| 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-a10 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{12} \cdot 11^{24} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$4.927934952$ |
0.841626760 |
\( \frac{60659117288902836130412254108283}{40344505040107975043939700736} a^{3} - \frac{86300110386401779908460150637033}{40344505040107975043939700736} a^{2} - \frac{97109108201994369690495482301689}{20172252520053987521969850368} a + \frac{184521640869151181066006363788207}{40344505040107975043939700736} \) |
\( \bigl[a + 1\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - a - 1\) , \( -115 a^{3} + 187 a^{2} + 362 a - 350\) , \( 926 a^{3} - 1541 a^{2} - 2691 a + 2732\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-115a^{3}+187a^{2}+362a-350\right){x}+926a^{3}-1541a^{2}-2691a+2732$ |
| 176.2-f8 |
176.2-f |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
176.2 |
\( 2^{4} \cdot 11 \) |
\( 2^{24} \cdot 11^{24} \) |
$8.98702$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{2} \) |
$4.052104082$ |
$2.024880922$ |
2.491220239 |
\( \frac{60659117288902836130412254108283}{40344505040107975043939700736} a^{3} - \frac{86300110386401779908460150637033}{40344505040107975043939700736} a^{2} - \frac{97109108201994369690495482301689}{20172252520053987521969850368} a + \frac{184521640869151181066006363788207}{40344505040107975043939700736} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 2 a + 1\) , \( a^{3} - 3 a\) , \( 186 a^{3} - 391 a^{2} - 156 a + 112\) , \( 2246 a^{3} - 1192 a^{2} - 7927 a - 4203\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+1\right){x}^{2}+\left(186a^{3}-391a^{2}-156a+112\right){x}+2246a^{3}-1192a^{2}-7927a-4203$ |
| 242.1-e1 |
242.1-e |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
242.1 |
\( 2 \cdot 11^{2} \) |
\( 2^{12} \cdot 11^{30} \) |
$9.35198$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$3.138538571$ |
2.858780215 |
\( \frac{60659117288902836130412254108283}{40344505040107975043939700736} a^{3} - \frac{86300110386401779908460150637033}{40344505040107975043939700736} a^{2} - \frac{97109108201994369690495482301689}{20172252520053987521969850368} a + \frac{184521640869151181066006363788207}{40344505040107975043939700736} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( -a^{2} + 2 a + 2\) , \( a^{2} - 2\) , \( 12 a^{3} - 138 a^{2} + 409 a - 250\) , \( 1543 a^{3} - 2755 a^{2} - 3591 a + 4038\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(12a^{3}-138a^{2}+409a-250\right){x}+1543a^{3}-2755a^{2}-3591a+4038$ |
| 704.3-d9 |
704.3-d |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
704.3 |
\( 2^{6} \cdot 11 \) |
\( 2^{30} \cdot 11^{24} \) |
$10.68742$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{5} \cdot 3 \) |
$0.484813603$ |
$3.891780691$ |
3.437213027 |
\( \frac{60659117288902836130412254108283}{40344505040107975043939700736} a^{3} - \frac{86300110386401779908460150637033}{40344505040107975043939700736} a^{2} - \frac{97109108201994369690495482301689}{20172252520053987521969850368} a + \frac{184521640869151181066006363788207}{40344505040107975043939700736} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - a - 2\) , \( 0\) , \( -82 a^{3} + 130 a^{2} + 338 a - 413\) , \( 330 a^{3} - 1614 a^{2} - 101 a + 4953\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-82a^{3}+130a^{2}+338a-413\right){x}+330a^{3}-1614a^{2}-101a+4953$ |
| 704.3-q7 |
704.3-q |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
704.3 |
\( 2^{6} \cdot 11 \) |
\( 2^{30} \cdot 11^{24} \) |
$10.68742$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$2.244040532$ |
2.044014604 |
\( \frac{60659117288902836130412254108283}{40344505040107975043939700736} a^{3} - \frac{86300110386401779908460150637033}{40344505040107975043939700736} a^{2} - \frac{97109108201994369690495482301689}{20172252520053987521969850368} a + \frac{184521640869151181066006363788207}{40344505040107975043939700736} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} + 3 a\) , \( a^{2} - a - 2\) , \( 88 a^{3} + 127 a^{2} - 399 a - 594\) , \( -18184 a^{3} + 3633 a^{2} + 74417 a + 41570\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(88a^{3}+127a^{2}-399a-594\right){x}-18184a^{3}+3633a^{2}+74417a+41570$ |
*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.
