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-a3 |
31.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.1 |
\( 31 \) |
\( 31^{4} \) |
$0.47148$ |
$(5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$6.438604964$ |
0.359928959 |
\( -\frac{156520379364360}{923521} a + \frac{253260256463213}{923521} \) |
\( \bigl[\phi\) , \( -1\) , \( \phi + 1\) , \( -12 \phi - 21\) , \( 42 \phi + 10\bigr] \) |
${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}-{x}^{2}+\left(-12\phi-21\right){x}+42\phi+10$ |
775.1-a3 |
775.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
775.1 |
\( 5^{2} \cdot 31 \) |
\( 5^{6} \cdot 31^{4} \) |
$1.05426$ |
$(-2a+1), (5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$6.746456922$ |
1.508553628 |
\( -\frac{156520379364360}{923521} a + \frac{253260256463213}{923521} \) |
\( \bigl[\phi + 1\) , \( \phi - 1\) , \( 1\) , \( 49 \phi - 152\) , \( -583 \phi + 688\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(49\phi-152\right){x}-583\phi+688$ |
961.2-c3 |
961.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
961.2 |
\( 31^{2} \) |
\( 31^{10} \) |
$1.11251$ |
$(5a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$7.096706804$ |
$0.861840578$ |
1.367630581 |
\( -\frac{156520379364360}{923521} a + \frac{253260256463213}{923521} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -474 \phi - 655\) , \( -6456 \phi - 6672\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-474\phi-655\right){x}-6456\phi-6672$ |
2511.1-f3 |
2511.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2511.1 |
\( 3^{4} \cdot 31 \) |
\( 3^{12} \cdot 31^{4} \) |
$1.41444$ |
$(5a-2), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$5.028512095$ |
1.124409487 |
\( -\frac{156520379364360}{923521} a + \frac{253260256463213}{923521} \) |
\( \bigl[\phi\) , \( -\phi - 1\) , \( 1\) , \( -98 \phi - 186\) , \( -1056 \phi - 98\bigr] \) |
${y}^2+\phi{x}{y}+{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(-98\phi-186\right){x}-1056\phi-98$ |
3751.4-b3 |
3751.4-b |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.4 |
\( 11^{2} \cdot 31 \) |
\( 11^{6} \cdot 31^{4} \) |
$1.56372$ |
$(-3a+2), (5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$6.103070773$ |
2.729376224 |
\( -\frac{156520379364360}{923521} a + \frac{253260256463213}{923521} \) |
\( \bigl[1\) , \( \phi - 1\) , \( 0\) , \( -47 \phi - 236\) , \( 185 \phi + 1360\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-47\phi-236\right){x}+185\phi+1360$ |
3751.6-a3 |
3751.6-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( 11^{6} \cdot 31^{4} \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{4} \) |
$1$ |
$1.446809802$ |
2.588132054 |
\( -\frac{156520379364360}{923521} a + \frac{253260256463213}{923521} \) |
\( \bigl[\phi + 1\) , \( \phi + 1\) , \( 0\) , \( -201 \phi - 238\) , \( -2077 \phi - 1767\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(-201\phi-238\right){x}-2077\phi-1767$ |
*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.