| 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 |
| 116964.1-a1 |
116964.1-a |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
116964.1 |
\( 2^{2} \cdot 3^{4} \cdot 19^{2} \) |
\( 2^{2} \cdot 3^{10} \cdot 19^{8} \) |
$2.86228$ |
$(-2a+1), (-5a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B |
$1$ |
\( 1 \) |
$1$ |
$0.319512217$ |
0.368940929 |
\( \frac{2317155}{2} a - \frac{43024347}{2} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 757 a + 1815\) , \( -44329 a + 39421\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(757a+1815\right){x}-44329a+39421$ |
| 116964.1-a2 |
116964.1-a |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
116964.1 |
\( 2^{2} \cdot 3^{4} \cdot 19^{2} \) |
\( 2^{14} \cdot 3^{10} \cdot 19^{8} \) |
$2.86228$ |
$(-2a+1), (-5a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B |
$1$ |
\( 1 \) |
$1$ |
$0.319512217$ |
0.368940929 |
\( -\frac{926085}{128} a + \frac{289917}{128} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 201 a - 693\) , \( -3864 a + 7056\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(201a-693\right){x}-3864a+7056$ |
| 116964.1-b1 |
116964.1-b |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
116964.1 |
\( 2^{2} \cdot 3^{4} \cdot 19^{2} \) |
\( 2^{2} \cdot 3^{4} \cdot 19^{2} \) |
$2.86228$ |
$(-2a+1), (-5a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B |
$1$ |
\( 1 \) |
$1$ |
$2.412264341$ |
2.785442933 |
\( \frac{2317155}{2} a - \frac{43024347}{2} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 5 a + 38\) , \( -87 a + 76\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(5a+38\right){x}-87a+76$ |
| 116964.1-b2 |
116964.1-b |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
116964.1 |
\( 2^{2} \cdot 3^{4} \cdot 19^{2} \) |
\( 2^{14} \cdot 3^{4} \cdot 19^{2} \) |
$2.86228$ |
$(-2a+1), (-5a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B |
$1$ |
\( 1 \) |
$1$ |
$2.412264341$ |
2.785442933 |
\( -\frac{926085}{128} a + \frac{289917}{128} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 6 a - 12\) , \( -16 a + 20\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(6a-12\right){x}-16a+20$ |
| 116964.1-c1 |
116964.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
116964.1 |
\( 2^{2} \cdot 3^{4} \cdot 19^{2} \) |
\( 2^{2} \cdot 3^{4} \cdot 19^{8} \) |
$2.86228$ |
$(-2a+1), (-5a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$1.198483745$ |
2.767779653 |
\( -\frac{101967}{722} a + \frac{786951}{722} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -8 a - 23\) , \( -58 a + 49\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-8a-23\right){x}-58a+49$ |
| 116964.1-d1 |
116964.1-d |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
116964.1 |
\( 2^{2} \cdot 3^{4} \cdot 19^{2} \) |
\( 2^{2} \cdot 3^{4} \cdot 19^{6} \) |
$2.86228$ |
$(-2a+1), (-5a+3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2 \) |
$1.562562219$ |
$0.746391894$ |
5.386834011 |
\( \frac{17268549}{2} a - \frac{246587109}{2} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( -588 a + 312\) , \( -3791 a + 5462\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-588a+312\right){x}-3791a+5462$ |
| 116964.1-d2 |
116964.1-d |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
116964.1 |
\( 2^{2} \cdot 3^{4} \cdot 19^{2} \) |
\( 2^{2} \cdot 3^{4} \cdot 19^{6} \) |
$2.86228$ |
$(-2a+1), (-5a+3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2 \) |
$1.562562219$ |
$0.746391894$ |
5.386834011 |
\( -\frac{17268549}{2} a - 114659280 \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -59 a + 538\) , \( 5135 a - 2103\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-59a+538\right){x}+5135a-2103$ |
| 116964.1-d3 |
116964.1-d |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
116964.1 |
\( 2^{2} \cdot 3^{4} \cdot 19^{2} \) |
\( 2^{2} \cdot 3^{12} \cdot 19^{6} \) |
$2.86228$ |
$(-2a+1), (-5a+3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2 \) |
$1.562562219$ |
$0.746391894$ |
5.386834011 |
\( -\frac{132651}{2} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -48 a - 153\) , \( 411 a + 662\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-48a-153\right){x}+411a+662$ |
| 116964.1-d4 |
116964.1-d |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
116964.1 |
\( 2^{2} \cdot 3^{4} \cdot 19^{2} \) |
\( 2^{18} \cdot 3^{4} \cdot 19^{6} \) |
$2.86228$ |
$(-2a+1), (-5a+3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.173618024$ |
$0.746391894$ |
5.386834011 |
\( -\frac{1167051}{512} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -24 a - 72\) , \( -160 a - 275\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-24a-72\right){x}-160a-275$ |
| 116964.1-d5 |
116964.1-d |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
116964.1 |
\( 2^{2} \cdot 3^{4} \cdot 19^{2} \) |
\( 2^{6} \cdot 3^{12} \cdot 19^{6} \) |
$2.86228$ |
$(-2a+1), (-5a+3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Cs[2] |
$1$ |
\( 2 \cdot 3 \) |
$0.520854073$ |
$0.746391894$ |
5.386834011 |
\( \frac{9261}{8} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 19 a + 63\) , \( -84 a - 110\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(19a+63\right){x}-84a-110$ |
*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.
