| 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 |
| 81.1-a1 |
81.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{16} \) |
$0.96657$ |
$(-a), (-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$4.298286986$ |
1.192130317 |
\( -\frac{24125}{27} a - \frac{1375}{27} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -4 a + 3\) , \( -4 a + 5\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-4a+3\right){x}-4a+5$ |
| 81.1-a2 |
81.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( - 3^{38} \) |
$0.96657$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.149143493$ |
1.192130317 |
\( -\frac{1794398270320625}{282429536481} a + \frac{1272952673786125}{94143178827} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( -43 a - 360\) , \( 1020 a + 3386\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-43a-360\right){x}+1020a+3386$ |
| 81.1-a3 |
81.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( - 3^{26} \) |
$0.96657$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.149143493$ |
1.192130317 |
\( -\frac{16961124145384625}{6561} a + \frac{13019221158502750}{2187} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( 2381 a - 5352\) , \( -81841 a + 187736\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(2381a-5352\right){x}-81841a+187736$ |
| 81.1-a4 |
81.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{16} \) |
$0.96657$ |
$(-a), (-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$4.298286986$ |
1.192130317 |
\( \frac{24125}{27} a - \frac{8500}{9} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 2 a\) , \( 3 a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+2a{x}+3a+2$ |
| 81.1-a5 |
81.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{20} \) |
$0.96657$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$4.298286986$ |
1.192130317 |
\( -\frac{1567304375}{729} a + \frac{1203684625}{243} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 2 a - 45\) , \( 39 a - 61\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(2a-45\right){x}+39a-61$ |
| 81.1-a6 |
81.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{28} \) |
$0.96657$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$4.298286986$ |
1.192130317 |
\( -\frac{449577713875}{531441} a + \frac{1037190880375}{531441} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( 131 a - 357\) , \( -1237 a + 2759\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(131a-357\right){x}-1237a+2759$ |
| 81.1-a7 |
81.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( - 3^{38} \) |
$0.96657$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.149143493$ |
1.192130317 |
\( \frac{1794398270320625}{282429536481} a + \frac{2024459751037750}{282429536481} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( 41 a - 402\) , \( -1021 a + 4406\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(41a-402\right){x}-1021a+4406$ |
| 81.1-a8 |
81.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{16} \) |
$0.96657$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.074571746$ |
1.192130317 |
\( -\frac{450190437580625}{27} a + \frac{345562524359500}{9} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 137 a - 585\) , \( 1686 a - 5677\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(137a-585\right){x}+1686a-5677$ |
| 81.1-a9 |
81.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{28} \) |
$0.96657$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$4.298286986$ |
1.192130317 |
\( \frac{449577713875}{531441} a + \frac{195871055500}{177147} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( -133 a - 225\) , \( 1236 a + 1523\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-133a-225\right){x}+1236a+1523$ |
| 81.1-a10 |
81.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{20} \) |
$0.96657$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$4.298286986$ |
1.192130317 |
\( \frac{1567304375}{729} a + \frac{2043749500}{729} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -4 a - 42\) , \( -40 a - 22\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-4a-42\right){x}-40a-22$ |
| 81.1-a11 |
81.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( - 3^{26} \) |
$0.96657$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.149143493$ |
1.192130317 |
\( \frac{16961124145384625}{6561} a + \frac{22096539330123625}{6561} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( -2383 a - 2970\) , \( 81840 a + 105896\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-2383a-2970\right){x}+81840a+105896$ |
| 81.1-a12 |
81.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{16} \) |
$0.96657$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.074571746$ |
1.192130317 |
\( \frac{450190437580625}{27} a + \frac{586497135497875}{27} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -139 a - 447\) , \( -1687 a - 3991\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-139a-447\right){x}-1687a-3991$ |
*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.
