| 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 |
| 392.1-a1 |
392.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{4} \) |
$2.10397$ |
$(a+3), (a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cn |
$1$ |
\( 2^{2} \) |
$1$ |
$8.636575703$ |
6.528637568 |
\( 48384 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -7\) , \( -7\bigr] \) |
${y}^2={x}^{3}-7{x}-7$ |
| 392.1-b1 |
392.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{4} \cdot 7^{8} \) |
$2.10397$ |
$(a+3), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$6.072068378$ |
1.147513062 |
\( \frac{432}{7} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 3 a + 10\) , \( 5 a + 8\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a+10\right){x}+5a+8$ |
| 392.1-b2 |
392.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{14} \) |
$2.10397$ |
$(a+3), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.036034189$ |
1.147513062 |
\( \frac{11090466}{2401} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 3 a - 95\) , \( -170 a - 97\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a-95\right){x}-170a-97$ |
| 392.1-b3 |
392.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{10} \) |
$2.10397$ |
$(a+3), (a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$6.072068378$ |
1.147513062 |
\( \frac{740772}{49} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 3 a - 25\) , \( 12 a - 27\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a-25\right){x}+12a-27$ |
| 392.1-b4 |
392.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{8} \) |
$2.10397$ |
$(a+3), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.036034189$ |
1.147513062 |
\( \frac{1443468546}{7} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 3 a - 515\) , \( 1482 a - 517\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a-515\right){x}+1482a-517$ |
| 392.1-c1 |
392.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{10} \) |
$2.10397$ |
$(a+3), (a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cn |
$1$ |
\( 2 \) |
$1$ |
$5.596061431$ |
2.115112409 |
\( 48384 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 2352 a - 6223\) , \( -99176 a + 262395\bigr] \) |
${y}^2={x}^{3}+\left(2352a-6223\right){x}-99176a+262395$ |
| 392.1-d1 |
392.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{8} \) |
$2.10397$ |
$(a+3), (a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cn |
$1$ |
\( 2 \cdot 3 \) |
$0.071541228$ |
$12.68156780$ |
2.057460805 |
\( 12544 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -16\) , \( 29\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-16{x}+29$ |
| 392.1-e1 |
392.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{8} \) |
$2.10397$ |
$(a+3), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.782382402$ |
$4.834724458$ |
3.257043758 |
\( -\frac{4}{7} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( a + 3\) , \( 4 a + 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(a+3\right){x}+4a+1$ |
| 392.1-e2 |
392.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{10} \) |
$2.10397$ |
$(a+3), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.891191201$ |
$4.834724458$ |
3.257043758 |
\( \frac{3543122}{49} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( a - 67\) , \( 74 a - 69\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(a-67\right){x}+74a-69$ |
| 392.1-f1 |
392.1-f |
$1$ |
$1$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{2} \) |
$2.10397$ |
$(a+3), (a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cn |
$1$ |
\( 2^{2} \) |
$0.091146142$ |
$22.48833051$ |
3.098892262 |
\( 12544 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 112 a - 294\) , \( 876 a - 2317\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(112a-294\right){x}+876a-2317$ |
| 392.1-g1 |
392.1-g |
$1$ |
$1$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{2} \) |
$2.10397$ |
$(a+3), (a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cn |
$1$ |
\( 2^{2} \) |
$0.091146142$ |
$22.48833051$ |
3.098892262 |
\( 12544 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -112 a - 294\) , \( -876 a - 2317\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-112a-294\right){x}-876a-2317$ |
| 392.1-h1 |
392.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{8} \) |
$2.10397$ |
$(a+3), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.782382402$ |
$4.834724458$ |
3.257043758 |
\( -\frac{4}{7} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -a + 3\) , \( -4 a + 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a+3\right){x}-4a+1$ |
| 392.1-h2 |
392.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{10} \) |
$2.10397$ |
$(a+3), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.891191201$ |
$4.834724458$ |
3.257043758 |
\( \frac{3543122}{49} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -a - 67\) , \( -74 a - 69\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a-67\right){x}-74a-69$ |
| 392.1-i1 |
392.1-i |
$1$ |
$1$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{8} \) |
$2.10397$ |
$(a+3), (a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cn |
$1$ |
\( 2 \) |
$1.004255008$ |
$5.696963580$ |
4.324823868 |
\( 12544 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -16\) , \( -29\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-16{x}-29$ |
| 392.1-j1 |
392.1-j |
$1$ |
$1$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{10} \) |
$2.10397$ |
$(a+3), (a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cn |
$1$ |
\( 2 \) |
$1$ |
$5.596061431$ |
2.115112409 |
\( 48384 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2352 a - 6223\) , \( 99176 a + 262395\bigr] \) |
${y}^2={x}^{3}+\left(-2352a-6223\right){x}+99176a+262395$ |
| 392.1-k1 |
392.1-k |
$4$ |
$4$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{4} \cdot 7^{8} \) |
$2.10397$ |
$(a+3), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$6.072068378$ |
1.147513062 |
\( \frac{432}{7} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 2 a + 6\) , \( a + 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a+6\right){x}+a+6$ |
| 392.1-k2 |
392.1-k |
$4$ |
$4$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{14} \) |
$2.10397$ |
$(a+3), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.036034189$ |
1.147513062 |
\( \frac{11090466}{2401} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 2 a - 99\) , \( 71 a - 99\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a-99\right){x}+71a-99$ |
| 392.1-k3 |
392.1-k |
$4$ |
$4$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{10} \) |
$2.10397$ |
$(a+3), (a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$6.072068378$ |
1.147513062 |
\( \frac{740772}{49} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 2 a - 29\) , \( -41 a - 29\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a-29\right){x}-41a-29$ |
| 392.1-k4 |
392.1-k |
$4$ |
$4$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{8} \) |
$2.10397$ |
$(a+3), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.036034189$ |
1.147513062 |
\( \frac{1443468546}{7} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 2 a - 519\) , \( -2001 a - 519\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a-519\right){x}-2001a-519$ |
| 392.1-l1 |
392.1-l |
$1$ |
$1$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{4} \) |
$2.10397$ |
$(a+3), (a)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cn |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.008372102$ |
$25.38174067$ |
1.927605462 |
\( 48384 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -7\) , \( 7\bigr] \) |
${y}^2={x}^{3}-7{x}+7$ |
*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.
