| 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 |
| 29241.8-a1 |
29241.8-a |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
29241.8 |
\( 3^{4} \cdot 19^{2} \) |
\( 3^{14} \cdot 19^{8} \) |
$3.30508$ |
$(-a-1), (a-1), (-3a+1), (3a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.053945323$ |
$0.544114833$ |
3.244020840 |
\( \frac{67419143}{390963} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 76\) , \( -790\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+76{x}-790$ |
| 29241.8-a2 |
29241.8-a |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
29241.8 |
\( 3^{4} \cdot 19^{2} \) |
\( 3^{14} \cdot 19^{2} \) |
$3.30508$ |
$(-a-1), (a-1), (-3a+1), (3a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.053945323$ |
$2.176459332$ |
3.244020840 |
\( \frac{389017}{57} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -14\) , \( 20\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-14{x}+20$ |
| 29241.8-a3 |
29241.8-a |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
29241.8 |
\( 3^{4} \cdot 19^{2} \) |
\( 3^{16} \cdot 19^{4} \) |
$3.30508$ |
$(-a-1), (a-1), (-3a+1), (3a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$0.526972661$ |
$1.088229666$ |
3.244020840 |
\( \frac{30664297}{3249} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -59\) , \( -142\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-59{x}-142$ |
| 29241.8-a4 |
29241.8-a |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
29241.8 |
\( 3^{4} \cdot 19^{2} \) |
\( 3^{20} \cdot 19^{2} \) |
$3.30508$ |
$(-a-1), (a-1), (-3a+1), (3a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.053945323$ |
$0.544114833$ |
3.244020840 |
\( \frac{115714886617}{1539} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -914\) , \( -10402\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-914{x}-10402$ |
| 29241.8-b1 |
29241.8-b |
$3$ |
$9$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
29241.8 |
\( 3^{4} \cdot 19^{2} \) |
\( 3^{12} \cdot 19^{2} \) |
$3.30508$ |
$(-a-1), (a-1), (-3a+1), (3a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B.1.1 |
$9$ |
\( 2^{2} \) |
$2.033701819$ |
$0.311769669$ |
3.586708887 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -6924\) , \( 221760\bigr] \) |
${y}^2+{y}={x}^{3}-6924{x}+221760$ |
| 29241.8-b2 |
29241.8-b |
$3$ |
$9$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
29241.8 |
\( 3^{4} \cdot 19^{2} \) |
\( 3^{12} \cdot 19^{6} \) |
$3.30508$ |
$(-a-1), (a-1), (-3a+1), (3a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.677900606$ |
$0.935309008$ |
3.586708887 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -84\) , \( 315\bigr] \) |
${y}^2+{y}={x}^{3}-84{x}+315$ |
| 29241.8-b3 |
29241.8-b |
$3$ |
$9$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
29241.8 |
\( 3^{4} \cdot 19^{2} \) |
\( 3^{12} \cdot 19^{2} \) |
$3.30508$ |
$(-a-1), (a-1), (-3a+1), (3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2^{2} \) |
$0.225966868$ |
$2.805927025$ |
3.586708887 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 6\) , \( 0\bigr] \) |
${y}^2+{y}={x}^{3}+6{x}$ |
| 29241.8-c1 |
29241.8-c |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
29241.8 |
\( 3^{4} \cdot 19^{2} \) |
\( 3^{17} \cdot 19^{3} \) |
$3.30508$ |
$(-a-1), (a-1), (-3a+1), (3a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.972489132$ |
$1.171350742$ |
9.848094932 |
\( -\frac{69965056}{3249} a - \frac{14926784}{3249} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 19 a - 56\) , \( -100 a + 181\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(19a-56\right){x}-100a+181$ |
| 29241.8-c2 |
29241.8-c |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
29241.8 |
\( 3^{4} \cdot 19^{2} \) |
\( 3^{19} \cdot 19^{3} \) |
$3.30508$ |
$(-a-1), (a-1), (-3a+1), (3a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.486244566$ |
$1.171350742$ |
9.848094932 |
\( -\frac{2546432}{29241} a - \frac{2656192}{29241} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( 10 a + 6\) , \( 4 a - 85\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(10a+6\right){x}+4a-85$ |
| 29241.8-d1 |
29241.8-d |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
29241.8 |
\( 3^{4} \cdot 19^{2} \) |
\( 3^{16} \cdot 19^{2} \) |
$3.30508$ |
$(-a-1), (a-1), (-3a+1), (3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.941130653$ |
$1.776214705$ |
9.456281006 |
\( -\frac{1404928}{171} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -21\) , \( -41\bigr] \) |
${y}^2+{y}={x}^{3}-21{x}-41$ |
| 29241.8-e1 |
29241.8-e |
$2$ |
$5$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
29241.8 |
\( 3^{4} \cdot 19^{2} \) |
\( 3^{16} \cdot 19^{10} \) |
$3.30508$ |
$(-a-1), (a-1), (-3a+1), (3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \) |
$19.80313271$ |
$0.090588610$ |
10.14804730 |
\( -\frac{9358714467168256}{22284891} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -39513\) , \( 3023145\bigr] \) |
${y}^2+{y}={x}^{3}-39513{x}+3023145$ |
| 29241.8-e2 |
29241.8-e |
$2$ |
$5$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
29241.8 |
\( 3^{4} \cdot 19^{2} \) |
\( 3^{32} \cdot 19^{2} \) |
$3.30508$ |
$(-a-1), (a-1), (-3a+1), (3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2^{2} \) |
$3.960626542$ |
$0.452943050$ |
10.14804730 |
\( \frac{841232384}{1121931} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 177\) , \( 1035\bigr] \) |
${y}^2+{y}={x}^{3}+177{x}+1035$ |
| 29241.8-f1 |
29241.8-f |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
29241.8 |
\( 3^{4} \cdot 19^{2} \) |
\( 3^{17} \cdot 19^{3} \) |
$3.30508$ |
$(-a-1), (a-1), (-3a+1), (3a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.972489132$ |
$1.171350742$ |
9.848094932 |
\( \frac{69965056}{3249} a - \frac{14926784}{3249} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -20 a - 56\) , \( 99 a + 181\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-20a-56\right){x}+99a+181$ |
| 29241.8-f2 |
29241.8-f |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
29241.8 |
\( 3^{4} \cdot 19^{2} \) |
\( 3^{19} \cdot 19^{3} \) |
$3.30508$ |
$(-a-1), (a-1), (-3a+1), (3a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.486244566$ |
$1.171350742$ |
9.848094932 |
\( \frac{2546432}{29241} a - \frac{2656192}{29241} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( -11 a + 6\) , \( -4 a - 85\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-11a+6\right){x}-4a-85$ |
*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.
