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 |
81.4-a1 |
81.4-a |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.4 |
\( 3^{4} \) |
\( 3^{6} \) |
$0.96657$ |
$(-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$5.586269779$ |
1.549352471 |
\( -3874871 a - 5047588 \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -7 a + 12\) , \( -16 a + 34\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-7a+12\right){x}-16a+34$ |
81.4-a2 |
81.4-a |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.4 |
\( 3^{4} \) |
\( 3^{10} \) |
$0.96657$ |
$(-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$5.586269779$ |
1.549352471 |
\( 175 a - 403 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 2 a + 2\) , \( a + 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a+2\right){x}+a+1$ |
81.4-b1 |
81.4-b |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.4 |
\( 3^{4} \) |
\( 3^{12} \) |
$0.96657$ |
$(-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2Cn, 3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$5.274169784$ |
1.462791507 |
\( -7081984 a + 16347136 \) |
\( \bigl[0\) , \( -a + 1\) , \( a\) , \( 20 a - 50\) , \( 89 a - 208\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(20a-50\right){x}+89a-208$ |
81.4-b2 |
81.4-b |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.4 |
\( 3^{4} \) |
\( 3^{4} \) |
$0.96657$ |
$(-a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2Cn, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$47.46752806$ |
1.462791507 |
\( 20480 a + 28672 \) |
\( \bigl[0\) , \( -a + 1\) , \( a\) , \( 0\) , \( -1\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a+1\right){x}^{2}-1$ |
81.4-c1 |
81.4-c |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.4 |
\( 3^{4} \) |
\( 3^{6} \) |
$0.96657$ |
$(-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2Cn, 3B |
$1$ |
\( 1 \) |
$0.073680404$ |
$24.79923923$ |
1.013558173 |
\( -7081984 a + 16347136 \) |
\( \bigl[0\) , \( a - 1\) , \( a\) , \( 3 a - 9\) , \( -10 a + 19\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(3a-9\right){x}-10a+19$ |
81.4-c2 |
81.4-c |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.4 |
\( 3^{4} \) |
\( 3^{10} \) |
$0.96657$ |
$(-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2Cn, 3B |
$1$ |
\( 3 \) |
$0.024560134$ |
$24.79923923$ |
1.013558173 |
\( 20480 a + 28672 \) |
\( \bigl[0\) , \( a - 1\) , \( a\) , \( 2 a - 5\) , \( -a + 1\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2a-5\right){x}-a+1$ |
81.4-d1 |
81.4-d |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.4 |
\( 3^{4} \) |
\( 3^{12} \) |
$0.96657$ |
$(-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$2.434315372$ |
0.675157607 |
\( -3874871 a - 5047588 \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( -19 a - 24\) , \( -94 a - 123\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-19a-24\right){x}-94a-123$ |
81.4-d2 |
81.4-d |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
81.4 |
\( 3^{4} \) |
\( 3^{4} \) |
$0.96657$ |
$(-a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$21.90883835$ |
0.675157607 |
\( 175 a - 403 \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( a + 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+1\right){x}$ |
*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.