| 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 |
| 961.8-a1 |
961.8-a |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.8 |
\( 31^{2} \) |
\( 31^{2} \) |
$7.07221$ |
$(-a^3+5a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.028323527$ |
$771.0390481$ |
2.604398559 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{3} - 3 a\) , \( -a + 2\) , \( a^{2} - a - 1\bigr] \) |
${y}^2+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(-a+2\right){x}+a^{2}-a-1$ |
| 961.8-a2 |
961.8-a |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.8 |
\( 31^{2} \) |
\( 31^{2} \) |
$7.07221$ |
$(-a^3+5a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.084970583$ |
$257.0130160$ |
2.604398559 |
\( 0 \) |
\( \bigl[0\) , \( a^{3} - 2 a + 1\) , \( a^{2} + a - 2\) , \( a^{3} + a^{2} - 3 a\) , \( -1139511 a^{3} - 942479 a^{2} + 2836049 a + 623674\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{3}-2a+1\right){x}^{2}+\left(a^{3}+a^{2}-3a\right){x}-1139511a^{3}-942479a^{2}+2836049a+623674$ |
| 961.8-b1 |
961.8-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.8 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-a^3+5a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2^{2} \) |
$0.055608631$ |
$388.2543833$ |
2.574792903 |
\( 85995 a^{3} - 257985 a - 52515 \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} + 2 a + 1\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 10406 a^{3} + 8606 a^{2} - 25904 a - 5696\) , \( -1039137 a^{3} - 859459 a^{2} + 2586237 a + 568739\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+2a+1\right){x}^{2}+\left(10406a^{3}+8606a^{2}-25904a-5696\right){x}-1039137a^{3}-859459a^{2}+2586237a+568739$ |
| 961.8-b2 |
961.8-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.8 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-a^3+5a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$0.111217262$ |
$388.2543833$ |
2.574792903 |
\( -16554983445 a^{3} + 49664950335 a + 10231546590 \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} + 2 a + 1\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -70144 a^{3} - 58044 a^{2} + 174511 a + 38379\) , \( -10627479 a^{3} - 8789766 a^{2} + 26450255 a + 5816663\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+2a+1\right){x}^{2}+\left(-70144a^{3}-58044a^{2}+174511a+38379\right){x}-10627479a^{3}-8789766a^{2}+26450255a+5816663$ |
| 961.8-b3 |
961.8-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.8 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-a^3+5a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$0.333651787$ |
$129.4181277$ |
2.574792903 |
\( 16554983445 a^{3} - 49664950335 a + 26786530035 \) |
\( \bigl[a^{2} + a - 1\) , \( -a^{2} - a + 1\) , \( a^{3} + a^{2} - 2 a - 2\) , \( -30 a^{3} - 5 a^{2} + 107 a - 63\) , \( -10 a^{3} - 73 a^{2} - 69 a + 248\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){x}{y}+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(-30a^{3}-5a^{2}+107a-63\right){x}-10a^{3}-73a^{2}-69a+248$ |
| 961.8-b4 |
961.8-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.8 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-a^3+5a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2^{2} \) |
$0.166825893$ |
$129.4181277$ |
2.574792903 |
\( -85995 a^{3} + 257985 a - 138510 \) |
\( \bigl[a^{2} + a - 1\) , \( -a^{2} - a + 1\) , \( a^{3} + a^{2} - 2 a - 2\) , \( -5 a^{3} + 12 a - 3\) , \( -2 a^{3} + 2 a + 2\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){x}{y}+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(-5a^{3}+12a-3\right){x}-2a^{3}+2a+2$ |
| 961.8-b5 |
961.8-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.8 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-a^3+5a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2^{2} \) |
$0.278043156$ |
$77.65087667$ |
2.574792903 |
\( -85995 a^{3} + 257985 a - 138510 \) |
\( \bigl[a^{3} - 2 a\) , \( a^{2} + a - 2\) , \( a^{2} + a - 1\) , \( -252118 a^{3} - 208524 a^{2} + 627478 a + 137990\) , \( 94587275 a^{3} + 78232276 a^{2} - 235411620 a - 51769332\bigr] \) |
${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(-252118a^{3}-208524a^{2}+627478a+137990\right){x}+94587275a^{3}+78232276a^{2}-235411620a-51769332$ |
| 961.8-b6 |
961.8-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.8 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-a^3+5a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$0.556086312$ |
$77.65087667$ |
2.574792903 |
\( 16554983445 a^{3} - 49664950335 a + 26786530035 \) |
\( \bigl[a^{3} - 2 a\) , \( a^{2} + a - 2\) , \( a^{2} + a - 1\) , \( -4038928 a^{3} - 3340559 a^{2} + 10052203 a + 2210575\) , \( 6056496764 a^{3} + 5009273452 a^{2} - 15073589041 a - 3314830541\bigr] \) |
${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(-4038928a^{3}-3340559a^{2}+10052203a+2210575\right){x}+6056496764a^{3}+5009273452a^{2}-15073589041a-3314830541$ |
| 961.8-b7 |
961.8-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.8 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-a^3+5a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$1.668258938$ |
$25.88362555$ |
2.574792903 |
\( -16554983445 a^{3} + 49664950335 a + 10231546590 \) |
\( \bigl[a + 1\) , \( -a^{3} - a^{2} + 3 a + 1\) , \( a^{2} - 1\) , \( -22 a^{3} - 7 a^{2} + 86 a - 51\) , \( 10 a^{3} + 29 a^{2} + 68 a - 169\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+1\right){x}^{2}+\left(-22a^{3}-7a^{2}+86a-51\right){x}+10a^{3}+29a^{2}+68a-169$ |
| 961.8-b8 |
961.8-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.8 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-a^3+5a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2^{2} \) |
$0.834129469$ |
$25.88362555$ |
2.574792903 |
\( 85995 a^{3} - 257985 a - 52515 \) |
\( \bigl[a + 1\) , \( -a^{3} - a^{2} + 3 a + 1\) , \( a^{2} - 1\) , \( 3 a^{3} - 2 a^{2} - 9 a + 9\) , \( -a^{3} + 4 a^{2} + 7 a - 14\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+1\right){x}^{2}+\left(3a^{3}-2a^{2}-9a+9\right){x}-a^{3}+4a^{2}+7a-14$ |
*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.
