| 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 |
| 343.2-a1 |
343.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
343.2 |
\( 7^{3} \) |
\( 7^{19} \) |
$0.66607$ |
$(-3a+1), (3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$0.640604886$ |
0.739706807 |
\( -\frac{2097781165791}{13841287201} a + \frac{1802695628925}{13841287201} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 6 a + 48\) , \( 416 a + 152\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(6a+48\right){x}+416a+152$ |
| 343.2-a2 |
343.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
343.2 |
\( 7^{3} \) |
\( 7^{19} \) |
$0.66607$ |
$(-3a+1), (3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$0.640604886$ |
0.739706807 |
\( \frac{2097781165791}{13841287201} a - \frac{295085536866}{13841287201} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( -44 a + 56\) , \( -540 a + 143\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-44a+56\right){x}-540a+143$ |
| 343.2-a3 |
343.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
343.2 |
\( 7^{3} \) |
\( 7^{10} \) |
$0.66607$ |
$(-3a+1), (3a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$2.562419546$ |
0.739706807 |
\( -\frac{988929}{343} a + \frac{2130273}{343} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( -4 a - 7\) , \( -9 a - 5\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-4a-7\right){x}-9a-5$ |
| 343.2-a4 |
343.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
343.2 |
\( 7^{3} \) |
\( 7^{10} \) |
$0.66607$ |
$(-3a+1), (3a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$2.562419546$ |
0.739706807 |
\( \frac{988929}{343} a + \frac{1141344}{343} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 11 a - 9\) , \( 15 a - 14\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(11a-9\right){x}+15a-14$ |
| 343.2-a5 |
343.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
343.2 |
\( 7^{3} \) |
\( 7^{14} \) |
$0.66607$ |
$(-3a+1), (3a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B[2] |
$1$ |
\( 2^{3} \) |
$1$ |
$1.281209773$ |
0.739706807 |
\( -\frac{854150427}{117649} a + \frac{1556711379}{117649} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 51 a - 34\) , \( -79 a - 41\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(51a-34\right){x}-79a-41$ |
| 343.2-a6 |
343.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
343.2 |
\( 7^{3} \) |
\( 7^{14} \) |
$0.66607$ |
$(-3a+1), (3a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B[2] |
$1$ |
\( 2^{3} \) |
$1$ |
$1.281209773$ |
0.739706807 |
\( \frac{854150427}{117649} a + \frac{702560952}{117649} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( -29 a - 22\) , \( 80 a + 19\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-29a-22\right){x}+80a+19$ |
| 343.2-a7 |
343.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
343.2 |
\( 7^{3} \) |
\( 7^{13} \) |
$0.66607$ |
$(-3a+1), (3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$0.640604886$ |
0.739706807 |
\( -\frac{308817493407}{2401} a + \frac{246921503922}{2401} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 786 a - 524\) , \( -6694 a - 433\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(786a-524\right){x}-6694a-433$ |
| 343.2-a8 |
343.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
343.2 |
\( 7^{3} \) |
\( 7^{13} \) |
$0.66607$ |
$(-3a+1), (3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$0.640604886$ |
0.739706807 |
\( \frac{308817493407}{2401} a - \frac{61895989485}{2401} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( -464 a - 332\) , \( 6180 a + 1082\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-464a-332\right){x}+6180a+1082$ |
*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.
