| 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 |
| 11.2-a1 |
11.2-a |
$4$ |
$15$ |
5.5.70601.1 |
$5$ |
$[5, 0]$ |
11.2 |
\( 11 \) |
\( 11^{5} \) |
$30.17750$ |
$(a^4-6a^2-3a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.2 |
$25$ |
\( 5 \) |
$1$ |
$3.026255417$ |
1.42367273 |
\( \frac{2419492877372724132926}{161051} a^{4} - \frac{4448947514199103019972}{161051} a^{3} - \frac{8274801601497550822783}{161051} a^{2} + \frac{11839841097468876790712}{161051} a - \frac{2833690910755968015329}{161051} \) |
\( \bigl[2 a^{4} - 2 a^{3} - 9 a^{2} + 4 a + 2\) , \( -a^{4} + 6 a^{2} + 2 a - 2\) , \( 3 a^{4} - 2 a^{3} - 15 a^{2} + 2 a + 6\) , \( -73 a^{4} + 29 a^{3} + 403 a^{2} + 60 a - 263\) , \( -329 a^{4} + 126 a^{3} + 1875 a^{2} + 263 a - 1434\bigr] \) |
${y}^2+\left(2a^{4}-2a^{3}-9a^{2}+4a+2\right){x}{y}+\left(3a^{4}-2a^{3}-15a^{2}+2a+6\right){y}={x}^{3}+\left(-a^{4}+6a^{2}+2a-2\right){x}^{2}+\left(-73a^{4}+29a^{3}+403a^{2}+60a-263\right){x}-329a^{4}+126a^{3}+1875a^{2}+263a-1434$ |
| 11.2-a2 |
11.2-a |
$4$ |
$15$ |
5.5.70601.1 |
$5$ |
$[5, 0]$ |
11.2 |
\( 11 \) |
\( 11^{15} \) |
$30.17750$ |
$(a^4-6a^2-3a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.2 |
$25$ |
\( 3 \cdot 5 \) |
$1$ |
$1.008751805$ |
1.42367273 |
\( -\frac{8205693349407499126130881172}{4177248169415651} a^{4} + \frac{5560827331783947650405359349}{4177248169415651} a^{3} + \frac{42820830025831262432181359037}{4177248169415651} a^{2} - \frac{2609321773754481822746200000}{4177248169415651} a - \frac{25458114683827965125866617974}{4177248169415651} \) |
\( \bigl[a^{4} - 6 a^{2} - a + 4\) , \( -1\) , \( 2 a^{4} - a^{3} - 11 a^{2} + 5\) , \( 111 a^{4} - 121 a^{3} - 452 a^{2} + 176 a - 91\) , \( 867 a^{4} - 1021 a^{3} - 3514 a^{2} + 1727 a - 475\bigr] \) |
${y}^2+\left(a^{4}-6a^{2}-a+4\right){x}{y}+\left(2a^{4}-a^{3}-11a^{2}+5\right){y}={x}^{3}-{x}^{2}+\left(111a^{4}-121a^{3}-452a^{2}+176a-91\right){x}+867a^{4}-1021a^{3}-3514a^{2}+1727a-475$ |
| 11.2-a3 |
11.2-a |
$4$ |
$15$ |
5.5.70601.1 |
$5$ |
$[5, 0]$ |
11.2 |
\( 11 \) |
\( 11 \) |
$30.17750$ |
$(a^4-6a^2-3a+3)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$9457.048180$ |
1.42367273 |
\( \frac{503992}{11} a^{4} - \frac{87885}{11} a^{3} - \frac{2613846}{11} a^{2} - \frac{1093926}{11} a + \frac{621783}{11} \) |
\( \bigl[2 a^{4} - a^{3} - 10 a^{2} + 4\) , \( -a^{4} + a^{3} + 5 a^{2} - 3 a - 2\) , \( a^{4} - a^{3} - 4 a^{2} + a\) , \( a^{4} - 5 a^{2} - 4 a + 2\) , \( 0\bigr] \) |
${y}^2+\left(2a^{4}-a^{3}-10a^{2}+4\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+a\right){y}={x}^{3}+\left(-a^{4}+a^{3}+5a^{2}-3a-2\right){x}^{2}+\left(a^{4}-5a^{2}-4a+2\right){x}$ |
| 11.2-a4 |
11.2-a |
$4$ |
$15$ |
5.5.70601.1 |
$5$ |
$[5, 0]$ |
11.2 |
\( 11 \) |
\( 11^{3} \) |
$30.17750$ |
$(a^4-6a^2-3a+3)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$3152.349393$ |
1.42367273 |
\( \frac{428200040}{1331} a^{4} + \frac{647357011}{1331} a^{3} - \frac{516344269}{1331} a^{2} - \frac{444536251}{1331} a + \frac{170612026}{1331} \) |
\( \bigl[3 a^{4} - 2 a^{3} - 15 a^{2} + a + 6\) , \( a^{3} - 2 a^{2} - 2 a + 3\) , \( a^{4} - a^{3} - 5 a^{2} + 3 a + 2\) , \( 3 a^{4} - 3 a^{3} - 16 a^{2} + 2 a + 13\) , \( -4 a^{4} + 4 a^{3} + 19 a^{2} - 3 a - 10\bigr] \) |
${y}^2+\left(3a^{4}-2a^{3}-15a^{2}+a+6\right){x}{y}+\left(a^{4}-a^{3}-5a^{2}+3a+2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-2a+3\right){x}^{2}+\left(3a^{4}-3a^{3}-16a^{2}+2a+13\right){x}-4a^{4}+4a^{3}+19a^{2}-3a-10$ |
| 11.2-b1 |
11.2-b |
$2$ |
$5$ |
5.5.70601.1 |
$5$ |
$[5, 0]$ |
11.2 |
\( 11 \) |
\( - 11^{2} \) |
$30.17750$ |
$(a^4-6a^2-3a+3)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$3139.012854$ |
0.945099765 |
\( \frac{403291}{121} a^{4} - \frac{737596}{121} a^{3} - \frac{1377634}{121} a^{2} + \frac{1964751}{121} a - \frac{468458}{121} \) |
\( \bigl[a^{4} - a^{3} - 5 a^{2} + 3 a + 3\) , \( a^{3} - 2 a^{2} - 3 a + 3\) , \( 2 a^{4} - a^{3} - 10 a^{2} - a + 3\) , \( -a^{2} + 5\) , \( 2 a^{4} - 12 a^{2} - 4 a + 6\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-5a^{2}+3a+3\right){x}{y}+\left(2a^{4}-a^{3}-10a^{2}-a+3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+3\right){x}^{2}+\left(-a^{2}+5\right){x}+2a^{4}-12a^{2}-4a+6$ |
| 11.2-b2 |
11.2-b |
$2$ |
$5$ |
5.5.70601.1 |
$5$ |
$[5, 0]$ |
11.2 |
\( 11 \) |
\( - 11^{10} \) |
$30.17750$ |
$(a^4-6a^2-3a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 2 \cdot 5 \) |
$1$ |
$1.004484113$ |
0.945099765 |
\( \frac{8307402570253990872274}{25937424601} a^{4} - \frac{1338226591259999802327}{25937424601} a^{3} - \frac{43272833962783971491091}{25937424601} a^{2} - \frac{18472674003862592516490}{25937424601} a + \frac{10404941001339037312187}{25937424601} \) |
\( \bigl[2 a^{4} - a^{3} - 10 a^{2} + 4\) , \( a^{4} - a^{3} - 5 a^{2} + a + 2\) , \( a^{4} - 6 a^{2} - 2 a + 3\) , \( -28 a^{4} + 4 a^{3} + 105 a^{2} - 20 a - 66\) , \( -172 a^{4} - 86 a^{3} + 571 a^{2} + 100 a - 379\bigr] \) |
${y}^2+\left(2a^{4}-a^{3}-10a^{2}+4\right){x}{y}+\left(a^{4}-6a^{2}-2a+3\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+a+2\right){x}^{2}+\left(-28a^{4}+4a^{3}+105a^{2}-20a-66\right){x}-172a^{4}-86a^{3}+571a^{2}+100a-379$ |
*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.
