| 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 |
| 81.5-a1 |
81.5-a |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.5 |
\( 3^{4} \) |
\( 3^{10} \) |
$0.96657$ |
$(-a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$5.586269779$ |
1.549352471 |
\( -175 a - 228 \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 0\) , \( -2\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}-2$ |
| 81.5-a2 |
81.5-a |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.5 |
\( 3^{4} \) |
\( 3^{6} \) |
$0.96657$ |
$(-a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$5.586269779$ |
1.549352471 |
\( 3874871 a - 8922459 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 5 a + 6\) , \( 15 a + 19\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(5a+6\right){x}+15a+19$ |
| 81.5-b1 |
81.5-b |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.5 |
\( 3^{4} \) |
\( 3^{4} \) |
$0.96657$ |
$(-a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2Cn, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$47.46752806$ |
1.462791507 |
\( -20480 a + 49152 \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( 0\) , \( -a - 1\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+a{x}^{2}-a-1$ |
| 81.5-b2 |
81.5-b |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.5 |
\( 3^{4} \) |
\( 3^{12} \) |
$0.96657$ |
$(-a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2Cn, 3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$5.274169784$ |
1.462791507 |
\( 7081984 a + 9265152 \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( -20 a - 30\) , \( -90 a - 119\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-20a-30\right){x}-90a-119$ |
| 81.5-c1 |
81.5-c |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.5 |
\( 3^{4} \) |
\( 3^{10} \) |
$0.96657$ |
$(-a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2Cn, 3B |
$1$ |
\( 3 \) |
$0.024560134$ |
$24.79923923$ |
1.013558173 |
\( -20480 a + 49152 \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( -2 a - 3\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-2a-3\right){x}$ |
| 81.5-c2 |
81.5-c |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.5 |
\( 3^{4} \) |
\( 3^{6} \) |
$0.96657$ |
$(-a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2Cn, 3B |
$1$ |
\( 1 \) |
$0.073680404$ |
$24.79923923$ |
1.013558173 |
\( 7081984 a + 9265152 \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( -3 a - 6\) , \( 9 a + 9\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-3a-6\right){x}+9a+9$ |
| 81.5-d1 |
81.5-d |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.5 |
\( 3^{4} \) |
\( 3^{4} \) |
$0.96657$ |
$(-a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$21.90883835$ |
0.675157607 |
\( -175 a - 228 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( a + 3\) , \( a\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(a+3\right){x}+a$ |
| 81.5-d2 |
81.5-d |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.5 |
\( 3^{4} \) |
\( 3^{12} \) |
$0.96657$ |
$(-a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$2.434315372$ |
0.675157607 |
\( 3874871 a - 8922459 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 21 a - 42\) , \( 50 a - 112\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(21a-42\right){x}+50a-112$ |
*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.
