| 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 |
| 8.1-a1 |
8.1-a |
$8$ |
$12$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{9} \) |
$2.24645$ |
$(a), (-a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$145.9386959$ |
0.684141088 |
\( \frac{33759}{2} a^{2} + 32803 a + 4371 \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} - a + 4\) , \( a^{2} - 2\) , \( -a^{2} - 8 a + 8\) , \( 5 a^{2} - 22 a + 11\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-a^{2}-8a+8\right){x}+5a^{2}-22a+11$ |
| 8.1-a2 |
8.1-a |
$8$ |
$12$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{11} \) |
$2.24645$ |
$(a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$5.405136888$ |
0.684141088 |
\( \frac{5335424117455903}{8} a^{2} + \frac{3582532102104931}{4} a - \frac{2277251078226861}{4} \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} - a + 4\) , \( a^{2} - 2\) , \( -126 a^{2} - 98 a + 78\) , \( -1365 a^{2} - 1026 a + 787\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-126a^{2}-98a+78\right){x}-1365a^{2}-1026a+787$ |
| 8.1-a3 |
8.1-a |
$8$ |
$12$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{11} \) |
$2.24645$ |
$(a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$5.405136888$ |
0.684141088 |
\( -\frac{596658698344264096415}{8} a^{2} + \frac{157910670855855346189}{4} a + \frac{1267643331281111416669}{4} \) |
\( \bigl[a\) , \( -a^{2} - a + 2\) , \( a\) , \( 696542 a^{2} - 368715 a - 2959749\) , \( 463057095 a^{2} - 245103850 a - 1967594769\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(696542a^{2}-368715a-2959749\right){x}+463057095a^{2}-245103850a-1967594769$ |
| 8.1-a4 |
8.1-a |
$8$ |
$12$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$2.24645$ |
$(a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$10.81027377$ |
0.684141088 |
\( -\frac{266664752495}{64} a^{2} + \frac{71121115277}{32} a + \frac{567573953693}{32} \) |
\( \bigl[a\) , \( -a^{2} - a + 2\) , \( a\) , \( 43522 a^{2} - 23060 a - 184974\) , \( 7250072 a^{2} - 3837642 a - 30806682\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(43522a^{2}-23060a-184974\right){x}+7250072a^{2}-3837642a-30806682$ |
| 8.1-a5 |
8.1-a |
$8$ |
$12$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{9} \) |
$2.24645$ |
$(a), (-a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$145.9386959$ |
0.684141088 |
\( \frac{2097153}{2} a^{2} - 6340607 a + 9585409 \) |
\( \bigl[a\) , \( -a^{2} - a + 2\) , \( a\) , \( 8597 a^{2} - 4555 a - 36539\) , \( 638631 a^{2} - 338020 a - 2713601\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(8597a^{2}-4555a-36539\right){x}+638631a^{2}-338020a-2713601$ |
| 8.1-a6 |
8.1-a |
$8$ |
$12$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{6} \) |
$2.24645$ |
$(a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$291.8773919$ |
0.684141088 |
\( \frac{113}{4} a^{2} - \frac{1939}{2} a + \frac{7773}{2} \) |
\( \bigl[a\) , \( -a^{2} - a + 2\) , \( a\) , \( 587 a^{2} - 310 a - 2494\) , \( 8248 a^{2} - 4366 a - 35048\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(587a^{2}-310a-2494\right){x}+8248a^{2}-4366a-35048$ |
| 8.1-a7 |
8.1-a |
$8$ |
$12$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{20} \) |
$2.24645$ |
$(a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$2.702568444$ |
0.684141088 |
\( \frac{532560105997425}{4096} a^{2} - \frac{749207151764755}{2048} a + \frac{293646661918461}{2048} \) |
\( \bigl[a\) , \( -a^{2} - a + 4\) , \( a\) , \( 420286700 a^{2} - 222464805 a - 1785857409\) , \( 6903318103765 a^{2} - 3654040730844 a - 29333168803431\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(420286700a^{2}-222464805a-1785857409\right){x}+6903318103765a^{2}-3654040730844a-29333168803431$ |
| 8.1-a8 |
8.1-a |
$8$ |
$12$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{12} \) |
$2.24645$ |
$(a), (-a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$72.96934799$ |
0.684141088 |
\( \frac{19825}{16} a^{2} - \frac{24115}{8} a + \frac{15581}{8} \) |
\( \bigl[a\) , \( -a^{2} - a + 4\) , \( a\) , \( -15155975 a^{2} + 8022315 a + 64399881\) , \( 50371899485 a^{2} - 26662681574 a - 214037280091\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-15155975a^{2}+8022315a+64399881\right){x}+50371899485a^{2}-26662681574a-214037280091$ |
*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.
