Special $L$-values of genus 2 curves over $\mathbb{Q}$

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Numbers

label

$r$

leading coefficient

169.a

0.09049039083242962911358975725801541226473

comment: Representative curve LMFDB 169.a.169.1, $y^2 + (x^3 + x + 1)y = x^5 + x^4$.

249.a

0.1315495070114787592134013430121752606933

comment: Representative curve LMFDB 249.a.249.1, $y^2 + (x^3 + 1)y = x^2 + x$.

277.a

0.1431366605510114901557152099923829602232

comment: Representative curve LMFDB 277.a.277.1, $y^2 + (x^3 + x^2 + x + 1)y = -x^2 - x$.

295.a

0.1492683688832182090588207138996158657800

comment: Representative curve LMFDB 295.a.295.1, $y^2 + (x^3 + 1)y = -x^2$.

349.a

0.1656123320984078287720952736293483346028

comment: Representative curve LMFDB 349.a.349.1, $y^2 + (x^3 + x^2 + x + 1)y = -x^3 - x^2$.

353.a

0.1859131786554872680710289056816858155417

comment: Representative curve LMFDB 353.a.353.1, $y^2 + (x^3 + x + 1)y = x^2$.

363.a

0.1897059599534884032426437150258731309382

comment: Representative curve LMFDB 363.a.11979.1, $y^2 + (x^2 + 1)y = x^5 + 2 x^3 + 4 x^2 + 2 x$.

389.a

0.1979862013008733330350209978381275022945

comment: Representative curve LMFDB 389.a.389.1, $y^2 + (x^3 + x)y = x^5 - 2 x^4 - 8 x^3 + 16 x + 7$.

427.a

0.1899303686308920817263923619718777416327

comment: Representative curve LMFDB 427.a.2989.1, $y^2 + (x^3 + 1)y = x^5 - x^4 - 5 x^3 + 4 x^2 + 4 x - 4$.

461.a

0.2458864264726567276756390789225886309417

comment: Representative curve LMFDB 461.a.461.1, $y^2 + x^3 y = x^5 - 3 x^3 + 3 x - 2$.

523.a

0.2481990412677571630041171344077874577287

comment: Representative curve LMFDB 523.a.523.1, $y^2 + (x + 1)y = x^5 - x^4 - x^3$.

529.a

0.2484318665905996812072503393142383906699

comment: Representative curve LMFDB 529.a.529.1, $y^2 + (x^3 + x + 1)y = -x^5$.

555.a

0.2569247202971451430840301627276747947597

comment: Representative curve LMFDB 555.a.8325.1, $y^2 + (x + 1)y = 3 x^5 - 2 x^4 - 4 x^3 + x^2 + x$.

587.a

0.1113515110642202373191027362087235867524

comment: Representative curve LMFDB 587.a.587.1, $y^2 + (x^3 + x + 1)y = -x^2 - x$.

597.a

0.2941146301882136359236849987869052563808

comment: Representative curve LMFDB 597.a.597.1, $y^2 + y = x^5 + 2 x^4 + 3 x^3 + 2 x^2 + x$.

603.a

0.2691001556363177690790850827729359332812

comment: Representative curve LMFDB 603.a.603.1, $y^2 + (x^2 + 1)y = x^5 + 8 x^4 + 4 x^3 + 4 x^2 + 2 x$.

691.a

0.2939463892598192424091021283086475110620

comment: Representative curve LMFDB 691.a.691.1, $y^2 + (x + 1)y = x^5 - x^3 - x^2$.

709.a

0.2868931505071272731308297049624563478404

comment: Representative curve LMFDB 709.a.709.1, $y^2 + xy = x^5 - 2 x^2 + x$.

713.a

0.1283949912472834590170543474773248575796

comment: Representative curve LMFDB 713.a.713.1, $y^2 + (x^3 + x + 1)y = -x^5 - x$.

713.b

0.2858010009469617170226097264141850097689

comment: Representative curve LMFDB 713.b.713.1, $y^2 + (x^3 + x + 1)y = -x^4$.

731.a

0.2985355886872414062863189873478987786819

comment: Representative curve LMFDB 731.a.12427.1, $y^2 + (x^3 + x^2)y = x^5 + 2 x^4 - x - 3$.

741.a

0.2930756651264371996923859602312386579243

comment: Representative curve LMFDB 741.a.28899.1, $y^2 + (x + 1)y = -3 x^5 - x^4 + 2 x^2 + x$.

743.a

0.1316557503846215116289169353731506251281

comment: Representative curve LMFDB 743.a.743.1, $y^2 + (x^3 + x + 1)y = -x^4 + x^2$.

745.a

0.3033683921052660240631287058786037771056

comment: Representative curve LMFDB 745.a.745.1, $y^2 + (x^3 + x + 1)y = -x$.

763.a

0.3048575004522535920085251501905636550504

comment: Representative curve LMFDB 763.a.763.1, $y^2 + (x^3 + x)y = -2 x^4 + 2 x^2 - x$.

797.a

0.3559385607403319429724872658598941145485

comment: Representative curve LMFDB 797.a.797.1, $y^2 + y = x^5 - x^4 + x^3$.

807.a

0.3050356332174254142944032925074203061351

comment: Representative curve LMFDB 807.a.2421.1, $y^2 + (x^3 + x)y = x^5 - 2 x^3 - x^2 + 2 x - 1$.

841.a

0.2915215656991529771681413103312969859954

comment: Representative curve LMFDB 841.a.841.1, $y^2 + (x^3 + x^2 + x)y = x^4 + x^3 + 3 x^2 + x + 2$.

847.a

0.1592441888803194513955584463536559869689

comment: Representative curve LMFDB 847.a.847.1, $y^2 + (x^3 + x^2 + x + 1)y = x^4 + x^3 + x^2$.

847.b

0.3365454249762917676170118858324423791181

comment: Representative curve LMFDB 847.b.9317.1, $y^2 + (x^2 + 1)y = x^5 + 2 x^4 - 3 x^3 + 2 x^2 - x$.

847.c

0.3119812484526625370039160119138202568854

comment: Representative curve LMFDB 847.c.9317.1, $y^2 + (x^3 + x^2)y = x^4 + x^3 - x - 2$.

847.d

0.2621188089061351721920136796692074600710

comment: Representative curve LMFDB 847.d.456533.1, $y^2 + y = -x^6 - 9 x^5 - 22 x^4 + 3 x^3 + 37 x^2 - 24 x + 4$.

893.a

0.1504585378200966582669018264044790617402

comment: Representative curve LMFDB 893.a.893.1, $y^2 + (x^3 + x + 1)y = -x^4 - x^2$.

909.a

0.3407116942606920617569877403142908280531

comment: Representative curve LMFDB 909.a.8181.1, $y^2 + xy = 3 x^5 - 7 x^4 + x^3 + 6 x^2 - 3 x$.

925.a

0.3262333436965224143091586365531384380133

comment: Representative curve LMFDB 925.a.23125.1, $y^2 + xy = 5 x^5 + x^4 - 19 x^3 + 18 x^2 - 5 x$.

953.a

0.1561935793455555266041051297046358534815

comment: Representative curve LMFDB 953.a.953.1, $y^2 + (x^3 + x + 1)y = x^3 + x^2$.

961.a

0.4492877238760407861133296681879239133411

comment: Representative curve LMFDB 961.a.923521.1, $y^2 + (x^3 + x^2 + 1)y = -5 x^4 + 4 x^3 + 3 x^2 - 2 x - 3$.

971.a

0.1769981337951764400226500892472379175742

comment: Representative curve LMFDB 971.a.971.1, $y^2 + y = x^5 - 2 x^3 + x$.

975.a

0.3987858446601027745461990966212669516994

comment: Representative curve LMFDB 975.a.63375.1, $y^2 + (x^3 + 1)y = -x^5 + x^3 + 2 x^2 + x - 1$.

997.a

0.3373378289933834996066688066442920811665

comment: Representative curve LMFDB 997.a.997.1, $y^2 + xy = x^5 - 8 x^4 + 16 x^3 - x$.

997.b

0.1799924859492394396660740920629892328536

comment: Representative curve LMFDB 997.b.997.1, $y^2 + y = x^5 - 2 x^4 + 2 x^3 - x^2$.

Definition

Let $C/\mathbb{Q}$ be a genus $2$ curve and let $A=\operatorname{Jac}(C)$. This table gives the leading Taylor coefficient $L^{(r)}(A,1)/r!$ [3] of the Hasse-Weil $L$-function $L(A,s)$ [1] in the arithmetic normalisation, where $r$ is the analytic rank of $A$.

Parameters

label

— LMFDB isogeny class label

$r$

— analytic rank ($r\geq 0$)

Formulas

(1)

If $L(A,s)=\sum_{m\geq r} c_m(s-1)^m$ with $c_r\neq 0$, then the table stores $c_r=L^{(r)}(A,1)/r!$.

(2)

At a prime $p$ of good reduction, the local factor is $P_p(p^{-s})^{-1}$, where $P_p(t)=\det(1-t\operatorname{Frob}_p\mid T_\ell A)$.

(3)

The Birch and Swinnerton-Dyer conjecture predicts $L^{(r)}(A,1)/r! = \Omega_A R_A |\text{Sha}(A)|\prod_p c_p / |A(\mathbb{Q})_{\mathrm{tors}}|^2$, where $\Omega_A$ is the real period, $R_A$ is the regulator, $c_p$ is the Tamagawa number at $p$, $\text{Sha}(A)$ is the Tate-Shafarevich group, and $A(\mathbb{Q})_{\mathrm{tors}}$ is the torsion subgroup [3].

Comments

(4)

The label is the LMFDB isogeny class label for the Jacobian of $C$ [2]. There is one row for each isogeny class, because $L(A,s)$ is an isogeny invariant; each entry comment names one representative curve and its model $y^2+h(x)y=f(x)$.

(5)

The arithmetic normalisation has centre $s=1$ and functional equation $s\leftrightarrow 2-s$, as in the LMFDB genus $2$ data and PARI's lfungenus2 [5].

(6)

When $r=0$ the stored value is $L(A,1)$, and when $r=1$ it is $L'(A,1)$. The factor $1/r!$ is part of the definition for every rank.

(7)

Only odd conductors are listed, because PARI's documentation for lfungenus2 says the model must be minimal at $2$ and, when the conductor is even, the conductor's valuation at $2$ may be incorrect [5].

Programs

(P1)

PARI/GP

default(realbitprecision, 320);
P = x^5 + x^4;
Q = x^3 + x + 1;
L = lfungenus2([P, Q]);        \\ L-function of 169.a.169.1
r = lfunorderzero(L, 1);
lfun(L, 1, r) / r!             \\ 0.090490390832429629113589757258...
lfuncheckfeq(L)                \\ about -320

Links

[1]

Wikipedia: Hasse-Weil zeta function

[2]

LMFDB: genus 2 curves over $\mathbb{Q}$

[3]

LMFDB knowl: BSD invariants for genus 2 curves

[4]

LMFDB JSON data: genus 2 curve 169.a.169.1

[5]

PARI/GP reference: L-functions

Similar tables

Special $L$-value of elliptic curves over $\mathbb{Q}$ of rank $1$ — stores the same leading Taylor coefficient for elliptic curves of rank $1$

Special $L$-value of elliptic curves over $\mathbb{Q}$ of rank $2$ — stores the same leading Taylor coefficient for elliptic curves of rank $2$

Special $L$-value of elliptic curves over $\mathbb{Q}$ of rank $3$ — stores the same leading Taylor coefficient for elliptic curves of rank $3$

Values of Dirichlet $L$-functions at positive integers — stores special values for degree-one $L$-functions

Zeros of Dirichlet L-series — stores zeros of another family of arithmetic $L$-functions

Equations satisfied by Igusa invariants of split Jacobians — stores equations that recognise genus $2$ curves whose Jacobians split

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds every LMFDB genus $2$ isogeny class of odd conductor $N\leq1000$ whose analytic rank and leading coefficient are recorded in the LMFDB genus $2$ curve data)

How they were obtained:

The generator uses the $41$ odd-conductor LMFDB genus $2$ isogeny classes with conductor $N\leq1000$, grouped from $56$ curve rows. For each class it builds the PARI L-function [5] with lfungenus2([P,Q]), where PARI's $[P,Q]$ is the model $[f,h]$ in $y^2+h(x)y=f(x)$.

It computes lfun(L,1,r,bitprec)/r! at $60$ and $80$ decimal working digits using explicit bit precision, and stores the digits that agree. Before the draft was created, PARI's lfunorderzero agreed with the LMFDB analytic rank on all $41$ classes, lfuncheckfeq(L,None,330) returned between $-339$ and $-331$, and the computed values agreed with the LMFDB leading_coeff fields, including [4]. Even-conductor classes were left out because their local data at $2$ need separate checking before the same computation is used. The BSD quotient from the LMFDB period, regulator, Tamagawa product, analytic order of $\text{Sha}$, and torsion order agreed to the precision those source fields carry.