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 |
13.1-a1 |
13.1-a |
$3$ |
$9$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
13.1 |
\( 13 \) |
\( 13 \) |
$12.25734$ |
$(-a^2+2a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1.079208123$ |
$441.3551652$ |
2.126649835 |
\( -\frac{7837506146388204859}{13} a^{3} - \frac{20303559893752613332}{13} a^{2} - \frac{5572637861243472815}{13} a + \frac{9076222771033149887}{13} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - a^{2} - 5 a\) , \( 0\) , \( -17 a^{3} - 3 a^{2} + 34 a - 30\) , \( 45 a^{3} + 106 a^{2} + 25 a - 23\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-17a^{3}-3a^{2}+34a-30\right){x}+45a^{3}+106a^{2}+25a-23$ |
13.1-a2 |
13.1-a |
$3$ |
$9$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
13.1 |
\( 13 \) |
\( 13^{3} \) |
$12.25734$ |
$(-a^2+2a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$0.359736041$ |
$441.3551652$ |
2.126649835 |
\( \frac{1140361876}{2197} a^{3} - \frac{2035650720}{2197} a^{2} - \frac{6076580550}{2197} a + \frac{3481904573}{2197} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - a^{2} - 5 a\) , \( 0\) , \( -2 a^{3} + 2 a^{2} + 9 a - 5\) , \( -14 a^{3} + 17 a^{2} + 64 a - 35\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-2a^{3}+2a^{2}+9a-5\right){x}-14a^{3}+17a^{2}+64a-35$ |
13.1-a3 |
13.1-a |
$3$ |
$9$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
13.1 |
\( 13 \) |
\( 13 \) |
$12.25734$ |
$(-a^2+2a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1.079208123$ |
$5.448829201$ |
2.126649835 |
\( \frac{54952780378700013073}{13} a^{3} - \frac{66356600984288283278}{13} a^{2} - \frac{249589735669688957882}{13} a + \frac{136526345323237627463}{13} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a\) , \( 0\) , \( 21 a^{3} + 8 a^{2} - 26 a - 9\) , \( 562 a^{3} + 1305 a^{2} + 277 a - 581\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}={x}^{3}+a{x}^{2}+\left(21a^{3}+8a^{2}-26a-9\right){x}+562a^{3}+1305a^{2}+277a-581$ |
13.1-b1 |
13.1-b |
$3$ |
$9$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
13.1 |
\( 13 \) |
\( 13^{3} \) |
$12.25734$ |
$(-a^2+2a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$0.277561838$ |
$403.6980252$ |
1.500860066 |
\( \frac{1140361876}{2197} a^{3} - \frac{2035650720}{2197} a^{2} - \frac{6076580550}{2197} a + \frac{3481904573}{2197} \) |
\( \bigl[a^{3} - 4 a - 2\) , \( a\) , \( a^{3} - 5 a - 1\) , \( 9 a^{3} - 18 a^{2} - 22 a + 14\) , \( -46 a^{3} + 74 a^{2} + 104 a - 66\bigr] \) |
${y}^2+\left(a^{3}-4a-2\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+a{x}^{2}+\left(9a^{3}-18a^{2}-22a+14\right){x}-46a^{3}+74a^{2}+104a-66$ |
13.1-b2 |
13.1-b |
$3$ |
$9$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
13.1 |
\( 13 \) |
\( 13 \) |
$12.25734$ |
$(-a^2+2a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$0.832685514$ |
$4.983926238$ |
1.500860066 |
\( -\frac{7837506146388204859}{13} a^{3} - \frac{20303559893752613332}{13} a^{2} - \frac{5572637861243472815}{13} a + \frac{9076222771033149887}{13} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( a^{3} - a^{2} - 5 a\) , \( a^{2} - a - 2\) , \( -130 a^{3} + 234 a^{2} + 353 a - 228\) , \( 477 a^{3} - 959 a^{2} - 995 a + 687\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-130a^{3}+234a^{2}+353a-228\right){x}+477a^{3}-959a^{2}-995a+687$ |
13.1-b3 |
13.1-b |
$3$ |
$9$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
13.1 |
\( 13 \) |
\( 13 \) |
$12.25734$ |
$(-a^2+2a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$0.832685514$ |
$403.6980252$ |
1.500860066 |
\( \frac{54952780378700013073}{13} a^{3} - \frac{66356600984288283278}{13} a^{2} - \frac{249589735669688957882}{13} a + \frac{136526345323237627463}{13} \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 4 a\) , \( 1\) , \( -220 a^{3} + 274 a^{2} + 1002 a - 570\) , \( 2785 a^{3} - 3453 a^{2} - 12765 a + 7022\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+4a\right){x}^{2}+\left(-220a^{3}+274a^{2}+1002a-570\right){x}+2785a^{3}-3453a^{2}-12765a+7022$ |
*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.