| 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 |
| 4.1-b1 |
4.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{20} \) |
$0.92001$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B.1.2 |
$4$ |
\( 2 \) |
$1$ |
$0.853190410$ |
0.937557727 |
\( -\frac{9814089221}{1024} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 1\) , \( 311 a - 1287\) , \( 6283 a - 26012\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(311a-1287\right){x}+6283a-26012$ |
| 196.2-g1 |
196.2-g |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
196.2 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{20} \cdot 7^{6} \) |
$2.43411$ |
$(-a-2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \) |
$1.394149711$ |
$1.650124506$ |
2.528006461 |
\( -\frac{9814089221}{1024} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( 400 a - 1689\) , \( 9263 a - 38278\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(400a-1689\right){x}+9263a-38278$ |
| 196.3-g1 |
196.3-g |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
196.3 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{20} \cdot 7^{6} \) |
$2.43411$ |
$(-a+3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \) |
$1.394149711$ |
$1.650124506$ |
2.528006461 |
\( -\frac{9814089221}{1024} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -400 a - 1289\) , \( -9263 a - 29015\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-400a-1289\right){x}-9263a-29015$ |
| 256.1-e1 |
256.1-e |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{44} \) |
$2.60217$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$5.585030015$ |
3.068651490 |
\( -\frac{9814089221}{1024} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 4996 a - 20692\) , \( -372172 a + 1540816\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(4996a-20692\right){x}-372172a+1540816$ |
| 256.1-z1 |
256.1-z |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{44} \) |
$2.60217$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$6.125642457$ |
$1.091454769$ |
3.673494921 |
\( -\frac{9814089221}{1024} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a - 709\) , \( -1779 a + 1244\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a-709\right){x}-1779a+1244$ |
| 256.1-bd1 |
256.1-bd |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{44} \) |
$2.60217$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$6.125642457$ |
$1.091454769$ |
3.673494921 |
\( -\frac{9814089221}{1024} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a - 709\) , \( 1779 a - 1244\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a-709\right){x}+1779a-1244$ |
| 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 |
\( -\frac{9814089221}{1024} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -2812 a - 8834\) , \( 155604 a + 488604\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2812a-8834\right){x}+155604a+488604$ |
| 484.2-a1 |
484.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
484.2 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{20} \cdot 11^{6} \) |
$3.05132$ |
$(a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$1.316343980$ |
3.616275029 |
\( -\frac{9814089221}{1024} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 1428 a - 5928\) , \( -57107 a + 236341\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(1428a-5928\right){x}-57107a+236341$ |
| 484.3-a1 |
484.3-a |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
484.3 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{20} \cdot 11^{6} \) |
$3.05132$ |
$(a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$1.316343980$ |
3.616275029 |
\( -\frac{9814089221}{1024} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -1427 a - 4501\) , \( 58533 a + 183735\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1427a-4501\right){x}+58533a+183735$ |
| 676.2-b1 |
676.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
676.2 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{20} \cdot 13^{6} \) |
$3.31714$ |
$(a), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \cdot 5 \) |
$9.265331654$ |
$0.236632444$ |
6.023200498 |
\( -\frac{9814089221}{1024} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( -40 a - 563\) , \( -1040 a - 6524\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-40a-563\right){x}-1040a-6524$ |
| 676.3-b1 |
676.3-b |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
676.3 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{20} \cdot 13^{6} \) |
$3.31714$ |
$(a-1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \cdot 5 \) |
$9.265331654$ |
$0.236632444$ |
6.023200498 |
\( -\frac{9814089221}{1024} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 47 a - 616\) , \( 424 a - 6379\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(47a-616\right){x}+424a-6379$ |
| 1156.2-a1 |
1156.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1156.2 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{20} \cdot 17^{6} \) |
$3.79329$ |
$(a+5), (2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$33.39194601$ |
$0.206929069$ |
7.593032991 |
\( -\frac{9814089221}{1024} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 1070 a - 4505\) , \( 37303 a - 154714\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(1070a-4505\right){x}+37303a-154714$ |
| 1156.3-a1 |
1156.3-a |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1156.3 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{20} \cdot 17^{6} \) |
$3.79329$ |
$(a-6), (2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$33.39194601$ |
$0.206929069$ |
7.593032991 |
\( -\frac{9814089221}{1024} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -1071 a - 3434\) , \( -37304 a - 117410\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-1071a-3434\right){x}-37304a-117410$ |
*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.
