| 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 |
| 6.4-a1 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{19} \cdot 3^{7} \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 7 \) |
$1$ |
$1.424700212$ |
1.039746854 |
\( \frac{133689253600013}{279936} a - \frac{9956234247227}{46656} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 206 a + 403\) , \( -1402 a + 7012\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-a-1\right){x}^2+\left(206a+403\right){x}-1402a+7012$ |
| 6.4-a2 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{26} \cdot 3^{14} \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$1.424700212$ |
1.039746854 |
\( -\frac{773243804465483}{78364164096} a - \frac{241910116865299}{13060694016} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 8 a - 178\) , \( -154 a + 999\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+a{x}^2+\left(8a-178\right){x}-154a+999$ |
| 6.4-a3 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{19} \cdot 3^{28} \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$0.712350106$ |
1.039746854 |
\( -\frac{413197302910024471}{2928229434235008} a + \frac{6518454992729281}{488038239039168} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 88 a - 98\) , \( 582 a + 1991\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+a{x}^2+\left(88a-98\right){x}+582a+1991$ |
| 6.4-a4 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{28} \cdot 3^{7} \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$1.424700212$ |
1.039746854 |
\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -4 a - 17\) , \( -9 a\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(-a-1\right){x}^2+\left(-4a-17\right){x}-9a$ |
| 6.4-a5 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{4} \cdot 3 \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2^{2} \) |
$1$ |
$9.972901485$ |
1.039746854 |
\( -\frac{9841}{48} a + \frac{8911}{8} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( a - 2\) , \( 1\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(-a-1\right){x}^2+\left(a-2\right){x}+1$ |
| 6.4-a6 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{14} \cdot 3^{2} \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2^{2} \) |
$1$ |
$9.972901485$ |
1.039746854 |
\( \frac{15457}{36} a + \frac{13391}{6} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -2 a + 2\) , \( 3\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+a{x}^2+\left(-2a+2\right){x}+3$ |
| 6.4-a7 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{13} \cdot 3 \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 1 \) |
$1$ |
$9.972901485$ |
1.039746854 |
\( -\frac{57217}{6} a + 27208 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( a - 2\) , \( -a + 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-a-1\right){x}^2+\left(a-2\right){x}-a+5$ |
| 6.4-a8 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{13} \cdot 3^{4} \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2 \) |
$1$ |
$4.986450742$ |
1.039746854 |
\( \frac{238419887}{162} a + \frac{18567478}{27} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -12 a + 22\) , \( 8 a - 45\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+a{x}^2+\left(-12a+22\right){x}+8a-45$ |
*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.
