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 |
71.5-a1 |
71.5-a |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.5 |
\( 71 \) |
\( - 71^{5} \) |
$69.83308$ |
$(-7a^5+2a^4+50a^3+22a^2-30a-10)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$2697.010303$ |
1.23075 |
\( \frac{18671790400300000}{1804229351} a^{5} - \frac{5223136808131864}{1804229351} a^{4} - \frac{134463355085824784}{1804229351} a^{3} - \frac{59498874802185092}{1804229351} a^{2} + \frac{87859926083703700}{1804229351} a + \frac{25940020939985535}{1804229351} \) |
\( \bigl[-6 a^{5} + 2 a^{4} + 43 a^{3} + 17 a^{2} - 29 a - 7\) , \( a^{5} - 8 a^{3} - 4 a^{2} + 7 a + 2\) , \( -6 a^{5} + a^{4} + 44 a^{3} + 23 a^{2} - 28 a - 10\) , \( 108 a^{5} - 25 a^{4} - 776 a^{3} - 378 a^{2} + 466 a + 140\) , \( 536 a^{5} - 123 a^{4} - 3847 a^{3} - 1892 a^{2} + 2295 a + 696\bigr] \) |
${y}^2+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-29a-7\right){x}{y}+\left(-6a^{5}+a^{4}+44a^{3}+23a^{2}-28a-10\right){y}={x}^{3}+\left(a^{5}-8a^{3}-4a^{2}+7a+2\right){x}^{2}+\left(108a^{5}-25a^{4}-776a^{3}-378a^{2}+466a+140\right){x}+536a^{5}-123a^{4}-3847a^{3}-1892a^{2}+2295a+696$ |
71.5-a2 |
71.5-a |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.5 |
\( 71 \) |
\( - 71^{10} \) |
$69.83308$ |
$(-7a^5+2a^4+50a^3+22a^2-30a-10)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1348.505151$ |
1.23075 |
\( -\frac{41014241389245672911233616}{3255243551009881201} a^{5} + \frac{31251665362953723702641564}{3255243551009881201} a^{4} + \frac{283323882933070565416889280}{3255243551009881201} a^{3} - \frac{8921536262849968349901544}{3255243551009881201} a^{2} - \frac{210325636400488563887402742}{3255243551009881201} a + \frac{49837787027906807246579723}{3255243551009881201} \) |
\( \bigl[-8 a^{5} + 2 a^{4} + 58 a^{3} + 27 a^{2} - 38 a - 12\) , \( -6 a^{5} + 2 a^{4} + 43 a^{3} + 17 a^{2} - 30 a - 6\) , \( -3 a^{5} + 23 a^{3} + 14 a^{2} - 17 a - 6\) , \( 10 a^{5} - 2 a^{4} - 72 a^{3} - 41 a^{2} + 36 a + 12\) , \( 15 a^{5} - 27 a^{4} - 174 a^{3} - 76 a^{2} + 101 a + 30\bigr] \) |
${y}^2+\left(-8a^{5}+2a^{4}+58a^{3}+27a^{2}-38a-12\right){x}{y}+\left(-3a^{5}+23a^{3}+14a^{2}-17a-6\right){y}={x}^{3}+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-30a-6\right){x}^{2}+\left(10a^{5}-2a^{4}-72a^{3}-41a^{2}+36a+12\right){x}+15a^{5}-27a^{4}-174a^{3}-76a^{2}+101a+30$ |
71.5-b1 |
71.5-b |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.5 |
\( 71 \) |
\( - 71^{2} \) |
$69.83308$ |
$(-7a^5+2a^4+50a^3+22a^2-30a-10)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.006077565$ |
$66963.54714$ |
2.22863 |
\( -\frac{3949307338048}{5041} a^{5} + \frac{657783939680}{5041} a^{4} + \frac{28702201019885}{5041} a^{3} + \frac{15411298823022}{5041} a^{2} - \frac{17627754558666}{5041} a - \frac{5427054812636}{5041} \) |
\( \bigl[-a^{5} + 8 a^{3} + 5 a^{2} - 7 a - 2\) , \( -6 a^{5} + 2 a^{4} + 43 a^{3} + 17 a^{2} - 28 a - 8\) , \( -3 a^{5} + 23 a^{3} + 14 a^{2} - 17 a - 5\) , \( 10 a^{5} - 2 a^{4} - 69 a^{3} - 40 a^{2} + 49 a + 14\) , \( 12 a^{5} + 4 a^{4} - 100 a^{3} - 51 a^{2} + 68 a + 20\bigr] \) |
${y}^2+\left(-a^{5}+8a^{3}+5a^{2}-7a-2\right){x}{y}+\left(-3a^{5}+23a^{3}+14a^{2}-17a-5\right){y}={x}^{3}+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-28a-8\right){x}^{2}+\left(10a^{5}-2a^{4}-69a^{3}-40a^{2}+49a+14\right){x}+12a^{5}+4a^{4}-100a^{3}-51a^{2}+68a+20$ |
71.5-b2 |
71.5-b |
$2$ |
$2$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.5 |
\( 71 \) |
\( -71 \) |
$69.83308$ |
$(-7a^5+2a^4+50a^3+22a^2-30a-10)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.003038782$ |
$267854.1885$ |
2.22863 |
\( \frac{5286451}{71} a^{5} - \frac{1105287}{71} a^{4} - \frac{38268478}{71} a^{3} - \frac{19161741}{71} a^{2} + \frac{24321390}{71} a + \frac{7597551}{71} \) |
\( \bigl[a^{5} - 7 a^{3} - 5 a^{2} + 2 a + 2\) , \( a^{5} - 7 a^{3} - 5 a^{2} + 3 a + 1\) , \( -3 a^{5} + 23 a^{3} + 14 a^{2} - 17 a - 5\) , \( -3 a^{5} + 2 a^{4} + 20 a^{3} + 4 a^{2} - 10 a - 4\) , \( -2 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} - 8 a - 2\bigr] \) |
${y}^2+\left(a^{5}-7a^{3}-5a^{2}+2a+2\right){x}{y}+\left(-3a^{5}+23a^{3}+14a^{2}-17a-5\right){y}={x}^{3}+\left(a^{5}-7a^{3}-5a^{2}+3a+1\right){x}^{2}+\left(-3a^{5}+2a^{4}+20a^{3}+4a^{2}-10a-4\right){x}-2a^{5}+a^{4}+14a^{3}+4a^{2}-8a-2$ |
71.5-c1 |
71.5-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.5 |
\( 71 \) |
\( - 71^{6} \) |
$69.83308$ |
$(-7a^5+2a^4+50a^3+22a^2-30a-10)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 2 \) |
$1$ |
$17.45399107$ |
1.29032 |
\( \frac{688385227545894441783394700691160040}{128100283921} a^{5} - \frac{1455011470820246681274446975813913565}{128100283921} a^{4} - \frac{3198310313219337512071267632384924449}{128100283921} a^{3} + \frac{4938596772879076842480778038133182698}{128100283921} a^{2} - \frac{681214996706145908023368640065836270}{128100283921} a - \frac{618129402248792677846156102081913073}{128100283921} \) |
\( \bigl[-8 a^{5} + 2 a^{4} + 58 a^{3} + 27 a^{2} - 39 a - 12\) , \( -4 a^{5} + a^{4} + 29 a^{3} + 14 a^{2} - 21 a - 6\) , \( 1\) , \( -944 a^{5} + 1869 a^{4} + 4540 a^{3} - 6026 a^{2} + 652 a + 593\) , \( -34009 a^{5} + 72266 a^{4} + 157382 a^{3} - 246003 a^{2} + 35333 a + 30593\bigr] \) |
${y}^2+\left(-8a^{5}+2a^{4}+58a^{3}+27a^{2}-39a-12\right){x}{y}+{y}={x}^{3}+\left(-4a^{5}+a^{4}+29a^{3}+14a^{2}-21a-6\right){x}^{2}+\left(-944a^{5}+1869a^{4}+4540a^{3}-6026a^{2}+652a+593\right){x}-34009a^{5}+72266a^{4}+157382a^{3}-246003a^{2}+35333a+30593$ |
71.5-c2 |
71.5-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.5 |
\( 71 \) |
\( -71 \) |
$69.83308$ |
$(-7a^5+2a^4+50a^3+22a^2-30a-10)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$25447.91898$ |
1.29032 |
\( \frac{3453103834}{71} a^{5} - \frac{966256218}{71} a^{4} - \frac{24868045898}{71} a^{3} - \frac{11002493289}{71} a^{2} + \frac{16250723867}{71} a + \frac{4795868035}{71} \) |
\( \bigl[-2 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} - 10 a - 1\) , \( -7 a^{5} + a^{4} + 51 a^{3} + 28 a^{2} - 32 a - 13\) , \( -a^{5} + 8 a^{3} + 4 a^{2} - 6 a - 1\) , \( 5 a^{5} - 38 a^{3} - 22 a^{2} + 26 a + 9\) , \( -3 a^{5} + a^{4} + 21 a^{3} + 8 a^{2} - 12 a - 4\bigr] \) |
${y}^2+\left(-2a^{5}+a^{4}+14a^{3}+4a^{2}-10a-1\right){x}{y}+\left(-a^{5}+8a^{3}+4a^{2}-6a-1\right){y}={x}^{3}+\left(-7a^{5}+a^{4}+51a^{3}+28a^{2}-32a-13\right){x}^{2}+\left(5a^{5}-38a^{3}-22a^{2}+26a+9\right){x}-3a^{5}+a^{4}+21a^{3}+8a^{2}-12a-4$ |
71.5-c3 |
71.5-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.5 |
\( 71 \) |
\( - 71^{3} \) |
$69.83308$ |
$(-7a^5+2a^4+50a^3+22a^2-30a-10)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 1 \) |
$1$ |
$34.90798215$ |
1.29032 |
\( -\frac{235353321428599978}{357911} a^{5} + \frac{580741577725442818}{357911} a^{4} + \frac{1162430939297012415}{357911} a^{3} - \frac{1879545228122345414}{357911} a^{2} + \frac{264656034786687147}{357911} a + \frac{239407872830503392}{357911} \) |
\( \bigl[a^{5} - a^{4} - 6 a^{3} + a^{2} + 3 a\) , \( -4 a^{5} + 30 a^{3} + 19 a^{2} - 18 a - 9\) , \( -6 a^{5} + 2 a^{4} + 43 a^{3} + 17 a^{2} - 29 a - 6\) , \( -36 a^{5} - 3 a^{4} + 278 a^{3} + 176 a^{2} - 156 a - 75\) , \( 1119 a^{5} - 361 a^{4} - 7802 a^{3} - 3654 a^{2} + 4623 a + 1358\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-6a^{3}+a^{2}+3a\right){x}{y}+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-29a-6\right){y}={x}^{3}+\left(-4a^{5}+30a^{3}+19a^{2}-18a-9\right){x}^{2}+\left(-36a^{5}-3a^{4}+278a^{3}+176a^{2}-156a-75\right){x}+1119a^{5}-361a^{4}-7802a^{3}-3654a^{2}+4623a+1358$ |
71.5-c4 |
71.5-c |
$4$ |
$6$ |
6.6.300125.1 |
$6$ |
$[6, 0]$ |
71.5 |
\( 71 \) |
\( - 71^{2} \) |
$69.83308$ |
$(-7a^5+2a^4+50a^3+22a^2-30a-10)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$12723.95949$ |
1.29032 |
\( \frac{6615986088905408509}{5041} a^{5} - \frac{1851862023389315072}{5041} a^{4} - \frac{47645414547979919968}{5041} a^{3} - \frac{21077149500839791045}{5041} a^{2} + \frac{31134395160454919709}{5041} a + \frac{9187684081459759389}{5041} \) |
\( \bigl[-2 a^{5} + 15 a^{3} + 10 a^{2} - 9 a - 5\) , \( a^{5} - 7 a^{3} - 5 a^{2} + a + 3\) , \( a + 1\) , \( 1534 a^{5} - 351 a^{4} - 11009 a^{3} - 5421 a^{2} + 6557 a + 1992\) , \( -18999 a^{5} + 4356 a^{4} + 136350 a^{3} + 67090 a^{2} - 81287 a - 24651\bigr] \) |
${y}^2+\left(-2a^{5}+15a^{3}+10a^{2}-9a-5\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{5}-7a^{3}-5a^{2}+a+3\right){x}^{2}+\left(1534a^{5}-351a^{4}-11009a^{3}-5421a^{2}+6557a+1992\right){x}-18999a^{5}+4356a^{4}+136350a^{3}+67090a^{2}-81287a-24651$ |
*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.