| 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 |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( - 11^{3} \) |
$1.02666$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$9.363938382$ |
1.351568086 |
\( -\frac{7660032}{121} a + \frac{13260992}{121} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 46 a - 74\) , \( 204 a - 350\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(46a-74\right){x}+204a-350$ |
| 121.1-a2 |
121.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( - 11^{9} \) |
$1.02666$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$9.363938382$ |
1.351568086 |
\( -\frac{8165337088}{1771561} a + \frac{14236882112}{1771561} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 251 a - 429\) , \( -2715 a + 4706\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(251a-429\right){x}-2715a+4706$ |
| 121.1-a3 |
121.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( - 11^{9} \) |
$1.02666$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$9.363938382$ |
1.351568086 |
\( \frac{8165337088}{1771561} a + \frac{14236882112}{1771561} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( -6 a + 4\) , \( -30 a + 53\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-6a+4\right){x}-30a+53$ |
| 121.1-a4 |
121.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( - 11^{3} \) |
$1.02666$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$9.363938382$ |
1.351568086 |
\( \frac{7660032}{121} a + \frac{13260992}{121} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( -a - 1\) , \( a - 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a-1\right){x}+a-3$ |
| 121.1-b1 |
121.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{2} \) |
$1.02666$ |
$(-2a+1), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$8.512583687$ |
2.457371241 |
\( -\frac{52893159101157376}{11} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( 437938 a - 758571\) , \( -207226626 a + 358927153\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(437938a-758571\right){x}-207226626a+358927153$ |
| 121.1-b2 |
121.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{10} \) |
$1.02666$ |
$(-2a+1), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5Cs.4.1 |
$1$ |
\( 1 \) |
$1$ |
$8.512583687$ |
2.457371241 |
\( -\frac{122023936}{161051} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( -10\) , \( 19\bigr] \) |
${y}^2+a{y}={x}^{3}+{x}^{2}-10{x}+19$ |
| 121.1-b3 |
121.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{2} \) |
$1.02666$ |
$(-2a+1), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$8.512583687$ |
2.457371241 |
\( -\frac{4096}{11} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( 18 a - 31\) , \( 154 a - 267\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(18a-31\right){x}+154a-267$ |
| 121.1-c1 |
121.1-c |
$3$ |
$25$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{2} \) |
$1.02666$ |
$(-2a+1), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$0.064435690$ |
0.465024539 |
\( -\frac{52893159101157376}{11} \) |
\( \bigl[0\) , \( a - 1\) , \( a\) , \( 437938 a - 758571\) , \( 207226626 a - 358927154\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(437938a-758571\right){x}+207226626a-358927154$ |
| 121.1-c2 |
121.1-c |
$3$ |
$25$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{10} \) |
$1.02666$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5Cs.1.1 |
$1$ |
\( 5^{2} \) |
$1$ |
$1.610892258$ |
0.465024539 |
\( -\frac{122023936}{161051} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -10\) , \( -20\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-10{x}-20$ |
| 121.1-c3 |
121.1-c |
$3$ |
$25$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{2} \) |
$1.02666$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$40.27230645$ |
0.465024539 |
\( -\frac{4096}{11} \) |
\( \bigl[0\) , \( a - 1\) , \( a\) , \( 18 a - 31\) , \( -154 a + 266\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(18a-31\right){x}-154a+266$ |
| 121.1-d1 |
121.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( - 11^{3} \) |
$1.02666$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$35.85281548$ |
0.574989796 |
\( -\frac{7660032}{121} a + \frac{13260992}{121} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 45 a - 76\) , \( -235 a + 408\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(45a-76\right){x}-235a+408$ |
| 121.1-d2 |
121.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( - 11^{9} \) |
$1.02666$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$3.983646165$ |
0.574989796 |
\( -\frac{8165337088}{1771561} a + \frac{14236882112}{1771561} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 250 a - 431\) , \( 2534 a - 4388\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(250a-431\right){x}+2534a-4388$ |
| 121.1-d3 |
121.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( - 11^{9} \) |
$1.02666$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$3.983646165$ |
0.574989796 |
\( \frac{8165337088}{1771561} a + \frac{14236882112}{1771561} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -5 a + 3\) , \( 25 a - 50\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-5a+3\right){x}+25a-50$ |
| 121.1-d4 |
121.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( - 11^{3} \) |
$1.02666$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$35.85281548$ |
0.574989796 |
\( \frac{7660032}{121} a + \frac{13260992}{121} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -2\) , \( -a + 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-2{x}-a+1$ |
*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.
