| 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 |
| 8.1-a1 |
8.1-a |
$6$ |
$8$ |
4.4.6224.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{8} \) |
$9.14239$ |
$(-a^3+a^2+4a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$451.9341989$ |
1.432123342 |
\( 524288 a^{3} + 1023968 a^{2} - 286688 a - 683152 \) |
\( \bigl[a^{3} - 5 a\) , \( a\) , \( a^{3} - a^{2} - 3 a + 3\) , \( 23 a^{3} - 45 a^{2} - 55 a + 66\) , \( -46 a^{3} + 85 a^{2} + 115 a - 121\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+a{x}^{2}+\left(23a^{3}-45a^{2}-55a+66\right){x}-46a^{3}+85a^{2}+115a-121$ |
| 8.1-a2 |
8.1-a |
$6$ |
$8$ |
4.4.6224.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{4} \) |
$9.14239$ |
$(-a^3+a^2+4a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$903.8683978$ |
1.432123342 |
\( -2048 a^{3} - 2048 a^{2} + 10240 a + 14336 \) |
\( \bigl[0\) , \( -a^{3} + 2 a^{2} + 3 a - 6\) , \( a^{3} - 4 a - 1\) , \( -a + 3\) , \( -a^{3} + 2 a^{2} + 2 a - 6\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-6\right){x}^{2}+\left(-a+3\right){x}-a^{3}+2a^{2}+2a-6$ |
| 8.1-a3 |
8.1-a |
$6$ |
$8$ |
4.4.6224.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{8} \) |
$9.14239$ |
$(-a^3+a^2+4a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$56.49177486$ |
1.432123342 |
\( -8484050373122 a^{3} - 5950704522102 a^{2} + 44647055214736 a + 50524147464286 \) |
\( \bigl[a + 1\) , \( a^{2} - 3\) , \( 0\) , \( 68 a^{3} + 51 a^{2} - 361 a - 422\) , \( 620 a^{3} + 504 a^{2} - 3336 a - 3927\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(68a^{3}+51a^{2}-361a-422\right){x}+620a^{3}+504a^{2}-3336a-3927$ |
| 8.1-a4 |
8.1-a |
$6$ |
$8$ |
4.4.6224.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{8} \) |
$9.14239$ |
$(-a^3+a^2+4a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$14.12294371$ |
1.432123342 |
\( 1537921833441718305282 a^{3} - 2107454190479225749642 a^{2} - 6339631844534479587472 a + 5611518334448113253746 \) |
\( \bigl[a + 1\) , \( a^{2} - 3\) , \( 0\) , \( -22 a^{3} + 41 a^{2} + 89 a - 122\) , \( -88 a^{3} + 144 a^{2} + 352 a - 423\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-22a^{3}+41a^{2}+89a-122\right){x}-88a^{3}+144a^{2}+352a-423$ |
| 8.1-a5 |
8.1-a |
$6$ |
$8$ |
4.4.6224.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{4} \) |
$9.14239$ |
$(-a^3+a^2+4a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$225.9670994$ |
1.432123342 |
\( 19246710912 a^{3} - 26376158916 a^{2} - 79338143744 a + 70236176888 \) |
\( \bigl[a + 1\) , \( a^{2} - 3\) , \( 0\) , \( 3 a^{3} + 6 a^{2} - 16 a - 32\) , \( 7 a^{3} + 8 a^{2} - 39 a - 55\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(3a^{3}+6a^{2}-16a-32\right){x}+7a^{3}+8a^{2}-39a-55$ |
| 8.1-a6 |
8.1-a |
$6$ |
$8$ |
4.4.6224.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{4} \) |
$9.14239$ |
$(-a^3+a^2+4a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$56.49177486$ |
1.432123342 |
\( 5668523114368 a^{3} + 13864000438468 a^{2} - 102746160640 a - 11588341595944 \) |
\( \bigl[a + 1\) , \( a^{2} - 3\) , \( a^{3} - a^{2} - 3 a + 3\) , \( -2 a^{3} - 5 a^{2} + 3 a + 9\) , \( -7 a^{3} - 18 a^{2} - 5 a + 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-2a^{3}-5a^{2}+3a+9\right){x}-7a^{3}-18a^{2}-5a+7$ |
| 8.1-b1 |
8.1-b |
$6$ |
$8$ |
4.4.6224.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{8} \) |
$9.14239$ |
$(-a^3+a^2+4a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.992443989$ |
$1047.332123$ |
1.646894014 |
\( 524288 a^{3} + 1023968 a^{2} - 286688 a - 683152 \) |
\( \bigl[a^{2} - 3\) , \( -a\) , \( a^{3} - 5 a\) , \( 11 a^{3} - 20 a^{2} - 29 a + 26\) , \( 19 a^{3} - 36 a^{2} - 45 a + 46\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}-a{x}^{2}+\left(11a^{3}-20a^{2}-29a+26\right){x}+19a^{3}-36a^{2}-45a+46$ |
| 8.1-b2 |
8.1-b |
$6$ |
$8$ |
4.4.6224.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{8} \) |
$9.14239$ |
$(-a^3+a^2+4a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.969775957$ |
$8.182282215$ |
1.646894014 |
\( 1537921833441718305282 a^{3} - 2107454190479225749642 a^{2} - 6339631844534479587472 a + 5611518334448113253746 \) |
\( \bigl[a^{2} - a - 2\) , \( 1\) , \( a^{2} - 3\) , \( 11 a^{3} + 17 a^{2} - 38 a - 56\) , \( 62 a^{3} + 93 a^{2} - 192 a - 280\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+{x}^{2}+\left(11a^{3}+17a^{2}-38a-56\right){x}+62a^{3}+93a^{2}-192a-280$ |
| 8.1-b3 |
8.1-b |
$6$ |
$8$ |
4.4.6224.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{4} \) |
$9.14239$ |
$(-a^3+a^2+4a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1.984887978$ |
$130.9165154$ |
1.646894014 |
\( 19246710912 a^{3} - 26376158916 a^{2} - 79338143744 a + 70236176888 \) |
\( \bigl[a^{2} - a - 2\) , \( 1\) , \( a^{2} - 3\) , \( 6 a^{3} + 2 a^{2} - 43 a - 46\) , \( 35 a^{3} + 32 a^{2} - 172 a - 206\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+{x}^{2}+\left(6a^{3}+2a^{2}-43a-46\right){x}+35a^{3}+32a^{2}-172a-206$ |
| 8.1-b4 |
8.1-b |
$6$ |
$8$ |
4.4.6224.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{8} \) |
$9.14239$ |
$(-a^3+a^2+4a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.969775957$ |
$8.182282215$ |
1.646894014 |
\( -8484050373122 a^{3} - 5950704522102 a^{2} + 44647055214736 a + 50524147464286 \) |
\( \bigl[a^{3} - 4 a - 1\) , \( a^{3} - 5 a - 2\) , \( a + 1\) , \( -74 a^{3} + 178 a^{2} + 262 a - 625\) , \( -810 a^{3} + 1616 a^{2} + 3050 a - 5278\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(-74a^{3}+178a^{2}+262a-625\right){x}-810a^{3}+1616a^{2}+3050a-5278$ |
| 8.1-b5 |
8.1-b |
$6$ |
$8$ |
4.4.6224.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{4} \) |
$9.14239$ |
$(-a^3+a^2+4a-2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.496221994$ |
$523.6660617$ |
1.646894014 |
\( 5668523114368 a^{3} + 13864000438468 a^{2} - 102746160640 a - 11588341595944 \) |
\( \bigl[a^{3} - 4 a - 1\) , \( a^{3} - 5 a - 2\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -4 a^{3} + 5 a^{2} + 17 a - 12\) , \( -31 a^{3} + 44 a^{2} + 127 a - 120\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(-4a^{3}+5a^{2}+17a-12\right){x}-31a^{3}+44a^{2}+127a-120$ |
| 8.1-b6 |
8.1-b |
$6$ |
$8$ |
4.4.6224.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{4} \) |
$9.14239$ |
$(-a^3+a^2+4a-2)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.496221994$ |
$2094.664247$ |
1.646894014 |
\( -2048 a^{3} - 2048 a^{2} + 10240 a + 14336 \) |
\( \bigl[0\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{2} - a - 2\) , \( -a^{2} - a + 2\) , \( a^{3} - 2 a\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(-a^{2}-a+2\right){x}+a^{3}-2a$ |
*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.
