| 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 |
| 31.4-a1 |
31.4-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.4 |
\( 31 \) |
\( 31^{2} \) |
$4.60400$ |
$(2a^3-8a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$14.14001553$ |
0.843147624 |
\( -\frac{2065276808971305476}{961} a^{3} - \frac{698632890478410244}{961} a^{2} + \frac{7326190029022112587}{961} a + \frac{1543360584318567026}{961} \) |
\( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( -71 a^{3} + 341 a^{2} + 296 a - 1252\) , \( -1813 a^{3} + 4283 a^{2} + 7065 a - 16149\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{3}+3a-1\right){x}^{2}+\left(-71a^{3}+341a^{2}+296a-1252\right){x}-1813a^{3}+4283a^{2}+7065a-16149$ |
| 31.4-a2 |
31.4-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.4 |
\( 31 \) |
\( 31^{4} \) |
$4.60400$ |
$(2a^3-8a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$226.2402485$ |
0.843147624 |
\( \frac{19386841577544688278603898209}{923521} a^{3} + \frac{16034680544860826249617094257}{923521} a^{2} - \frac{48250547157163713883980478227}{923521} a - \frac{10610770058214958430895584744}{923521} \) |
\( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( -341 a^{3} + 156 a^{2} + 1091 a - 1022\) , \( 4485 a^{3} - 2644 a^{2} - 14997 a + 14364\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{3}+3a-1\right){x}^{2}+\left(-341a^{3}+156a^{2}+1091a-1022\right){x}+4485a^{3}-2644a^{2}-14997a+14364$ |
| 31.4-a3 |
31.4-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.4 |
\( 31 \) |
\( 31^{8} \) |
$4.60400$ |
$(2a^3-8a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$226.2402485$ |
0.843147624 |
\( \frac{63679535248100158844676}{852891037441} a^{3} + \frac{52668771245192606365444}{852891037441} a^{2} - \frac{158487521534002991503115}{852891037441} a - \frac{34852964110250777095970}{852891037441} \) |
\( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( -21 a^{3} + 21 a^{2} + 66 a - 102\) , \( 51 a^{3} + 19 a^{2} - 157 a - 1\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{3}+3a-1\right){x}^{2}+\left(-21a^{3}+21a^{2}+66a-102\right){x}+51a^{3}+19a^{2}-157a-1$ |
| 31.4-a4 |
31.4-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.4 |
\( 31 \) |
\( 31^{4} \) |
$4.60400$ |
$(2a^3-8a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$226.2402485$ |
0.843147624 |
\( -\frac{26935491035136}{923521} a^{3} - \frac{9133676668039}{923521} a^{2} + \frac{95674789534220}{923521} a + \frac{20462261904636}{923521} \) |
\( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( -6 a^{3} + 21 a^{2} + 21 a - 77\) , \( -21 a^{3} + 61 a^{2} + 84 a - 229\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{3}+3a-1\right){x}^{2}+\left(-6a^{3}+21a^{2}+21a-77\right){x}-21a^{3}+61a^{2}+84a-229$ |
| 31.4-a5 |
31.4-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.4 |
\( 31 \) |
\( 31^{2} \) |
$4.60400$ |
$(2a^3-8a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$226.2402485$ |
0.843147624 |
\( \frac{2900297}{961} a^{3} + \frac{1134908}{961} a^{2} - \frac{10585572}{961} a - \frac{2345377}{961} \) |
\( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( -a^{3} + a^{2} + a - 2\) , \( -a^{3} + a^{2} + 4 a - 6\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{3}+3a-1\right){x}^{2}+\left(-a^{3}+a^{2}+a-2\right){x}-a^{3}+a^{2}+4a-6$ |
| 31.4-a6 |
31.4-a |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.4 |
\( 31 \) |
\( 31^{16} \) |
$4.60400$ |
$(2a^3-8a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$14.14001553$ |
0.843147624 |
\( -\frac{102107545967007124835404235745}{727423121747185263828481} a^{3} - \frac{84890779633537182060967675121}{727423121747185263828481} a^{2} + \frac{253895730793338645242467504787}{727423121747185263828481} a + \frac{57110378827362524396580356024}{727423121747185263828481} \) |
\( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( 59 a^{3} - 114 a^{2} - 239 a + 418\) , \( 45 a^{3} + 134 a^{2} - 121 a - 414\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{3}+3a-1\right){x}^{2}+\left(59a^{3}-114a^{2}-239a+418\right){x}+45a^{3}+134a^{2}-121a-414$ |
| 31.4-b1 |
31.4-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.4 |
\( 31 \) |
\( 31^{2} \) |
$4.60400$ |
$(2a^3-8a+1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$757.6135842$ |
0.705864781 |
\( -\frac{2065276808971305476}{961} a^{3} - \frac{698632890478410244}{961} a^{2} + \frac{7326190029022112587}{961} a + \frac{1543360584318567026}{961} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 0\) , \( 20 a^{3} + 20 a^{2} - 55 a - 94\) , \( -92 a^{3} - 99 a^{2} + 265 a + 402\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}={x}^{3}+\left(a^{3}+a^{2}-3a-3\right){x}^{2}+\left(20a^{3}+20a^{2}-55a-94\right){x}-92a^{3}-99a^{2}+265a+402$ |
| 31.4-b2 |
31.4-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.4 |
\( 31 \) |
\( 31^{4} \) |
$4.60400$ |
$(2a^3-8a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$757.6135842$ |
0.705864781 |
\( -\frac{26935491035136}{923521} a^{3} - \frac{9133676668039}{923521} a^{2} + \frac{95674789534220}{923521} a + \frac{20462261904636}{923521} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 0\) , \( -4\) , \( -4 a^{3} - 4 a^{2} + 11 a + 10\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}={x}^{3}+\left(a^{3}+a^{2}-3a-3\right){x}^{2}-4{x}-4a^{3}-4a^{2}+11a+10$ |
| 31.4-b3 |
31.4-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.4 |
\( 31 \) |
\( 31^{2} \) |
$4.60400$ |
$(2a^3-8a+1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$757.6135842$ |
0.705864781 |
\( \frac{2900297}{961} a^{3} + \frac{1134908}{961} a^{2} - \frac{10585572}{961} a - \frac{2345377}{961} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}={x}^{3}+\left(a^{3}+a^{2}-3a-3\right){x}^{2}+{x}$ |
| 31.4-b4 |
31.4-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.4 |
\( 31 \) |
\( 31^{8} \) |
$4.60400$ |
$(2a^3-8a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$47.35084901$ |
0.705864781 |
\( \frac{63679535248100158844676}{852891037441} a^{3} + \frac{52668771245192606365444}{852891037441} a^{2} - \frac{158487521534002991503115}{852891037441} a - \frac{34852964110250777095970}{852891037441} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 0\) , \( -20 a^{3} - 20 a^{2} + 55 a + 6\) , \( -92 a^{3} - 85 a^{2} + 241 a + 58\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}={x}^{3}+\left(a^{3}+a^{2}-3a-3\right){x}^{2}+\left(-20a^{3}-20a^{2}+55a+6\right){x}-92a^{3}-85a^{2}+241a+58$ |
| 31.4-b5 |
31.4-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.4 |
\( 31 \) |
\( 31^{16} \) |
$4.60400$ |
$(2a^3-8a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$2.959428063$ |
0.705864781 |
\( -\frac{102107545967007124835404235745}{727423121747185263828481} a^{3} - \frac{84890779633537182060967675121}{727423121747185263828481} a^{2} + \frac{253895730793338645242467504787}{727423121747185263828481} a + \frac{57110378827362524396580356024}{727423121747185263828481} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 0\) , \( -20 a^{3} - 25 a^{2} + 45 a + 41\) , \( -93 a^{3} - 101 a^{2} + 280 a + 46\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}={x}^{3}+\left(a^{3}+a^{2}-3a-3\right){x}^{2}+\left(-20a^{3}-25a^{2}+45a+41\right){x}-93a^{3}-101a^{2}+280a+46$ |
| 31.4-b6 |
31.4-b |
$6$ |
$8$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
31.4 |
\( 31 \) |
\( 31^{4} \) |
$4.60400$ |
$(2a^3-8a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$2.959428063$ |
0.705864781 |
\( \frac{19386841577544688278603898209}{923521} a^{3} + \frac{16034680544860826249617094257}{923521} a^{2} - \frac{48250547157163713883980478227}{923521} a - \frac{10610770058214958430895584744}{923521} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 0\) , \( -340 a^{3} - 335 a^{2} + 945 a + 131\) , \( -5083 a^{3} - 4753 a^{2} + 13402 a + 2742\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}={x}^{3}+\left(a^{3}+a^{2}-3a-3\right){x}^{2}+\left(-340a^{3}-335a^{2}+945a+131\right){x}-5083a^{3}-4753a^{2}+13402a+2742$ |
*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.
