| 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 |
$2$ |
$2$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
31.1 |
\( 31 \) |
\( 31^{2} \) |
$91.58448$ |
$(-a^5+5a^3-5a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$494.6028815$ |
1.28494 |
\( -\frac{14206653307178619495}{961} a^{5} - \frac{5497977302510641703}{961} a^{4} + \frac{63407570964288615284}{961} a^{3} + \frac{31119698730479344528}{961} a^{2} - \frac{899039953178525727}{31} a - \frac{10242719039868077233}{961} \) |
\( \bigl[a^{5} - 5 a^{3} + a^{2} + 4 a - 2\) , \( a^{5} - 5 a^{3} + a^{2} + 3 a - 3\) , \( a^{5} - a^{4} - 5 a^{3} + 5 a^{2} + 4 a - 3\) , \( 17 a^{5} - 24 a^{4} - 67 a^{3} + 89 a^{2} + 21 a - 30\) , \( -32 a^{5} + 64 a^{4} + 107 a^{3} - 253 a^{2} + 71 a + 24\bigr] \) |
${y}^2+\left(a^{5}-5a^{3}+a^{2}+4a-2\right){x}{y}+\left(a^{5}-a^{4}-5a^{3}+5a^{2}+4a-3\right){y}={x}^{3}+\left(a^{5}-5a^{3}+a^{2}+3a-3\right){x}^{2}+\left(17a^{5}-24a^{4}-67a^{3}+89a^{2}+21a-30\right){x}-32a^{5}+64a^{4}+107a^{3}-253a^{2}+71a+24$ |
| 31.1-a2 |
31.1-a |
$2$ |
$2$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
31.1 |
\( 31 \) |
\( -31 \) |
$91.58448$ |
$(-a^5+5a^3-5a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$3956.823052$ |
1.28494 |
\( \frac{1524420667}{31} a^{5} + \frac{589876778}{31} a^{4} - \frac{6803520577}{31} a^{3} - \frac{3338194444}{31} a^{2} + 96463426 a + \frac{1098598601}{31} \) |
\( \bigl[2 a^{5} - a^{4} - 9 a^{3} + 4 a^{2} + 5 a - 2\) , \( 2 a^{5} - 3 a^{4} - 9 a^{3} + 13 a^{2} + 6 a - 7\) , \( a^{5} - 4 a^{3} + 2 a - 1\) , \( -3 a^{5} - a^{4} + 20 a^{3} - 5 a^{2} - 15 a + 3\) , \( -3 a^{5} + 5 a^{4} + a^{3} + 3 a^{2} - 6\bigr] \) |
${y}^2+\left(2a^{5}-a^{4}-9a^{3}+4a^{2}+5a-2\right){x}{y}+\left(a^{5}-4a^{3}+2a-1\right){y}={x}^{3}+\left(2a^{5}-3a^{4}-9a^{3}+13a^{2}+6a-7\right){x}^{2}+\left(-3a^{5}-a^{4}+20a^{3}-5a^{2}-15a+3\right){x}-3a^{5}+5a^{4}+a^{3}+3a^{2}-6$ |
| 31.1-b1 |
31.1-b |
$2$ |
$2$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
31.1 |
\( 31 \) |
\( -31 \) |
$91.58448$ |
$(-a^5+5a^3-5a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$5001.214246$ |
1.62410 |
\( \frac{2995464392}{31} a^{5} + \frac{1293428394}{31} a^{4} - \frac{8899729189}{31} a^{3} + \frac{17678994}{31} a^{2} + 72760745 a + \frac{420249726}{31} \) |
\( \bigl[2 a^{5} - a^{4} - 9 a^{3} + 5 a^{2} + 6 a - 5\) , \( 2 a^{5} - 3 a^{4} - 9 a^{3} + 12 a^{2} + 6 a - 4\) , \( a^{4} - 4 a^{2} + 1\) , \( -a^{5} + 7 a^{3} - a^{2} - 5 a + 4\) , \( 4 a^{5} - 4 a^{4} - 18 a^{3} + 19 a^{2} + 12 a - 10\bigr] \) |
${y}^2+\left(2a^{5}-a^{4}-9a^{3}+5a^{2}+6a-5\right){x}{y}+\left(a^{4}-4a^{2}+1\right){y}={x}^{3}+\left(2a^{5}-3a^{4}-9a^{3}+12a^{2}+6a-4\right){x}^{2}+\left(-a^{5}+7a^{3}-a^{2}-5a+4\right){x}+4a^{5}-4a^{4}-18a^{3}+19a^{2}+12a-10$ |
| 31.1-b2 |
31.1-b |
$2$ |
$2$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
31.1 |
\( 31 \) |
\( - 31^{2} \) |
$91.58448$ |
$(-a^5+5a^3-5a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$2500.607123$ |
1.62410 |
\( \frac{139573028727470950776}{961} a^{5} + \frac{146091909064081505603}{961} a^{4} - \frac{398857975268896914941}{961} a^{3} - \frac{258052846331700559100}{961} a^{2} + \frac{5474415068525499730}{31} a + \frac{68193984513407074549}{961} \) |
\( \bigl[a^{5} - a^{4} - 5 a^{3} + 5 a^{2} + 4 a - 4\) , \( -a^{5} + 2 a^{4} + 5 a^{3} - 9 a^{2} - 5 a + 6\) , \( a^{5} - 4 a^{3} + a - 1\) , \( -18 a^{5} + 35 a^{4} + 48 a^{3} - 86 a^{2} - 25 a + 31\) , \( -51 a^{5} + 152 a^{4} - 32 a^{3} - 169 a^{2} + 64 a + 14\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-5a^{3}+5a^{2}+4a-4\right){x}{y}+\left(a^{5}-4a^{3}+a-1\right){y}={x}^{3}+\left(-a^{5}+2a^{4}+5a^{3}-9a^{2}-5a+6\right){x}^{2}+\left(-18a^{5}+35a^{4}+48a^{3}-86a^{2}-25a+31\right){x}-51a^{5}+152a^{4}-32a^{3}-169a^{2}+64a+14$ |
| 31.1-c1 |
31.1-c |
$4$ |
$4$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
31.1 |
\( 31 \) |
\( -31 \) |
$91.58448$ |
$(-a^5+5a^3-5a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$12318.19278$ |
1.00006 |
\( -\frac{526618}{31} a^{5} + \frac{9219416}{31} a^{4} + \frac{18276705}{31} a^{3} - \frac{15857144}{31} a^{2} - 975716 a - \frac{7676519}{31} \) |
\( \bigl[a^{2} - 2\) , \( a^{5} - 5 a^{3} - a^{2} + 3 a + 1\) , \( a^{2} + a - 2\) , \( -4 a^{5} + a^{4} + 19 a^{3} - 2 a^{2} - 14 a - 2\) , \( -6 a^{5} + 6 a^{4} + 29 a^{3} - 23 a^{2} - 26 a + 6\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{5}-5a^{3}-a^{2}+3a+1\right){x}^{2}+\left(-4a^{5}+a^{4}+19a^{3}-2a^{2}-14a-2\right){x}-6a^{5}+6a^{4}+29a^{3}-23a^{2}-26a+6$ |
| 31.1-c2 |
31.1-c |
$4$ |
$4$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
31.1 |
\( 31 \) |
\( 31^{2} \) |
$91.58448$ |
$(-a^5+5a^3-5a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$1539.774098$ |
1.00006 |
\( \frac{1264802592419037}{961} a^{5} + \frac{1051692343665191}{961} a^{4} - \frac{4077796430304573}{961} a^{3} - \frac{1864472573084891}{961} a^{2} + \frac{75867838088346}{31} a + \frac{820784281275104}{961} \) |
\( \bigl[a^{2} - 2\) , \( a^{5} - 5 a^{3} - a^{2} + 3 a + 1\) , \( a^{2} + a - 2\) , \( -14 a^{5} - 14 a^{4} + 84 a^{3} + 83 a^{2} - 109 a - 102\) , \( -189 a^{5} + 106 a^{4} + 961 a^{3} - 316 a^{2} - 926 a - 97\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{5}-5a^{3}-a^{2}+3a+1\right){x}^{2}+\left(-14a^{5}-14a^{4}+84a^{3}+83a^{2}-109a-102\right){x}-189a^{5}+106a^{4}+961a^{3}-316a^{2}-926a-97$ |
| 31.1-c3 |
31.1-c |
$4$ |
$4$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
31.1 |
\( 31 \) |
\( 31^{4} \) |
$91.58448$ |
$(-a^5+5a^3-5a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$64$ |
\( 2 \) |
$1$ |
$24.05897029$ |
1.00006 |
\( \frac{148599407106618710023544430465}{923521} a^{5} - \frac{51162067164523519616985613951}{923521} a^{4} - \frac{776544246975688503097892288522}{923521} a^{3} + \frac{85213862179278656941786180477}{923521} a^{2} + \frac{25770070044101319817576470702}{29791} a + \frac{226625478544070566211027844328}{923521} \) |
\( \bigl[a^{5} - 4 a^{3} + a^{2} + a - 2\) , \( 2 a^{5} - 2 a^{4} - 9 a^{3} + 8 a^{2} + 4 a - 4\) , \( 0\) , \( 29 a^{5} - 43 a^{4} - 105 a^{3} + 123 a^{2} + 61 a - 61\) , \( 78 a^{5} - 175 a^{4} - 139 a^{3} + 291 a^{2} + 77 a - 115\bigr] \) |
${y}^2+\left(a^{5}-4a^{3}+a^{2}+a-2\right){x}{y}={x}^{3}+\left(2a^{5}-2a^{4}-9a^{3}+8a^{2}+4a-4\right){x}^{2}+\left(29a^{5}-43a^{4}-105a^{3}+123a^{2}+61a-61\right){x}+78a^{5}-175a^{4}-139a^{3}+291a^{2}+77a-115$ |
| 31.1-c4 |
31.1-c |
$4$ |
$4$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
31.1 |
\( 31 \) |
\( -31 \) |
$91.58448$ |
$(-a^5+5a^3-5a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 1 \) |
$1$ |
$192.4717623$ |
1.00006 |
\( \frac{959737921004117219213596641}{31} a^{5} + \frac{1004563318667521336749896161}{31} a^{4} - \frac{2742643960681810144259812298}{31} a^{3} - \frac{1774433818463953428848303523}{31} a^{2} + 37643402810618935845636590 a + \frac{468918340227326770581313560}{31} \) |
\( \bigl[2 a^{5} - a^{4} - 9 a^{3} + 4 a^{2} + 6 a - 3\) , \( -a^{4} + a^{3} + 4 a^{2} - 4 a\) , \( a^{4} - 4 a^{2} + a + 2\) , \( -74 a^{5} + 74 a^{4} + 339 a^{3} - 244 a^{2} - 243 a - 44\) , \( -500 a^{5} + 472 a^{4} + 2228 a^{3} - 1549 a^{2} - 1348 a - 204\bigr] \) |
${y}^2+\left(2a^{5}-a^{4}-9a^{3}+4a^{2}+6a-3\right){x}{y}+\left(a^{4}-4a^{2}+a+2\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}-4a\right){x}^{2}+\left(-74a^{5}+74a^{4}+339a^{3}-244a^{2}-243a-44\right){x}-500a^{5}+472a^{4}+2228a^{3}-1549a^{2}-1348a-204$ |
*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.
