| 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 |
| 31.1-a5 |
31.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.1 |
\( 31 \) |
\( 31^{2} \) |
$0.47148$ |
$(5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$25.75441985$ |
0.359928959 |
\( \frac{9029272560}{961} a + \frac{5599830233}{961} \) |
\( \bigl[1\) , \( \phi + 1\) , \( \phi\) , \( \phi - 5\) , \( 3 \phi - 5\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(\phi-5\right){x}+3\phi-5$ |
| 775.1-a5 |
775.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
775.1 |
\( 5^{2} \cdot 31 \) |
\( 5^{6} \cdot 31^{2} \) |
$1.05426$ |
$(-2a+1), (5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$6.746456922$ |
1.508553628 |
\( \frac{9029272560}{961} a + \frac{5599830233}{961} \) |
\( \bigl[\phi + 1\) , \( \phi - 1\) , \( 1\) , \( -26 \phi - 27\) , \( -133 \phi - 62\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-26\phi-27\right){x}-133\phi-62$ |
| 961.2-c5 |
961.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
961.2 |
\( 31^{2} \) |
\( 31^{8} \) |
$1.11251$ |
$(5a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$3.548353402$ |
$1.723681156$ |
1.367630581 |
\( \frac{9029272560}{961} a + \frac{5599830233}{961} \) |
\( \bigl[\phi + 1\) , \( -\phi\) , \( 0\) , \( -18 \phi - 168\) , \( -140 \phi - 845\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}={x}^{3}-\phi{x}^{2}+\left(-18\phi-168\right){x}-140\phi-845$ |
| 2511.1-f5 |
2511.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2511.1 |
\( 3^{4} \cdot 31 \) |
\( 3^{12} \cdot 31^{2} \) |
$1.41444$ |
$(5a-2), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$5.028512095$ |
1.124409487 |
\( \frac{9029272560}{961} a + \frac{5599830233}{961} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 3 \phi - 53\) , \( -124 \phi + 85\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(3\phi-53\right){x}-124\phi+85$ |
| 3751.4-b5 |
3751.4-b |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.4 |
\( 11^{2} \cdot 31 \) |
\( 11^{6} \cdot 31^{2} \) |
$1.56372$ |
$(-3a+2), (5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$12.20614154$ |
2.729376224 |
\( \frac{9029272560}{961} a + \frac{5599830233}{961} \) |
\( \bigl[\phi + 1\) , \( \phi\) , \( \phi + 1\) , \( 21 \phi - 77\) , \( -121 \phi + 257\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\phi{x}^{2}+\left(21\phi-77\right){x}-121\phi+257$ |
| 3751.6-a5 |
3751.6-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( 11^{6} \cdot 31^{2} \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$2.893619604$ |
2.588132054 |
\( \frac{9029272560}{961} a + \frac{5599830233}{961} \) |
\( \bigl[\phi\) , \( 1\) , \( 0\) , \( -13 \phi - 57\) , \( -75 \phi - 192\bigr] \) |
${y}^2+\phi{x}{y}={x}^{3}+{x}^{2}+\left(-13\phi-57\right){x}-75\phi-192$ |
*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.
