| 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-a7 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{4} \cdot 11^{8} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$399.1627311$ |
0.841626760 |
\( \frac{158950703525365495771}{3429742096} a^{3} - \frac{266808256244810209657}{3429742096} a^{2} - \frac{227450195191568192265}{1714871048} a + \frac{468083456791050062719}{3429742096} \) |
\( \bigl[a^{2} - 1\) , \( a^{3} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -29 a^{3} - 37 a^{2} + 29 a + 24\) , \( 247 a^{3} + 345 a^{2} - 177 a - 202\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-3a-1\right){x}^{2}+\left(-29a^{3}-37a^{2}+29a+24\right){x}+247a^{3}+345a^{2}-177a-202$ |
| 176.2-f9 |
176.2-f |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
176.2 |
\( 2^{4} \cdot 11 \) |
\( 2^{16} \cdot 11^{8} \) |
$8.98702$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{2} \) |
$1.350701360$ |
$6.074642767$ |
2.491220239 |
\( \frac{158950703525365495771}{3429742096} a^{3} - \frac{266808256244810209657}{3429742096} a^{2} - \frac{227450195191568192265}{1714871048} a + \frac{468083456791050062719}{3429742096} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 2 a + 1\) , \( a^{3} - 3 a\) , \( 111 a^{3} - 291 a^{2} + 4 a + 152\) , \( 1232 a^{3} - 3270 a^{2} + 58 a + 1863\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+1\right){x}^{2}+\left(111a^{3}-291a^{2}+4a+152\right){x}+1232a^{3}-3270a^{2}+58a+1863$ |
| 242.1-e5 |
242.1-e |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
242.1 |
\( 2 \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{14} \) |
$9.35198$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{4} \) |
$1$ |
$9.415615714$ |
2.858780215 |
\( \frac{158950703525365495771}{3429742096} a^{3} - \frac{266808256244810209657}{3429742096} a^{2} - \frac{227450195191568192265}{1714871048} a + \frac{468083456791050062719}{3429742096} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{3} + 3 a + 1\) , \( a + 1\) , \( -80 a^{3} - 132 a^{2} + 78 a + 83\) , \( -1572 a^{3} - 2289 a^{2} + 1189 a + 1379\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(-80a^{3}-132a^{2}+78a+83\right){x}-1572a^{3}-2289a^{2}+1189a+1379$ |
| 704.3-d10 |
704.3-d |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
704.3 |
\( 2^{6} \cdot 11 \) |
\( 2^{22} \cdot 11^{8} \) |
$10.68742$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{5} \) |
$0.161604534$ |
$35.02602622$ |
3.437213027 |
\( \frac{158950703525365495771}{3429742096} a^{3} - \frac{266808256244810209657}{3429742096} a^{2} - \frac{227450195191568192265}{1714871048} a + \frac{468083456791050062719}{3429742096} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - a - 2\) , \( 0\) , \( -107 a^{3} + 150 a^{2} + 403 a - 363\) , \( 1045 a^{3} - 1548 a^{2} - 3544 a + 3243\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-107a^{3}+150a^{2}+403a-363\right){x}+1045a^{3}-1548a^{2}-3544a+3243$ |
| 704.3-q8 |
704.3-q |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
704.3 |
\( 2^{6} \cdot 11 \) |
\( 2^{22} \cdot 11^{8} \) |
$10.68742$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{4} \) |
$1$ |
$6.732121598$ |
2.044014604 |
\( \frac{158950703525365495771}{3429742096} a^{3} - \frac{266808256244810209657}{3429742096} a^{2} - \frac{227450195191568192265}{1714871048} a + \frac{468083456791050062719}{3429742096} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} + 3 a\) , \( a^{2} - a - 2\) , \( -92 a^{3} + 122 a^{2} + 281 a - 174\) , \( 175 a^{3} + 692 a^{2} - 1295 a - 2930\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(-92a^{3}+122a^{2}+281a-174\right){x}+175a^{3}+692a^{2}-1295a-2930$ |
*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.
