| 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 |
| 31.3-a1 |
31.3-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.3 |
\( 31 \) |
\( 31^{4} \) |
$4.60400$ |
$(-2a^3-2a^2+6a+3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$226.2402485$ |
0.843147624 |
\( -\frac{3352161032683862028986803952}{923521} a^{3} + \frac{9909977575470350951831216400}{923521} a^{2} - \frac{5978197446809240162656682401}{923521} a - \frac{1713525152117869864310632982}{923521} \) |
\( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 495 a^{3} + 69 a^{2} - 1642 a - 353\) , \( -7129 a^{3} - 1543 a^{2} + 24030 a + 5034\bigr] \) |
${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(495a^{3}+69a^{2}-1642a-353\right){x}-7129a^{3}-1543a^{2}+24030a+5034$ |
| 31.3-a2 |
31.3-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.3 |
\( 31 \) |
\( 31^{2} \) |
$4.60400$ |
$(-2a^3-2a^2+6a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$14.14001553$ |
0.843147624 |
\( \frac{1366643918492895232}{961} a^{3} + \frac{1130359602108196159}{961} a^{2} - \frac{3401298865000275452}{961} a - \frac{747980482361750548}{961} \) |
\( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 410 a^{3} + 84 a^{2} - 1572 a - 328\) , \( 6096 a^{3} + 1625 a^{2} - 22572 a - 4736\bigr] \) |
${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(410a^{3}+84a^{2}-1572a-328\right){x}+6096a^{3}+1625a^{2}-22572a-4736$ |
| 31.3-a3 |
31.3-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.3 |
\( 31 \) |
\( 31^{8} \) |
$4.60400$ |
$(-2a^3-2a^2+6a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$226.2402485$ |
0.843147624 |
\( -\frac{11010764002907552479232}{852891037441} a^{3} + \frac{32551084210297485030913}{852891037441} a^{2} - \frac{19636479236469948927748}{852891037441} a - \frac{5628354043368086906140}{852891037441} \) |
\( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 40 a^{3} + 4 a^{2} - 142 a - 28\) , \( -32 a^{3} - 5 a^{2} + 76 a + 16\bigr] \) |
${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(40a^{3}+4a^{2}-142a-28\right){x}-32a^{3}-5a^{2}+76a+16$ |
| 31.3-a4 |
31.3-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.3 |
\( 31 \) |
\( 31^{4} \) |
$4.60400$ |
$(-2a^3-2a^2+6a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$226.2402485$ |
0.843147624 |
\( \frac{17801814367097}{923521} a^{3} + \frac{14868316428812}{923521} a^{2} - \frac{44271766433252}{923521} a - \frac{9739909921969}{923521} \) |
\( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 25 a^{3} + 4 a^{2} - 97 a - 18\) , \( 82 a^{3} + 20 a^{2} - 308 a - 64\bigr] \) |
${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(25a^{3}+4a^{2}-97a-18\right){x}+82a^{3}+20a^{2}-308a-64$ |
| 31.3-a5 |
31.3-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.3 |
\( 31 \) |
\( 31^{2} \) |
$4.60400$ |
$(-2a^3-2a^2+6a+3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$226.2402485$ |
0.843147624 |
\( -\frac{1765389}{961} a^{3} - \frac{1884681}{961} a^{2} + \frac{4161259}{961} a + \frac{1928412}{961} \) |
\( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -a^{2} - 2 a + 2\) , \( 2 a^{3} - 8 a - 1\bigr] \) |
${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(-a^{2}-2a+2\right){x}+2a^{3}-8a-1$ |
| 31.3-a6 |
31.3-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.3 |
\( 31 \) |
\( 31^{16} \) |
$4.60400$ |
$(-2a^3-2a^2+6a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$14.14001553$ |
0.843147624 |
\( \frac{17216766333469942774436560624}{727423121747185263828481} a^{3} - \frac{52426907107682729263745202448}{727423121747185263828481} a^{2} + \frac{33240480633127353737657993249}{727423121747185263828481} a + \frac{9399400109123561576571971302}{727423121747185263828481} \) |
\( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -175 a^{3} - 61 a^{2} + 638 a + 137\) , \( 89 a^{3} + 13 a^{2} - 402 a - 82\bigr] \) |
${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(-175a^{3}-61a^{2}+638a+137\right){x}+89a^{3}+13a^{2}-402a-82$ |
| 31.3-b1 |
31.3-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.3 |
\( 31 \) |
\( 31^{4} \) |
$4.60400$ |
$(-2a^3-2a^2+6a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$2.959428063$ |
0.705864781 |
\( -\frac{3352161032683862028986803952}{923521} a^{3} + \frac{9909977575470350951831216400}{923521} a^{2} - \frac{5978197446809240162656682401}{923521} a - \frac{1713525152117869864310632982}{923521} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 5 a^{3} - 75 a^{2} + 320 a - 384\) , \( 330 a^{3} - 1847 a^{2} + 3763 a - 2740\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(5a^{3}-75a^{2}+320a-384\right){x}+330a^{3}-1847a^{2}+3763a-2740$ |
| 31.3-b2 |
31.3-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.3 |
\( 31 \) |
\( 31^{2} \) |
$4.60400$ |
$(-2a^3-2a^2+6a+3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$757.6135842$ |
0.705864781 |
\( \frac{1366643918492895232}{961} a^{3} + \frac{1130359602108196159}{961} a^{2} - \frac{3401298865000275452}{961} a - \frac{747980482361750548}{961} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 5 a^{2} - 20 a - 64\) , \( -7 a^{3} - 11 a^{2} + 120 a + 219\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(5a^{2}-20a-64\right){x}-7a^{3}-11a^{2}+120a+219$ |
| 31.3-b3 |
31.3-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.3 |
\( 31 \) |
\( 31^{8} \) |
$4.60400$ |
$(-2a^3-2a^2+6a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$47.35084901$ |
0.705864781 |
\( -\frac{11010764002907552479232}{852891037441} a^{3} + \frac{32551084210297485030913}{852891037441} a^{2} - \frac{19636479236469948927748}{852891037441} a - \frac{5628354043368086906140}{852891037441} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -5 a^{2} + 20 a - 24\) , \( 7 a^{3} - 35 a^{2} + 64 a - 35\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5a^{2}+20a-24\right){x}+7a^{3}-35a^{2}+64a-35$ |
| 31.3-b4 |
31.3-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.3 |
\( 31 \) |
\( 31^{4} \) |
$4.60400$ |
$(-2a^3-2a^2+6a+3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$757.6135842$ |
0.705864781 |
\( \frac{17801814367097}{923521} a^{3} + \frac{14868316428812}{923521} a^{2} - \frac{44271766433252}{923521} a - \frac{9739909921969}{923521} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -4\) , \( -a^{2} + 4 a + 4\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}-4{x}-a^{2}+4a+4$ |
| 31.3-b5 |
31.3-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.3 |
\( 31 \) |
\( 31^{2} \) |
$4.60400$ |
$(-2a^3-2a^2+6a+3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$757.6135842$ |
0.705864781 |
\( -\frac{1765389}{961} a^{3} - \frac{1884681}{961} a^{2} + \frac{4161259}{961} a + \frac{1928412}{961} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+{x}$ |
| 31.3-b6 |
31.3-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.3 |
\( 31 \) |
\( 31^{16} \) |
$4.60400$ |
$(-2a^3-2a^2+6a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$2.959428063$ |
0.705864781 |
\( \frac{17216766333469942774436560624}{727423121747185263828481} a^{3} - \frac{52426907107682729263745202448}{727423121747185263828481} a^{2} + \frac{33240480633127353737657993249}{727423121747185263828481} a + \frac{9399400109123561576571971302}{727423121747185263828481} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -5 a^{3} - 15 a^{2} + 40 a + 16\) , \( -8 a^{3} + a^{2} + 125 a - 166\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5a^{3}-15a^{2}+40a+16\right){x}-8a^{3}+a^{2}+125a-166$ |
*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.
