| 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 |
| 32193.2-a1 |
32193.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32193.2 |
\( 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 3^{7} \cdot 7^{7} \cdot 73^{2} \) |
$2.07319$ |
$(-2a+1), (-3a+1), (9a-8)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{5} \) |
$1$ |
$0.132410580$ |
1.223156549 |
\( \frac{234281345534853355}{111909} a - \frac{62564148987073151}{111909} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -41918 a + 11695\) , \( -2986848 a + 2620271\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-41918a+11695\right){x}-2986848a+2620271$ |
| 32193.2-a2 |
32193.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32193.2 |
\( 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 3^{8} \cdot 7^{8} \cdot 73^{4} \) |
$2.07319$ |
$(-2a+1), (-3a+1), (9a-8)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$0.264821161$ |
1.223156549 |
\( -\frac{1339415391745625}{1391513809} a - \frac{2527435047310744}{4174541427} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -2618 a + 730\) , \( -48168 a + 42275\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-2618a+730\right){x}-48168a+42275$ |
| 32193.2-a3 |
32193.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32193.2 |
\( 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 3^{8} \cdot 7^{8} \cdot 73 \) |
$2.07319$ |
$(-2a+1), (-3a+1), (9a-8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.059284644$ |
1.223156549 |
\( -\frac{46553449}{3577} a - \frac{262310864}{10731} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -23 a - 65\) , \( 66 a + 185\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-23a-65\right){x}+66a+185$ |
| 32193.2-a4 |
32193.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32193.2 |
\( 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 3^{10} \cdot 7^{10} \cdot 73^{2} \) |
$2.07319$ |
$(-2a+1), (-3a+1), (9a-8)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$0.529642322$ |
1.223156549 |
\( \frac{42509139235}{38384787} a - \frac{28989387209}{115154361} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -143 a + 10\) , \( -828 a + 1001\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-143a+10\right){x}-828a+1001$ |
| 32193.2-a5 |
32193.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32193.2 |
\( 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 3^{7} \cdot 7^{7} \cdot 73^{8} \) |
$2.07319$ |
$(-2a+1), (-3a+1), (9a-8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.132410580$ |
1.223156549 |
\( -\frac{18382515174811826875}{16935661929775701} a + \frac{28855701603949641503}{16935661929775701} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -2918 a + 1285\) , \( -50508 a + 24995\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-2918a+1285\right){x}-50508a+24995$ |
| 32193.2-a6 |
32193.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32193.2 |
\( 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 3^{14} \cdot 7^{14} \cdot 73 \) |
$2.07319$ |
$(-2a+1), (-3a+1), (9a-8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.264821161$ |
1.223156549 |
\( -\frac{39237205025653}{11362422771} a + \frac{116367310463512}{34087268313} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 412 a + 490\) , \( -7524 a + 7391\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(412a+490\right){x}-7524a+7391$ |
| 32193.2-b1 |
32193.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32193.2 |
\( 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 3^{6} \cdot 7^{6} \cdot 73^{3} \) |
$2.07319$ |
$(-2a+1), (-3a+1), (9a-8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$0.707535304$ |
2.450974190 |
\( -\frac{60988685561}{389017} a - \frac{108786941280}{389017} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -103 a - 205\) , \( -1178 a - 1036\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-103a-205\right){x}-1178a-1036$ |
| 32193.2-b2 |
32193.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32193.2 |
\( 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 3^{6} \cdot 7^{6} \cdot 73^{2} \) |
$2.07319$ |
$(-2a+1), (-3a+1), (9a-8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{3} \) |
$1$ |
$1.061302956$ |
2.450974190 |
\( \frac{927841113}{5329} a - \frac{1323774856}{5329} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -43 a + 125\) , \( -416 a + 68\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-43a+125\right){x}-416a+68$ |
| 32193.2-b3 |
32193.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32193.2 |
\( 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 3^{6} \cdot 7^{6} \cdot 73^{6} \) |
$2.07319$ |
$(-2a+1), (-3a+1), (9a-8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$0.353767652$ |
2.450974190 |
\( \frac{55816089234767}{151334226289} a + \frac{51536736771337}{151334226289} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 17 a - 280\) , \( -2657 a + 17\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(17a-280\right){x}-2657a+17$ |
| 32193.2-b4 |
32193.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32193.2 |
\( 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 3^{6} \cdot 7^{6} \cdot 73 \) |
$2.07319$ |
$(-2a+1), (-3a+1), (9a-8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$2.122605913$ |
2.450974190 |
\( -\frac{9927}{73} a + \frac{30887}{73} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 2 a + 5\) , \( -2 a - 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a+5\right){x}-2a-7$ |
*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.
