| 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 |
| 324.1-a1 |
324.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{30} \) |
$2.76002$ |
$(2), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$9$ |
\( 2 \) |
$1$ |
$0.305477747$ |
0.755290721 |
\( -\frac{6362477477}{39366} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -2434 a - 7646\) , \( -124602 a - 391272\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2434a-7646\right){x}-124602a-391272$ |
| 324.1-b1 |
324.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{16} \) |
$2.76002$ |
$(2), (3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.341781253$ |
$12.59499794$ |
4.730405727 |
\( -\frac{20833}{18} a - \frac{65509}{18} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -6 a - 12\) , \( 14 a + 48\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6a-12\right){x}+14a+48$ |
| 324.1-c1 |
324.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{22} \cdot 3^{14} \) |
$2.76002$ |
$(2), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$9$ |
\( 2 \) |
$1$ |
$1.001585182$ |
2.476409499 |
\( -\frac{1953125}{6144} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -166 a - 518\) , \( -5494 a - 17250\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-166a-518\right){x}-5494a-17250$ |
| 324.1-d1 |
324.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{34} \cdot 3^{12} \) |
$2.76002$ |
$(2), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$9$ |
\( 2^{2} \) |
$1$ |
$0.624031290$ |
3.085822437 |
\( -\frac{25153757}{131072} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -386 a - 1211\) , \( -24677 a - 77489\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-386a-1211\right){x}-24677a-77489$ |
| 324.1-e1 |
324.1-e |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{20} \cdot 3^{12} \) |
$2.76002$ |
$(2), (3)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$0.608550812$ |
$7.446706687$ |
4.979814287 |
\( -\frac{9814089221}{1024} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -2812 a - 8834\) , \( 155604 a + 488604\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2812a-8834\right){x}+155604a+488604$ |
| 324.1-e2 |
324.1-e |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{4} \cdot 3^{12} \) |
$2.76002$ |
$(2), (3)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \) |
$0.608550812$ |
$7.446706687$ |
4.979814287 |
\( \frac{6859}{4} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 23 a + 76\) , \( 3 a + 12\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(23a+76\right){x}+3a+12$ |
| 324.1-f1 |
324.1-f |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{16} \) |
$2.76002$ |
$(2), (3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.341781253$ |
$12.59499794$ |
4.730405727 |
\( \frac{20833}{18} a - \frac{43171}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 6 a - 19\) , \( -8 a + 43\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(6a-19\right){x}-8a+43$ |
| 324.1-g1 |
324.1-g |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{16} \) |
$2.76002$ |
$(2), (3)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2^{2} \) |
$2.165104934$ |
$1.077364688$ |
5.126532733 |
\( -\frac{3307949}{18} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -197 a - 617\) , \( -2800 a - 8792\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-197a-617\right){x}-2800a-8792$ |
| 324.1-g2 |
324.1-g |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{10} \cdot 3^{32} \) |
$2.76002$ |
$(2), (3)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \) |
$2.165104934$ |
$1.077364688$ |
5.126532733 |
\( \frac{1160935651}{1889568} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( 1378 a + 4333\) , \( 71279 a + 223831\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1378a+4333\right){x}+71279a+223831$ |
*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.
