| 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-a1 |
12.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{50} \cdot 3^{2} \cdot 7^{12} \) |
$1.75225$ |
$(2,a), (2,a+1), (3,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \cdot 5^{2} \) |
$0.261617015$ |
$0.624987031$ |
3.103884118 |
\( -\frac{136511322949}{100663296} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( -857 a - 2262\) , \( -43379 a + 24135\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(a+1\right){x}^2+\left(-857a-2262\right){x}-43379a+24135$ |
| 12.2-b1 |
12.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{50} \cdot 3^{2} \cdot 7^{12} \) |
$1.75225$ |
$(2,a), (2,a+1), (3,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \cdot 5^{2} \) |
$0.261617015$ |
$0.624987031$ |
3.103884118 |
\( -\frac{136511322949}{100663296} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( 859 a - 3120\) , \( 42521 a - 16124\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(-a-1\right){x}^2+\left(859a-3120\right){x}+42521a-16124$ |
| 288.2-a1 |
288.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{74} \cdot 3^{8} \) |
$3.87836$ |
$(2,a), (2,a+1), (3,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B |
$1$ |
\( 2^{2} \) |
$7.154187014$ |
$0.360836430$ |
1.960194492 |
\( -\frac{136511322949}{100663296} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 5140\) , \( -38996 a + 22068\bigr] \) |
${y}^2={x}^3+\left(a+1\right){x}^2+\left(a+5140\right){x}-38996a+22068$ |
| 288.2-f1 |
288.2-f |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{62} \cdot 3^{8} \cdot 5^{12} \) |
$3.87836$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B |
$1$ |
\( 2^{3} \cdot 5^{2} \) |
$1$ |
$0.360836430$ |
6.849815668 |
\( -\frac{136511322949}{100663296} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 5469 a + 11620\) , \( 46311 a + 3262872\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(5469a+11620\right){x}+46311a+3262872$ |
| 288.5-a1 |
288.5-a |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
288.5 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{74} \cdot 3^{8} \) |
$3.87836$ |
$(2,a), (2,a+1), (3,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B |
$1$ |
\( 2^{2} \) |
$7.154187014$ |
$0.360836430$ |
1.960194492 |
\( -\frac{136511322949}{100663296} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 5140\) , \( 38996 a - 22068\bigr] \) |
${y}^2={x}^3+\left(-a-1\right){x}^2+\left(a+5140\right){x}+38996a-22068$ |
| 288.5-f1 |
288.5-f |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
288.5 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{62} \cdot 3^{8} \cdot 5^{12} \) |
$3.87836$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B |
$1$ |
\( 2^{3} \cdot 5^{2} \) |
$1$ |
$0.360836430$ |
6.849815668 |
\( -\frac{136511322949}{100663296} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -5471 a + 17089\) , \( -46312 a + 3309183\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-5471a+17089\right){x}-46312a+3309183$ |
| 768.5-g1 |
768.5-g |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
768.5 |
\( 2^{8} \cdot 3 \) |
\( 2^{74} \cdot 3^{2} \cdot 7^{12} \) |
$4.95610$ |
$(2,a), (2,a+1), (3,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B |
$1$ |
\( 2^{4} \) |
$7.597072013$ |
$0.312493515$ |
7.210672021 |
\( -\frac{136511322949}{100663296} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 13731 a - 49783\) , \( -2698978 a + 1466382\bigr] \) |
${y}^2={x}^3+a{x}^2+\left(13731a-49783\right){x}-2698978a+1466382$ |
| 768.5-l1 |
768.5-l |
$2$ |
$5$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
768.5 |
\( 2^{8} \cdot 3 \) |
\( 2^{74} \cdot 3^{2} \cdot 7^{12} \) |
$4.95610$ |
$(2,a), (2,a+1), (3,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B |
$1$ |
\( 2^{4} \) |
$7.597072013$ |
$0.312493515$ |
7.210672021 |
\( -\frac{136511322949}{100663296} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -13731 a - 36052\) , \( 2698978 a - 1232596\bigr] \) |
${y}^2={x}^3+\left(-a+1\right){x}^2+\left(-13731a-36052\right){x}+2698978a-1232596$ |
*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.
