Label for a local field ec.q.lmfdb_label Label for an elliptic curve over $\mathbb Q$ g2c.q.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 lfunction.level Level of an L-function crystals.littelmann_path Littelmann path model for crystals doc.lmfdb.improve LMFDB backend lf.invariants Local field invariants M mf.maass.mwf Maass form on GL2(R) 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 g2c.minimal_equation Minimal equation of a genus 2 curve mf.elliptic.minimal Minimal modular form ec.q.minimal_weierstrass_equation Minimal Weierstrass equation over $\mathbb Q$ demo.aim.mission_statement Mission statement of AIM mf.siegel.lift.miyawaki Miyawaki lift 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 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 mf.maass.mwf.dimension Multiplicity of an eigenvalue N 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.q.non-surjective_prime Non-surjective prime 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 character.dirichlet.related_fields Number fields related to a Dirichlet character O demo.grassy_knoll obsolete ec.q.optimal Optimal elliptic curve over $\mathbb Q$ character.dirichlet.order Order of a Dirichlet character group.order Order of a group 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 character.dirichlet.primitive Primitive Dirichlet character 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 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 character.dirichlet.real Real Dirichlet character g2c.real_end_alg Real endomorphism algebra ec.q.real_period Real period 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 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 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.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 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 nf.separable Separable extension doc.lmfdb.share Sharing goal of the LMFDB 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 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.curve.smooth Smoothness of an algebraic curve 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 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.q.tamagawa_numbers Tamagawa numbers for an elliptic curve 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 ec.q.torsion_generators Torsion generators of an elliptic curve over $\mathbb Q$ 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_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 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 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/coefficients mf.hilbert.weight_vector Weight of a Hilbert modular form mf.elliptic.weight Weight of an elliptic modular form 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 ? mf.1-12-0-a.formulas mf.elliptic.coefficient_field nf.17.1.604462909807314587353088.1.bottom lfunction.riemann.bottom mf.maass.mwf.plot ec.q.11a3.top users.d_yasaki.note mf.2.8.0.a.formulas mf.3.6.0.a.formulas mf.4.5.1.a.formulas mf.4.6.0.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 doc.related_to "related to" field specialfunction.gammarandc $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ g2c.gl2type $\textrm{GL}_2$-type ec.padic_tate_module $p$-adic Tate module of an elliptic curve