| 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 |
| 20.4-b3 |
20.4-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{34} \cdot 5 \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$3.414954535$ |
1.226687881 |
\( -\frac{7985051}{1310720} a - \frac{24684557}{163840} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -3\) , \( 5 a + 30\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-a{x}^2-3{x}+5a+30$ |
| 100.6-b3 |
100.6-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
100.6 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{22} \cdot 5^{7} \) |
$1.57333$ |
$(2,a), (2,a+1), (5,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \cdot 3^{2} \) |
$0.107464068$ |
$1.527214096$ |
4.244678959 |
\( -\frac{7985051}{1310720} a - \frac{24684557}{163840} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -3 a - 1\) , \( -9 a + 25\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-3a-1\right){x}-9a+25$ |
| 500.6-b3 |
500.6-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
500.6 |
\( 2^{2} \cdot 5^{3} \) |
\( 2^{34} \cdot 5^{7} \) |
$2.35267$ |
$(2,a), (2,a+1), (5,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.527214096$ |
1.097182995 |
\( -\frac{7985051}{1310720} a - \frac{24684557}{163840} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 5 a - 37\) , \( -61 a + 293\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-a+1\right){x}^2+\left(5a-37\right){x}-61a+293$ |
| 640.14-c3 |
640.14-c |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
640.14 |
\( 2^{7} \cdot 5 \) |
\( 2^{52} \cdot 5 \) |
$2.50244$ |
$(2,a), (2,a+1), (5,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1.539795620$ |
$1.207368754$ |
2.671235346 |
\( -\frac{7985051}{1310720} a - \frac{24684557}{163840} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 12 a + 40\) , \( 128 a - 496\bigr] \) |
${y}^2={x}^3+\left(-a+1\right){x}^2+\left(12a+40\right){x}+128a-496$ |
| 640.4-h3 |
640.4-h |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
640.4 |
\( 2^{7} \cdot 5 \) |
\( 2^{52} \cdot 5 \) |
$2.50244$ |
$(2,a), (2,a+1), (5,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.794893457$ |
$1.207368754$ |
6.205410396 |
\( -\frac{7985051}{1310720} a - \frac{24684557}{163840} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -3 a + 67\) , \( -210 a - 67\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(-3a+67\right){x}-210a-67$ |
| 1280.10-d3 |
1280.10-d |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1280.10 |
\( 2^{8} \cdot 5 \) |
\( 2^{58} \cdot 5 \) |
$2.97592$ |
$(2,a), (2,a+1), (5,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1.983540999$ |
$0.853738633$ |
4.866371412 |
\( -\frac{7985051}{1310720} a - \frac{24684557}{163840} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -11 a - 109\) , \( -205 a - 1499\bigr] \) |
${y}^2={x}^3-{x}^2+\left(-11a-109\right){x}-205a-1499$ |
| 1620.4-a3 |
1620.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1620.4 |
\( 2^{2} \cdot 3^{4} \cdot 5 \) |
\( 2^{34} \cdot 3^{12} \cdot 5 \) |
$3.15644$ |
$(2,a), (2,a+1), (5,a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{4} \) |
$1.718214029$ |
$1.138318178$ |
5.620566207 |
\( -\frac{7985051}{1310720} a - \frac{24684557}{163840} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -5 a - 56\) , \( -66 a - 596\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a-1\right){x}^2+\left(-5a-56\right){x}-66a-596$ |
| 2500.8-g3 |
2500.8-g |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
2500.8 |
\( 2^{2} \cdot 5^{4} \) |
\( 2^{34} \cdot 5^{13} \) |
$3.51807$ |
$(2,a), (2,a+1), (5,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{6} \) |
$1$ |
$0.682990907$ |
3.925401220 |
\( -\frac{7985051}{1310720} a - \frac{24684557}{163840} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -17 a - 168\) , \( 460 a + 3024\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(-17a-168\right){x}+460a+3024$ |
| 3200.21-d3 |
3200.21-d |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
3200.21 |
\( 2^{7} \cdot 5^{2} \) |
\( 2^{40} \cdot 5^{7} \) |
$3.74203$ |
$(2,a), (2,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{5} \) |
$1$ |
$0.539951721$ |
6.206604293 |
\( -\frac{7985051}{1310720} a - \frac{24684557}{163840} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 22 a - 36\) , \( 272 a + 55\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(22a-36\right){x}+272a+55$ |
| 3200.6-i3 |
3200.6-i |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
3200.6 |
\( 2^{7} \cdot 5^{2} \) |
\( 2^{40} \cdot 5^{7} \) |
$3.74203$ |
$(2,a), (2,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \cdot 3^{2} \) |
$1$ |
$0.539951721$ |
6.982429830 |
\( -\frac{7985051}{1310720} a - \frac{24684557}{163840} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 25 a + 7\) , \( -71 a - 881\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(a+1\right){x}^2+\left(25a+7\right){x}-71a-881$ |
*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.
