| 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 |
| 121.1-a1 |
121.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{2} \) |
$2.15761$ |
$(a+1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$35.02801964$ |
1.202867133 |
\( -\frac{125}{11} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 2 a + 8\) , \( 9 a + 28\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(2a+8\right){x}+9a+28$ |
| 121.1-a2 |
121.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{5} \) |
$2.15761$ |
$(a+1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$17.51400982$ |
1.202867133 |
\( -\frac{29343598513250}{14641} a + \frac{121484137321625}{14641} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -58 a - 182\) , \( -299 a - 938\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-58a-182\right){x}-299a-938$ |
| 121.1-a3 |
121.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{4} \) |
$2.15761$ |
$(a+1), (a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$35.02801964$ |
1.202867133 |
\( \frac{14706125}{121} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -33 a - 102\) , \( 123 a + 386\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-33a-102\right){x}+123a+386$ |
| 121.1-a4 |
121.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{5} \) |
$2.15761$ |
$(a+1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$17.51400982$ |
1.202867133 |
\( \frac{29343598513250}{14641} a + \frac{92140538808375}{14641} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 67 a - 233\) , \( 118 a - 434\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(67a-233\right){x}+118a-434$ |
| 121.1-b1 |
121.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{2} \) |
$2.15761$ |
$(a+1), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.2 |
$100$ |
\( 1 \) |
$1$ |
$0.064435690$ |
0.885092276 |
\( -\frac{52893159101157376}{11} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -7820\) , \( -263580\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-7820{x}-263580$ |
| 121.1-b2 |
121.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{10} \) |
$2.15761$ |
$(a+1), (a-2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5Cs.1.1 |
$4$ |
\( 5^{2} \) |
$1$ |
$1.610892258$ |
0.885092276 |
\( -\frac{122023936}{161051} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -10\) , \( -20\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-10{x}-20$ |
| 121.1-b3 |
121.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{2} \) |
$2.15761$ |
$(a+1), (a-2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.1 |
$4$ |
\( 1 \) |
$1$ |
$40.27230645$ |
0.885092276 |
\( -\frac{4096}{11} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}$ |
| 121.1-c1 |
121.1-c |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{8} \) |
$2.15761$ |
$(a+1), (a-2)$ |
$0 \le r \le 2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$7$ |
7B |
$16$ |
\( 1 \) |
$1$ |
$1.814354014$ |
3.987531049 |
\( -\frac{75036161277952}{19487171} a - \frac{235604626407424}{19487171} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( -51 a - 155\) , \( -449 a - 1412\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(-51a-155\right){x}-449a-1412$ |
| 121.1-c2 |
121.1-c |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{8} \) |
$2.15761$ |
$(a+1), (a-2)$ |
$0 \le r \le 2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$7$ |
7B |
$16$ |
\( 1 \) |
$1$ |
$1.814354014$ |
3.987531049 |
\( \frac{75036161277952}{19487171} a - \frac{310640787685376}{19487171} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( 51 a - 206\) , \( 449 a - 1861\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(51a-206\right){x}+449a-1861$ |
| 121.1-d1 |
121.1-d |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{16} \) |
$2.15761$ |
$(a+1), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$7$ |
7B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.160687108$ |
0.637730543 |
\( -\frac{341414959364206000}{379749833583241} a - \frac{1598907453972972625}{379749833583241} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 107 a - 471\) , \( 1778 a - 7344\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(107a-471\right){x}+1778a-7344$ |
| 121.1-d2 |
121.1-d |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{16} \) |
$2.15761$ |
$(a+1), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$7$ |
7B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.160687108$ |
0.637730543 |
\( \frac{341414959364206000}{379749833583241} a - \frac{1940322413337178625}{379749833583241} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( -99 a - 357\) , \( -2242 a - 6908\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-99a-357\right){x}-2242a-6908$ |
*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.
