| 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 |
| 2.1-a1 |
2.1-a |
$8$ |
$12$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( 2^{9} \) |
$35.19903$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$3.190434005$ |
$13.54900121$ |
1.32318471 |
\( -\frac{9453960236739132558727861787495}{8} a^{4} - \frac{2538198883159343121643145799747}{8} a^{3} + \frac{11012536690340190808726426019389}{2} a^{2} + \frac{18060887533903400257685370034437}{8} a - \frac{14905967017989550227913584754415}{8} \) |
\( \bigl[a^{3} - 3 a + 1\) , \( a^{4} - a^{3} - 5 a^{2} + 3 a + 2\) , \( a^{3} - 4 a + 1\) , \( -227 a^{4} + 136 a^{3} + 1158 a^{2} - 329 a - 1156\) , \( -3084 a^{4} + 1598 a^{3} + 16542 a^{2} - 5075 a - 15482\bigr] \) |
${y}^2+\left(a^{3}-3a+1\right){x}{y}+\left(a^{3}-4a+1\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+3a+2\right){x}^{2}+\left(-227a^{4}+136a^{3}+1158a^{2}-329a-1156\right){x}-3084a^{4}+1598a^{3}+16542a^{2}-5075a-15482$ |
| 2.1-a2 |
2.1-a |
$8$ |
$12$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( 2^{18} \) |
$35.19903$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2 \) |
$1.595217002$ |
$54.19600485$ |
1.32318471 |
\( -\frac{30492494674395553}{64} a^{4} - \frac{4090354265569063}{32} a^{3} + \frac{142074194389859355}{64} a^{2} + \frac{14557189787994341}{16} a - \frac{48066807833590099}{64} \) |
\( \bigl[a^{4} - 4 a^{2} + 1\) , \( a^{2} + a - 1\) , \( a^{3} - 3 a\) , \( -85 a^{4} + 89 a^{3} + 478 a^{2} - 321 a - 564\) , \( -858 a^{4} + 786 a^{3} + 4705 a^{2} - 2796 a - 5180\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-85a^{4}+89a^{3}+478a^{2}-321a-564\right){x}-858a^{4}+786a^{3}+4705a^{2}-2796a-5180$ |
| 2.1-a3 |
2.1-a |
$8$ |
$12$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{36} \) |
$35.19903$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$0.797608501$ |
$27.09800242$ |
1.32318471 |
\( \frac{146574081421}{512} a^{4} - \frac{3578721076161}{4096} a^{3} + \frac{265694297377}{1024} a^{2} + \frac{802266854809}{1024} a - \frac{1308242726357}{4096} \) |
\( \bigl[1\) , \( a^{2} - a - 3\) , \( a^{4} + a^{3} - 4 a^{2} - 4 a + 2\) , \( 13 a^{4} + 22 a^{3} - 45 a^{2} - 14 a + 15\) , \( -118 a^{4} - 77 a^{3} + 242 a^{2} + 120 a - 85\bigr] \) |
${y}^2+{x}{y}+\left(a^{4}+a^{3}-4a^{2}-4a+2\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(13a^{4}+22a^{3}-45a^{2}-14a+15\right){x}-118a^{4}-77a^{3}+242a^{2}+120a-85$ |
| 2.1-a4 |
2.1-a |
$8$ |
$12$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{12} \) |
$35.19903$ |
$(a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$0.265869500$ |
$6584.814590$ |
1.32318471 |
\( -350 a^{4} - \frac{6093}{16} a^{3} + \frac{3921}{2} a^{2} + \frac{1613}{2} a - \frac{10717}{16} \) |
\( \bigl[1\) , \( a^{2} - a - 3\) , \( a^{4} + a^{3} - 4 a^{2} - 4 a + 2\) , \( -2 a^{4} - 3 a^{3} + 5 a^{2} + 6 a\) , \( 7 a^{4} + 7 a^{3} - 20 a^{2} - 10 a + 6\bigr] \) |
${y}^2+{x}{y}+\left(a^{4}+a^{3}-4a^{2}-4a+2\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-2a^{4}-3a^{3}+5a^{2}+6a\right){x}+7a^{4}+7a^{3}-20a^{2}-10a+6$ |
| 2.1-a5 |
2.1-a |
$8$ |
$12$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( 2^{6} \) |
$35.19903$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2 \) |
$0.531739000$ |
$13169.62918$ |
1.32318471 |
\( \frac{702837}{4} a^{4} + \frac{333247}{2} a^{3} - \frac{1897105}{4} a^{2} - \frac{268947}{2} a + \frac{1099289}{4} \) |
\( \bigl[a^{4} - 4 a^{2} + 1\) , \( a^{2} + a - 1\) , \( a^{3} - 3 a\) , \( -20 a^{4} + 14 a^{3} + 108 a^{2} - 41 a - 99\) , \( 85 a^{4} - 46 a^{3} - 441 a^{2} + 145 a + 395\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-20a^{4}+14a^{3}+108a^{2}-41a-99\right){x}+85a^{4}-46a^{3}-441a^{2}+145a+395$ |
| 2.1-a6 |
2.1-a |
$8$ |
$12$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( 2^{3} \) |
$35.19903$ |
$(a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1.063478001$ |
$3292.407295$ |
1.32318471 |
\( 5016614959 a^{4} - 2898765869 a^{3} - \frac{52633975911}{2} a^{2} + \frac{17942542315}{2} a + \frac{47787291711}{2} \) |
\( \bigl[a^{4} + a^{3} - 5 a^{2} - 4 a + 3\) , \( a^{3} - 5 a\) , \( a^{3} - 4 a + 1\) , \( 4 a^{4} + 2 a^{3} - 18 a^{2} - 14 a + 5\) , \( 104 a^{4} + 22 a^{3} - 478 a^{2} - 187 a + 166\bigr] \) |
${y}^2+\left(a^{4}+a^{3}-5a^{2}-4a+3\right){x}{y}+\left(a^{3}-4a+1\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(4a^{4}+2a^{3}-18a^{2}-14a+5\right){x}+104a^{4}+22a^{3}-478a^{2}-187a+166$ |
| 2.1-a7 |
2.1-a |
$8$ |
$12$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( 2^{3} \) |
$35.19903$ |
$(a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1.063478001$ |
$3292.407295$ |
1.32318471 |
\( 537901203529 a^{4} + 622410006495 a^{3} - \frac{2698409411353}{2} a^{2} - \frac{1531643690819}{2} a + \frac{995369683225}{2} \) |
\( \bigl[a^{4} - 4 a^{2} + 1\) , \( a^{2} + a - 1\) , \( a^{3} - 3 a\) , \( -25 a^{4} + 24 a^{3} + 138 a^{2} - 81 a - 149\) , \( 9 a^{4} + 27 a^{3} - 23 a^{2} - 121 a - 77\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-25a^{4}+24a^{3}+138a^{2}-81a-149\right){x}+9a^{4}+27a^{3}-23a^{2}-121a-77$ |
| 2.1-a8 |
2.1-a |
$8$ |
$12$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( 2^{9} \) |
$35.19903$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$3.190434005$ |
$13.54900121$ |
1.32318471 |
\( -\frac{132887811460648998459913}{8} a^{4} + \frac{251927835108746278903907}{8} a^{3} + \frac{109690941712723804359691}{2} a^{2} - \frac{924592929709332117530341}{8} a + \frac{296692992719903596467759}{8} \) |
\( \bigl[a^{4} - 4 a^{2} + 1\) , \( a^{2} + a - 3\) , \( a^{3} + a^{2} - 4 a - 2\) , \( -47 a^{4} - 46 a^{3} + 135 a^{2} + 55 a - 76\) , \( -554 a^{4} - 661 a^{3} + 1331 a^{2} + 850 a - 440\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+1\right){x}{y}+\left(a^{3}+a^{2}-4a-2\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(-47a^{4}-46a^{3}+135a^{2}+55a-76\right){x}-554a^{4}-661a^{3}+1331a^{2}+850a-440$ |
| 4.1-a1 |
4.1-a |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$37.72538$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$560.8113928$ |
1.52590583 |
\( 33665007625 a^{4} - 99143139329 a^{3} + 24506351566 a^{2} + 87008280558 a - 34604383470 \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} + 5 a - 1\) , \( a^{3} + a^{2} - 4 a - 2\) , \( -50 a^{4} - 11 a^{3} + 233 a^{2} + 89 a - 78\) , \( -46 a^{4} - 11 a^{3} + 216 a^{2} + 86 a - 75\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}+a^{2}-4a-2\right){y}={x}^{3}+\left(-a^{3}+5a-1\right){x}^{2}+\left(-50a^{4}-11a^{3}+233a^{2}+89a-78\right){x}-46a^{4}-11a^{3}+216a^{2}+86a-75$ |
| 4.1-a2 |
4.1-a |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$37.72538$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$2243.245571$ |
1.52590583 |
\( -19594657821 a^{4} - 5078605824 a^{3} + 91740987570 a^{2} + 37533442058 a - 31028215264 \) |
\( \bigl[a^{4} - 5 a^{2} + a + 4\) , \( -a^{2} + a + 1\) , \( a^{3} - 4 a\) , \( 12 a^{4} - 3 a^{3} - 67 a^{2} + 5 a + 73\) , \( 512 a^{4} - 312 a^{3} - 2660 a^{2} + 979 a + 2344\bigr] \) |
${y}^2+\left(a^{4}-5a^{2}+a+4\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(12a^{4}-3a^{3}-67a^{2}+5a+73\right){x}+512a^{4}-312a^{3}-2660a^{2}+979a+2344$ |
| 4.1-a3 |
4.1-a |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$37.72538$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$8972.982285$ |
1.52590583 |
\( -53219 a^{4} - 14895 a^{3} + 250802 a^{2} + 118664 a - 64050 \) |
\( \bigl[a^{4} - 5 a^{2} + 4\) , \( -a^{3} + 4 a - 1\) , \( 0\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( 0\bigr] \) |
${y}^2+\left(a^{4}-5a^{2}+4\right){x}{y}={x}^{3}+\left(-a^{3}+4a-1\right){x}^{2}+\left(-a^{3}+a^{2}+4a-3\right){x}$ |
| 4.1-a4 |
4.1-a |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$37.72538$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$8972.982285$ |
1.52590583 |
\( 31641 a^{4} - 93154 a^{3} + 24182 a^{2} + 78106 a - 28044 \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{4} + a^{3} + 6 a^{2} - 5 a - 5\) , \( a^{4} - 5 a^{2} + 4\) , \( 37 a^{4} + 9 a^{3} - 173 a^{2} - 68 a + 65\) , \( -12 a^{4} - a^{3} + 54 a^{2} + 17 a - 19\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{4}-5a^{2}+4\right){y}={x}^{3}+\left(-a^{4}+a^{3}+6a^{2}-5a-5\right){x}^{2}+\left(37a^{4}+9a^{3}-173a^{2}-68a+65\right){x}-12a^{4}-a^{3}+54a^{2}+17a-19$ |
| 4.1-a5 |
4.1-a |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{8} \) |
$37.72538$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$2243.245571$ |
1.52590583 |
\( 35483 a^{4} + 84245 a^{3} + 15450 a^{2} - 26382 a + 1878 \) |
\( \bigl[a\) , \( a^{3} + a^{2} - 3 a - 2\) , \( a^{2} + a - 2\) , \( 12 a^{4} + 3 a^{3} - 55 a^{2} - 19 a + 21\) , \( -36 a^{4} - 9 a^{3} + 169 a^{2} + 68 a - 59\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-2\right){x}^{2}+\left(12a^{4}+3a^{3}-55a^{2}-19a+21\right){x}-36a^{4}-9a^{3}+169a^{2}+68a-59$ |
| 4.1-a6 |
4.1-a |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{4} \) |
$37.72538$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$2243.245571$ |
1.52590583 |
\( 151403617 a^{4} - 81479148 a^{3} - 803119086 a^{2} + 247195958 a + 753004456 \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + 4 a\) , \( a^{3} - 3 a\) , \( -a^{4} + 5 a^{2} - 2 a - 2\) , \( a^{4} - 5 a^{2} - a + 1\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+4a\right){x}^{2}+\left(-a^{4}+5a^{2}-2a-2\right){x}+a^{4}-5a^{2}-a+1$ |
| 4.2-a1 |
4.2-a |
$4$ |
$4$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{2} \) |
$37.72538$ |
$(-a^4+a^3+4a^2-3a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$0.159788041$ |
$1071.092828$ |
2.32837160 |
\( -1333357625169412059 a^{4} - \frac{715959608688771237}{2} a^{3} + 6212697917356441608 a^{2} + 2547252316594307138 a - 2102291979590761752 \) |
\( \bigl[1\) , \( a^{3} + a^{2} - 3 a - 2\) , \( a^{4} - 5 a^{2} + a + 3\) , \( 43 a^{4} + 11 a^{3} - 198 a^{2} - 80 a + 60\) , \( -283 a^{4} - 79 a^{3} + 1318 a^{2} + 549 a - 453\bigr] \) |
${y}^2+{x}{y}+\left(a^{4}-5a^{2}+a+3\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-2\right){x}^{2}+\left(43a^{4}+11a^{3}-198a^{2}-80a+60\right){x}-283a^{4}-79a^{3}+1318a^{2}+549a-453$ |
| 4.2-a2 |
4.2-a |
$4$ |
$4$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{8} \) |
$37.72538$ |
$(-a^4+a^3+4a^2-3a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.039947010$ |
$8568.742628$ |
2.32837160 |
\( \frac{284199609173}{8} a^{4} - \frac{1673854492703}{16} a^{3} + \frac{103352543439}{4} a^{2} + \frac{734787875991}{8} a - \frac{292257868217}{8} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( -a^{3} - a^{2} + 5 a + 2\) , \( a^{4} - 4 a^{2} + 1\) , \( -17 a^{4} + 31 a^{3} + 54 a^{2} - 113 a + 46\) , \( 15 a^{4} - 29 a^{3} - 50 a^{2} + 107 a - 29\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+\left(a^{4}-4a^{2}+1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+5a+2\right){x}^{2}+\left(-17a^{4}+31a^{3}+54a^{2}-113a+46\right){x}+15a^{4}-29a^{3}-50a^{2}+107a-29$ |
| 4.2-a3 |
4.2-a |
$4$ |
$4$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{4} \) |
$37.72538$ |
$(-a^4+a^3+4a^2-3a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.079894020$ |
$17137.48525$ |
2.32837160 |
\( -\frac{1012004261}{2} a^{4} - \frac{543840139}{4} a^{3} + 2357863114 a^{2} + \frac{1933667123}{2} a - \frac{1595759059}{2} \) |
\( \bigl[1\) , \( a^{3} + a^{2} - 3 a - 2\) , \( a^{4} - 5 a^{2} + a + 3\) , \( 3 a^{4} + a^{3} - 13 a^{2} - 5 a + 5\) , \( -6 a^{4} - a^{3} + 28 a^{2} + 10 a - 11\bigr] \) |
${y}^2+{x}{y}+\left(a^{4}-5a^{2}+a+3\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-2\right){x}^{2}+\left(3a^{4}+a^{3}-13a^{2}-5a+5\right){x}-6a^{4}-a^{3}+28a^{2}+10a-11$ |
| 4.2-a4 |
4.2-a |
$4$ |
$4$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.2 |
\( 2^{2} \) |
\( - 2^{2} \) |
$37.72538$ |
$(-a^4+a^3+4a^2-3a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.039947010$ |
$17137.48525$ |
2.32837160 |
\( 25441880 a^{4} + \frac{58705661}{2} a^{3} - 64179600 a^{2} - 36488805 a + 23722764 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 2\) , \( a^{4} - 5 a^{2} + a + 3\) , \( 3 a^{4} + 3 a^{3} - 13 a^{2} - 3 a + 10\) , \( 4 a^{4} - a^{3} - 6 a^{2} + 1\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+\left(a^{4}-5a^{2}+a+3\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+3a+2\right){x}^{2}+\left(3a^{4}+3a^{3}-13a^{2}-3a+10\right){x}+4a^{4}-a^{3}-6a^{2}+1$ |
| 4.2-b1 |
4.2-b |
$4$ |
$6$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.2 |
\( 2^{2} \) |
\( - 2^{18} \) |
$37.72538$ |
$(-a^4+a^3+4a^2-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 3^{2} \) |
$1$ |
$23.68361070$ |
1.30491987 |
\( \frac{15397719183967089}{128} a^{4} - \frac{181376071060914879}{512} a^{3} + \frac{2799761622472495}{32} a^{2} + \frac{79620370510126325}{256} a - \frac{31668516161419889}{256} \) |
\( \bigl[a^{3} - 4 a\) , \( a^{4} - 5 a^{2} - a + 3\) , \( a^{2} - 1\) , \( 21 a^{4} - 75 a^{3} - 14 a^{2} + 274 a - 267\) , \( -336 a^{4} + 398 a^{3} + 1469 a^{2} - 1422 a - 589\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{4}-5a^{2}-a+3\right){x}^{2}+\left(21a^{4}-75a^{3}-14a^{2}+274a-267\right){x}-336a^{4}+398a^{3}+1469a^{2}-1422a-589$ |
| 4.2-b2 |
4.2-b |
$4$ |
$6$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.2 |
\( 2^{2} \) |
\( - 2^{36} \) |
$37.72538$ |
$(-a^4+a^3+4a^2-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$11.84180535$ |
1.30491987 |
\( -\frac{359368410501823}{131072} a^{4} - \frac{191346035043539}{262144} a^{3} + \frac{836549249443483}{65536} a^{2} + \frac{685398991052723}{131072} a - \frac{565954847155821}{131072} \) |
\( \bigl[a\) , \( -a^{3} - a^{2} + 3 a + 2\) , \( a^{4} + a^{3} - 4 a^{2} - 3 a + 1\) , \( 157 a^{4} + 45 a^{3} - 735 a^{2} - 306 a + 246\) , \( 1904 a^{4} + 518 a^{3} - 8879 a^{2} - 3657 a + 2995\bigr] \) |
${y}^2+a{x}{y}+\left(a^{4}+a^{3}-4a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+2\right){x}^{2}+\left(157a^{4}+45a^{3}-735a^{2}-306a+246\right){x}+1904a^{4}+518a^{3}-8879a^{2}-3657a+2995$ |
| 4.2-b3 |
4.2-b |
$4$ |
$6$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.2 |
\( 2^{2} \) |
\( - 2^{12} \) |
$37.72538$ |
$(-a^4+a^3+4a^2-3a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$2877.558700$ |
1.30491987 |
\( -\frac{342899}{64} a^{4} + \frac{609873}{64} a^{3} + \frac{571513}{32} a^{2} - \frac{274447}{8} a + \frac{398567}{32} \) |
\( \bigl[a\) , \( -a^{3} - a^{2} + 3 a + 2\) , \( a^{4} + a^{3} - 4 a^{2} - 3 a + 1\) , \( 2 a^{4} - 10 a^{2} - a + 6\) , \( 4 a^{4} - 21 a^{2} - 7 a + 8\bigr] \) |
${y}^2+a{x}{y}+\left(a^{4}+a^{3}-4a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+2\right){x}^{2}+\left(2a^{4}-10a^{2}-a+6\right){x}+4a^{4}-21a^{2}-7a+8$ |
| 4.2-b4 |
4.2-b |
$4$ |
$6$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.2 |
\( 2^{2} \) |
\( - 2^{6} \) |
$37.72538$ |
$(-a^4+a^3+4a^2-3a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$5755.117400$ |
1.30491987 |
\( -\frac{1896656913}{8} a^{4} + \frac{898908179}{2} a^{3} + \frac{3131230247}{4} a^{2} - \frac{6598271179}{4} a + 529331411 \) |
\( \bigl[a^{4} - 4 a^{2} + 2\) , \( -a^{3} - a^{2} + 4 a + 1\) , \( 1\) , \( -6 a^{4} - 6 a^{3} + 17 a^{2} + 8 a - 5\) , \( -10 a^{4} - 11 a^{3} + 26 a^{2} + 14 a - 10\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}-a^{2}+4a+1\right){x}^{2}+\left(-6a^{4}-6a^{3}+17a^{2}+8a-5\right){x}-10a^{4}-11a^{3}+26a^{2}+14a-10$ |
| 4.2-c1 |
4.2-c |
$4$ |
$10$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.2 |
\( 2^{2} \) |
\( - 2^{20} \) |
$37.72538$ |
$(-a^4+a^3+4a^2-3a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 2 \) |
$1.918832945$ |
$3.405819625$ |
1.11134713 |
\( -\frac{740019557895315941311935}{512} a^{4} - \frac{397360845260720523017779}{1024} a^{3} + \frac{1724037827284625591052763}{256} a^{2} + \frac{1413736642988691675515731}{512} a - \frac{1166781628696534563202829}{512} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{2} - a + 3\) , \( a^{4} + a^{3} - 4 a^{2} - 4 a + 1\) , \( 514 a^{4} - 330 a^{3} - 2655 a^{2} + 1045 a + 2288\) , \( -11965 a^{4} + 6709 a^{3} + 63063 a^{2} - 20600 a - 58068\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-4a+1\right){y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(514a^{4}-330a^{3}-2655a^{2}+1045a+2288\right){x}-11965a^{4}+6709a^{3}+63063a^{2}-20600a-58068$ |
| 4.2-c2 |
4.2-c |
$4$ |
$10$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.2 |
\( 2^{2} \) |
\( - 2^{4} \) |
$37.72538$ |
$(-a^4+a^3+4a^2-3a-1)$ |
$1$ |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2 \) |
$0.383766589$ |
$10643.18632$ |
1.11134713 |
\( -\frac{123471}{2} a^{4} + \frac{232381}{4} a^{3} + 282635 a^{2} - \frac{392669}{2} a - \frac{279965}{2} \) |
\( \bigl[a^{4} + a^{3} - 4 a^{2} - 3 a + 2\) , \( -a^{4} - a^{3} + 4 a^{2} + 4 a - 1\) , \( a^{4} + a^{3} - 4 a^{2} - 3 a + 1\) , \( -2 a^{4} + 12 a^{2} + 6 a - 4\) , \( 3 a^{3} + 7 a^{2} + a - 2\bigr] \) |
${y}^2+\left(a^{4}+a^{3}-4a^{2}-3a+2\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{4}-a^{3}+4a^{2}+4a-1\right){x}^{2}+\left(-2a^{4}+12a^{2}+6a-4\right){x}+3a^{3}+7a^{2}+a-2$ |
| 4.2-c3 |
4.2-c |
$4$ |
$10$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.2 |
\( 2^{2} \) |
\( - 2^{2} \) |
$37.72538$ |
$(-a^4+a^3+4a^2-3a-1)$ |
$1$ |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 1 \) |
$0.767533178$ |
$10643.18632$ |
1.11134713 |
\( 6391993330 a^{4} - \frac{7406636871}{2} a^{3} - 33517698568 a^{2} + 11469365389 a + 30392352151 \) |
\( \bigl[a^{4} + a^{3} - 4 a^{2} - 4 a + 2\) , \( a^{4} + a^{3} - 5 a^{2} - 4 a + 2\) , \( a^{2} - 1\) , \( 32 a^{4} - 57 a^{3} - 104 a^{2} + 214 a - 69\) , \( 60 a^{4} - 108 a^{3} - 197 a^{2} + 402 a - 128\bigr] \) |
${y}^2+\left(a^{4}+a^{3}-4a^{2}-4a+2\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{4}+a^{3}-5a^{2}-4a+2\right){x}^{2}+\left(32a^{4}-57a^{3}-104a^{2}+214a-69\right){x}+60a^{4}-108a^{3}-197a^{2}+402a-128$ |
| 4.2-c4 |
4.2-c |
$4$ |
$10$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
4.2 |
\( 2^{2} \) |
\( - 2^{10} \) |
$37.72538$ |
$(-a^4+a^3+4a^2-3a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 1 \) |
$3.837665891$ |
$3.405819625$ |
1.11134713 |
\( -\frac{8517265234167}{8} a^{4} + \frac{1529274209840145}{32} a^{3} + \frac{30380157421603}{2} a^{2} - \frac{2134375507546139}{16} a + \frac{798411274673343}{16} \) |
\( \bigl[a^{4} - 4 a^{2} + 2\) , \( -a^{3} - a^{2} + 3 a + 2\) , \( a^{4} + a^{3} - 5 a^{2} - 3 a + 3\) , \( -96 a^{4} - 163 a^{3} + 879 a^{2} + 672 a - 1851\) , \( -828 a^{4} - 3179 a^{3} + 10081 a^{2} + 12795 a - 24800\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+2\right){x}{y}+\left(a^{4}+a^{3}-5a^{2}-3a+3\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+2\right){x}^{2}+\left(-96a^{4}-163a^{3}+879a^{2}+672a-1851\right){x}-828a^{4}-3179a^{3}+10081a^{2}+12795a-24800$ |
| 8.1-a1 |
8.1-a |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{11} \) |
$40.43306$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$0.489730859$ |
$1537.222331$ |
2.56044277 |
\( 77494575090 a^{4} - 228210387023 a^{3} + 56363527918 a^{2} + 200359342781 a - 79691826938 \) |
\( \bigl[a\) , \( -a^{4} + a^{3} + 4 a^{2} - 3 a\) , \( a^{4} - 4 a^{2} + a + 2\) , \( -5 a^{4} + 2 a^{3} + 18 a^{2} - 3\) , \( 45 a^{4} + 9 a^{3} - 206 a^{2} - 85 a + 70\bigr] \) |
${y}^2+a{x}{y}+\left(a^{4}-4a^{2}+a+2\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}-3a\right){x}^{2}+\left(-5a^{4}+2a^{3}+18a^{2}-3\right){x}+45a^{4}+9a^{3}-206a^{2}-85a+70$ |
| 8.1-a2 |
8.1-a |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{4} \) |
$40.43306$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.061216357$ |
$6148.889324$ |
2.56044277 |
\( -1212363 a^{4} - 335844 a^{3} + 5661974 a^{2} + 2341590 a - 1923192 \) |
\( \bigl[a^{2} + a - 2\) , \( a^{3} + a^{2} - 3 a - 2\) , \( a^{4} - 5 a^{2} + a + 4\) , \( 5 a^{4} - 16 a^{2} + 8 a + 4\) , \( 2 a^{4} + 8 a^{3} - 2 a^{2} - 17 a + 3\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{4}-5a^{2}+a+4\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-2\right){x}^{2}+\left(5a^{4}-16a^{2}+8a+4\right){x}+2a^{4}+8a^{3}-2a^{2}-17a+3$ |
| 8.1-a3 |
8.1-a |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$40.43306$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.244865429$ |
$6148.889324$ |
2.56044277 |
\( 1031892 a^{4} + 992717 a^{3} - 2442519 a^{2} - 1279950 a + 880638 \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} - a^{2} + 3 a + 1\) , \( a^{3} - 3 a\) , \( 7 a^{4} + 3 a^{3} - 35 a^{2} - 17 a + 11\) , \( 25 a^{4} + 5 a^{3} - 114 a^{2} - 46 a + 38\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+1\right){x}^{2}+\left(7a^{4}+3a^{3}-35a^{2}-17a+11\right){x}+25a^{4}+5a^{3}-114a^{2}-46a+38$ |
| 8.1-a4 |
8.1-a |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{8} \) |
$40.43306$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.122432714$ |
$12297.77864$ |
2.56044277 |
\( -32968927 a^{4} + 62502895 a^{3} + 108855000 a^{2} - 229389554 a + 73613506 \) |
\( \bigl[a^{4} + a^{3} - 5 a^{2} - 3 a + 4\) , \( a^{2} - a - 2\) , \( a^{2} + a - 2\) , \( 48 a^{4} + 16 a^{3} - 221 a^{2} - 103 a + 69\) , \( -203 a^{4} - 52 a^{3} + 949 a^{2} + 382 a - 327\bigr] \) |
${y}^2+\left(a^{4}+a^{3}-5a^{2}-3a+4\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(48a^{4}+16a^{3}-221a^{2}-103a+69\right){x}-203a^{4}-52a^{3}+949a^{2}+382a-327$ |
| 8.1-a5 |
8.1-a |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{11} \) |
$40.43306$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$4$ |
\( 1 \) |
$0.489730859$ |
$384.3055827$ |
2.56044277 |
\( 18560530350058 a^{4} + 21373162192979 a^{3} - 46817401864278 a^{2} - 26487272767469 a + 17253397824258 \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} - a^{2} + 3 a + 1\) , \( a^{3} - 3 a\) , \( 122 a^{4} + 43 a^{3} - 565 a^{2} - 277 a + 161\) , \( 1442 a^{4} + 368 a^{3} - 6728 a^{2} - 2678 a + 2338\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+1\right){x}^{2}+\left(122a^{4}+43a^{3}-565a^{2}-277a+161\right){x}+1442a^{4}+368a^{3}-6728a^{2}-2678a+2338$ |
| 8.1-a6 |
8.1-a |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$40.43306$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.244865429$ |
$1537.222331$ |
2.56044277 |
\( -9900431112339440 a^{4} + 18769171904102413 a^{3} + 32688855357226927 a^{2} - 68884185154748454 a + 22104273569186590 \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{4} + a^{3} + 5 a^{2} - 4 a - 4\) , \( a^{4} + a^{3} - 4 a^{2} - 4 a + 2\) , \( -41 a^{4} - 22 a^{3} + 195 a^{2} + 120 a - 76\) , \( -284 a^{4} - 64 a^{3} + 1315 a^{2} + 490 a - 428\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-4a+2\right){y}={x}^{3}+\left(-a^{4}+a^{3}+5a^{2}-4a-4\right){x}^{2}+\left(-41a^{4}-22a^{3}+195a^{2}+120a-76\right){x}-284a^{4}-64a^{3}+1315a^{2}+490a-428$ |
| 8.1-b1 |
8.1-b |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{8} \) |
$40.43306$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$17.41748236$ |
0.758256717 |
\( 26632420060253126 a^{4} - 78428649152468163 a^{3} + 19370350420989990 a^{2} + 68857132133829512 a - 27387545582998666 \) |
\( \bigl[a^{4} - 5 a^{2} + a + 4\) , \( a^{4} - 5 a^{2} + a + 2\) , \( a^{3} + a^{2} - 4 a - 2\) , \( 16 a^{4} - 56 a^{3} - 46 a^{2} + 194 a - 98\) , \( 85 a^{4} - 331 a^{3} - 242 a^{2} + 1116 a - 574\bigr] \) |
${y}^2+\left(a^{4}-5a^{2}+a+4\right){x}{y}+\left(a^{3}+a^{2}-4a-2\right){y}={x}^{3}+\left(a^{4}-5a^{2}+a+2\right){x}^{2}+\left(16a^{4}-56a^{3}-46a^{2}+194a-98\right){x}+85a^{4}-331a^{3}-242a^{2}+1116a-574$ |
| 8.1-b2 |
8.1-b |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{4} \) |
$40.43306$ |
$(a)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8917.750970$ |
0.758256717 |
\( -20226 a^{4} - 4635 a^{3} + 93864 a^{2} + 35590 a - 30538 \) |
\( \bigl[a^{4} - 5 a^{2} + a + 4\) , \( a^{3} - 5 a + 1\) , \( a^{4} + a^{3} - 5 a^{2} - 4 a + 4\) , \( 10 a^{4} + 13 a^{3} - 22 a^{2} - 18 a + 9\) , \( 76 a^{4} + 85 a^{3} - 194 a^{2} - 106 a + 70\bigr] \) |
${y}^2+\left(a^{4}-5a^{2}+a+4\right){x}{y}+\left(a^{4}+a^{3}-5a^{2}-4a+4\right){y}={x}^{3}+\left(a^{3}-5a+1\right){x}^{2}+\left(10a^{4}+13a^{3}-22a^{2}-18a+9\right){x}+76a^{4}+85a^{3}-194a^{2}-106a+70$ |
| 8.1-b3 |
8.1-b |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{8} \) |
$40.43306$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$8917.750970$ |
0.758256717 |
\( 1146 a^{4} + 857 a^{3} - 5732 a^{2} - 11346 a + 20814 \) |
\( \bigl[a^{4} - 5 a^{2} + 4\) , \( -a^{4} + 5 a^{2} - a - 2\) , \( a^{3} + a^{2} - 4 a - 2\) , \( a^{4} - 5 a^{3} - 6 a^{2} + 12 a + 2\) , \( 4 a^{4} - 2 a^{3} - 11 a^{2} + 12 a - 6\bigr] \) |
${y}^2+\left(a^{4}-5a^{2}+4\right){x}{y}+\left(a^{3}+a^{2}-4a-2\right){y}={x}^{3}+\left(-a^{4}+5a^{2}-a-2\right){x}^{2}+\left(a^{4}-5a^{3}-6a^{2}+12a+2\right){x}+4a^{4}-2a^{3}-11a^{2}+12a-6$ |
| 8.1-b4 |
8.1-b |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{4} \) |
$40.43306$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$557.3594356$ |
0.758256717 |
\( -44400618 a^{4} + 53862389 a^{3} + 261873552 a^{2} - 432299994 a + 132891494 \) |
\( \bigl[a^{4} - 5 a^{2} + a + 4\) , \( a^{4} - 5 a^{2} + a + 2\) , \( a^{3} + a^{2} - 4 a - 2\) , \( 31 a^{4} - 56 a^{3} - 106 a^{2} + 204 a - 53\) , \( 157 a^{4} - 300 a^{3} - 518 a^{2} + 1099 a - 358\bigr] \) |
${y}^2+\left(a^{4}-5a^{2}+a+4\right){x}{y}+\left(a^{3}+a^{2}-4a-2\right){y}={x}^{3}+\left(a^{4}-5a^{2}+a+2\right){x}^{2}+\left(31a^{4}-56a^{3}-106a^{2}+204a-53\right){x}+157a^{4}-300a^{3}-518a^{2}+1099a-358$ |
| 8.1-b5 |
8.1-b |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{4} \) |
$40.43306$ |
$(a)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8917.750970$ |
0.758256717 |
\( 45416486 a^{4} - 25905713 a^{3} - 237509244 a^{2} + 80734850 a + 215043810 \) |
\( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( a^{3} - 5 a + 1\) , \( a^{4} + a^{3} - 5 a^{2} - 3 a + 4\) , \( -2 a^{2} - 5 a + 7\) , \( a^{4} - 2 a^{3} - 4 a^{2} + 5 a - 2\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-2\right){x}{y}+\left(a^{4}+a^{3}-5a^{2}-3a+4\right){y}={x}^{3}+\left(a^{3}-5a+1\right){x}^{2}+\left(-2a^{2}-5a+7\right){x}+a^{4}-2a^{3}-4a^{2}+5a-2$ |
| 8.1-b6 |
8.1-b |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{8} \) |
$40.43306$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$17.41748236$ |
0.758256717 |
\( -48257426918163398 a^{4} + 91485793982106307 a^{3} + 159334679541267930 a^{2} - 335759183157168520 a + 107741756513224282 \) |
\( \bigl[a^{4} - 5 a^{2} + a + 4\) , \( a^{4} - 5 a^{2} + a + 2\) , \( a^{3} + a^{2} - 4 a - 2\) , \( 466 a^{4} - 876 a^{3} - 1546 a^{2} + 3214 a - 1008\) , \( 11157 a^{4} - 21151 a^{3} - 36840 a^{2} + 77624 a - 24906\bigr] \) |
${y}^2+\left(a^{4}-5a^{2}+a+4\right){x}{y}+\left(a^{3}+a^{2}-4a-2\right){y}={x}^{3}+\left(a^{4}-5a^{2}+a+2\right){x}^{2}+\left(466a^{4}-876a^{3}-1546a^{2}+3214a-1008\right){x}+11157a^{4}-21151a^{3}-36840a^{2}+77624a-24906$ |
| 8.1-c1 |
8.1-c |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{8} \) |
$40.43306$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.026313655$ |
$6368.045616$ |
2.27965043 |
\( -6387897 a^{4} - 2021991 a^{3} + 30276582 a^{2} + 12631780 a - 10319650 \) |
\( \bigl[a^{4} - 5 a^{2} + 4\) , \( a^{4} - a^{3} - 4 a^{2} + 4 a + 2\) , \( a^{4} + a^{3} - 5 a^{2} - 4 a + 4\) , \( -a^{4} + 4 a^{3} - a^{2} - 16 a + 14\) , \( -3 a^{4} + 9 a^{3} + 6 a^{2} - 34 a + 18\bigr] \) |
${y}^2+\left(a^{4}-5a^{2}+4\right){x}{y}+\left(a^{4}+a^{3}-5a^{2}-4a+4\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+4a+2\right){x}^{2}+\left(-a^{4}+4a^{3}-a^{2}-16a+14\right){x}-3a^{4}+9a^{3}+6a^{2}-34a+18$ |
| 8.1-c2 |
8.1-c |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{10} \) |
$40.43306$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.210509242$ |
$1592.011404$ |
2.27965043 |
\( -26874886 a^{4} + 16955439 a^{3} + 110061999 a^{2} - 47698422 a + 158150 \) |
\( \bigl[a\) , \( a^{2} - 2\) , \( a^{4} - 4 a^{2} + 2\) , \( 2 a^{4} + 5 a^{3} - 5 a^{2} - 25 a - 16\) , \( 38 a^{4} - 9 a^{3} - 179 a^{2} - 4 a + 109\bigr] \) |
${y}^2+a{x}{y}+\left(a^{4}-4a^{2}+2\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(2a^{4}+5a^{3}-5a^{2}-25a-16\right){x}+38a^{4}-9a^{3}-179a^{2}-4a+109$ |
| 8.1-c3 |
8.1-c |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{4} \) |
$40.43306$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.052627310$ |
$25472.18246$ |
2.27965043 |
\( 7166493 a^{4} + 8250422 a^{3} - 18081098 a^{2} - 10222758 a + 6673620 \) |
\( \bigl[a\) , \( -a^{4} + a^{3} + 5 a^{2} - 3 a - 3\) , \( a^{4} - 4 a^{2} + 2\) , \( 15 a^{4} + 3 a^{3} - 70 a^{2} - 25 a + 25\) , \( -37 a^{4} - 10 a^{3} + 172 a^{2} + 70 a - 59\bigr] \) |
${y}^2+a{x}{y}+\left(a^{4}-4a^{2}+2\right){y}={x}^{3}+\left(-a^{4}+a^{3}+5a^{2}-3a-3\right){x}^{2}+\left(15a^{4}+3a^{3}-70a^{2}-25a+25\right){x}-37a^{4}-10a^{3}+172a^{2}+70a-59$ |
| 8.1-c4 |
8.1-c |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{8} \) |
$40.43306$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.105254621$ |
$12736.09123$ |
2.27965043 |
\( 32036669 a^{4} - 18562781 a^{3} - 167985144 a^{2} + 57484142 a + 152331306 \) |
\( \bigl[a\) , \( a^{3} - a^{2} - 4 a + 2\) , \( a^{2} + a - 2\) , \( -a^{4} + a^{3} + 2 a^{2} - 3 a + 1\) , \( -a^{3} + a^{2} + a - 1\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+2\right){x}^{2}+\left(-a^{4}+a^{3}+2a^{2}-3a+1\right){x}-a^{3}+a^{2}+a-1$ |
| 8.1-c5 |
8.1-c |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{8} \) |
$40.43306$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.105254621$ |
$3184.022808$ |
2.27965043 |
\( 979660506644517 a^{4} + 1128116323631105 a^{3} - 2471117395902660 a^{2} - 1398061018677258 a + 910660662690686 \) |
\( \bigl[a\) , \( -a^{4} + a^{3} + 5 a^{2} - 3 a - 3\) , \( a^{4} - 4 a^{2} + 2\) , \( -20 a^{4} - 17 a^{3} + 100 a^{2} + 85 a - 50\) , \( -250 a^{4} - 46 a^{3} + 1151 a^{2} + 390 a - 357\bigr] \) |
${y}^2+a{x}{y}+\left(a^{4}-4a^{2}+2\right){y}={x}^{3}+\left(-a^{4}+a^{3}+5a^{2}-3a-3\right){x}^{2}+\left(-20a^{4}-17a^{3}+100a^{2}+85a-50\right){x}-250a^{4}-46a^{3}+1151a^{2}+390a-357$ |
| 8.1-c6 |
8.1-c |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$40.43306$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.210509242$ |
$1592.011404$ |
2.27965043 |
\( 4690637708984362 a^{4} - 2717606932742159 a^{3} - 24596299833901223 a^{2} + 8416567498539054 a + 22302825599075082 \) |
\( \bigl[a\) , \( a^{3} - a^{2} - 4 a + 2\) , \( a^{2} + a - 2\) , \( -6 a^{4} + 11 a^{3} + 7 a^{2} - 13 a + 1\) , \( 15 a^{4} - 44 a^{3} + 8 a^{2} + 37 a - 13\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+2\right){x}^{2}+\left(-6a^{4}+11a^{3}+7a^{2}-13a+1\right){x}+15a^{4}-44a^{3}+8a^{2}+37a-13$ |
| 8.2-a1 |
8.2-a |
$2$ |
$2$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{38} \) |
$40.43306$ |
$(a), (-a^4+a^3+4a^2-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$821.0981839$ |
2.23411743 |
\( \frac{519721311677}{262144} a^{4} - \frac{332132344695}{262144} a^{3} - \frac{1332877916535}{131072} a^{2} + \frac{125522017123}{32768} a + \frac{1174375611055}{131072} \) |
\( \bigl[a^{4} + a^{3} - 5 a^{2} - 4 a + 3\) , \( -a^{3} - a^{2} + 5 a + 3\) , \( a^{4} - 5 a^{2} + a + 3\) , \( -32 a^{4} + a^{3} + 180 a^{2} + 13 a - 190\) , \( 80 a^{4} - 10 a^{3} - 482 a^{2} + 3 a + 607\bigr] \) |
${y}^2+\left(a^{4}+a^{3}-5a^{2}-4a+3\right){x}{y}+\left(a^{4}-5a^{2}+a+3\right){y}={x}^{3}+\left(-a^{3}-a^{2}+5a+3\right){x}^{2}+\left(-32a^{4}+a^{3}+180a^{2}+13a-190\right){x}+80a^{4}-10a^{3}-482a^{2}+3a+607$ |
| 8.2-a2 |
8.2-a |
$2$ |
$2$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{22} \) |
$40.43306$ |
$(a), (-a^4+a^3+4a^2-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$821.0981839$ |
2.23411743 |
\( -\frac{11138664729995}{512} a^{4} + \frac{2640166169917}{64} a^{3} + \frac{18389207506285}{256} a^{2} - \frac{38756215289445}{256} a + \frac{388651940119}{8} \) |
\( \bigl[a^{4} - 4 a^{2} + a + 1\) , \( a^{2} - 3\) , \( a^{4} + a^{3} - 4 a^{2} - 3 a + 1\) , \( -a^{4} + 11 a^{2} + 4 a - 23\) , \( 14 a^{4} + a^{3} - 78 a^{2} - 4 a + 87\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+a+1\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-3a+1\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-a^{4}+11a^{2}+4a-23\right){x}+14a^{4}+a^{3}-78a^{2}-4a+87$ |
| 8.2-b1 |
8.2-b |
$2$ |
$5$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{15} \) |
$40.43306$ |
$(a), (-a^4+a^3+4a^2-3a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$625$ |
\( 5 \) |
$1$ |
$0.225946325$ |
1.92117183 |
\( \frac{15036425161345293015684423}{32} a^{4} - \frac{28505955559700134315153085}{32} a^{3} - \frac{24823339543336714873353281}{16} a^{2} + \frac{6538679260608748572495303}{2} a - \frac{16785595236289982205609819}{16} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( -a^{4} + a^{3} + 5 a^{2} - 3 a - 4\) , \( 0\) , \( -6573 a^{4} - 7877 a^{3} + 16493 a^{2} + 10242 a - 6435\) , \( -953587 a^{4} - 1101967 a^{3} + 2404253 a^{2} + 1371692 a - 890501\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}={x}^{3}+\left(-a^{4}+a^{3}+5a^{2}-3a-4\right){x}^{2}+\left(-6573a^{4}-7877a^{3}+16493a^{2}+10242a-6435\right){x}-953587a^{4}-1101967a^{3}+2404253a^{2}+1371692a-890501$ |
| 8.2-b2 |
8.2-b |
$2$ |
$5$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{27} \) |
$40.43306$ |
$(a), (-a^4+a^3+4a^2-3a-1)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5^{2} \) |
$1$ |
$706.0822679$ |
1.92117183 |
\( \frac{4735193}{512} a^{4} - \frac{8725221}{512} a^{3} - \frac{15740273}{512} a^{2} + \frac{16087109}{256} a - \frac{5348079}{256} \) |
\( \bigl[a^{4} + a^{3} - 4 a^{2} - 4 a + 1\) , \( a^{3} - 5 a - 1\) , \( a^{4} - 4 a^{2} + a + 1\) , \( -a^{4} - a^{3} + 6 a^{2} + 3 a - 10\) , \( -5 a^{3} + 6 a^{2} + 19 a - 23\bigr] \) |
${y}^2+\left(a^{4}+a^{3}-4a^{2}-4a+1\right){x}{y}+\left(a^{4}-4a^{2}+a+1\right){y}={x}^{3}+\left(a^{3}-5a-1\right){x}^{2}+\left(-a^{4}-a^{3}+6a^{2}+3a-10\right){x}-5a^{3}+6a^{2}+19a-23$ |
| 8.2-c1 |
8.2-c |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{5} \) |
$40.43306$ |
$(a), (-a^4+a^3+4a^2-3a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$0.077690911$ |
$790.9120897$ |
1.67189629 |
\( \frac{14245650079897091793}{4} a^{4} - \frac{41951392040178417467}{4} a^{3} + 2590294421750350214 a^{2} + 9207899717692489572 a - \frac{7324782945359101081}{2} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{4} - 4 a^{2} + a + 1\) , \( a^{4} + a^{3} - 5 a^{2} - 3 a + 4\) , \( 24 a^{4} - 32 a^{3} - 69 a^{2} + 129 a - 65\) , \( 381 a^{4} - 332 a^{3} - 1779 a^{2} + 1152 a + 1075\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+\left(a^{4}+a^{3}-5a^{2}-3a+4\right){y}={x}^{3}+\left(a^{4}-4a^{2}+a+1\right){x}^{2}+\left(24a^{4}-32a^{3}-69a^{2}+129a-65\right){x}+381a^{4}-332a^{3}-1779a^{2}+1152a+1075$ |
| 8.2-c2 |
8.2-c |
$6$ |
$8$ |
5.5.135076.1 |
$5$ |
$[5, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{10} \) |
$40.43306$ |
$(a), (-a^4+a^3+4a^2-3a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.038845455$ |
$12654.59343$ |
1.67189629 |
\( \frac{5283434917}{16} a^{4} - \frac{7779112811}{8} a^{3} + \frac{1920453169}{8} a^{2} + \frac{6829940129}{8} a - \frac{1357845841}{4} \) |
\( \bigl[a^{4} + a^{3} - 5 a^{2} - 3 a + 3\) , \( a^{4} + a^{3} - 4 a^{2} - 4 a + 1\) , \( a^{3} - 3 a\) , \( 157 a^{4} + 45 a^{3} - 727 a^{2} - 305 a + 241\) , \( 1152 a^{4} + 309 a^{3} - 5364 a^{2} - 2189 a + 1823\bigr] \) |
${y}^2+\left(a^{4}+a^{3}-5a^{2}-3a+3\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{4}+a^{3}-4a^{2}-4a+1\right){x}^{2}+\left(157a^{4}+45a^{3}-727a^{2}-305a+241\right){x}+1152a^{4}+309a^{3}-5364a^{2}-2189a+1823$ |
*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.
