| 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 |
| 22.1-a3 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{16} \cdot 11^{2} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$399.1627311$ |
0.841626760 |
\( \frac{22702786953003143813}{7929856} a^{3} + \frac{30928910991628706537}{7929856} a^{2} - \frac{8873274696408279559}{3964928} a - \frac{19220593607745698479}{7929856} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( a^{2} - a - 2\) , \( -20 a^{3} + 35 a^{2} + 56 a - 64\) , \( 299 a^{3} - 501 a^{2} - 856 a + 876\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-3\right){x}^{2}+\left(-20a^{3}+35a^{2}+56a-64\right){x}+299a^{3}-501a^{2}-856a+876$ |
| 176.2-f12 |
176.2-f |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
176.2 |
\( 2^{4} \cdot 11 \) |
\( 2^{28} \cdot 11^{2} \) |
$8.98702$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1.350701360$ |
$12.14928553$ |
2.491220239 |
\( \frac{22702786953003143813}{7929856} a^{3} + \frac{30928910991628706537}{7929856} a^{2} - \frac{8873274696408279559}{3964928} a - \frac{19220593607745698479}{7929856} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{2} - a - 2\) , \( -21 a^{3} - 22 a^{2} + 13 a + 14\) , \( -135 a^{3} - 300 a^{2} + 121 a + 178\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(-21a^{3}-22a^{2}+13a+14\right){x}-135a^{3}-300a^{2}+121a+178$ |
| 242.1-e8 |
242.1-e |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
242.1 |
\( 2 \cdot 11^{2} \) |
\( 2^{16} \cdot 11^{8} \) |
$9.35198$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{5} \) |
$1$ |
$18.83123142$ |
2.858780215 |
\( \frac{22702786953003143813}{7929856} a^{3} + \frac{30928910991628706537}{7929856} a^{2} - \frac{8873274696408279559}{3964928} a - \frac{19220593607745698479}{7929856} \) |
\( \bigl[a + 1\) , \( a^{2} - a - 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -28 a^{3} + 49 a^{2} + 75 a - 87\) , \( 393 a^{3} - 671 a^{2} - 1105 a + 1159\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-28a^{3}+49a^{2}+75a-87\right){x}+393a^{3}-671a^{2}-1105a+1159$ |
| 704.3-d2 |
704.3-d |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
704.3 |
\( 2^{6} \cdot 11 \) |
\( 2^{34} \cdot 11^{2} \) |
$10.68742$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$0.646418137$ |
$35.02602622$ |
3.437213027 |
\( \frac{22702786953003143813}{7929856} a^{3} + \frac{30928910991628706537}{7929856} a^{2} - \frac{8873274696408279559}{3964928} a - \frac{19220593607745698479}{7929856} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{3} + 3 a\) , \( 0\) , \( -3 a^{3} + 41 a^{2} - 20 a - 132\) , \( 311 a^{3} - 592 a^{2} - 833 a + 1166\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(-3a^{3}+41a^{2}-20a-132\right){x}+311a^{3}-592a^{2}-833a+1166$ |
| 704.3-q11 |
704.3-q |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
704.3 |
\( 2^{6} \cdot 11 \) |
\( 2^{34} \cdot 11^{2} \) |
$10.68742$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$53.85697278$ |
2.044014604 |
\( \frac{22702786953003143813}{7929856} a^{3} + \frac{30928910991628706537}{7929856} a^{2} - \frac{8873274696408279559}{3964928} a - \frac{19220593607745698479}{7929856} \) |
\( \bigl[a\) , \( a^{2} - a - 2\) , \( a^{2} - a - 2\) , \( 27 a^{3} + a^{2} - 122 a - 77\) , \( -103 a^{3} - 32 a^{2} + 512 a + 327\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(27a^{3}+a^{2}-122a-77\right){x}-103a^{3}-32a^{2}+512a+327$ |
*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.
