| 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 |
| 37.1-a1 |
37.1-a |
$2$ |
$3$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
37.1 |
\( 37 \) |
\( - 37^{3} \) |
$24.49556$ |
$(a^4-2a^3-2a^2+4a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 3 \) |
$1$ |
$10.08159607$ |
1.42483467 |
\( -\frac{1272945793794372}{50653} a^{4} + \frac{859591705080800}{50653} a^{3} + \frac{4922533480405137}{50653} a^{2} + \frac{233798144853843}{50653} a - \frac{948239678172729}{50653} \) |
\( \bigl[a\) , \( 2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 3\) , \( 2 a^{4} - 3 a^{3} - 6 a^{2} + 7 a + 2\) , \( 32 a^{4} - 13 a^{3} - 148 a^{2} - 4 a + 30\) , \( 154 a^{4} - 69 a^{3} - 676 a^{2} - 48 a + 128\bigr] \) |
${y}^2+a{x}{y}+\left(2a^{4}-3a^{3}-6a^{2}+7a+2\right){y}={x}^{3}+\left(2a^{4}-3a^{3}-7a^{2}+7a+3\right){x}^{2}+\left(32a^{4}-13a^{3}-148a^{2}-4a+30\right){x}+154a^{4}-69a^{3}-676a^{2}-48a+128$ |
| 37.1-a2 |
37.1-a |
$2$ |
$3$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
37.1 |
\( 37 \) |
\( -37 \) |
$24.49556$ |
$(a^4-2a^3-2a^2+4a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$2449.827847$ |
1.42483467 |
\( \frac{434197}{37} a^{4} - \frac{726502}{37} a^{3} - \frac{1586537}{37} a^{2} + \frac{1678795}{37} a + \frac{1145975}{37} \) |
\( \bigl[a\) , \( 2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 3\) , \( 2 a^{4} - 3 a^{3} - 6 a^{2} + 7 a + 2\) , \( 2 a^{4} - 3 a^{3} - 8 a^{2} + 6 a + 5\) , \( -a^{3} + 2 a - 1\bigr] \) |
${y}^2+a{x}{y}+\left(2a^{4}-3a^{3}-6a^{2}+7a+2\right){y}={x}^{3}+\left(2a^{4}-3a^{3}-7a^{2}+7a+3\right){x}^{2}+\left(2a^{4}-3a^{3}-8a^{2}+6a+5\right){x}-a^{3}+2a-1$ |
| 37.1-b1 |
37.1-b |
$1$ |
$1$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
37.1 |
\( 37 \) |
\( -37 \) |
$24.49556$ |
$(a^4-2a^3-2a^2+4a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$288.0518460$ |
1.50779424 |
\( -\frac{12336731}{37} a^{4} - \frac{600304}{37} a^{3} + \frac{27788818}{37} a^{2} + \frac{2658172}{37} a - \frac{5310421}{37} \) |
\( \bigl[2 a^{4} - 3 a^{3} - 7 a^{2} + 8 a + 4\) , \( a^{2} - a - 1\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 1\) , \( a^{4} - a^{3} - a^{2} + a - 2\) , \( -4 a^{4} + 5 a^{3} + 15 a^{2} - 12 a - 11\bigr] \) |
${y}^2+\left(2a^{4}-3a^{3}-7a^{2}+8a+4\right){x}{y}+\left(a^{4}-2a^{3}-3a^{2}+6a+1\right){y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}+\left(a^{4}-a^{3}-a^{2}+a-2\right){x}-4a^{4}+5a^{3}+15a^{2}-12a-11$ |
| 37.1-c1 |
37.1-c |
$1$ |
$1$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
37.1 |
\( 37 \) |
\( -37 \) |
$24.49556$ |
$(a^4-2a^3-2a^2+4a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.002017156$ |
$31970.22748$ |
1.68782263 |
\( -\frac{12336731}{37} a^{4} - \frac{600304}{37} a^{3} + \frac{27788818}{37} a^{2} + \frac{2658172}{37} a - \frac{5310421}{37} \) |
\( \bigl[a^{4} - a^{3} - 3 a^{2} + a\) , \( a^{3} - a^{2} - 2 a + 1\) , \( 2 a^{4} - 3 a^{3} - 7 a^{2} + 8 a + 3\) , \( a^{4} - 2 a^{3} - 2 a^{2} + 4 a - 1\) , \( -2 a^{4} + 3 a^{3} + 7 a^{2} - 6 a - 5\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-3a^{2}+a\right){x}{y}+\left(2a^{4}-3a^{3}-7a^{2}+8a+3\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+1\right){x}^{2}+\left(a^{4}-2a^{3}-2a^{2}+4a-1\right){x}-2a^{4}+3a^{3}+7a^{2}-6a-5$ |
| 37.1-d1 |
37.1-d |
$2$ |
$3$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
37.1 |
\( 37 \) |
\( - 37^{3} \) |
$24.49556$ |
$(a^4-2a^3-2a^2+4a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.009869798$ |
$6342.573603$ |
1.63838223 |
\( -\frac{1272945793794372}{50653} a^{4} + \frac{859591705080800}{50653} a^{3} + \frac{4922533480405137}{50653} a^{2} + \frac{233798144853843}{50653} a - \frac{948239678172729}{50653} \) |
\( \bigl[a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 2\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - 2\) , \( 7 a^{4} - 48 a^{3} - 16 a^{2} + 121 a - 21\) , \( -2 a^{4} + 130 a^{3} - 11 a^{2} - 300 a + 131\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-3a^{2}+6a+2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(7a^{4}-48a^{3}-16a^{2}+121a-21\right){x}-2a^{4}+130a^{3}-11a^{2}-300a+131$ |
| 37.1-d2 |
37.1-d |
$2$ |
$3$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
37.1 |
\( 37 \) |
\( -37 \) |
$24.49556$ |
$(a^4-2a^3-2a^2+4a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.003289932$ |
$19027.72081$ |
1.63838223 |
\( \frac{434197}{37} a^{4} - \frac{726502}{37} a^{3} - \frac{1586537}{37} a^{2} + \frac{1678795}{37} a + \frac{1145975}{37} \) |
\( \bigl[a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 2\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - 2\) , \( 2 a^{4} - 8 a^{3} - 6 a^{2} + 26 a + 14\) , \( -4 a^{3} + a^{2} + 17 a + 9\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-3a^{2}+6a+2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(2a^{4}-8a^{3}-6a^{2}+26a+14\right){x}-4a^{3}+a^{2}+17a+9$ |
*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.
