| 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 |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{2} \) |
$1.35814$ |
$(11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$36.82322769$ |
2.008871764 |
\( \frac{19683}{11} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( a + 2\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(a+2\right){x}$ |
| 121.1-a2 |
121.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{4} \) |
$1.35814$ |
$(11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$18.41161384$ |
2.008871764 |
\( \frac{19034163}{121} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -4 a - 8\) , \( -20 a - 35\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-4a-8\right){x}-20a-35$ |
| 121.1-b1 |
121.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{2} \) |
$1.35814$ |
$(11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 1 \) |
$66.01014894$ |
$0.064435690$ |
1.856340101 |
\( -\frac{52893159101157376}{11} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -7820\) , \( -263580\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-7820{x}-263580$ |
| 121.1-b2 |
121.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{10} \) |
$1.35814$ |
$(11)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5Cs.1.1 |
$1$ |
\( 5 \) |
$13.20202978$ |
$1.610892258$ |
1.856340101 |
\( -\frac{122023936}{161051} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -10\) , \( -20\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-10{x}-20$ |
| 121.1-b3 |
121.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{2} \) |
$1.35814$ |
$(11)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 1 \) |
$2.640405957$ |
$40.27230645$ |
1.856340101 |
\( -\frac{4096}{11} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}$ |
| 121.1-c1 |
121.1-c |
$3$ |
$25$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{2} \) |
$1.35814$ |
$(11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$0.555680735$ |
$8.512583687$ |
2.064462905 |
\( -\frac{52893159101157376}{11} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 39102 a - 109483\) , \( -6365015 a + 17769325\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(39102a-109483\right){x}-6365015a+17769325$ |
| 121.1-c2 |
121.1-c |
$3$ |
$25$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{10} \) |
$1.35814$ |
$(11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5Cs.4.1 |
$1$ |
\( 5 \) |
$0.111136147$ |
$8.512583687$ |
2.064462905 |
\( -\frac{122023936}{161051} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 52 a - 143\) , \( -525 a + 1465\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(52a-143\right){x}-525a+1465$ |
| 121.1-c3 |
121.1-c |
$3$ |
$25$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{2} \) |
$1.35814$ |
$(11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$0.555680735$ |
$8.512583687$ |
2.064462905 |
\( -\frac{4096}{11} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( -2 a - 1\) , \( -5 a - 10\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2a-1\right){x}-5a-10$ |
| 121.1-d1 |
121.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{2} \) |
$1.35814$ |
$(11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$36.82322769$ |
2.008871764 |
\( \frac{19683}{11} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 4 a + 5\) , \( 3 a + 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a+5\right){x}+3a+6$ |
| 121.1-d2 |
121.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{4} \) |
$1.35814$ |
$(11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$18.41161384$ |
2.008871764 |
\( \frac{19034163}{121} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 9 a - 10\) , \( 13 a - 24\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(9a-10\right){x}+13a-24$ |
*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.
