| 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 |
| 9.1-a1 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$0.55805$ |
$(-a), (-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$27.39950531$ |
0.474953467 |
\( -\frac{24125}{27} a - \frac{1375}{27} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+{x}$ |
| 9.1-a2 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{26} \) |
$0.55805$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.712469082$ |
0.474953467 |
\( -\frac{1794398270320625}{282429536481} a + \frac{1272952673786125}{94143178827} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -5 a - 40\) , \( -56 a - 157\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-5a-40\right){x}-56a-157$ |
| 9.1-a3 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{14} \) |
$0.55805$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.712469082$ |
0.474953467 |
\( -\frac{16961124145384625}{6561} a + \frac{13019221158502750}{2187} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 265 a - 594\) , \( 3141 a - 7218\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(265a-594\right){x}+3141a-7218$ |
| 9.1-a4 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$0.55805$ |
$(-a), (-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$27.39950531$ |
0.474953467 |
\( \frac{24125}{27} a - \frac{8500}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}$ |
| 9.1-a5 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$0.55805$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$27.39950531$ |
0.474953467 |
\( -\frac{1567304375}{729} a + \frac{1203684625}{243} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -5\) , \( -3 a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}-5{x}-3a-1$ |
| 9.1-a6 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{16} \) |
$0.55805$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$6.849876328$ |
0.474953467 |
\( -\frac{449577713875}{531441} a + \frac{1037190880375}{531441} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 15 a - 39\) , \( 54 a - 117\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(15a-39\right){x}+54a-117$ |
| 9.1-a7 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{26} \) |
$0.55805$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.712469082$ |
0.474953467 |
\( \frac{1794398270320625}{282429536481} a + \frac{2024459751037750}{282429536481} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 5 a - 44\) , \( 51 a - 168\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(5a-44\right){x}+51a-168$ |
| 9.1-a8 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$0.55805$ |
$(-a), (-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$27.39950531$ |
0.474953467 |
\( -\frac{450190437580625}{27} a + \frac{345562524359500}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 15 a - 65\) , \( -69 a + 182\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(15a-65\right){x}-69a+182$ |
| 9.1-a9 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{16} \) |
$0.55805$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$6.849876328$ |
0.474953467 |
\( \frac{449577713875}{531441} a + \frac{195871055500}{177147} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -15 a - 25\) , \( -69 a - 88\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-15a-25\right){x}-69a-88$ |
| 9.1-a10 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$0.55805$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$27.39950531$ |
0.474953467 |
\( \frac{1567304375}{729} a + \frac{2043749500}{729} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -4\) , \( 3 a + 1\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}-4{x}+3a+1$ |
| 9.1-a11 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{14} \) |
$0.55805$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.712469082$ |
0.474953467 |
\( \frac{16961124145384625}{6561} a + \frac{22096539330123625}{6561} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -265 a - 330\) , \( -3406 a - 4407\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-265a-330\right){x}-3406a-4407$ |
| 9.1-a12 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$0.55805$ |
$(-a), (-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$27.39950531$ |
0.474953467 |
\( \frac{450190437580625}{27} a + \frac{586497135497875}{27} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -15 a - 49\) , \( 84 a + 163\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-15a-49\right){x}+84a+163$ |
*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.
