| 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-a4 |
31.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.2 |
\( 31 \) |
\( 31^{2} \) |
$0.47148$ |
$(5a-3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$25.75441985$ |
0.359928959 |
\( -\frac{9029272560}{961} a + \frac{14629102793}{961} \) |
\( \bigl[1\) , \( -\phi - 1\) , \( \phi\) , \( -5\) , \( -3 \phi + 3\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}-5{x}-3\phi+3$ |
| 775.2-a4 |
775.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
775.2 |
\( 5^{2} \cdot 31 \) |
\( 5^{6} \cdot 31^{2} \) |
$1.05426$ |
$(-2a+1), (5a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$6.746456922$ |
1.508553628 |
\( -\frac{9029272560}{961} a + \frac{14629102793}{961} \) |
\( \bigl[\phi\) , \( \phi + 1\) , \( \phi\) , \( 27 \phi - 53\) , \( 106 \phi - 169\bigr] \) |
${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(27\phi-53\right){x}+106\phi-169$ |
| 961.3-c4 |
961.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
961.3 |
\( 31^{2} \) |
\( 31^{8} \) |
$1.11251$ |
$(5a-3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$3.548353402$ |
$1.723681156$ |
1.367630581 |
\( -\frac{9029272560}{961} a + \frac{14629102793}{961} \) |
\( \bigl[\phi\) , \( 0\) , \( 0\) , \( 18 \phi - 186\) , \( 140 \phi - 985\bigr] \) |
${y}^2+\phi{x}{y}={x}^{3}+\left(18\phi-186\right){x}+140\phi-985$ |
| 2511.2-f4 |
2511.2-f |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2511.2 |
\( 3^{4} \cdot 31 \) |
\( 3^{12} \cdot 31^{2} \) |
$1.41444$ |
$(5a-3), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$5.028512095$ |
1.124409487 |
\( -\frac{9029272560}{961} a + \frac{14629102793}{961} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -3 \phi - 50\) , \( 124 \phi - 39\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-3\phi-50\right){x}+124\phi-39$ |
| 3751.3-a4 |
3751.3-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.3 |
\( 11^{2} \cdot 31 \) |
\( 11^{6} \cdot 31^{2} \) |
$1.56372$ |
$(-3a+2), (5a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$2.893619604$ |
2.588132054 |
\( -\frac{9029272560}{961} a + \frac{14629102793}{961} \) |
\( \bigl[\phi + 1\) , \( -\phi + 1\) , \( 0\) , \( 13 \phi - 70\) , \( 75 \phi - 267\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(13\phi-70\right){x}+75\phi-267$ |
| 3751.5-b4 |
3751.5-b |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.5 |
\( 11^{2} \cdot 31 \) |
\( 11^{6} \cdot 31^{2} \) |
$1.56372$ |
$(-3a+1), (5a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$12.20614154$ |
2.729376224 |
\( -\frac{9029272560}{961} a + \frac{14629102793}{961} \) |
\( \bigl[\phi\) , \( \phi - 1\) , \( \phi + 1\) , \( -22 \phi - 56\) , \( 64 \phi + 171\bigr] \) |
${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-22\phi-56\right){x}+64\phi+171$ |
*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.
