| 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-a9 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2^{6} \cdot 11^{3} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$36$ |
\( 2 \) |
$1$ |
$2.463967476$ |
0.841626760 |
\( -\frac{5006065775663608963358467}{85184} a^{3} + \frac{12557622075673248231038785}{85184} a^{2} + \frac{540663280751702976925425}{42592} a - \frac{6637226716732893866394967}{85184} \) |
\( \bigl[a^{2} - 1\) , \( 0\) , \( a^{3} - 4 a - 1\) , \( 210 a^{3} - 374 a^{2} - 201 a - 214\) , \( 1496 a^{3} - 6224 a^{2} + 4609 a + 4096\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(210a^{3}-374a^{2}-201a-214\right){x}+1496a^{3}-6224a^{2}+4609a+4096$ |
| 176.2-f1 |
176.2-f |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
176.2 |
\( 2^{4} \cdot 11 \) |
\( - 2^{18} \cdot 11^{3} \) |
$8.98702$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$2.026052041$ |
$129.5923790$ |
2.491220239 |
\( -\frac{5006065775663608963358467}{85184} a^{3} + \frac{12557622075673248231038785}{85184} a^{2} + \frac{540663280751702976925425}{42592} a - \frac{6637226716732893866394967}{85184} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + 2 a^{2} + a - 3\) , \( a^{3} - 4 a\) , \( -725 a^{3} + 1148 a^{2} + 2368 a - 2515\) , \( 20645 a^{3} - 33727 a^{2} - 62573 a + 64056\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+a-3\right){x}^{2}+\left(-725a^{3}+1148a^{2}+2368a-2515\right){x}+20645a^{3}-33727a^{2}-62573a+64056$ |
| 242.1-e10 |
242.1-e |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
242.1 |
\( 2 \cdot 11^{2} \) |
\( - 2^{6} \cdot 11^{9} \) |
$9.35198$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$12.55415428$ |
2.858780215 |
\( -\frac{5006065775663608963358467}{85184} a^{3} + \frac{12557622075673248231038785}{85184} a^{2} + \frac{540663280751702976925425}{42592} a - \frac{6637226716732893866394967}{85184} \) |
\( \bigl[a + 1\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{2} - a - 2\) , \( -1447 a^{3} + 4370 a^{2} + 2445 a - 11248\) , \( -72557 a^{3} + 186607 a^{2} + 152304 a - 443880\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(-1447a^{3}+4370a^{2}+2445a-11248\right){x}-72557a^{3}+186607a^{2}+152304a-443880$ |
| 704.3-d4 |
704.3-d |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
704.3 |
\( 2^{6} \cdot 11 \) |
\( - 2^{24} \cdot 11^{3} \) |
$10.68742$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \cdot 3 \) |
$3.878508824$ |
$1.945890345$ |
3.437213027 |
\( -\frac{5006065775663608963358467}{85184} a^{3} + \frac{12557622075673248231038785}{85184} a^{2} + \frac{540663280751702976925425}{42592} a - \frac{6637226716732893866394967}{85184} \) |
\( \bigl[a^{2} - 2\) , \( a + 1\) , \( a^{3} - 4 a\) , \( -542 a^{3} - 397 a^{2} + 210 a - 660\) , \( 18600 a^{3} + 29412 a^{2} - 17792 a - 29952\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-542a^{3}-397a^{2}+210a-660\right){x}+18600a^{3}+29412a^{2}-17792a-29952$ |
| 704.3-q2 |
704.3-q |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
704.3 |
\( 2^{6} \cdot 11 \) |
\( - 2^{24} \cdot 11^{3} \) |
$10.68742$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$8.976162131$ |
2.044014604 |
\( -\frac{5006065775663608963358467}{85184} a^{3} + \frac{12557622075673248231038785}{85184} a^{2} + \frac{540663280751702976925425}{42592} a - \frac{6637226716732893866394967}{85184} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{3} + 2 a^{2} + a - 2\) , \( a\) , \( -108 a^{3} - 94 a^{2} + 1028 a - 848\) , \( 1989 a^{3} + 506 a^{2} - 19238 a + 13851\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+2a^{2}+a-2\right){x}^{2}+\left(-108a^{3}-94a^{2}+1028a-848\right){x}+1989a^{3}+506a^{2}-19238a+13851$ |
*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.
