| 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 |
| 4032.7-a1 |
4032.7-a |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{16} \cdot 3^{8} \cdot 7 \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.882282413$ |
$1.552362152$ |
2.070673566 |
\( \frac{48284377}{189} a - \frac{108521789}{567} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -4 a + 56\) , \( 148 a - 16\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-4a+56\right){x}+148a-16$ |
| 4032.7-a2 |
4032.7-a |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{16} \cdot 3^{2} \cdot 7 \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.882282413$ |
$3.104724304$ |
2.070673566 |
\( -\frac{219127}{7} a - \frac{566221}{21} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 4 a - 13\) , \( -11 a + 10\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(4a-13\right){x}-11a+10$ |
| 4032.7-a3 |
4032.7-a |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{14} \cdot 3^{4} \cdot 7^{2} \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.441141206$ |
$3.104724304$ |
2.070673566 |
\( -\frac{4009}{21} a + \frac{2191}{9} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( a + 1\) , \( 4 a\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+1\right){x}+4a$ |
| 4032.7-a4 |
4032.7-a |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{10} \cdot 3^{2} \cdot 7^{4} \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.220570603$ |
$3.104724304$ |
2.070673566 |
\( \frac{166231}{49} a + \frac{660025}{147} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 2 a - 6\) , \( 2 a - 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2a-6\right){x}+2a-2$ |
| 4032.7-b1 |
4032.7-b |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{10} \cdot 3^{4} \cdot 7 \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.983354618$ |
1.499275166 |
\( -\frac{863944673}{63} a - \frac{1616364293}{63} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 38 a + 22\) , \( 87 a - 245\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(38a+22\right){x}+87a-245$ |
| 4032.7-b2 |
4032.7-b |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{17} \cdot 3^{2} \cdot 7^{8} \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.991677309$ |
1.499275166 |
\( \frac{70011793}{7203} a - \frac{221078161}{7203} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 21 a + 73\) , \( 214 a - 242\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(21a+73\right){x}+214a-242$ |
| 4032.7-b3 |
4032.7-b |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{16} \cdot 3^{4} \cdot 7^{4} \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$1.983354618$ |
1.499275166 |
\( \frac{172799}{441} a - \frac{2545}{9} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 6 a - 2\) , \( 13 a + 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(6a-2\right){x}+13a+1$ |
| 4032.7-b4 |
4032.7-b |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{14} \cdot 3^{8} \cdot 7^{2} \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$1.983354618$ |
1.499275166 |
\( -\frac{839201}{189} a + \frac{2555873}{567} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 8 a - 17\) , \( 12 a - 23\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(8a-17\right){x}+12a-23$ |
| 4032.7-b5 |
4032.7-b |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{16} \cdot 3^{16} \cdot 7 \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.991677309$ |
1.499275166 |
\( \frac{78717967}{15309} a - \frac{9092939}{45927} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 13 a - 72\) , \( -80 a + 205\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(13a-72\right){x}-80a+205$ |
| 4032.7-b6 |
4032.7-b |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{17} \cdot 3^{2} \cdot 7^{2} \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.983354618$ |
1.499275166 |
\( -\frac{13784383}{21} a + \frac{8018911}{21} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( a - 39\) , \( 112\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a-39\right){x}+112$ |
| 4032.7-c1 |
4032.7-c |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{18} \cdot 3^{2} \cdot 7^{16} \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.304790221$ |
1.843198006 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 170 a - 238\) , \( 3689 a - 2387\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(170a-238\right){x}+3689a-2387$ |
| 4032.7-c2 |
4032.7-c |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{18} \cdot 3^{4} \cdot 7^{2} \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.438321771$ |
1.843198006 |
\( \frac{103823}{63} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -5 a + 7\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-5a+7\right){x}$ |
| 4032.7-c3 |
4032.7-c |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{18} \cdot 3^{8} \cdot 7^{4} \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.219160885$ |
1.843198006 |
\( \frac{7189057}{3969} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 20 a - 28\) , \( 17 a - 11\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(20a-28\right){x}+17a-11$ |
| 4032.7-c4 |
4032.7-c |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{18} \cdot 3^{16} \cdot 7^{2} \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.609580442$ |
1.843198006 |
\( \frac{6570725617}{45927} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 195 a - 273\) , \( -1530 a + 990\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(195a-273\right){x}-1530a+990$ |
| 4032.7-c5 |
4032.7-c |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{18} \cdot 3^{2} \cdot 7 \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.438321771$ |
1.843198006 |
\( -\frac{2940226}{21} a + \frac{5920433}{21} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 6 a + 18\) , \( 21 a - 39\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(6a+18\right){x}+21a-39$ |
| 4032.7-c6 |
4032.7-c |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{18} \cdot 3^{2} \cdot 7 \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.438321771$ |
1.843198006 |
\( \frac{2940226}{21} a + \frac{2980207}{21} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -13 a - 10\) , \( 16 a + 13\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-13a-10\right){x}+16a+13$ |
| 4032.7-c7 |
4032.7-c |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{18} \cdot 3^{4} \cdot 7^{8} \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$0.609580442$ |
1.843198006 |
\( \frac{13027640977}{21609} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 245 a - 343\) , \( 2312 a - 1496\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(245a-343\right){x}+2312a-1496$ |
| 4032.7-c8 |
4032.7-c |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4032.7 |
\( 2^{6} \cdot 3^{2} \cdot 7 \) |
\( 2^{18} \cdot 3^{2} \cdot 7^{4} \) |
$1.88394$ |
$(-a+1), (-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$0.304790221$ |
1.843198006 |
\( \frac{53297461115137}{147} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 3920 a - 5488\) , \( 144755 a - 93665\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(3920a-5488\right){x}+144755a-93665$ |
*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.
