| 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 |
$2$ |
$3$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31 \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$2.427779910$ |
$3.844472540$ |
3.584089492 |
\( \frac{10077453354340891857}{31} a^{3} - \frac{17253366899431201762}{31} a^{2} - \frac{48179010153890934085}{31} a + \frac{14152214278679135626}{31} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 10 a\) , \( a^{3} - a^{2} - 6 a - 1\) , \( a\) , \( 37 a^{3} - 88 a^{2} - 96 a + 12\) , \( 300 a^{3} - 579 a^{2} - 1016 a - 309\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-10a\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-6a-1\right){x}^{2}+\left(37a^{3}-88a^{2}-96a+12\right){x}+300a^{3}-579a^{2}-1016a-309$ |
| 31.1-a2 |
31.1-a |
$2$ |
$3$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31^{3} \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$0.809259970$ |
$311.4022757$ |
3.584089492 |
\( \frac{8753568431}{29791} a^{3} - \frac{14981602414}{29791} a^{2} - \frac{41870763685}{29791} a + \frac{12298529441}{29791} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 10 a\) , \( a^{3} - a^{2} - 6 a - 1\) , \( a\) , \( 7 a^{3} - 13 a^{2} - 31 a + 12\) , \( 2 a^{3} - 8 a^{2} - 3 a + 24\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-10a\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-6a-1\right){x}^{2}+\left(7a^{3}-13a^{2}-31a+12\right){x}+2a^{3}-8a^{2}-3a+24$ |
| 31.1-b1 |
31.1-b |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31^{3} \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$167.6906818$ |
1.788706965 |
\( \frac{6213409058807}{29791} a^{3} - \frac{8633170361643}{29791} a^{2} - \frac{36475930058120}{29791} a + \frac{10416320461402}{29791} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 1\) , \( a^{3} - 2 a^{2} - 3 a + 1\) , \( 6 a^{3} - 15 a^{2} - 10 a + 3\) , \( -15 a^{3} + 41 a^{2} + 24 a - 11\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+\left(a^{3}-2a^{2}-3a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+1\right){x}^{2}+\left(6a^{3}-15a^{2}-10a+3\right){x}-15a^{3}+41a^{2}+24a-11$ |
| 31.1-c1 |
31.1-c |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31^{3} \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$4$ |
\( 1 \) |
$1$ |
$42.42759390$ |
1.810250442 |
\( \frac{41710021098}{29791} a^{3} - \frac{91407542611}{29791} a^{2} - \frac{228254055529}{29791} a + \frac{67866582299}{29791} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 1\) , \( -a^{3} + a^{2} + 5 a + 3\) , \( a + 1\) , \( -a^{3} - 3 a^{2} + 14 a + 8\) , \( 4 a^{3} - 13 a^{2} - a + 4\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+3\right){x}^{2}+\left(-a^{3}-3a^{2}+14a+8\right){x}+4a^{3}-13a^{2}-a+4$ |
| 31.1-d1 |
31.1-d |
$4$ |
$4$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31^{6} \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$492.1641382$ |
0.656221184 |
\( \frac{25290147472452905}{887503681} a^{3} + \frac{53256018737476624}{887503681} a^{2} + \frac{13664823639496413}{887503681} a - \frac{8138544755072017}{887503681} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( -a - 1\) , \( 0\) , \( 48 a^{3} - 128 a^{2} - 76 a + 28\) , \( 46 a^{3} - 121 a^{2} - 73 a + 26\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(48a^{3}-128a^{2}-76a+28\right){x}+46a^{3}-121a^{2}-73a+26$ |
| 31.1-d2 |
31.1-d |
$4$ |
$4$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31^{3} \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$246.0820691$ |
0.656221184 |
\( \frac{1248919832943643893409}{29791} a^{3} + \frac{2629997049374852711936}{29791} a^{2} + \frac{674771461425427867735}{29791} a - \frac{402122756071306404146}{29791} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( -a - 1\) , \( 0\) , \( 508 a^{3} - 1353 a^{2} - 821 a + 283\) , \( -12729 a^{3} + 33998 a^{2} + 19826 a - 7716\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(508a^{3}-1353a^{2}-821a+283\right){x}-12729a^{3}+33998a^{2}+19826a-7716$ |
| 31.1-d3 |
31.1-d |
$4$ |
$4$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( - 31^{12} \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$30.76025864$ |
0.656221184 |
\( -\frac{28701736354825279403647}{787662783788549761} a^{3} + \frac{83371699993555035031040}{787662783788549761} a^{2} + \frac{32340922878974239852151}{787662783788549761} a - \frac{13979020961410432424738}{787662783788549761} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 10 a\) , \( 2 a^{3} - 3 a^{2} - 10 a - 2\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( -242 a^{3} + 404 a^{2} + 1171 a - 287\) , \( -3175 a^{3} + 5490 a^{2} + 15103 a - 4738\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-10a\right){x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-10a-2\right){x}^{2}+\left(-242a^{3}+404a^{2}+1171a-287\right){x}-3175a^{3}+5490a^{2}+15103a-4738$ |
| 31.1-d4 |
31.1-d |
$4$ |
$4$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( - 31^{3} \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$984.3282764$ |
0.656221184 |
\( -\frac{95127527}{29791} a^{3} + \frac{13279296}{29791} a^{2} + \frac{424673785}{29791} a + \frac{282223088}{29791} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 2\) , \( a^{3} - a^{2} - 7 a - 2\) , \( a^{3} - a^{2} - 6 a - 1\) , \( a^{3} - 4 a^{2} - 7 a + 8\) , \( 8 a^{3} - 7 a^{2} - 31 a + 5\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+2\right){x}{y}+\left(a^{3}-a^{2}-6a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-7a-2\right){x}^{2}+\left(a^{3}-4a^{2}-7a+8\right){x}+8a^{3}-7a^{2}-31a+5$ |
*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.
