| 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 |
| 81.1-a1 |
81.1-a |
$2$ |
$2$ |
4.4.14272.1 |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{14} \) |
$18.49020$ |
$(-a), (a^3-2a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$87.56040464$ |
0.732934797 |
\( -\frac{21181467867805696}{177147} a^{3} + \frac{72820918233614336}{177147} a^{2} + \frac{1208734651678976}{177147} a - \frac{44161394733457472}{177147} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( 2 a^{3} - 5 a^{2} - 6 a + 4\) , \( a^{3} - 3 a^{2} - a + 3\) , \( 41 a^{3} - 91 a^{2} - 131 a - 28\) , \( 74 a^{3} - 245 a^{2} - 29 a + 133\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-3a^{2}-a+3\right){y}={x}^{3}+\left(2a^{3}-5a^{2}-6a+4\right){x}^{2}+\left(41a^{3}-91a^{2}-131a-28\right){x}+74a^{3}-245a^{2}-29a+133$ |
| 81.1-a2 |
81.1-a |
$2$ |
$2$ |
4.4.14272.1 |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{28} \) |
$18.49020$ |
$(-a), (a^3-2a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$21.89010116$ |
0.732934797 |
\( \frac{2654964903950336}{31381059609} a^{3} - \frac{12244791555920896}{31381059609} a^{2} + \frac{7551611381343488}{31381059609} a + \frac{13539530555930944}{31381059609} \) |
\( \bigl[2 a^{3} - 5 a^{2} - 5 a + 5\) , \( a^{3} - 2 a^{2} - 5 a + 1\) , \( 2 a^{3} - 5 a^{2} - 5 a + 4\) , \( -29 a^{3} + 78 a^{2} + 90 a - 124\) , \( 56 a^{3} - 154 a^{2} - 167 a + 230\bigr] \) |
${y}^2+\left(2a^{3}-5a^{2}-5a+5\right){x}{y}+\left(2a^{3}-5a^{2}-5a+4\right){y}={x}^{3}+\left(a^{3}-2a^{2}-5a+1\right){x}^{2}+\left(-29a^{3}+78a^{2}+90a-124\right){x}+56a^{3}-154a^{2}-167a+230$ |
| 81.1-b1 |
81.1-b |
$2$ |
$2$ |
4.4.14272.1 |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{16} \) |
$18.49020$ |
$(-a), (a^3-2a^2-5a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.040338144$ |
$463.3279682$ |
3.128902350 |
\( -\frac{490020352}{59049} a^{3} + \frac{1353206720}{59049} a^{2} + \frac{1412786048}{59049} a - \frac{1912129280}{59049} \) |
\( \bigl[a^{3} - 3 a^{2} - a + 4\) , \( -a^{3} + 3 a^{2} + 2 a - 3\) , \( a\) , \( 6 a^{3} - 21 a^{2} + 14\) , \( 34 a^{3} - 117 a^{2} - 2 a + 71\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-a+4\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+3a^{2}+2a-3\right){x}^{2}+\left(6a^{3}-21a^{2}+14\right){x}+34a^{3}-117a^{2}-2a+71$ |
| 81.1-b2 |
81.1-b |
$2$ |
$2$ |
4.4.14272.1 |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{8} \) |
$18.49020$ |
$(-a), (a^3-2a^2-5a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 5 \) |
$0.080676288$ |
$926.6559364$ |
3.128902350 |
\( \frac{66181349632}{243} a^{3} - \frac{179765938112}{243} a^{2} - \frac{202122759680}{243} a + \frac{277150255040}{243} \) |
\( \bigl[a^{3} - 3 a^{2} - a + 4\) , \( -2 a^{3} + 5 a^{2} + 5 a - 3\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 2 a^{3} + 8 a^{2} - 39 a - 38\) , \( 20 a^{3} - 14 a^{2} - 143 a - 86\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-a+4\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(-2a^{3}+5a^{2}+5a-3\right){x}^{2}+\left(2a^{3}+8a^{2}-39a-38\right){x}+20a^{3}-14a^{2}-143a-86$ |
| 81.1-c1 |
81.1-c |
$2$ |
$2$ |
4.4.14272.1 |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{14} \) |
$18.49020$ |
$(-a), (a^3-2a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 11 \) |
$1$ |
$27.64501416$ |
2.545464728 |
\( -\frac{21181467867805696}{177147} a^{3} + \frac{72820918233614336}{177147} a^{2} + \frac{1208734651678976}{177147} a - \frac{44161394733457472}{177147} \) |
\( \bigl[a^{3} - 3 a^{2} - a + 4\) , \( 0\) , \( a^{3} - 3 a^{2} - a + 3\) , \( 21 a^{3} - 79 a^{2} - 3 a + 43\) , \( 129 a^{3} - 480 a^{2} + 291\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-a+4\right){x}{y}+\left(a^{3}-3a^{2}-a+3\right){y}={x}^{3}+\left(21a^{3}-79a^{2}-3a+43\right){x}+129a^{3}-480a^{2}+291$ |
| 81.1-c2 |
81.1-c |
$2$ |
$2$ |
4.4.14272.1 |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{28} \) |
$18.49020$ |
$(-a), (a^3-2a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 11 \) |
$1$ |
$27.64501416$ |
2.545464728 |
\( \frac{2654964903950336}{31381059609} a^{3} - \frac{12244791555920896}{31381059609} a^{2} + \frac{7551611381343488}{31381059609} a + \frac{13539530555930944}{31381059609} \) |
\( \bigl[2 a^{3} - 5 a^{2} - 5 a + 5\) , \( -2 a^{3} + 5 a^{2} + 6 a - 3\) , \( 1\) , \( 83 a^{3} - 82 a^{2} - 520 a - 304\) , \( -787 a^{3} + 877 a^{2} + 4684 a + 2609\bigr] \) |
${y}^2+\left(2a^{3}-5a^{2}-5a+5\right){x}{y}+{y}={x}^{3}+\left(-2a^{3}+5a^{2}+6a-3\right){x}^{2}+\left(83a^{3}-82a^{2}-520a-304\right){x}-787a^{3}+877a^{2}+4684a+2609$ |
| 81.1-d1 |
81.1-d |
$2$ |
$2$ |
4.4.14272.1 |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{8} \) |
$18.49020$ |
$(-a), (a^3-2a^2-5a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.059430007$ |
$1099.772439$ |
2.188397414 |
\( \frac{8192}{9} a^{3} + \frac{7744}{3} a^{2} - \frac{11648}{9} a - \frac{17408}{9} \) |
\( \bigl[a^{3} - 3 a^{2} - a + 4\) , \( -a^{3} + 3 a^{2} + 3 a - 2\) , \( 2 a^{3} - 5 a^{2} - 6 a + 5\) , \( -5 a^{3} + 15 a^{2} + 15 a - 17\) , \( 24 a^{3} - 63 a^{2} - 75 a + 99\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-a+4\right){x}{y}+\left(2a^{3}-5a^{2}-6a+5\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+3a-2\right){x}^{2}+\left(-5a^{3}+15a^{2}+15a-17\right){x}+24a^{3}-63a^{2}-75a+99$ |
| 81.1-d2 |
81.1-d |
$2$ |
$2$ |
4.4.14272.1 |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{4} \) |
$18.49020$ |
$(-a), (a^3-2a^2-5a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.118860015$ |
$2199.544878$ |
2.188397414 |
\( -73728 a^{3} + \frac{138304}{3} a^{2} + 497664 a + \frac{1329344}{3} \) |
\( \bigl[a^{3} - 3 a^{2} - a + 4\) , \( a^{3} - 3 a^{2} - a + 4\) , \( 2 a^{3} - 5 a^{2} - 6 a + 4\) , \( 3 a^{3} - 9 a^{2} + 5\) , \( 3 a^{3} - 9 a^{2} + 2\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-a+4\right){x}{y}+\left(2a^{3}-5a^{2}-6a+4\right){y}={x}^{3}+\left(a^{3}-3a^{2}-a+4\right){x}^{2}+\left(3a^{3}-9a^{2}+5\right){x}+3a^{3}-9a^{2}+2$ |
| 81.1-e1 |
81.1-e |
$2$ |
$2$ |
4.4.14272.1 |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{16} \) |
$18.49020$ |
$(-a), (a^3-2a^2-5a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.885324704$ |
$156.6419309$ |
4.643315395 |
\( -\frac{490020352}{59049} a^{3} + \frac{1353206720}{59049} a^{2} + \frac{1412786048}{59049} a - \frac{1912129280}{59049} \) |
\( \bigl[a^{3} - 3 a^{2} - a + 4\) , \( 2 a^{3} - 5 a^{2} - 5 a + 4\) , \( a\) , \( -62 a^{3} + 63 a^{2} + 384 a + 234\) , \( -236 a^{3} + 258 a^{2} + 1420 a + 807\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-a+4\right){x}{y}+a{y}={x}^{3}+\left(2a^{3}-5a^{2}-5a+4\right){x}^{2}+\left(-62a^{3}+63a^{2}+384a+234\right){x}-236a^{3}+258a^{2}+1420a+807$ |
| 81.1-e2 |
81.1-e |
$2$ |
$2$ |
4.4.14272.1 |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{8} \) |
$18.49020$ |
$(-a), (a^3-2a^2-5a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1.770649408$ |
$313.2838618$ |
4.643315395 |
\( \frac{66181349632}{243} a^{3} - \frac{179765938112}{243} a^{2} - \frac{202122759680}{243} a + \frac{277150255040}{243} \) |
\( \bigl[2 a^{3} - 5 a^{2} - 5 a + 5\) , \( a^{3} - 3 a^{2} - 3 a + 4\) , \( a^{3} - 3 a^{2} - 2 a + 3\) , \( 6 a^{3} - 25 a^{2} - 23 a + 33\) , \( -a^{3} - 43 a^{2} - 21 a + 54\bigr] \) |
${y}^2+\left(2a^{3}-5a^{2}-5a+5\right){x}{y}+\left(a^{3}-3a^{2}-2a+3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-3a+4\right){x}^{2}+\left(6a^{3}-25a^{2}-23a+33\right){x}-a^{3}-43a^{2}-21a+54$ |
| 81.1-f1 |
81.1-f |
$2$ |
$2$ |
4.4.14272.1 |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{8} \) |
$18.49020$ |
$(-a), (a^3-2a^2-5a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.738109799$ |
$268.4351752$ |
6.634038209 |
\( \frac{8192}{9} a^{3} + \frac{7744}{3} a^{2} - \frac{11648}{9} a - \frac{17408}{9} \) |
\( \bigl[a^{3} - 3 a^{2} - a + 4\) , \( -a\) , \( 2 a^{3} - 5 a^{2} - 6 a + 5\) , \( -2 a^{3} + 5 a^{2} + 5 a - 6\) , \( -2 a^{3} + 6 a^{2} + 4 a - 10\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-a+4\right){x}{y}+\left(2a^{3}-5a^{2}-6a+5\right){y}={x}^{3}-a{x}^{2}+\left(-2a^{3}+5a^{2}+5a-6\right){x}-2a^{3}+6a^{2}+4a-10$ |
| 81.1-f2 |
81.1-f |
$2$ |
$2$ |
4.4.14272.1 |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{4} \) |
$18.49020$ |
$(-a), (a^3-2a^2-5a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1.476219599$ |
$536.8703504$ |
6.634038209 |
\( -73728 a^{3} + \frac{138304}{3} a^{2} + 497664 a + \frac{1329344}{3} \) |
\( \bigl[2 a^{3} - 5 a^{2} - 5 a + 5\) , \( -a^{2} + a + 3\) , \( a + 1\) , \( 6 a^{3} - 20 a^{2} - 11 a + 29\) , \( 11 a^{3} - 35 a^{2} - 15 a + 35\bigr] \) |
${y}^2+\left(2a^{3}-5a^{2}-5a+5\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(6a^{3}-20a^{2}-11a+29\right){x}+11a^{3}-35a^{2}-15a+35$ |
*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.
