| 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 |
| 31.2-a1 |
31.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{2} \) |
$4.60400$ |
$(-a^3+5a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$14.14001553$ |
0.843147624 |
\( \frac{934917206863109317}{961} a^{3} - \frac{1130359602108196159}{961} a^{2} - \frac{3503384511067738195}{961} a + \frac{4472090816549444332}{961} \) |
\( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( -15 a^{3} - 85 a^{2} + 385 a - 331\) , \( 187 a^{3} - 1626 a^{2} + 3723 a - 2518\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(-15a^{3}-85a^{2}+385a-331\right){x}+187a^{3}-1626a^{2}+3723a-2518$ |
| 31.2-a2 |
31.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{4} \) |
$4.60400$ |
$(-a^3+5a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$226.2402485$ |
0.843147624 |
\( -\frac{29296819153015039230435114609}{923521} a^{3} - \frac{9909977575470350951831216400}{923521} a^{2} + \frac{103925138003905943940922438084}{923521} a + \frac{21891704604902707693397138361}{923521} \) |
\( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( 270 a^{3} - 70 a^{2} - 655 a - 231\) , \( -2943 a^{3} + 1542 a^{2} + 6186 a + 1507\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(270a^{3}-70a^{2}-655a-231\right){x}-2943a^{3}+1542a^{2}+6186a+1507$ |
| 31.2-a3 |
31.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{8} \) |
$4.60400$ |
$(-a^3+5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$226.2402485$ |
0.843147624 |
\( -\frac{96230619458397643875589}{852891037441} a^{3} - \frac{32551084210297485030913}{852891037441} a^{2} + \frac{341360629620385537992211}{852891037441} a + \frac{71907211552629246852068}{852891037441} \) |
\( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( 15 a^{3} - 5 a^{2} - 25 a - 31\) , \( -47 a^{3} + 4 a^{2} + 161 a - 22\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(15a^{3}-5a^{2}-25a-31\right){x}-47a^{3}+4a^{2}+161a-22$ |
| 31.2-a4 |
31.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{4} \) |
$4.60400$ |
$(-a^3+5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$226.2402485$ |
0.843147624 |
\( \frac{12067174606324}{923521} a^{3} - \frac{14868316428812}{923521} a^{2} - \frac{45335200487011}{923521} a + \frac{58867032461318}{923521} \) |
\( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( -5 a^{2} + 20 a - 21\) , \( -21 a^{2} + 62 a - 44\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(-5a^{2}+20a-21\right){x}-21a^{2}+62a-44$ |
| 31.2-a5 |
31.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{2} \) |
$4.60400$ |
$(-a^3+5a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$226.2402485$ |
0.843147624 |
\( -\frac{1015616}{961} a^{3} + \frac{1884681}{961} a^{2} + \frac{4181756}{961} a - \frac{6745220}{961} \) |
\( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( -1\) , \( -a^{2} + 2 a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}-{x}-a^{2}+2a-1$ |
| 31.2-a6 |
31.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{16} \) |
$4.60400$ |
$(-a^3+5a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$14.14001553$ |
0.843147624 |
\( \frac{154534453074689854099149438193}{727423121747185263828481} a^{3} + \frac{52426907107682729263745202448}{727423121747185263828481} a^{2} - \frac{548494138857606744358415989700}{727423121747185263828481} a - \frac{115417448688070173417441163369}{727423121747185263828481} \) |
\( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( 60 a^{2} - 115 a + 9\) , \( -59 a^{3} - 14 a^{2} + 312 a - 163\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(60a^{2}-115a+9\right){x}-59a^{3}-14a^{2}+312a-163$ |
| 31.2-b1 |
31.2-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{4} \) |
$4.60400$ |
$(-a^3+5a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$2.959428063$ |
0.705864781 |
\( -\frac{29296819153015039230435114609}{923521} a^{3} - \frac{9909977575470350951831216400}{923521} a^{2} + \frac{103925138003905943940922438084}{923521} a + \frac{21891704604902707693397138361}{923521} \) |
\( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( 416 a^{3} + 76 a^{2} - 1585 a - 348\) , \( 6396 a^{3} + 1587 a^{2} - 23865 a - 5050\bigr] \) |
${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(416a^{3}+76a^{2}-1585a-348\right){x}+6396a^{3}+1587a^{2}-23865a-5050$ |
| 31.2-b2 |
31.2-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{2} \) |
$4.60400$ |
$(-a^3+5a-2)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$757.6135842$ |
0.705864781 |
\( \frac{934917206863109317}{961} a^{3} - \frac{1130359602108196159}{961} a^{2} - \frac{3503384511067738195}{961} a + \frac{4472090816549444332}{961} \) |
\( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( -24 a^{3} - 4 a^{2} + 90 a - 63\) , \( 49 a^{3} + 26 a^{2} - 250 a + 169\bigr] \) |
${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(-24a^{3}-4a^{2}+90a-63\right){x}+49a^{3}+26a^{2}-250a+169$ |
| 31.2-b3 |
31.2-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{8} \) |
$4.60400$ |
$(-a^3+5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$47.35084901$ |
0.705864781 |
\( -\frac{96230619458397643875589}{852891037441} a^{3} - \frac{32551084210297485030913}{852891037441} a^{2} + \frac{341360629620385537992211}{852891037441} a + \frac{71907211552629246852068}{852891037441} \) |
\( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( 26 a^{3} + 6 a^{2} - 100 a - 23\) , \( 93 a^{3} + 20 a^{2} - 358 a - 75\bigr] \) |
${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(26a^{3}+6a^{2}-100a-23\right){x}+93a^{3}+20a^{2}-358a-75$ |
| 31.2-b4 |
31.2-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{16} \) |
$4.60400$ |
$(-a^3+5a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$2.959428063$ |
0.705864781 |
\( \frac{154534453074689854099149438193}{727423121747185263828481} a^{3} + \frac{52426907107682729263745202448}{727423121747185263828481} a^{2} - \frac{548494138857606744358415989700}{727423121747185263828481} a - \frac{115417448688070173417441163369}{727423121747185263828481} \) |
\( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( 36 a^{3} + 16 a^{2} - 135 a - 18\) , \( 78 a^{3} - 11 a^{2} - 319 a - 36\bigr] \) |
${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(36a^{3}+16a^{2}-135a-18\right){x}+78a^{3}-11a^{2}-319a-36$ |
| 31.2-b5 |
31.2-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{4} \) |
$4.60400$ |
$(-a^3+5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$757.6135842$ |
0.705864781 |
\( \frac{12067174606324}{923521} a^{3} - \frac{14868316428812}{923521} a^{2} - \frac{45335200487011}{923521} a + \frac{58867032461318}{923521} \) |
\( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a^{3} + a^{2} - 6 a - 1\bigr] \) |
${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(a^{3}+a^{2}-5a-3\right){x}+a^{3}+a^{2}-6a-1$ |
| 31.2-b6 |
31.2-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{2} \) |
$4.60400$ |
$(-a^3+5a-2)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$757.6135842$ |
0.705864781 |
\( -\frac{1015616}{961} a^{3} + \frac{1884681}{961} a^{2} + \frac{4181756}{961} a - \frac{6745220}{961} \) |
\( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( a^{3} + a^{2} - 5 a + 2\) , \( a^{3} - 2 a\bigr] \) |
${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(a^{3}+a^{2}-5a+2\right){x}+a^{3}-2a$ |
*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.
