| 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 |
| 2.2-a4 |
2.2-a |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
2.2 |
\( 2 \) |
\( - 2^{4} \) |
$1.78301$ |
$(-a+1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$171.4772930$ |
0.602896961 |
\( \frac{32251567931279}{16} a^{2} + \frac{21655756210331}{8} a - \frac{13765401110581}{8} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{2} + 2\) , \( a\) , \( 469933301874402734133 a^{2} - 248743488236812365957 a - 1996812643211595590900\) , \( -8108483751005890586477675293469 a^{2} + 4291954889112701686585656294515 a + 34454087009163876680154219600725\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(469933301874402734133a^{2}-248743488236812365957a-1996812643211595590900\right){x}-8108483751005890586477675293469a^{2}+4291954889112701686585656294515a+34454087009163876680154219600725$ |
| 16.5-b7 |
16.5-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
16.5 |
\( 2^{4} \) |
\( - 2^{16} \) |
$2.52156$ |
$(-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$6.268450466$ |
0.705255777 |
\( \frac{32251567931279}{16} a^{2} + \frac{21655756210331}{8} a - \frac{13765401110581}{8} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a^{2} - 3\) , \( 64001862107159 a^{2} - 33877246772437 a - 271952906795303\) , \( 407543787751541638838 a^{2} - 215719683985399604073 a - 1731710829597940478002\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(64001862107159a^{2}-33877246772437a-271952906795303\right){x}+407543787751541638838a^{2}-215719683985399604073a-1731710829597940478002$ |
| 32.1-c7 |
32.1-c |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
32.1 |
\( 2^{5} \) |
\( - 2^{16} \) |
$2.83035$ |
$(a), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.264164657$ |
$30.31256343$ |
1.351372542 |
\( \frac{32251567931279}{16} a^{2} + \frac{21655756210331}{8} a - \frac{13765401110581}{8} \) |
\( \bigl[a\) , \( a - 1\) , \( a^{2} - 2\) , \( 60199270876066245436 a^{2} - 31864472186369026517 a - 255795162245334633462\) , \( 371768135944838763461274232817 a^{2} - 196783038319195503754985629908 a - 1579695057228071638067406296651\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(60199270876066245436a^{2}-31864472186369026517a-255795162245334633462\right){x}+371768135944838763461274232817a^{2}-196783038319195503754985629908a-1579695057228071638067406296651$ |
| 64.7-a2 |
64.7-a |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
64.7 |
\( 2^{6} \) |
\( - 2^{22} \) |
$3.17696$ |
$(-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$57.28306679$ |
1.611212134 |
\( \frac{32251567931279}{16} a^{2} + \frac{21655756210331}{8} a - \frac{13765401110581}{8} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} - a + 4\) , \( a^{2} - 3\) , \( 1698582370190 a^{2} - 899087811257 a - 7217515206526\) , \( -1762034893335112832 a^{2} + 932674283807826223 a + 7487133894869004409\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(1698582370190a^{2}-899087811257a-7217515206526\right){x}-1762034893335112832a^{2}+932674283807826223a+7487133894869004409$ |
| 64.7-d6 |
64.7-d |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
64.7 |
\( 2^{6} \) |
\( - 2^{22} \) |
$3.17696$ |
$(-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.241056509$ |
$4.691163453$ |
2.565930457 |
\( \frac{32251567931279}{16} a^{2} + \frac{21655756210331}{8} a - \frac{13765401110581}{8} \) |
\( \bigl[a + 1\) , \( a^{2} - 3\) , \( 0\) , \( 1698582370192 a^{2} - 899087811255 a - 7217515206531\) , \( 1762036591917483023 a^{2} - 932675182895637479 a - 7487141112384210938\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(1698582370192a^{2}-899087811255a-7217515206531\right){x}+1762036591917483023a^{2}-932675182895637479a-7487141112384210938$ |
| 242.3-b5 |
242.3-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( - 2^{4} \cdot 11^{6} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$10.69172574$ |
1.202913127 |
\( \frac{32251567931279}{16} a^{2} + \frac{21655756210331}{8} a - \frac{13765401110581}{8} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 20203900862562389334443 a^{2} - 10694259709833632274695 a - 85849214183464417082380\) , \( -2285799641956889659257421153575487 a^{2} + 1209911649340346379381685404090182 a + 9712683920681029332844506518986769\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(20203900862562389334443a^{2}-10694259709833632274695a-85849214183464417082380\right){x}-2285799641956889659257421153575487a^{2}+1209911649340346379381685404090182a+9712683920681029332844506518986769$ |
*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.
