| 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 |
| 961.9-a1 |
961.9-a |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.9 |
\( 31^{2} \) |
\( 31^{2} \) |
$7.07221$ |
$(2a^3-8a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.028323527$ |
$771.0390481$ |
2.604398559 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} + 2 a - 1\) , \( a^{3} - 3 a + 1\) , \( a^{3} + a^{2} - 3 a\) , \( -a^{2} - a + 3\bigr] \) |
${y}^2+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(-a^{3}+2a-1\right){x}^{2}+\left(a^{3}+a^{2}-3a\right){x}-a^{2}-a+3$ |
| 961.9-a2 |
961.9-a |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.9 |
\( 31^{2} \) |
\( 31^{2} \) |
$7.07221$ |
$(2a^3-8a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.084970583$ |
$257.0130160$ |
2.604398559 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} - a^{2} + 4 a + 3\) , \( a^{3} + a^{2} - 3 a - 2\) , \( -2 a^{3} - a^{2} + 7 a + 3\) , \( -6908 a^{3} - 5713 a^{2} + 17194 a + 3781\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+3\right){x}^{2}+\left(-2a^{3}-a^{2}+7a+3\right){x}-6908a^{3}-5713a^{2}+17194a+3781$ |
| 961.9-b1 |
961.9-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.9 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(2a^3-8a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2^{2} \) |
$0.166825893$ |
$129.4181277$ |
2.574792903 |
\( 85995 a^{3} - 257985 a - 52515 \) |
\( \bigl[a^{2} - 2\) , \( a^{3} + a^{2} - 2 a - 2\) , \( a^{2} - 2\) , \( 511247 a^{3} + 422844 a^{2} - 1272415 a - 279818\) , \( 359892277 a^{3} + 297663641 a^{2} - 895710571 a - 196975569\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-2a-2\right){x}^{2}+\left(511247a^{3}+422844a^{2}-1272415a-279818\right){x}+359892277a^{3}+297663641a^{2}-895710571a-196975569$ |
| 961.9-b2 |
961.9-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.9 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(2a^3-8a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$0.333651787$ |
$129.4181277$ |
2.574792903 |
\( -16554983445 a^{3} + 49664950335 a + 10231546590 \) |
\( \bigl[a^{2} - 2\) , \( a^{3} + a^{2} - 2 a - 2\) , \( a^{2} - 2\) , \( -3449313 a^{3} - 2852961 a^{2} + 8584610 a + 1887842\) , \( 3646894359 a^{3} + 3016313673 a^{2} - 9076497964 a - 1996011239\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-2a-2\right){x}^{2}+\left(-3449313a^{3}-2852961a^{2}+8584610a+1887842\right){x}+3646894359a^{3}+3016313673a^{2}-9076497964a-1996011239$ |
| 961.9-b3 |
961.9-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.9 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(2a^3-8a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$0.556086312$ |
$77.65087667$ |
2.574792903 |
\( -16554983445 a^{3} + 49664950335 a + 10231546590 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( a^{3} - 3 a + 1\) , \( 62 a^{3} + 19 a^{2} - 234 a - 62\) , \( 439 a^{3} + 129 a^{2} - 1605 a - 346\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-1\right){x}^{2}+\left(62a^{3}+19a^{2}-234a-62\right){x}+439a^{3}+129a^{2}-1605a-346$ |
| 961.9-b4 |
961.9-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.9 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(2a^3-8a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2^{2} \) |
$0.278043156$ |
$77.65087667$ |
2.574792903 |
\( 85995 a^{3} - 257985 a - 52515 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( a^{3} - 3 a + 1\) , \( 7 a^{3} + 4 a^{2} - 24 a - 2\) , \( 10 a^{3} + 5 a^{2} - 36 a - 11\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-1\right){x}^{2}+\left(7a^{3}+4a^{2}-24a-2\right){x}+10a^{3}+5a^{2}-36a-11$ |
| 961.9-b5 |
961.9-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.9 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(2a^3-8a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$0.111217262$ |
$388.2543833$ |
2.574792903 |
\( 16554983445 a^{3} - 49664950335 a + 26786530035 \) |
\( \bigl[a^{3} + a^{2} - 2 a - 1\) , \( -a + 1\) , \( a^{3} + a^{2} - 3 a - 2\) , \( -917 a^{3} - 463 a^{2} + 2463 a - 272\) , \( 16332 a^{3} + 10420 a^{2} - 42556 a - 853\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-1\right){x}{y}+\left(a^{3}+a^{2}-3a-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-917a^{3}-463a^{2}+2463a-272\right){x}+16332a^{3}+10420a^{2}-42556a-853$ |
| 961.9-b6 |
961.9-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.9 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(2a^3-8a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2^{2} \) |
$0.055608631$ |
$388.2543833$ |
2.574792903 |
\( -85995 a^{3} + 257985 a - 138510 \) |
\( \bigl[a^{3} + a^{2} - 2 a - 1\) , \( -a + 1\) , \( a^{3} + a^{2} - 3 a - 2\) , \( -57 a^{3} - 28 a^{2} + 153 a - 17\) , \( 195 a^{3} + 119 a^{2} - 512 a + 4\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-1\right){x}{y}+\left(a^{3}+a^{2}-3a-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-57a^{3}-28a^{2}+153a-17\right){x}+195a^{3}+119a^{2}-512a+4$ |
| 961.9-b7 |
961.9-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.9 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(2a^3-8a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2^{2} \) |
$0.834129469$ |
$25.88362555$ |
2.574792903 |
\( -85995 a^{3} + 257985 a - 138510 \) |
\( \bigl[a + 1\) , \( -a^{3} + a^{2} + 2 a - 1\) , \( 1\) , \( -263712 a^{3} - 218103 a^{2} + 656349 a + 144337\) , \( -101761255 a^{3} - 84165827 a^{2} + 253266390 a + 55695772\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+2a-1\right){x}^{2}+\left(-263712a^{3}-218103a^{2}+656349a+144337\right){x}-101761255a^{3}-84165827a^{2}+253266390a+55695772$ |
| 961.9-b8 |
961.9-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.9 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(2a^3-8a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$1.668258938$ |
$25.88362555$ |
2.574792903 |
\( 16554983445 a^{3} - 49664950335 a + 26786530035 \) |
\( \bigl[a + 1\) , \( -a^{3} + a^{2} + 2 a - 1\) , \( 1\) , \( -4224272 a^{3} - 3493908 a^{2} + 10513374 a + 2311997\) , \( -6487761039 a^{3} - 5365968522 a^{2} + 16146931708 a + 3550869173\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+2a-1\right){x}^{2}+\left(-4224272a^{3}-3493908a^{2}+10513374a+2311997\right){x}-6487761039a^{3}-5365968522a^{2}+16146931708a+3550869173$ |
*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.
