| 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 |
| 31.1-a1 |
31.1-a |
$6$ |
$8$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( - 31^{2} \) |
$3.69596$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$2.037764955$ |
0.605445524 |
\( -\frac{684671267177826448818939504010641}{961} a^{3} + \frac{179630144675804716369884860197999}{961} a^{2} + \frac{2186516236280323395350819631179492}{961} a + \frac{928191236738378773754859996135354}{961} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( a^{3} - a^{2} - a + 1\) , \( 1\) , \( 396 a^{3} - 114 a^{2} - 1262 a - 535\) , \( 5675 a^{3} - 1590 a^{2} - 18077 a - 7642\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-a+1\right){x}^{2}+\left(396a^{3}-114a^{2}-1262a-535\right){x}+5675a^{3}-1590a^{2}-18077a-7642$ |
| 31.1-a2 |
31.1-a |
$6$ |
$8$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( -31 \) |
$3.69596$ |
$(a^3-4a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$260.8339143$ |
0.605445524 |
\( \frac{124454632427233023090075}{31} a^{3} - \frac{183851849860629664577947}{31} a^{2} - \frac{285618764065472969753522}{31} a + \frac{260769042758034501218421}{31} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( a^{3} - a^{2} - a + 1\) , \( 1\) , \( -24 a^{3} + 26 a^{2} + 68 a - 60\) , \( 61 a^{3} - 64 a^{2} - 197 a + 150\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-a+1\right){x}^{2}+\left(-24a^{3}+26a^{2}+68a-60\right){x}+61a^{3}-64a^{2}-197a+150$ |
| 31.1-a3 |
31.1-a |
$6$ |
$8$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31^{4} \) |
$3.69596$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$32.60423929$ |
0.605445524 |
\( -\frac{419511683908480280709}{923521} a^{3} + \frac{110062957460434799749}{923521} a^{2} + \frac{1339721925491697657614}{923521} a + \frac{568721206322809065061}{923521} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( a^{3} - a^{2} - a + 1\) , \( 1\) , \( 26 a^{3} - 24 a^{2} - 72 a - 30\) , \( 81 a^{3} - 76 a^{2} - 269 a - 102\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-a+1\right){x}^{2}+\left(26a^{3}-24a^{2}-72a-30\right){x}+81a^{3}-76a^{2}-269a-102$ |
| 31.1-a4 |
31.1-a |
$6$ |
$8$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31^{2} \) |
$3.69596$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$521.6678287$ |
0.605445524 |
\( \frac{36738001081029}{961} a^{3} - \frac{54284970454237}{961} a^{2} - \frac{84302445013370}{961} a + \frac{77014952470666}{961} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( a^{3} - a^{2} - a + 1\) , \( 1\) , \( a^{3} + a^{2} - 2 a - 5\) , \( a^{3} - 5 a - 2\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-a+1\right){x}^{2}+\left(a^{3}+a^{2}-2a-5\right){x}+a^{3}-5a-2$ |
| 31.1-a5 |
31.1-a |
$6$ |
$8$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( -31 \) |
$3.69596$ |
$(a^3-4a+1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$1043.335657$ |
0.605445524 |
\( -\frac{3718699}{31} a^{3} + \frac{5432555}{31} a^{2} + \frac{8575846}{31} a - \frac{7570141}{31} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( a^{3} - a^{2} - a + 1\) , \( 1\) , \( a^{3} + a^{2} - 2 a\) , \( a^{3} - a\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-a+1\right){x}^{2}+\left(a^{3}+a^{2}-2a\right){x}+a^{3}-a$ |
| 31.1-a6 |
31.1-a |
$6$ |
$8$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( - 31^{8} \) |
$3.69596$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$2.037764955$ |
0.605445524 |
\( \frac{206065140764354323915985}{852891037441} a^{3} + \frac{249452633351127967773393}{852891037441} a^{2} - \frac{151495593455929061592036}{852891037441} a - \frac{106721268617817034471018}{852891037441} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( a^{3} - a^{2} - a + 1\) , \( 1\) , \( 56 a^{3} - 334 a^{2} - 2 a + 75\) , \( 107 a^{3} - 3446 a^{2} + 603 a + 1078\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-a+1\right){x}^{2}+\left(56a^{3}-334a^{2}-2a+75\right){x}+107a^{3}-3446a^{2}+603a+1078$ |
*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.
