| 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 |
| 12.2-a8 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{32} \cdot 3^{4} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{3} \cdot 5 \) |
$1$ |
$0.347310172$ |
1.112282735 |
\( \frac{18013780041269221}{9216} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -5462 a - 10917\) , \( 389122 a + 96183\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+\left(-5462a-10917\right){x}+389122a+96183$ |
| 12.2-b8 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{32} \cdot 3^{4} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{3} \cdot 5 \) |
$1$ |
$0.347310172$ |
1.112282735 |
\( \frac{18013780041269221}{9216} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 5460 a - 16379\) , \( -389123 a + 485305\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a+1\right){x}^2+\left(5460a-16379\right){x}-389123a+485305$ |
| 288.2-b8 |
288.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{44} \cdot 3^{10} \) |
$2.29889$ |
$(2,a), (2,a+1), (3,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{5} \) |
$6.607473266$ |
$0.100259810$ |
3.394532497 |
\( \frac{18013780041269221}{9216} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 262138\) , \( -16631674 a + 8184768\bigr] \) |
${y}^2={x}^3+\left(-a-1\right){x}^2+\left(a+262138\right){x}-16631674a+8184768$ |
| 288.2-c8 |
288.2-c |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{44} \cdot 3^{10} \) |
$2.29889$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{5} \cdot 5 \) |
$1$ |
$0.100259810$ |
1.284353474 |
\( \frac{18013780041269221}{9216} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 81917 a - 98301\) , \( 14788921 a + 14992366\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+a{x}^2+\left(81917a-98301\right){x}+14788921a+14992366$ |
| 288.5-b8 |
288.5-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
288.5 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{44} \cdot 3^{10} \) |
$2.29889$ |
$(2,a), (2,a+1), (3,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{5} \) |
$6.607473266$ |
$0.100259810$ |
3.394532497 |
\( \frac{18013780041269221}{9216} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 262138\) , \( 16631674 a - 8184768\bigr] \) |
${y}^2={x}^3+\left(a+1\right){x}^2+\left(a+262138\right){x}+16631674a-8184768$ |
| 288.5-c8 |
288.5-c |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
288.5 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{44} \cdot 3^{10} \) |
$2.29889$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{5} \cdot 5 \) |
$1$ |
$0.100259810$ |
1.284353474 |
\( \frac{18013780041269221}{9216} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -81921 a - 16389\) , \( -14887227 a + 30600477\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(-81921a-16389\right){x}-14887227a+30600477$ |
| 768.5-f8 |
768.5-f |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
768.5 |
\( 2^{8} \cdot 3 \) |
\( 2^{56} \cdot 3^{4} \) |
$2.93773$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cs, 5B |
$25$ |
\( 2^{6} \) |
$1$ |
$0.086827543$ |
2.780706839 |
\( \frac{18013780041269221}{9216} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 87380 a - 262144\) , \( 24729060 a - 30535092\bigr] \) |
${y}^2={x}^3+\left(a-1\right){x}^2+\left(87380a-262144\right){x}+24729060a-30535092$ |
| 768.5-i8 |
768.5-i |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
768.5 |
\( 2^{8} \cdot 3 \) |
\( 2^{56} \cdot 3^{4} \) |
$2.93773$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cs, 5B |
$25$ |
\( 2^{6} \) |
$1$ |
$0.086827543$ |
2.780706839 |
\( \frac{18013780041269221}{9216} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -87380 a - 174764\) , \( -24729060 a - 5806032\bigr] \) |
${y}^2={x}^3-a{x}^2+\left(-87380a-174764\right){x}-24729060a-5806032$ |
*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.
