| 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 |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{4} \cdot 7^{2} \) |
$1.37737$ |
$(a+1), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$10.54517411$ |
1.522064778 |
\( \frac{432}{7} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -a + 4\) , \( 5 a - 8\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-a+4\right){x}+5a-8$ |
| 392.1-a2 |
392.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{8} \) |
$1.37737$ |
$(a+1), (7)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$10.54517411$ |
1.522064778 |
\( \frac{11090466}{2401} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 59 a - 101\) , \( -310 a + 537\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(59a-101\right){x}-310a+537$ |
| 392.1-a3 |
392.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{4} \) |
$1.37737$ |
$(a+1), (7)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$10.54517411$ |
1.522064778 |
\( \frac{740772}{49} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 19 a - 31\) , \( 40 a - 69\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(19a-31\right){x}+40a-69$ |
| 392.1-a4 |
392.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{2} \) |
$1.37737$ |
$(a+1), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$2.636293528$ |
1.522064778 |
\( \frac{1443468546}{7} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 299 a - 521\) , \( 3470 a - 6019\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(299a-521\right){x}+3470a-6019$ |
| 392.1-b1 |
392.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{2} \) |
$1.37737$ |
$(a+1), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$7.189921948$ |
2.075551686 |
\( -\frac{4}{7} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -a + 1\) , \( -8 a - 13\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a+1\right){x}-8a-13$ |
| 392.1-b2 |
392.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{4} \) |
$1.37737$ |
$(a+1), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$7.189921948$ |
2.075551686 |
\( \frac{3543122}{49} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -41 a - 69\) , \( -198 a - 343\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-41a-69\right){x}-198a-343$ |
| 392.1-c1 |
392.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{2} \) |
$1.37737$ |
$(a+1), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.239919898$ |
$22.75712104$ |
1.576133376 |
\( -\frac{4}{7} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -a - 2\) , \( 7 a + 12\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-2\right){x}+7a+12$ |
| 392.1-c2 |
392.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{4} \) |
$1.37737$ |
$(a+1), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.119959949$ |
$22.75712104$ |
1.576133376 |
\( \frac{3543122}{49} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -41 a - 72\) , \( 157 a + 272\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-41a-72\right){x}+157a+272$ |
| 392.1-d1 |
392.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{4} \cdot 7^{2} \) |
$1.37737$ |
$(a+1), (7)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.137828769$ |
$24.47471212$ |
2.009758566 |
\( \frac{432}{7} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -2 a + 2\) , \( -3 a + 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2a+2\right){x}-3a+4$ |
| 392.1-d2 |
392.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{8} \) |
$1.37737$ |
$(a+1), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.284457192$ |
$6.118678030$ |
2.009758566 |
\( \frac{11090466}{2401} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 58 a - 103\) , \( 207 a - 361\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(58a-103\right){x}+207a-361$ |
| 392.1-d3 |
392.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{4} \) |
$1.37737$ |
$(a+1), (7)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.568914384$ |
$24.47471212$ |
2.009758566 |
\( \frac{740772}{49} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 18 a - 33\) , \( -73 a + 125\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(18a-33\right){x}-73a+125$ |
| 392.1-d4 |
392.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{2} \) |
$1.37737$ |
$(a+1), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.284457192$ |
$24.47471212$ |
2.009758566 |
\( \frac{1443468546}{7} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 298 a - 523\) , \( -3993 a + 6915\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(298a-523\right){x}-3993a+6915$ |
*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.
