| 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-a1 |
2.2-a |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
2.2 |
\( 2 \) |
\( 2^{4} \) |
$1.78301$ |
$(-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$171.4772930$ |
0.602896961 |
\( \frac{200481}{16} a^{2} - \frac{325027}{8} a + \frac{233973}{8} \) |
\( \bigl[a^{2} + a - 3\) , \( -a - 1\) , \( a^{2} - 3\) , \( 971347319857519 a^{2} - 514150241464297 a - 4127391273410075\) , \( 22425156860537519167317 a^{2} - 11870007338272500196506 a - 95287643090023949417928\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(971347319857519a^{2}-514150241464297a-4127391273410075\right){x}+22425156860537519167317a^{2}-11870007338272500196506a-95287643090023949417928$ |
| 2.2-a2 |
2.2-a |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
2.2 |
\( 2 \) |
\( 2^{16} \) |
$1.78301$ |
$(-a+1)$ |
0 |
$\Z/16\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$171.4772930$ |
0.602896961 |
\( \frac{4108077233}{65536} a^{2} - \frac{5810733523}{32768} a + \frac{2294990397}{32768} \) |
\( \bigl[a^{2} + a - 3\) , \( -a\) , \( a\) , \( -608022265892705467026233 a^{2} + 321836266424514724186808 a + 2583572058090078441238362\) , \( -627730820434767876188418741768082419 a^{2} + 332268331114001902947589285516929027 a + 2667316476142693080354081387519995775\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-608022265892705467026233a^{2}+321836266424514724186808a+2583572058090078441238362\right){x}-627730820434767876188418741768082419a^{2}+332268331114001902947589285516929027a+2667316476142693080354081387519995775$ |
| 2.2-a3 |
2.2-a |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
2.2 |
\( 2 \) |
\( 2 \) |
$1.78301$ |
$(-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 1 \) |
$1$ |
$2.679332704$ |
0.602896961 |
\( \frac{89069256294443128323}{2} a^{2} - 125302919348047971845 a + 49111676741552886754 \) |
\( \bigl[a^{2} + a - 3\) , \( -a + 1\) , \( a^{2} - 3\) , \( -310 a^{2} + 882 a - 364\) , \( -6966 a^{2} + 19620 a - 7728\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-310a^{2}+882a-364\right){x}-6966a^{2}+19620a-7728$ |
| 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 |
\( \frac{32251567931279}{16} a^{2} + \frac{21655756210331}{8} a - \frac{13765401110581}{8} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{2} + 2\) , \( a\) , \( 469933301874402734133 a^{2} - 248743488236812365957 a - 1996812643211595590900\) , \( -8108483751005890586477675293469 a^{2} + 4291954889112701686585656294515 a + 34454087009163876680154219600725\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(469933301874402734133a^{2}-248743488236812365957a-1996812643211595590900\right){x}-8108483751005890586477675293469a^{2}+4291954889112701686585656294515a+34454087009163876680154219600725$ |
| 2.2-a5 |
2.2-a |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
2.2 |
\( 2 \) |
\( 2^{8} \) |
$1.78301$ |
$(-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$342.9545861$ |
0.602896961 |
\( \frac{130050481}{256} a^{2} + \frac{87514925}{128} a - \frac{54929731}{128} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{2} + 2\) , \( a\) , \( 30965253611602296633 a^{2} - 16390422144515520442 a - 131575714394521753905\) , \( -112172935746444460857621319378 a^{2} + 59374994732320022090359795265 a + 476638568561218377704701696552\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(30965253611602296633a^{2}-16390422144515520442a-131575714394521753905\right){x}-112172935746444460857621319378a^{2}+59374994732320022090359795265a+476638568561218377704701696552$ |
| 2.2-a6 |
2.2-a |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
2.2 |
\( 2 \) |
\( - 2^{2} \) |
$1.78301$ |
$(-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$85.73864654$ |
0.602896961 |
\( \frac{427}{4} a^{2} + \frac{383}{2} a - \frac{47}{2} \) |
\( \bigl[1\) , \( -a^{2} + 2\) , \( a^{2} - 2\) , \( -397482296655616833 a^{2} + 210393969927507918 a + 1688958139908216880\) , \( 849554077046907793046157006 a^{2} - 449683058696981763869697464 a - 3609874668113250400329290553\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-397482296655616833a^{2}+210393969927507918a+1688958139908216880\right){x}+849554077046907793046157006a^{2}-449683058696981763869697464a-3609874668113250400329290553$ |
| 2.2-a7 |
2.2-a |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
2.2 |
\( 2 \) |
\( -2 \) |
$1.78301$ |
$(-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 1 \) |
$1$ |
$2.679332704$ |
0.602896961 |
\( -\frac{10154621765056003}{2} a^{2} + 2687505089603077 a + 21574207961612782 \) |
\( \bigl[1\) , \( -a^{2} + a + 3\) , \( 1\) , \( 3795999062 a^{2} - 2009285242 a - 16129733497\) , \( 186167522763990 a^{2} - 98541556480493 a - 791051967859626\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(3795999062a^{2}-2009285242a-16129733497\right){x}+186167522763990a^{2}-98541556480493a-791051967859626$ |
| 2.2-a8 |
2.2-a |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
2.2 |
\( 2 \) |
\( 2^{2} \) |
$1.78301$ |
$(-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$21.43466163$ |
0.602896961 |
\( \frac{8525185037}{4} a^{2} - \frac{12150636535}{2} a + \frac{5069473275}{2} \) |
\( \bigl[1\) , \( -a^{2} + a + 3\) , \( 1\) , \( 237250327 a^{2} - 125580532 a - 1008109982\) , \( 2909065137552 a^{2} - 1539816409981 a - 12361026604032\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(237250327a^{2}-125580532a-1008109982\right){x}+2909065137552a^{2}-1539816409981a-12361026604032$ |
*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.
