| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 31.2-a1 |
31.2-a |
$1$ |
$1$ |
4.4.4400.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{7} \) |
$9.10511$ |
$(a^3+a^2-4a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$155.2018111$ |
2.339755338 |
\( -\frac{20658244006503}{27512614111} a^{3} + \frac{47471524533857}{27512614111} a^{2} + \frac{114350909406398}{27512614111} a - \frac{152350807490055}{27512614111} \) |
\( \bigl[1\) , \( -a^{2} - a + 4\) , \( 0\) , \( 9 a^{3} + 13 a^{2} - 42 a - 59\) , \( -11 a^{3} - 17 a^{2} + 51 a + 80\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(9a^{3}+13a^{2}-42a-59\right){x}-11a^{3}-17a^{2}+51a+80$ |
| 31.2-b1 |
31.2-b |
$1$ |
$1$ |
4.4.4400.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31 \) |
$9.10511$ |
$(a^3+a^2-4a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$106.8964377$ |
1.611524434 |
\( -\frac{206255355828299}{31} a^{3} - \frac{318326661270219}{31} a^{2} + \frac{952494243687014}{31} a + \frac{1470043341319301}{31} \) |
\( \bigl[a^{3} - 3 a + 1\) , \( a^{3} - a^{2} - 5 a + 5\) , \( a^{2} - 3\) , \( 16 a^{3} - 23 a^{2} - 49 a + 56\) , \( 66 a^{3} - 130 a^{2} - 167 a + 307\bigr] \) |
${y}^2+\left(a^{3}-3a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+5\right){x}^{2}+\left(16a^{3}-23a^{2}-49a+56\right){x}+66a^{3}-130a^{2}-167a+307$ |
| 31.2-c1 |
31.2-c |
$4$ |
$4$ |
4.4.4400.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( -31 \) |
$9.10511$ |
$(a^3+a^2-4a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.301614233$ |
$440.5254452$ |
2.003071688 |
\( -\frac{77286524488566078}{31} a^{3} + \frac{166085782662265724}{31} a^{2} + \frac{184093874850639127}{31} a - \frac{395610688650363197}{31} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 4\) , \( -a^{3} + a^{2} + 5 a - 4\) , \( a^{3} - 3 a\) , \( -116 a^{3} + 88 a^{2} + 840 a - 1067\) , \( 2720 a^{3} - 3047 a^{2} - 16831 a + 23245\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-4\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-4\right){x}^{2}+\left(-116a^{3}+88a^{2}+840a-1067\right){x}+2720a^{3}-3047a^{2}-16831a+23245$ |
| 31.2-c2 |
31.2-c |
$4$ |
$4$ |
4.4.4400.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( -31 \) |
$9.10511$ |
$(a^3+a^2-4a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.075403558$ |
$1762.101780$ |
2.003071688 |
\( \frac{155772}{31} a^{3} - \frac{267325}{31} a^{2} - \frac{634876}{31} a + \frac{1052962}{31} \) |
\( \bigl[a^{2} + a - 4\) , \( -a^{2} + 4\) , \( a^{3} - 4 a\) , \( a^{3} - 3 a^{2} - 5 a + 12\) , \( 30 a^{3} + 64 a^{2} - 72 a - 153\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(a^{3}-3a^{2}-5a+12\right){x}+30a^{3}+64a^{2}-72a-153$ |
| 31.2-c3 |
31.2-c |
$4$ |
$4$ |
4.4.4400.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{4} \) |
$9.10511$ |
$(a^3+a^2-4a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.075403558$ |
$440.5254452$ |
2.003071688 |
\( \frac{59370852200354013489666}{923521} a^{3} - \frac{91630712350575600915204}{923521} a^{2} - \frac{274176613402283668981641}{923521} a + \frac{423153744048330031118195}{923521} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} + a^{2} + 5 a - 4\) , \( a^{3} + a^{2} - 4 a - 4\) , \( 80 a^{3} - 151 a^{2} - 203 a + 339\) , \( -712 a^{3} + 1513 a^{2} + 1703 a - 3582\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}+a^{2}-4a-4\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-4\right){x}^{2}+\left(80a^{3}-151a^{2}-203a+339\right){x}-712a^{3}+1513a^{2}+1703a-3582$ |
| 31.2-c4 |
31.2-c |
$4$ |
$4$ |
4.4.4400.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{2} \) |
$9.10511$ |
$(a^3+a^2-4a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.150807116$ |
$1762.101780$ |
2.003071688 |
\( \frac{77272583940}{961} a^{3} - \frac{109867494715}{961} a^{2} - \frac{391445107350}{961} a + \frac{581776625313}{961} \) |
\( \bigl[1\) , \( a^{2} - a - 3\) , \( a^{3} + a^{2} - 4 a - 4\) , \( 1477 a^{3} + 2281 a^{2} - 6825 a - 10537\) , \( 49010 a^{3} + 75642 a^{2} - 226327 a - 349309\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}+a^{2}-4a-4\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(1477a^{3}+2281a^{2}-6825a-10537\right){x}+49010a^{3}+75642a^{2}-226327a-349309$ |
| 31.2-d1 |
31.2-d |
$4$ |
$4$ |
4.4.4400.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( -31 \) |
$9.10511$ |
$(a^3+a^2-4a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$1515.810669$ |
1.428231603 |
\( \frac{155772}{31} a^{3} - \frac{267325}{31} a^{2} - \frac{634876}{31} a + \frac{1052962}{31} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 4\) , \( 1\) , \( a\) , \( -2 a^{3} - 3 a^{2} + 9 a + 13\) , \( 6 a^{3} + 14 a^{2} - 12 a - 32\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-4\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-2a^{3}-3a^{2}+9a+13\right){x}+6a^{3}+14a^{2}-12a-32$ |
| 31.2-d2 |
31.2-d |
$4$ |
$4$ |
4.4.4400.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{4} \) |
$9.10511$ |
$(a^3+a^2-4a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$47.36908341$ |
1.428231603 |
\( \frac{59370852200354013489666}{923521} a^{3} - \frac{91630712350575600915204}{923521} a^{2} - \frac{274176613402283668981641}{923521} a + \frac{423153744048330031118195}{923521} \) |
\( \bigl[a^{2} - 4\) , \( a^{3} + a^{2} - 4 a - 3\) , \( a^{3} - 3 a + 1\) , \( 197 a^{3} - 416 a^{2} - 474 a + 985\) , \( -3002 a^{3} + 6453 a^{2} + 7149 a - 15377\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-3\right){x}^{2}+\left(197a^{3}-416a^{2}-474a+985\right){x}-3002a^{3}+6453a^{2}+7149a-15377$ |
| 31.2-d3 |
31.2-d |
$4$ |
$4$ |
4.4.4400.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( -31 \) |
$9.10511$ |
$(a^3+a^2-4a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$378.9526672$ |
1.428231603 |
\( -\frac{77286524488566078}{31} a^{3} + \frac{166085782662265724}{31} a^{2} + \frac{184093874850639127}{31} a - \frac{395610688650363197}{31} \) |
\( \bigl[a^{2} + a - 4\) , \( a^{3} + a^{2} - 4 a - 4\) , \( a^{2} - 4\) , \( 260 a^{3} - 624 a^{2} - 403 a + 1158\) , \( -9182 a^{3} + 19073 a^{2} + 24315 a - 49206\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-4\right){x}^{2}+\left(260a^{3}-624a^{2}-403a+1158\right){x}-9182a^{3}+19073a^{2}+24315a-49206$ |
| 31.2-d4 |
31.2-d |
$4$ |
$4$ |
4.4.4400.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{2} \) |
$9.10511$ |
$(a^3+a^2-4a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$757.9053345$ |
1.428231603 |
\( \frac{77272583940}{961} a^{3} - \frac{109867494715}{961} a^{2} - \frac{391445107350}{961} a + \frac{581776625313}{961} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 3 a\) , \( a^{3} - 3 a + 1\) , \( 560 a^{3} + 866 a^{2} - 2597 a - 4013\) , \( -11564 a^{3} - 17840 a^{2} + 53415 a + 82420\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(560a^{3}+866a^{2}-2597a-4013\right){x}-11564a^{3}-17840a^{2}+53415a+82420$ |
| 31.2-e1 |
31.2-e |
$1$ |
$1$ |
4.4.4400.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31 \) |
$9.10511$ |
$(a^3+a^2-4a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.195376744$ |
$161.8086715$ |
1.906374917 |
\( -\frac{206255355828299}{31} a^{3} - \frac{318326661270219}{31} a^{2} + \frac{952494243687014}{31} a + \frac{1470043341319301}{31} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 3\) , \( -a^{3} - a^{2} + 4 a + 5\) , \( 1\) , \( 29 a^{3} - 62 a^{2} - 67 a + 150\) , \( 242 a^{3} - 518 a^{2} - 576 a + 1234\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-3\right){x}{y}+{y}={x}^{3}+\left(-a^{3}-a^{2}+4a+5\right){x}^{2}+\left(29a^{3}-62a^{2}-67a+150\right){x}+242a^{3}-518a^{2}-576a+1234$ |
| 31.2-f1 |
31.2-f |
$1$ |
$1$ |
4.4.4400.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{7} \) |
$9.10511$ |
$(a^3+a^2-4a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 7 \) |
$0.008173367$ |
$374.2505338$ |
1.291204819 |
\( -\frac{20658244006503}{27512614111} a^{3} + \frac{47471524533857}{27512614111} a^{2} + \frac{114350909406398}{27512614111} a - \frac{152350807490055}{27512614111} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} - a^{2} + 3 a + 4\) , \( a^{3} - 3 a\) , \( a^{3} + 4 a^{2} - 9 a - 20\) , \( 5 a^{3} + 12 a^{2} - 25 a - 49\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+4\right){x}^{2}+\left(a^{3}+4a^{2}-9a-20\right){x}+5a^{3}+12a^{2}-25a-49$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
