| 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 |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.2 |
\( 31 \) |
\( - 31^{8} \) |
$0.47148$ |
$(5a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1.609651241$ |
0.359928959 |
\( -\frac{11889611722383394}{852891037441} a + \frac{3260226660263703}{852891037441} \) |
\( \bigl[1\) , \( -\phi - 1\) , \( \phi\) , \( -30 \phi - 45\) , \( -111 \phi - 117\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(-30\phi-45\right){x}-111\phi-117$ |
| 775.2-a1 |
775.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
775.2 |
\( 5^{2} \cdot 31 \) |
\( - 5^{6} \cdot 31^{8} \) |
$1.05426$ |
$(-2a+1), (5a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.373228461$ |
1.508553628 |
\( -\frac{11889611722383394}{852891037441} a + \frac{3260226660263703}{852891037441} \) |
\( \bigl[\phi\) , \( \phi + 1\) , \( \phi\) , \( 77 \phi - 303\) , \( -674 \phi + 1871\bigr] \) |
${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(77\phi-303\right){x}-674\phi+1871$ |
| 961.3-c1 |
961.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
961.3 |
\( 31^{2} \) |
\( - 31^{14} \) |
$1.11251$ |
$(5a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.548353402$ |
$0.430920289$ |
1.367630581 |
\( -\frac{11889611722383394}{852891037441} a + \frac{3260226660263703}{852891037441} \) |
\( \bigl[\phi\) , \( 0\) , \( 0\) , \( -652 \phi - 1396\) , \( -27054 \phi - 5575\bigr] \) |
${y}^2+\phi{x}{y}={x}^{3}+\left(-652\phi-1396\right){x}-27054\phi-5575$ |
| 2511.2-f1 |
2511.2-f |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2511.2 |
\( 3^{4} \cdot 31 \) |
\( - 3^{12} \cdot 31^{8} \) |
$1.41444$ |
$(5a-3), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.514256047$ |
1.124409487 |
\( -\frac{11889611722383394}{852891037441} a + \frac{3260226660263703}{852891037441} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -273 \phi - 410\) , \( 3940 \phi + 3831\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-273\phi-410\right){x}+3940\phi+3831$ |
| 3751.3-a1 |
3751.3-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.3 |
\( 11^{2} \cdot 31 \) |
\( - 11^{6} \cdot 31^{8} \) |
$1.56372$ |
$(-3a+2), (5a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.723404901$ |
2.588132054 |
\( -\frac{11889611722383394}{852891037441} a + \frac{3260226660263703}{852891037441} \) |
\( \bigl[\phi + 1\) , \( -\phi + 1\) , \( 0\) , \( -167 \phi - 500\) , \( -5143 \phi - 565\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-167\phi-500\right){x}-5143\phi-565$ |
| 3751.5-b1 |
3751.5-b |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.5 |
\( 11^{2} \cdot 31 \) |
\( - 11^{6} \cdot 31^{8} \) |
$1.56372$ |
$(-3a+1), (5a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.525767693$ |
2.729376224 |
\( -\frac{11889611722383394}{852891037441} a + \frac{3260226660263703}{852891037441} \) |
\( \bigl[\phi\) , \( \phi - 1\) , \( \phi + 1\) , \( -532 \phi - 546\) , \( 9674 \phi + 4939\bigr] \) |
${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-532\phi-546\right){x}+9674\phi+4939$ |
*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.
