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

   comment-rank: 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.-  comment-even-conductors: Only odd conductors are listed. PARI's documentation for-    `lfungenus2` says that when the conductor is even, its valuation at $2$ may be-    incorrect CITE{PARILfun}; those classes need the local Euler factor at $2$ supplied-    separately before their values can be computed by this method.+  comment-even-conductors: 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 CITE{PARILfun}. Formulas:   formula-leading-coefficient: If $L(A,s)=\sum_{m\geq r} c_m(s-1)^m$ with $c_r\neq

     where $P_p(t)=\det(1-t\operatorname{Frob}_p\mid T_\ell A)$.   formula-bsd: The Birch and Swinnerton-Dyer conjecture predicts $L^{(r)}(A,1)/r!-    = \Omega_A R_A |\Sha(A)|\prod_p c_p / |A(\mathbb{Q})_{\mathrm{tors}}|^2$, with-    the factors named as in the LMFDB BSD invariants CITE{LMFDBBSD}.+    = \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 CITE{LMFDBBSD}. Programs:   program-pari:

   rigour details: 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 CITE{PARILfun} with `lfungenus2([P,Q])` from the-    representative curve $y^2+Q(x)y=P(x)$, 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 CITE{LMFDBData169}. The BSD quotient from the LMFDB period,-    regulator, Tamagawa product, analytic order of $\Sha$, and torsion order agreed-    to the precision those source fields carry.+    it builds the PARI L-function CITE{PARILfun} 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 CITE{LMFDBData169}. 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. Display properties:   number-header: leading coefficient

   - - rank Numbers:-- params:-    label: 169.a-    rank: '0'-  number: '0.09049039083242962911358975725801541226473'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/169/a/169/1}[LMFDB-    169.a.169.1], $y^2 + (x^3 + x + 1)y = x^5 + x^4$.-- params:-    label: 249.a-    rank: '0'-  number: '0.1315495070114787592134013430121752606933'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/249/a/249/1}[LMFDB-    249.a.249.1], $y^2 + (x^3 + 1)y = x^2 + x$. The LMFDB class has 2 curves in this-    range.-- params:-    label: 277.a-    rank: '0'-  number: '0.1431366605510114901557152099923829602232'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/277/a/277/1}[LMFDB-    277.a.277.1], $y^2 + (x^3 + x^2 + x + 1)y = -x^2 - x$. The LMFDB class has 2 curves-    in this range.-- params:-    label: 295.a-    rank: '0'-  number: '0.1492683688832182090588207138996158657800'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/295/a/295/1}[LMFDB-    295.a.295.1], $y^2 + (x^3 + 1)y = -x^2$. The LMFDB class has 2 curves in this-    range.-- params:-    label: 349.a-    rank: '0'-  number: '0.1656123320984078287720952736293483346028'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/349/a/349/1}[LMFDB-    349.a.349.1], $y^2 + (x^3 + x^2 + x + 1)y = -x^3 - x^2$.-- params:-    label: 353.a-    rank: '0'-  number: '0.1859131786554872680710289056816858155417'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/353/a/353/1}[LMFDB-    353.a.353.1], $y^2 + (x^3 + x + 1)y = x^2$.-- params:-    label: 363.a-    rank: '0'-  number: '0.1897059599534884032426437150258731309382'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/363/a/11979/1}[LMFDB-    363.a.11979.1], $y^2 + (x^2 + 1)y = x^5 + 2 x^3 + 4 x^2 + 2 x$. The LMFDB class-    has 2 curves in this range.-- params:-    label: 389.a-    rank: '0'-  number: '0.1979862013008733330350209978381275022945'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/389/a/389/1}[LMFDB-    389.a.389.1], $y^2 + (x^3 + x)y = x^5 - 2 x^4 - 8 x^3 + 16 x + 7$. The LMFDB class-    has 2 curves in this range.-- params:-    label: 427.a-    rank: '0'-  number: '0.1899303686308920817263923619718777416327'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/427/a/2989/1}[LMFDB-    427.a.2989.1], $y^2 + (x^3 + 1)y = x^5 - x^4 - 5 x^3 + 4 x^2 + 4 x - 4$.-- params:-    label: 461.a-    rank: '0'-  number: '0.2458864264726567276756390789225886309417'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/461/a/461/1}[LMFDB-    461.a.461.1], $y^2 + (x^3)y = x^5 - 3 x^3 + 3 x - 2$. The LMFDB class has 2 curves-    in this range.-- params:-    label: 523.a-    rank: '0'-  number: '0.2481990412677571630041171344077874577287'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/523/a/523/1}[LMFDB-    523.a.523.1], $y^2 + (x + 1)y = x^5 - x^4 - x^3$. The LMFDB class has 2 curves-    in this range.-- params:-    label: 529.a-    rank: '0'-  number: '0.2484318665905996812072503393142383906699'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/529/a/529/1}[LMFDB-    529.a.529.1], $y^2 + (x^3 + x + 1)y = -x^5$.-- params:-    label: 555.a-    rank: '0'-  number: '0.2569247202971451430840301627276747947597'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/555/a/8325/1}[LMFDB-    555.a.8325.1], $y^2 + (x + 1)y = 3 x^5 - 2 x^4 - 4 x^3 + x^2 + x$.-- params:-    label: 587.a-    rank: '1'-  number: '0.1113515110642202373191027362087235867524'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/587/a/587/1}[LMFDB-    587.a.587.1], $y^2 + (x^3 + x + 1)y = -x^2 - x$.-- params:-    label: 597.a-    rank: '0'-  number: '0.2941146301882136359236849987869052563808'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/597/a/597/1}[LMFDB-    597.a.597.1], $y^2 + (1)y = x^5 + 2 x^4 + 3 x^3 + 2 x^2 + x$.-- params:-    label: 603.a-    rank: '0'-  number: '0.2691001556363177690790850827729359332812'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/603/a/603/1}[LMFDB-    603.a.603.1], $y^2 + (x^2 + 1)y = x^5 + 8 x^4 + 4 x^3 + 4 x^2 + 2 x$. The LMFDB-    class has 2 curves in this range.-- params:-    label: 691.a-    rank: '0'-  number: '0.2939463892598192424091021283086475110620'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/691/a/691/1}[LMFDB-    691.a.691.1], $y^2 + (x + 1)y = x^5 - x^3 - x^2$.-- params:-    label: 709.a-    rank: '0'-  number: '0.2868931505071272731308297049624563478404'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/709/a/709/1}[LMFDB-    709.a.709.1], $y^2 + (x)y = x^5 - 2 x^2 + x$.-- params:-    label: 713.a-    rank: '1'-  number: '0.1283949912472834590170543474773248575796'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/713/a/713/1}[LMFDB-    713.a.713.1], $y^2 + (x^3 + x + 1)y = -x^5 - x$.-- params:-    label: 713.b-    rank: '0'-  number: '0.2858010009469617170226097264141850097689'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/713/b/713/1}[LMFDB-    713.b.713.1], $y^2 + (x^3 + x + 1)y = -x^4$.-- params:-    label: 731.a-    rank: '0'-  number: '0.2985355886872414062863189873478987786819'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/731/a/12427/1}[LMFDB-    731.a.12427.1], $y^2 + (x^3 + x^2)y = x^5 + 2 x^4 - x - 3$.-- params:-    label: 741.a-    rank: '0'-  number: '0.2930756651264371996923859602312386579243'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/741/a/28899/1}[LMFDB-    741.a.28899.1], $y^2 + (x + 1)y = -3 x^5 - x^4 + 2 x^2 + x$.-- params:-    label: 743.a-    rank: '1'-  number: '0.1316557503846215116289169353731506251281'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/743/a/743/1}[LMFDB-    743.a.743.1], $y^2 + (x^3 + x + 1)y = -x^4 + x^2$.-- params:-    label: 745.a-    rank: '0'-  number: '0.3033683921052660240631287058786037771056'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/745/a/745/1}[LMFDB-    745.a.745.1], $y^2 + (x^3 + x + 1)y = -x$.-- params:-    label: 763.a-    rank: '0'-  number: '0.3048575004522535920085251501905636550504'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/763/a/763/1}[LMFDB-    763.a.763.1], $y^2 + (x^3 + x)y = -2 x^4 + 2 x^2 - x$.-- params:-    label: 797.a-    rank: '0'-  number: '0.3559385607403319429724872658598941145485'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/797/a/797/1}[LMFDB-    797.a.797.1], $y^2 + (1)y = x^5 - x^4 + x^3$.-- params:-    label: 807.a-    rank: '0'-  number: '0.3050356332174254142944032925074203061351'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/807/a/2421/1}[LMFDB-    807.a.2421.1], $y^2 + (x^3 + x)y = x^5 - 2 x^3 - x^2 + 2 x - 1$.-- params:-    label: 841.a-    rank: '0'-  number: '0.2915215656991529771681413103312969859954'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/841/a/841/1}[LMFDB-    841.a.841.1], $y^2 + (x^3 + x^2 + x)y = x^4 + x^3 + 3 x^2 + x + 2$.-- params:-    label: 847.a-    rank: '1'-  number: '0.1592441888803194513955584463536559869689'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/847/a/847/1}[LMFDB-    847.a.847.1], $y^2 + (x^3 + x^2 + x + 1)y = x^4 + x^3 + x^2$.-- params:-    label: 847.b-    rank: '0'-  number: '0.3365454249762917676170118858324423791181'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/847/b/9317/1}[LMFDB-    847.b.9317.1], $y^2 + (x^2 + 1)y = x^5 + 2 x^4 - 3 x^3 + 2 x^2 - x$.-- params:-    label: 847.c-    rank: '0'-  number: '0.3119812484526625370039160119138202568854'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/847/c/9317/1}[LMFDB-    847.c.9317.1], $y^2 + (x^3 + x^2)y = x^4 + x^3 - x - 2$.-- params:-    label: 847.d-    rank: '0'-  number: '0.2621188089061351721920136796692074600710'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/847/d/456533/1}[LMFDB-    847.d.456533.1], $y^2 + (1)y = -x^6 - 9 x^5 - 22 x^4 + 3 x^3 + 37 x^2 - 24 x +-    4$. The LMFDB class has 2 curves in this range.-- params:-    label: 893.a-    rank: '1'-  number: '0.1504585378200966582669018264044790617402'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/893/a/893/1}[LMFDB-    893.a.893.1], $y^2 + (x^3 + x + 1)y = -x^4 - x^2$.-- params:-    label: 909.a-    rank: '0'-  number: '0.3407116942606920617569877403142908280531'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/909/a/8181/1}[LMFDB-    909.a.8181.1], $y^2 + (x)y = 3 x^5 - 7 x^4 + x^3 + 6 x^2 - 3 x$. The LMFDB class-    has 2 curves in this range.-- params:-    label: 925.a-    rank: '0'-  number: '0.3262333436965224143091586365531384380133'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/925/a/23125/1}[LMFDB-    925.a.23125.1], $y^2 + (x)y = 5 x^5 + x^4 - 19 x^3 + 18 x^2 - 5 x$. The LMFDB-    class has 2 curves in this range.-- params:-    label: 953.a-    rank: '1'-  number: '0.1561935793455555266041051297046358534815'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/953/a/953/1}[LMFDB-    953.a.953.1], $y^2 + (x^3 + x + 1)y = x^3 + x^2$.-- params:-    label: 961.a-    rank: '0'-  number: '0.4492877238760407861133296681879239133411'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/961/a/923521/1}[LMFDB-    961.a.923521.1], $y^2 + (x^3 + x^2 + 1)y = -5 x^4 + 4 x^3 + 3 x^2 - 2 x - 3$.-    The LMFDB class has 4 curves in this range.-- params:-    label: 971.a-    rank: '1'-  number: '0.1769981337951764400226500892472379175742'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/971/a/971/1}[LMFDB-    971.a.971.1], $y^2 + (1)y = x^5 - 2 x^3 + x$.-- params:-    label: 975.a-    rank: '0'-  number: '0.3987858446601027745461990966212669516994'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/975/a/63375/1}[LMFDB-    975.a.63375.1], $y^2 + (x^3 + 1)y = -x^5 + x^3 + 2 x^2 + x - 1$.-- params:-    label: 997.a-    rank: '0'-  number: '0.3373378289933834996066688066442920811665'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/997/a/997/1}[LMFDB-    997.a.997.1], $y^2 + (x)y = x^5 - 8 x^4 + 16 x^3 - x$. The LMFDB class has 2 curves-    in this range.-- params:-    label: 997.b-    rank: '1'-  number: '0.1799924859492394396660740920629892328536'-  comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/997/b/997/1}[LMFDB-    997.b.997.1], $y^2 + (1)y = x^5 - 2 x^4 + 2 x^3 - x^2$.+  169.a:+    '0':+      number: '0.09049039083242962911358975725801541226473'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/169/a/169/1}[LMFDB+        169.a.169.1], $y^2 + (x^3 + x + 1)y = x^5 + x^4$.+  249.a:+    '0':+      number: '0.1315495070114787592134013430121752606933'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/249/a/249/1}[LMFDB+        249.a.249.1], $y^2 + (x^3 + 1)y = x^2 + x$.+  277.a:+    '0':+      number: '0.1431366605510114901557152099923829602232'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/277/a/277/1}[LMFDB+        277.a.277.1], $y^2 + (x^3 + x^2 + x + 1)y = -x^2 - x$.+  295.a:+    '0':+      number: '0.1492683688832182090588207138996158657800'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/295/a/295/1}[LMFDB+        295.a.295.1], $y^2 + (x^3 + 1)y = -x^2$.+  349.a:+    '0':+      number: '0.1656123320984078287720952736293483346028'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/349/a/349/1}[LMFDB+        349.a.349.1], $y^2 + (x^3 + x^2 + x + 1)y = -x^3 - x^2$.+  353.a:+    '0':+      number: '0.1859131786554872680710289056816858155417'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/353/a/353/1}[LMFDB+        353.a.353.1], $y^2 + (x^3 + x + 1)y = x^2$.+  363.a:+    '0':+      number: '0.1897059599534884032426437150258731309382'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/363/a/11979/1}[LMFDB+        363.a.11979.1], $y^2 + (x^2 + 1)y = x^5 + 2 x^3 + 4 x^2 + 2 x$.+  389.a:+    '0':+      number: '0.1979862013008733330350209978381275022945'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/389/a/389/1}[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':+      number: '0.1899303686308920817263923619718777416327'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/427/a/2989/1}[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':+      number: '0.2458864264726567276756390789225886309417'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/461/a/461/1}[LMFDB+        461.a.461.1], $y^2 + x^3 y = x^5 - 3 x^3 + 3 x - 2$.+  523.a:+    '0':+      number: '0.2481990412677571630041171344077874577287'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/523/a/523/1}[LMFDB+        523.a.523.1], $y^2 + (x + 1)y = x^5 - x^4 - x^3$.+  529.a:+    '0':+      number: '0.2484318665905996812072503393142383906699'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/529/a/529/1}[LMFDB+        529.a.529.1], $y^2 + (x^3 + x + 1)y = -x^5$.+  555.a:+    '0':+      number: '0.2569247202971451430840301627276747947597'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/555/a/8325/1}[LMFDB+        555.a.8325.1], $y^2 + (x + 1)y = 3 x^5 - 2 x^4 - 4 x^3 + x^2 + x$.+  587.a:+    '1':+      number: '0.1113515110642202373191027362087235867524'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/587/a/587/1}[LMFDB+        587.a.587.1], $y^2 + (x^3 + x + 1)y = -x^2 - x$.+  597.a:+    '0':+      number: '0.2941146301882136359236849987869052563808'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/597/a/597/1}[LMFDB+        597.a.597.1], $y^2 + y = x^5 + 2 x^4 + 3 x^3 + 2 x^2 + x$.+  603.a:+    '0':+      number: '0.2691001556363177690790850827729359332812'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/603/a/603/1}[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':+      number: '0.2939463892598192424091021283086475110620'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/691/a/691/1}[LMFDB+        691.a.691.1], $y^2 + (x + 1)y = x^5 - x^3 - x^2$.+  709.a:+    '0':+      number: '0.2868931505071272731308297049624563478404'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/709/a/709/1}[LMFDB+        709.a.709.1], $y^2 + xy = x^5 - 2 x^2 + x$.+  713.a:+    '1':+      number: '0.1283949912472834590170543474773248575796'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/713/a/713/1}[LMFDB+        713.a.713.1], $y^2 + (x^3 + x + 1)y = -x^5 - x$.+  713.b:+    '0':+      number: '0.2858010009469617170226097264141850097689'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/713/b/713/1}[LMFDB+        713.b.713.1], $y^2 + (x^3 + x + 1)y = -x^4$.+  731.a:+    '0':+      number: '0.2985355886872414062863189873478987786819'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/731/a/12427/1}[LMFDB+        731.a.12427.1], $y^2 + (x^3 + x^2)y = x^5 + 2 x^4 - x - 3$.+  741.a:+    '0':+      number: '0.2930756651264371996923859602312386579243'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/741/a/28899/1}[LMFDB+        741.a.28899.1], $y^2 + (x + 1)y = -3 x^5 - x^4 + 2 x^2 + x$.+  743.a:+    '1':+      number: '0.1316557503846215116289169353731506251281'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/743/a/743/1}[LMFDB+        743.a.743.1], $y^2 + (x^3 + x + 1)y = -x^4 + x^2$.+  745.a:+    '0':+      number: '0.3033683921052660240631287058786037771056'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/745/a/745/1}[LMFDB+        745.a.745.1], $y^2 + (x^3 + x + 1)y = -x$.+  763.a:+    '0':+      number: '0.3048575004522535920085251501905636550504'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/763/a/763/1}[LMFDB+        763.a.763.1], $y^2 + (x^3 + x)y = -2 x^4 + 2 x^2 - x$.+  797.a:+    '0':+      number: '0.3559385607403319429724872658598941145485'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/797/a/797/1}[LMFDB+        797.a.797.1], $y^2 + y = x^5 - x^4 + x^3$.+  807.a:+    '0':+      number: '0.3050356332174254142944032925074203061351'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/807/a/2421/1}[LMFDB+        807.a.2421.1], $y^2 + (x^3 + x)y = x^5 - 2 x^3 - x^2 + 2 x - 1$.+  841.a:+    '0':+      number: '0.2915215656991529771681413103312969859954'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/841/a/841/1}[LMFDB+        841.a.841.1], $y^2 + (x^3 + x^2 + x)y = x^4 + x^3 + 3 x^2 + x + 2$.+  847.a:+    '1':+      number: '0.1592441888803194513955584463536559869689'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/847/a/847/1}[LMFDB+        847.a.847.1], $y^2 + (x^3 + x^2 + x + 1)y = x^4 + x^3 + x^2$.+  847.b:+    '0':+      number: '0.3365454249762917676170118858324423791181'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/847/b/9317/1}[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':+      number: '0.3119812484526625370039160119138202568854'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/847/c/9317/1}[LMFDB+        847.c.9317.1], $y^2 + (x^3 + x^2)y = x^4 + x^3 - x - 2$.+  847.d:+    '0':+      number: '0.2621188089061351721920136796692074600710'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/847/d/456533/1}[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:+    '1':+      number: '0.1504585378200966582669018264044790617402'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/893/a/893/1}[LMFDB+        893.a.893.1], $y^2 + (x^3 + x + 1)y = -x^4 - x^2$.+  909.a:+    '0':+      number: '0.3407116942606920617569877403142908280531'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/909/a/8181/1}[LMFDB+        909.a.8181.1], $y^2 + xy = 3 x^5 - 7 x^4 + x^3 + 6 x^2 - 3 x$.+  925.a:+    '0':+      number: '0.3262333436965224143091586365531384380133'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/925/a/23125/1}[LMFDB+        925.a.23125.1], $y^2 + xy = 5 x^5 + x^4 - 19 x^3 + 18 x^2 - 5 x$.+  953.a:+    '1':+      number: '0.1561935793455555266041051297046358534815'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/953/a/953/1}[LMFDB+        953.a.953.1], $y^2 + (x^3 + x + 1)y = x^3 + x^2$.+  961.a:+    '0':+      number: '0.4492877238760407861133296681879239133411'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/961/a/923521/1}[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:+    '1':+      number: '0.1769981337951764400226500892472379175742'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/971/a/971/1}[LMFDB+        971.a.971.1], $y^2 + y = x^5 - 2 x^3 + x$.+  975.a:+    '0':+      number: '0.3987858446601027745461990966212669516994'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/975/a/63375/1}[LMFDB+        975.a.63375.1], $y^2 + (x^3 + 1)y = -x^5 + x^3 + 2 x^2 + x - 1$.+  997.a:+    '0':+      number: '0.3373378289933834996066688066442920811665'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/997/a/997/1}[LMFDB+        997.a.997.1], $y^2 + xy = x^5 - 8 x^4 + 16 x^3 - x$.+  997.b:+    '1':+      number: '0.1799924859492394396660740920629892328536'+      comment: Representative curve HREF{https://www.lmfdb.org/Genus2Curve/Q/997/b/997/1}[LMFDB+        997.b.997.1], $y^2 + y = x^5 - 2 x^4 + 2 x^3 - x^2$.