| 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 |
| 6.1-a1 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{26} \cdot 3^{14} \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$1.424700212$ |
1.039746854 |
\( \frac{773243804465483}{78364164096} a - \frac{2224704505657277}{78364164096} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -10 a - 173\) , \( -27 a + 899\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(-10a-173\right){x}-27a+899$ |
| 6.1-a2 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{19} \cdot 3^{28} \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$0.712350106$ |
1.039746854 |
\( \frac{413197302910024471}{2928229434235008} a - \frac{374086572953648785}{2928229434235008} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -90 a - 13\) , \( -683 a + 3107\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(-90a-13\right){x}-683a+3107$ |
| 6.1-a3 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{28} \cdot 3^{7} \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$1.424700212$ |
1.039746854 |
\( -\frac{291660246420647}{587068342272} a + \frac{520440745417985}{587068342272} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( 6 a - 22\) , \( 14 a - 31\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(a+1\right){x}^2+\left(6a-22\right){x}+14a-31$ |
| 6.1-a4 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{4} \cdot 3 \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$9.972901485$ |
1.039746854 |
\( \frac{9841}{48} a + \frac{43625}{48} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( a - 2\) , \( -1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(a+1\right){x}^2+\left(a-2\right){x}-1$ |
| 6.1-a5 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{14} \cdot 3^{2} \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$9.972901485$ |
1.039746854 |
\( -\frac{15457}{36} a + \frac{95803}{36} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -3\) , \( -a - 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2-3{x}-a-3$ |
| 6.1-a6 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{13} \cdot 3 \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 1 \) |
$1$ |
$9.972901485$ |
1.039746854 |
\( \frac{57217}{6} a + \frac{106031}{6} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -a - 1\) , \( a + 4\bigr] \) |
${y}^2+a{x}{y}={x}^3-{x}^2+\left(-a-1\right){x}+a+4$ |
| 6.1-a7 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{13} \cdot 3^{4} \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2 \) |
$1$ |
$4.986450742$ |
1.039746854 |
\( -\frac{238419887}{162} a + \frac{349824755}{162} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 10 a + 7\) , \( 11 a - 103\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(10a+7\right){x}+11a-103$ |
| 6.1-a8 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{19} \cdot 3^{7} \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 7 \) |
$1$ |
$1.424700212$ |
1.039746854 |
\( -\frac{133689253600013}{279936} a + \frac{73951848116651}{279936} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -206 a + 609\) , \( 1402 a + 5610\bigr] \) |
${y}^2+a{x}{y}={x}^3-{x}^2+\left(-206a+609\right){x}+1402a+5610$ |
*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.
