| 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-a4 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{48} \cdot 11^{6} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$4.927934952$ |
0.841626760 |
\( -\frac{1502268147076192175039019336779}{498650091216506454016} a^{3} + \frac{261719896030458847171340772313}{498650091216506454016} a^{2} + \frac{3112599543413066339294706687465}{249325045608253227008} a + \frac{3638403917729845697875388025793}{498650091216506454016} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( a^{2} - a - 2\) , \( 255 a^{3} - 345 a^{2} - 799 a + 451\) , \( -6627 a^{3} + 11701 a^{2} + 18463 a - 21549\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-3\right){x}^{2}+\left(255a^{3}-345a^{2}-799a+451\right){x}-6627a^{3}+11701a^{2}+18463a-21549$ |
| 176.2-f11 |
176.2-f |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
176.2 |
\( 2^{4} \cdot 11 \) |
\( 2^{60} \cdot 11^{6} \) |
$8.98702$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$4.052104082$ |
$4.049761845$ |
2.491220239 |
\( -\frac{1502268147076192175039019336779}{498650091216506454016} a^{3} + \frac{261719896030458847171340772313}{498650091216506454016} a^{2} + \frac{3112599543413066339294706687465}{249325045608253227008} a + \frac{3638403917729845697875388025793}{498650091216506454016} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{2} - a - 2\) , \( 34 a^{3} - 82 a^{2} - 132 a - 36\) , \( -783 a^{3} + 1758 a^{2} - 458 a - 1364\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(34a^{3}-82a^{2}-132a-36\right){x}-783a^{3}+1758a^{2}-458a-1364$ |
| 242.1-e9 |
242.1-e |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
242.1 |
\( 2 \cdot 11^{2} \) |
\( 2^{48} \cdot 11^{12} \) |
$9.35198$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{5} \cdot 3 \) |
$1$ |
$6.277077143$ |
2.858780215 |
\( -\frac{1502268147076192175039019336779}{498650091216506454016} a^{3} + \frac{261719896030458847171340772313}{498650091216506454016} a^{2} + \frac{3112599543413066339294706687465}{249325045608253227008} a + \frac{3638403917729845697875388025793}{498650091216506454016} \) |
\( \bigl[a + 1\) , \( a^{2} - a - 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 372 a^{3} - 456 a^{2} - 1195 a + 458\) , \( -10981 a^{3} + 16821 a^{2} + 32400 a - 25660\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(372a^{3}-456a^{2}-1195a+458\right){x}-10981a^{3}+16821a^{2}+32400a-25660$ |
| 704.3-d6 |
704.3-d |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
704.3 |
\( 2^{6} \cdot 11 \) |
\( 2^{66} \cdot 11^{6} \) |
$10.68742$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1.939254412$ |
$3.891780691$ |
3.437213027 |
\( -\frac{1502268147076192175039019336779}{498650091216506454016} a^{3} + \frac{261719896030458847171340772313}{498650091216506454016} a^{2} + \frac{3112599543413066339294706687465}{249325045608253227008} a + \frac{3638403917729845697875388025793}{498650091216506454016} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + 3 a + 2\) , \( a^{2} - a - 2\) , \( 19 a^{3} - 80 a^{2} - 108 a + 43\) , \( -279 a^{3} - 197 a^{2} + 851 a - 191\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+3a+2\right){x}^{2}+\left(19a^{3}-80a^{2}-108a+43\right){x}-279a^{3}-197a^{2}+851a-191$ |
| 704.3-q1 |
704.3-q |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
704.3 |
\( 2^{6} \cdot 11 \) |
\( 2^{66} \cdot 11^{6} \) |
$10.68742$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$17.95232426$ |
2.044014604 |
\( -\frac{1502268147076192175039019336779}{498650091216506454016} a^{3} + \frac{261719896030458847171340772313}{498650091216506454016} a^{2} + \frac{3112599543413066339294706687465}{249325045608253227008} a + \frac{3638403917729845697875388025793}{498650091216506454016} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a - 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 678 a^{3} - 1671 a^{2} - 249 a + 958\) , \( -157478 a^{3} + 395491 a^{2} + 32545 a - 207862\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(678a^{3}-1671a^{2}-249a+958\right){x}-157478a^{3}+395491a^{2}+32545a-207862$ |
*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.
