| 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 |
$6$ |
$8$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( - 31^{8} \) |
$3.69596$ |
$(a^2-2a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$2.037764955$ |
0.605445524 |
\( \frac{922217602952616201845297}{852891037441} a^{3} - \frac{1377735377068098493534675}{852891037441} a^{2} - \frac{2105069893978011989930528}{852891037441} a + \frac{1975984515916891718526428}{852891037441} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( -a^{3} + a^{2} + a - 1\) , \( 0\) , \( -115 a^{3} + 395 a^{2} + 120 a - 934\) , \( -1763 a^{3} + 4933 a^{2} + 2430 a - 10526\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}={x}^{3}+\left(-a^{3}+a^{2}+a-1\right){x}^{2}+\left(-115a^{3}+395a^{2}+120a-934\right){x}-1763a^{3}+4933a^{2}+2430a-10526$ |
| 31.2-a2 |
31.2-a |
$6$ |
$8$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{4} \) |
$3.69596$ |
$(a^2-2a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$32.60423929$ |
0.605445524 |
\( -\frac{228261852681788665473}{923521} a^{3} + \frac{537710579129834146433}{923521} a^{2} - \frac{44174852311159765250}{923521} a - \frac{168375141794635762583}{923521} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( -a^{3} + a^{2} + a - 1\) , \( 0\) , \( 5 a^{3} - 5 a^{2} + 10 a - 49\) , \( 26 a^{3} - 50 a^{2} + 56 a - 129\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}={x}^{3}+\left(-a^{3}+a^{2}+a-1\right){x}^{2}+\left(5a^{3}-5a^{2}+10a-49\right){x}+26a^{3}-50a^{2}+56a-129$ |
| 31.2-a3 |
31.2-a |
$6$ |
$8$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( -31 \) |
$3.69596$ |
$(a^2-2a-3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$260.8339143$ |
0.605445524 |
\( \frac{28347915782829458028831}{31} a^{3} + \frac{31049301650567183459041}{31} a^{2} - \frac{19986332354651992484290}{31} a - \frac{13529326186558988306439}{31} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( -a^{3} + a^{2} + a - 1\) , \( 0\) , \( -5 a^{3} + 5 a^{2} - 10 a - 39\) , \( 20 a^{3} + 4 a^{2} + 36 a + 83\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}={x}^{3}+\left(-a^{3}+a^{2}+a-1\right){x}^{2}+\left(-5a^{3}+5a^{2}-10a-39\right){x}+20a^{3}+4a^{2}+36a+83$ |
| 31.2-a4 |
31.2-a |
$6$ |
$8$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{2} \) |
$3.69596$ |
$(a^2-2a-3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$521.6678287$ |
0.605445524 |
\( \frac{8364588856509}{961} a^{3} + \frac{9182380516699}{961} a^{2} - \frac{5902734861706}{961} a - \frac{3999367873478}{961} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( -a^{3} + a^{2} + a - 1\) , \( 0\) , \( -4\) , \( a^{3} - a^{2} + 2 a - 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}={x}^{3}+\left(-a^{3}+a^{2}+a-1\right){x}^{2}-4{x}+a^{3}-a^{2}+2a-1$ |
| 31.2-a5 |
31.2-a |
$6$ |
$8$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( -31 \) |
$3.69596$ |
$(a^2-2a-3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$1043.335657$ |
0.605445524 |
\( -\frac{866395}{31} a^{3} - \frac{847461}{31} a^{2} + \frac{594342}{31} a + \frac{423731}{31} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( -a^{3} + a^{2} + a - 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}={x}^{3}+\left(-a^{3}+a^{2}+a-1\right){x}^{2}+{x}$ |
| 31.2-a6 |
31.2-a |
$6$ |
$8$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( - 31^{2} \) |
$3.69596$ |
$(a^2-2a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$2.037764955$ |
0.605445524 |
\( -\frac{372538687755177683555053524665073}{961} a^{3} + \frac{877579810257199416004108168477715}{961} a^{2} - \frac{72096326414315130602833573828064}{961} a - \frac{274799551345037658328417955957004}{961} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 2\) , \( -a^{3} + a^{2} + a - 1\) , \( 0\) , \( 205 a^{3} - 485 a^{2} + 60 a + 116\) , \( 3195 a^{3} - 7469 a^{2} + 618 a + 2116\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}={x}^{3}+\left(-a^{3}+a^{2}+a-1\right){x}^{2}+\left(205a^{3}-485a^{2}+60a+116\right){x}+3195a^{3}-7469a^{2}+618a+2116$ |
*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.
