| 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 |
| 961.3-a1 |
961.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
961.3 |
\( 31^{2} \) |
\( 31^{2} \) |
$1.11251$ |
$(5a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2Cn, 5Nn.3.2[2] |
$1$ |
\( 1 \) |
$0.046233252$ |
$35.44911584$ |
1.465901688 |
\( -122880 a + 200704 \) |
\( \bigl[0\) , \( \phi + 1\) , \( 1\) , \( 2 \phi - 2\) , \( -\phi + 1\bigr] \) |
${y}^2+{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(2\phi-2\right){x}-\phi+1$ |
| 961.3-b1 |
961.3-b |
$1$ |
$1$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
961.3 |
\( 31^{2} \) |
\( 31^{8} \) |
$1.11251$ |
$(5a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2Cn, 5Nn.2.2[2] |
$1$ |
\( 1 \) |
$1$ |
$3.193224251$ |
1.428053298 |
\( -122880 a + 200704 \) |
\( \bigl[0\) , \( \phi - 1\) , \( 1\) , \( 42 \phi - 95\) , \( 192 \phi - 332\bigr] \) |
${y}^2+{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(42\phi-95\right){x}+192\phi-332$ |
| 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 |
\( -\frac{11889611722383394}{852891037441} a + \frac{3260226660263703}{852891037441} \) |
\( \bigl[\phi\) , \( 0\) , \( 0\) , \( -652 \phi - 1396\) , \( -27054 \phi - 5575\bigr] \) |
${y}^2+\phi{x}{y}={x}^{3}+\left(-652\phi-1396\right){x}-27054\phi-5575$ |
| 961.3-c2 |
961.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
961.3 |
\( 31^{2} \) |
\( - 31^{7} \) |
$1.11251$ |
$(5a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$7.096706804$ |
$0.430920289$ |
1.367630581 |
\( -\frac{6130703730739448}{31} a + \frac{9919687011293045}{31} \) |
\( \bigl[\phi + 1\) , \( 0\) , \( 0\) , \( 497 \phi - 153\) , \( -20816 \phi - 16670\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}={x}^{3}+\left(497\phi-153\right){x}-20816\phi-16670$ |
| 961.3-c3 |
961.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
961.3 |
\( 31^{2} \) |
\( - 31^{7} \) |
$1.11251$ |
$(5a-3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.774176701$ |
$3.447362312$ |
1.367630581 |
\( \frac{106208}{31} a - \frac{54753}{31} \) |
\( \bigl[\phi\) , \( 0\) , \( 0\) , \( -7 \phi - 16\) , \( 24 \phi - 4\bigr] \) |
${y}^2+\phi{x}{y}={x}^{3}+\left(-7\phi-16\right){x}+24\phi-4$ |
| 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 |
\( -\frac{9029272560}{961} a + \frac{14629102793}{961} \) |
\( \bigl[\phi\) , \( 0\) , \( 0\) , \( 18 \phi - 186\) , \( 140 \phi - 985\bigr] \) |
${y}^2+\phi{x}{y}={x}^{3}+\left(18\phi-186\right){x}+140\phi-985$ |
| 961.3-c5 |
961.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
961.3 |
\( 31^{2} \) |
\( 31^{10} \) |
$1.11251$ |
$(5a-3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$7.096706804$ |
$0.861840578$ |
1.367630581 |
\( \frac{156520379364360}{923521} a + \frac{96739877098853}{923521} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 474 \phi - 1129\) , \( 6456 \phi - 13128\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(474\phi-1129\right){x}+6456\phi-13128$ |
| 961.3-c6 |
961.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
961.3 |
\( 31^{2} \) |
\( - 31^{8} \) |
$1.11251$ |
$(5a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$14.19341360$ |
$0.215460144$ |
1.367630581 |
\( \frac{61725871986044215714}{961} a + \frac{38148686872600722809}{961} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -3856 \phi + 446\) , \( -31874 \phi - 118625\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-3856\phi+446\right){x}-31874\phi-118625$ |
*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.
