| 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 |
| 961.10-a1 |
961.10-a |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.10 |
\( 31^{2} \) |
\( 31^{2} \) |
$7.07221$ |
$(-2a^3-2a^2+6a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.084970583$ |
$257.0130160$ |
2.604398559 |
\( 0 \) |
\( \bigl[0\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 1\) , \( a^{3} + a^{2} - 2 a\) , \( -a^{3} - a^{2} + 2 a\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(a^{3}+a^{2}-2a\right){x}-a^{3}-a^{2}+2a$ |
| 961.10-a2 |
961.10-a |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.10 |
\( 31^{2} \) |
\( 31^{2} \) |
$7.07221$ |
$(-2a^3-2a^2+6a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.028323527$ |
$771.0390481$ |
2.604398559 |
\( 0 \) |
\( \bigl[0\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - 3 a\) , \( -a + 2\) , \( -2365813 a^{3} - 1956743 a^{2} + 5888106 a + 1294853\bigr] \) |
${y}^2+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(-a+2\right){x}-2365813a^{3}-1956743a^{2}+5888106a+1294853$ |
| 961.10-b1 |
961.10-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.10 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-2a^3-2a^2+6a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2^{2} \) |
$0.166825893$ |
$129.4181277$ |
2.574792903 |
\( -85995 a^{3} + 257985 a - 138510 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} - 3\) , \( a + 1\) , \( -826861 a^{3} - 683889 a^{2} + 2057918 a + 452559\) , \( 563758230 a^{3} + 466279311 a^{2} - 1403098227 a - 308555106\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-826861a^{3}-683889a^{2}+2057918a+452559\right){x}+563758230a^{3}+466279311a^{2}-1403098227a-308555106$ |
| 961.10-b2 |
961.10-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.10 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-2a^3-2a^2+6a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$0.333651787$ |
$129.4181277$ |
2.574792903 |
\( 16554983445 a^{3} - 49664950335 a + 26786530035 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} - 3\) , \( a + 1\) , \( -13246251 a^{3} - 10955859 a^{2} + 32967653 a + 7249914\) , \( 36003341430 a^{3} + 29778036624 a^{2} - 89606185527 a - 19705281836\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-13246251a^{3}-10955859a^{2}+32967653a+7249914\right){x}+36003341430a^{3}+29778036624a^{2}-89606185527a-19705281836$ |
| 961.10-b3 |
961.10-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.10 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-2a^3-2a^2+6a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2^{2} \) |
$0.055608631$ |
$388.2543833$ |
2.574792903 |
\( 85995 a^{3} - 257985 a - 52515 \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 6 a^{3} + 4 a^{2} - 15 a - 2\) , \( 15 a^{3} + 13 a^{2} - 37 a - 9\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(6a^{3}+4a^{2}-15a-2\right){x}+15a^{3}+13a^{2}-37a-9$ |
| 961.10-b4 |
961.10-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.10 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-2a^3-2a^2+6a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$0.111217262$ |
$388.2543833$ |
2.574792903 |
\( -16554983445 a^{3} + 49664950335 a + 10231546590 \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -34 a^{3} - 41 a^{2} + 90 a + 18\) , \( 123 a^{3} + 135 a^{2} - 322 a - 72\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-34a^{3}-41a^{2}+90a+18\right){x}+123a^{3}+135a^{2}-322a-72$ |
| 961.10-b5 |
961.10-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.10 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-2a^3-2a^2+6a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2^{2} \) |
$0.834129469$ |
$25.88362555$ |
2.574792903 |
\( 85995 a^{3} - 257985 a - 52515 \) |
\( \bigl[a^{3} - 2 a\) , \( a^{3} - 2 a\) , \( 1\) , \( 34159 a^{3} + 28236 a^{2} - 85053 a - 18703\) , \( -6148692 a^{3} - 5085587 a^{2} + 15302918 a + 3365266\bigr] \) |
${y}^2+\left(a^{3}-2a\right){x}{y}+{y}={x}^{3}+\left(a^{3}-2a\right){x}^{2}+\left(34159a^{3}+28236a^{2}-85053a-18703\right){x}-6148692a^{3}-5085587a^{2}+15302918a+3365266$ |
| 961.10-b6 |
961.10-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.10 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-2a^3-2a^2+6a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$1.668258938$ |
$25.88362555$ |
2.574792903 |
\( -16554983445 a^{3} + 49664950335 a + 10231546590 \) |
\( \bigl[a^{3} - 2 a\) , \( a^{3} - 2 a\) , \( 1\) , \( -229681 a^{3} - 190239 a^{2} + 571047 a + 125597\) , \( -63266176 a^{3} - 52330761 a^{2} + 157450333 a + 34625138\bigr] \) |
${y}^2+\left(a^{3}-2a\right){x}{y}+{y}={x}^{3}+\left(a^{3}-2a\right){x}^{2}+\left(-229681a^{3}-190239a^{2}+571047a+125597\right){x}-63266176a^{3}-52330761a^{2}+157450333a+34625138$ |
| 961.10-b7 |
961.10-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.10 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-2a^3-2a^2+6a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2^{2} \) |
$0.278043156$ |
$77.65087667$ |
2.574792903 |
\( -85995 a^{3} + 257985 a - 138510 \) |
\( \bigl[a + 1\) , \( -a^{3} + 4 a\) , \( a^{3} - 2 a\) , \( -7 a^{3} - 3 a^{2} + 20 a + 4\) , \( -9 a^{3} - 9 a^{2} + 27 a + 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{3}+4a\right){x}^{2}+\left(-7a^{3}-3a^{2}+20a+4\right){x}-9a^{3}-9a^{2}+27a+6$ |
| 961.10-b8 |
961.10-b |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.10 |
\( 31^{2} \) |
\( 31^{6} \) |
$7.07221$ |
$(-2a^3-2a^2+6a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 2 \) |
$0.556086312$ |
$77.65087667$ |
2.574792903 |
\( 16554983445 a^{3} - 49664950335 a + 26786530035 \) |
\( \bigl[a + 1\) , \( -a^{3} + 4 a\) , \( a^{3} - 2 a\) , \( -47 a^{3} - 48 a^{2} + 125 a + 24\) , \( -314 a^{3} - 286 a^{2} + 788 a + 178\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{3}+4a\right){x}^{2}+\left(-47a^{3}-48a^{2}+125a+24\right){x}-314a^{3}-286a^{2}+788a+178$ |
*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.
