| 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 |
| 7.2-a1 |
7.2-a |
$1$ |
$1$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
7.2 |
\( 7 \) |
\( -7 \) |
$10.99109$ |
$(a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.026207781$ |
$1494.446755$ |
1.624448239 |
\( -\frac{73728}{7} a^{3} + \frac{12288}{7} a^{2} + \frac{364544}{7} a + \frac{225280}{7} \) |
\( \bigl[0\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{3} - 5 a - 3\) , \( 2 a^{3} - 3 a^{2} - 10 a + 2\) , \( 2 a^{2} + a - 4\bigr] \) |
${y}^2+\left(a^{3}-5a-3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-4\right){x}^{2}+\left(2a^{3}-3a^{2}-10a+2\right){x}+2a^{2}+a-4$ |
| 15.1-a1 |
15.1-a |
$6$ |
$8$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{6} \cdot 5^{24} \) |
$12.08968$ |
$(-a), (-a^3+a^2+4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{4} \cdot 3 \) |
$1.705449550$ |
$2.795036312$ |
2.372481019 |
\( -\frac{1838899427339088360320392720528}{43451786041259765625} a^{3} + \frac{239083282688250681080233210279}{43451786041259765625} a^{2} + \frac{9402497034463545997974102358322}{43451786041259765625} a + \frac{6341137901789312627182533751898}{43451786041259765625} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 5 a + 1\) , \( a^{3} - a^{2} - 4 a\) , \( 115 a^{3} - 44 a^{2} - 545 a - 335\) , \( 1371 a^{3} - 299 a^{2} - 6868 a - 4524\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+1\right){x}^{2}+\left(115a^{3}-44a^{2}-545a-335\right){x}+1371a^{3}-299a^{2}-6868a-4524$ |
| 15.1-a2 |
15.1-a |
$6$ |
$8$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{3} \cdot 5^{3} \) |
$12.08968$ |
$(-a), (-a^3+a^2+4a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 3 \) |
$0.852724775$ |
$357.7646480$ |
2.372481019 |
\( \frac{2135315017580596621}{3375} a^{3} - \frac{3808789312094362603}{3375} a^{2} - \frac{7686059730886656329}{3375} a + \frac{8163497292155544289}{3375} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 5 a + 1\) , \( a^{3} - a^{2} - 4 a\) , \( -15 a^{3} + 26 a^{2} + 40 a - 75\) , \( 69 a^{3} - 120 a^{2} - 229 a + 267\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+1\right){x}^{2}+\left(-15a^{3}+26a^{2}+40a-75\right){x}+69a^{3}-120a^{2}-229a+267$ |
| 15.1-a3 |
15.1-a |
$6$ |
$8$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{12} \cdot 5^{12} \) |
$12.08968$ |
$(-a), (-a^3+a^2+4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3Ns |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.852724775$ |
$44.72058100$ |
2.372481019 |
\( -\frac{5608173175704549785027}{129746337890625} a^{3} + \frac{15053308221079393467461}{129746337890625} a^{2} + \frac{2624391126667291079623}{129746337890625} a - \frac{9911446235169938034143}{129746337890625} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 5 a + 1\) , \( a^{3} - a^{2} - 4 a\) , \( 15 a^{3} - 24 a^{2} - 30 a - 5\) , \( 69 a^{3} - 106 a^{2} - 175 a - 33\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+1\right){x}^{2}+\left(15a^{3}-24a^{2}-30a-5\right){x}+69a^{3}-106a^{2}-175a-33$ |
| 15.1-a4 |
15.1-a |
$6$ |
$8$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{6} \cdot 5^{6} \) |
$12.08968$ |
$(-a), (-a^3+a^2+4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3Ns |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.426362387$ |
$715.5292960$ |
2.372481019 |
\( \frac{128092594571557}{11390625} a^{3} - \frac{225657720048376}{11390625} a^{2} - \frac{458845597722218}{11390625} a + \frac{485659032605788}{11390625} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 5 a + 1\) , \( a^{3} - a^{2} - 4 a\) , \( a^{2} + 5 a\) , \( 3 a^{3} - 3 a^{2} - 8 a + 3\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+1\right){x}^{2}+\left(a^{2}+5a\right){x}+3a^{3}-3a^{2}-8a+3$ |
| 15.1-a5 |
15.1-a |
$6$ |
$8$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{3} \cdot 5^{3} \) |
$12.08968$ |
$(-a), (-a^3+a^2+4a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 3 \) |
$0.213181193$ |
$1431.058592$ |
2.372481019 |
\( -\frac{6637246}{3375} a^{3} + \frac{11349478}{3375} a^{2} + \frac{20884829}{3375} a - \frac{16775914}{3375} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 5 a + 1\) , \( a^{3} - a^{2} - 4 a\) , \( a^{2} + 5 a + 5\) , \( 2 a^{2} + 4 a + 1\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+1\right){x}^{2}+\left(a^{2}+5a+5\right){x}+2a^{2}+4a+1$ |
| 15.1-a6 |
15.1-a |
$6$ |
$8$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{24} \cdot 5^{6} \) |
$12.08968$ |
$(-a), (-a^3+a^2+4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$4$ |
\( 2^{2} \cdot 3 \) |
$1.705449550$ |
$2.795036312$ |
2.372481019 |
\( -\frac{81577402532451076166045520603248}{4412961507515625} a^{3} + \frac{219331599661591677554721948716489}{4412961507515625} a^{2} + \frac{37516695901889429460362076037102}{4412961507515625} a - \frac{144929288746881365623568260159082}{4412961507515625} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 5 a + 1\) , \( a^{3} - a^{2} - 4 a\) , \( 155 a^{3} - 404 a^{2} - 75 a + 245\) , \( 2371 a^{3} - 6265 a^{2} - 1250 a + 3994\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+1\right){x}^{2}+\left(155a^{3}-404a^{2}-75a+245\right){x}+2371a^{3}-6265a^{2}-1250a+3994$ |
| 16.1-a1 |
16.1-a |
$2$ |
$3$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{24} \) |
$12.18761$ |
$(2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.027664224$ |
$228.6471171$ |
2.098791920 |
\( -\frac{13849113}{64} a^{3} + \frac{6178847}{16} a^{2} + \frac{24941015}{32} a - \frac{13231507}{16} \) |
\( \bigl[a^{3} - 4 a - 3\) , \( -a^{3} + a^{2} + 5 a + 1\) , \( a^{2} - a - 3\) , \( -9 a^{3} + 9 a^{2} + 48 a + 13\) , \( -4 a^{3} - 5 a^{2} + 43 a + 67\bigr] \) |
${y}^2+\left(a^{3}-4a-3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+1\right){x}^{2}+\left(-9a^{3}+9a^{2}+48a+13\right){x}-4a^{3}-5a^{2}+43a+67$ |
| 16.1-a2 |
16.1-a |
$2$ |
$3$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$12.18761$ |
$(2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.003073802$ |
$2057.824054$ |
2.098791920 |
\( -\frac{435}{2} a^{3} - \frac{3137}{4} a^{2} - \frac{2943}{4} a - \frac{325}{4} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( -a^{3} + 2 a^{2} + 4 a - 2\) , \( a^{3} - 5 a - 2\) , \( 8 a^{3} - 12 a^{2} - 30 a + 28\) , \( 59 a^{3} - 103 a^{2} - 213 a + 219\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-2\right){x}^{2}+\left(8a^{3}-12a^{2}-30a+28\right){x}+59a^{3}-103a^{2}-213a+219$ |
| 21.1-a1 |
21.1-a |
$4$ |
$6$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{24} \cdot 7^{2} \) |
$12.60900$ |
$(-a), (a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.107036923$ |
$59.73912353$ |
3.182503427 |
\( \frac{12139958654477371634599}{13839047287569} a^{3} - \frac{21663517130798003313736}{13839047287569} a^{2} - \frac{6243629356318396630157}{1977006755367} a + \frac{46425880180445182133308}{13839047287569} \) |
\( \bigl[a^{3} - 5 a - 2\) , \( a^{3} - 6 a - 3\) , \( a + 1\) , \( -866 a^{3} + 1538 a^{2} + 3134 a - 3315\) , \( 21757 a^{3} - 38844 a^{2} - 78267 a + 83164\bigr] \) |
${y}^2+\left(a^{3}-5a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-6a-3\right){x}^{2}+\left(-866a^{3}+1538a^{2}+3134a-3315\right){x}+21757a^{3}-38844a^{2}-78267a+83164$ |
| 21.1-a2 |
21.1-a |
$4$ |
$6$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{12} \cdot 7 \) |
$12.60900$ |
$(-a), (a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.214073847$ |
$119.4782470$ |
3.182503427 |
\( \frac{142974922238248}{3720087} a^{3} + \frac{229371570224327}{3720087} a^{2} - \frac{16869411841703}{531441} a - \frac{165012873824636}{3720087} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{3} + 6 a + 3\) , \( a^{2} - 2\) , \( -13 a^{3} + 10 a^{2} + 58 a + 6\) , \( 3 a^{3} - 22 a^{2} + 3 a + 82\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+6a+3\right){x}^{2}+\left(-13a^{3}+10a^{2}+58a+6\right){x}+3a^{3}-22a^{2}+3a+82$ |
| 21.1-a3 |
21.1-a |
$4$ |
$6$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{8} \cdot 7^{6} \) |
$12.60900$ |
$(-a), (a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$0.035678974$ |
$537.6521118$ |
3.182503427 |
\( -\frac{1074405343192445}{771895089} a^{3} + \frac{234555270075566}{771895089} a^{2} + \frac{760113922146847}{110270727} a + \frac{3512945019740809}{771895089} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( a^{3} - 5 a - 2\) , \( a^{3} - a^{2} - 4 a\) , \( 6 a^{3} - 30 a - 23\) , \( -18 a^{3} + 3 a^{2} + 92 a + 60\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(6a^{3}-30a-23\right){x}-18a^{3}+3a^{2}+92a+60$ |
| 21.1-a4 |
21.1-a |
$4$ |
$6$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{4} \cdot 7^{3} \) |
$12.60900$ |
$(-a), (a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.071357949$ |
$1075.304223$ |
3.182503427 |
\( \frac{18355138}{27783} a^{3} - \frac{10539937}{27783} a^{2} - \frac{10358057}{3969} a - \frac{9284621}{27783} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( a^{3} - 5 a - 2\) , \( a^{3} - a^{2} - 4 a\) , \( a^{3} - 5 a - 3\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(a^{3}-5a-3\right){x}$ |
| 21.1-b1 |
21.1-b |
$2$ |
$2$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3 \cdot 7^{2} \) |
$12.60900$ |
$(-a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$669.5968752$ |
3.471511437 |
\( -\frac{117760324691}{147} a^{3} + \frac{14692717250}{147} a^{2} + \frac{85065399970}{21} a + \frac{427610123113}{147} \) |
\( \bigl[a^{3} - 5 a - 3\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - 13 a^{2} + 11 a + 23\) , \( 4 a^{3} - 20 a^{2} + 13 a + 25\bigr] \) |
${y}^2+\left(a^{3}-5a-3\right){x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+4\right){x}^{2}+\left(a^{3}-13a^{2}+11a+23\right){x}+4a^{3}-20a^{2}+13a+25$ |
| 21.1-b2 |
21.1-b |
$2$ |
$2$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7 \) |
$12.60900$ |
$(-a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$669.5968752$ |
3.471511437 |
\( \frac{1309228}{63} a^{3} + \frac{1116638}{63} a^{2} - \frac{490421}{9} a - \frac{3546203}{63} \) |
\( \bigl[a^{3} - 5 a - 3\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a^{3} - a^{2} - 4 a + 1\) , \( -4 a^{3} - 3 a^{2} + 21 a + 23\) , \( -8 a^{3} - a^{2} + 38 a + 28\bigr] \) |
${y}^2+\left(a^{3}-5a-3\right){x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+4\right){x}^{2}+\left(-4a^{3}-3a^{2}+21a+23\right){x}-8a^{3}-a^{2}+38a+28$ |
| 21.2-a1 |
21.2-a |
$2$ |
$3$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( 3^{18} \cdot 7^{6} \) |
$12.60900$ |
$(-a), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.051110412$ |
$48.62112098$ |
3.710496496 |
\( \frac{118218511942319081}{45579633110361} a^{3} - \frac{333005923573368578}{45579633110361} a^{2} - \frac{42455169182260615}{45579633110361} a + \frac{211096262657702147}{45579633110361} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a^{3} + 5 a + 3\) , \( a^{3} - 5 a - 3\) , \( -2 a^{3} + a^{2} + 10 a + 1\) , \( 18 a^{3} - 6 a^{2} - 88 a - 48\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a^{3}-5a-3\right){y}={x}^{3}+\left(-a^{3}+5a+3\right){x}^{2}+\left(-2a^{3}+a^{2}+10a+1\right){x}+18a^{3}-6a^{2}-88a-48$ |
| 21.2-a2 |
21.2-a |
$2$ |
$3$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( 3^{6} \cdot 7^{2} \) |
$12.60900$ |
$(-a), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.017036804$ |
$437.5900888$ |
3.710496496 |
\( -\frac{15284366469079}{35721} a^{3} - \frac{24502690450736}{35721} a^{2} + \frac{12638284830671}{35721} a + \frac{17614650371666}{35721} \) |
\( \bigl[1\) , \( a^{3} - 5 a - 3\) , \( a^{3} - 4 a - 2\) , \( -3 a^{3} + 12 a + 8\) , \( 2 a^{3} - 10 a - 7\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-4a-2\right){y}={x}^{3}+\left(a^{3}-5a-3\right){x}^{2}+\left(-3a^{3}+12a+8\right){x}+2a^{3}-10a-7$ |
| 21.2-b1 |
21.2-b |
$1$ |
$1$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( 3^{6} \cdot 7^{2} \) |
$12.60900$ |
$(-a), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.016289712$ |
$819.1538591$ |
2.213777913 |
\( -\frac{200982595}{35721} a^{3} + \frac{355439194}{35721} a^{2} + \frac{729940112}{35721} a - \frac{760490623}{35721} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} - 2 a^{2} - 2 a + 2\) , \( a^{2} - 3\) , \( -8 a^{3} + 3 a^{2} + 37 a + 24\) , \( 45 a^{3} - 7 a^{2} - 228 a - 153\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-2a+2\right){x}^{2}+\left(-8a^{3}+3a^{2}+37a+24\right){x}+45a^{3}-7a^{2}-228a-153$ |
| 21.2-c1 |
21.2-c |
$2$ |
$2$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{4} \cdot 7^{2} \) |
$12.60900$ |
$(-a), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.195751684$ |
$174.8702172$ |
2.839530393 |
\( -\frac{85939510025255}{3969} a^{3} + \frac{231497549184008}{3969} a^{2} + \frac{39485343160999}{3969} a - \frac{152977768925705}{3969} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( -a^{3} + 2 a^{2} + 3 a - 2\) , \( a^{2} - 2\) , \( 5 a^{2} - 9 a - 20\) , \( 24 a^{3} - 4 a^{2} - 128 a - 85\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-2\right){x}^{2}+\left(5a^{2}-9a-20\right){x}+24a^{3}-4a^{2}-128a-85$ |
| 21.2-c2 |
21.2-c |
$2$ |
$2$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7 \) |
$12.60900$ |
$(-a), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.391503368$ |
$349.7404344$ |
2.839530393 |
\( -\frac{1506950}{63} a^{3} - \frac{7460401}{63} a^{2} + \frac{21143365}{63} a + \frac{20539483}{63} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - a^{2} - 5 a\) , \( a^{2} - 3\) , \( 8 a^{3} - 15 a^{2} - 29 a + 31\) , \( 5 a^{3} - 9 a^{2} - 19 a + 18\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(8a^{3}-15a^{2}-29a+31\right){x}+5a^{3}-9a^{2}-19a+18$ |
| 23.1-a1 |
23.1-a |
$2$ |
$2$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( -23 \) |
$12.75321$ |
$(a^3-2a^2-3a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$699.3167136$ |
1.812796668 |
\( \frac{2498171}{23} a^{3} - \frac{4445065}{23} a^{2} - \frac{9012902}{23} a + \frac{9556518}{23} \) |
\( \bigl[a^{3} - 4 a - 3\) , \( -a^{2} + 3\) , \( 0\) , \( -4 a^{3} + 21 a + 16\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-4a-3\right){x}{y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-4a^{3}+21a+16\right){x}$ |
| 23.1-a2 |
23.1-a |
$2$ |
$2$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( - 23^{2} \) |
$12.75321$ |
$(a^3-2a^2-3a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$87.41458920$ |
1.812796668 |
\( \frac{31669540033177}{529} a^{3} - \frac{56513557506917}{529} a^{2} - \frac{114014117433480}{529} a + \frac{121111124789691}{529} \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - 2 a^{2} - 2 a + 2\) , \( a^{3} - a^{2} - 4 a\) , \( -19 a^{3} + 40 a^{2} + 54 a - 67\) , \( -67 a^{3} + 122 a^{2} + 234 a - 255\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(a^{3}-2a^{2}-2a+2\right){x}^{2}+\left(-19a^{3}+40a^{2}+54a-67\right){x}-67a^{3}+122a^{2}+234a-255$ |
| 35.1-a1 |
35.1-a |
$2$ |
$2$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( 5 \cdot 7^{6} \) |
$13.44039$ |
$(-a^3+a^2+4a+1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$320.7568951$ |
1.662957626 |
\( -\frac{1104456139558171}{588245} a^{3} + \frac{136298756772478}{588245} a^{2} + \frac{5653879195095484}{588245} a + \frac{3840606837039936}{588245} \) |
\( \bigl[a\) , \( a^{3} - a^{2} - 4 a\) , \( 0\) , \( -4 a - 8\) , \( -9 a^{3} + 2 a^{2} + 36 a + 12\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a^{3}-a^{2}-4a\right){x}^{2}+\left(-4a-8\right){x}-9a^{3}+2a^{2}+36a+12$ |
| 35.1-a2 |
35.1-a |
$2$ |
$2$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( - 5^{2} \cdot 7^{3} \) |
$13.44039$ |
$(-a^3+a^2+4a+1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$320.7568951$ |
1.662957626 |
\( -\frac{148667534}{8575} a^{3} + \frac{38455962}{8575} a^{2} + \frac{740896341}{8575} a + \frac{439033194}{8575} \) |
\( \bigl[a\) , \( a^{3} - a^{2} - 4 a\) , \( 0\) , \( a + 2\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a^{3}-a^{2}-4a\right){x}^{2}+\left(a+2\right){x}$ |
| 35.1-b1 |
35.1-b |
$2$ |
$3$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( 5^{3} \cdot 7^{6} \) |
$13.44039$ |
$(-a^3+a^2+4a+1), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$0.153914602$ |
$102.1460461$ |
3.912441037 |
\( -\frac{149303585836}{14706125} a^{3} + \frac{198050877473}{14706125} a^{2} + \frac{477202818614}{14706125} a - \frac{467649393349}{14706125} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + 4\) , \( a^{2} - 3\) , \( 24 a^{3} - 5 a^{2} - 123 a - 77\) , \( 699 a^{3} - 92 a^{2} - 3574 a - 2407\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(24a^{3}-5a^{2}-123a-77\right){x}+699a^{3}-92a^{2}-3574a-2407$ |
| 35.1-b2 |
35.1-b |
$2$ |
$3$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( 5 \cdot 7^{2} \) |
$13.44039$ |
$(-a^3+a^2+4a+1), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.051304867$ |
$919.3144152$ |
3.912441037 |
\( \frac{102096}{245} a^{3} - \frac{159018}{245} a^{2} - \frac{249574}{245} a + \frac{284909}{245} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - a^{2} - 5 a\) , \( a^{2} - 3\) , \( -a^{2} + 3\) , \( -a - 1\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-a^{2}+3\right){x}-a-1$ |
| 35.1-c1 |
35.1-c |
$1$ |
$1$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( 5 \cdot 7^{2} \) |
$13.44039$ |
$(-a^3+a^2+4a+1), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$36.15937190$ |
0.749870124 |
\( -\frac{1511373427094}{245} a^{3} + \frac{2697024241997}{245} a^{2} + \frac{5441114386131}{245} a - \frac{5779872146466}{245} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( a^{3} - 5 a - 3\) , \( a^{2} - 2\) , \( a^{3} - 2 a^{2} - 2 a - 1\) , \( 9 a^{3} - 25 a^{2} - 3 a + 15\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-5a-3\right){x}^{2}+\left(a^{3}-2a^{2}-2a-1\right){x}+9a^{3}-25a^{2}-3a+15$ |
| 35.2-a1 |
35.2-a |
$1$ |
$1$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
35.2 |
\( 5 \cdot 7 \) |
\( 5 \cdot 7^{3} \) |
$13.44039$ |
$(-a^3+a^2+4a+1), (a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.078469383$ |
$310.7155721$ |
3.033749274 |
\( \frac{42498}{1715} a^{3} - \frac{33766219}{1715} a^{2} + \frac{408374}{245} a + \frac{22553322}{1715} \) |
\( \bigl[a\) , \( a^{3} - 2 a^{2} - 3 a + 3\) , \( a^{2} - 2\) , \( -5 a^{3} + 8 a^{2} + 17 a - 17\) , \( 22 a^{3} - 40 a^{2} - 80 a + 85\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+3\right){x}^{2}+\left(-5a^{3}+8a^{2}+17a-17\right){x}+22a^{3}-40a^{2}-80a+85$ |
| 35.2-b1 |
35.2-b |
$2$ |
$3$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
35.2 |
\( 5 \cdot 7 \) |
\( 5^{9} \cdot 7^{3} \) |
$13.44039$ |
$(-a^3+a^2+4a+1), (a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3^{2} \) |
$0.052099374$ |
$121.8003249$ |
2.368746754 |
\( -\frac{432927023773}{669921875} a^{3} + \frac{482070734064}{669921875} a^{2} + \frac{285452037011}{95703125} a - \frac{661689248807}{669921875} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + 4\) , \( a^{2} - a - 3\) , \( a^{3} - a^{2} - a + 6\) , \( -2 a^{3} - 5 a^{2} + 2 a + 7\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(a^{3}-a^{2}-a+6\right){x}-2a^{3}-5a^{2}+2a+7$ |
| 35.2-b2 |
35.2-b |
$2$ |
$3$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
35.2 |
\( 5 \cdot 7 \) |
\( 5^{3} \cdot 7 \) |
$13.44039$ |
$(-a^3+a^2+4a+1), (a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.017366458$ |
$1096.202924$ |
2.368746754 |
\( \frac{1657567}{875} a^{3} - \frac{440056}{875} a^{2} - \frac{1208119}{125} a - \frac{4490372}{875} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - a^{2} - 5 a\) , \( a^{2} - 2\) , \( -a^{2} - a + 4\) , \( -2 a\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-a^{2}-a+4\right){x}-2a$ |
| 45.1-a1 |
45.1-a |
$6$ |
$8$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{9} \cdot 5^{3} \) |
$13.86931$ |
$(-a), (-a^3+a^2+4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{2} \) |
$2.264687853$ |
$36.35000671$ |
3.414349822 |
\( \frac{2135315017580596621}{3375} a^{3} - \frac{3808789312094362603}{3375} a^{2} - \frac{7686059730886656329}{3375} a + \frac{8163497292155544289}{3375} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - a^{2} - 3 a\) , \( a^{3} - 4 a - 3\) , \( -224 a^{3} + 400 a^{2} + 791 a - 861\) , \( -2984 a^{3} + 5303 a^{2} + 10852 a - 11488\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-4a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(-224a^{3}+400a^{2}+791a-861\right){x}-2984a^{3}+5303a^{2}+10852a-11488$ |
| 45.1-a2 |
45.1-a |
$6$ |
$8$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{12} \cdot 5^{6} \) |
$13.86931$ |
$(-a), (-a^3+a^2+4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3Ns |
$1$ |
\( 2^{3} \) |
$1.132343926$ |
$145.4000268$ |
3.414349822 |
\( \frac{128092594571557}{11390625} a^{3} - \frac{225657720048376}{11390625} a^{2} - \frac{458845597722218}{11390625} a + \frac{485659032605788}{11390625} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - a^{2} - 3 a\) , \( a^{3} - 4 a - 3\) , \( -9 a^{3} + 15 a^{2} + 41 a - 51\) , \( -47 a^{3} + 80 a^{2} + 185 a - 199\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-4a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(-9a^{3}+15a^{2}+41a-51\right){x}-47a^{3}+80a^{2}+185a-199$ |
| 45.1-a3 |
45.1-a |
$6$ |
$8$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{9} \cdot 5^{3} \) |
$13.86931$ |
$(-a), (-a^3+a^2+4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2 \) |
$0.566171963$ |
$290.8000537$ |
3.414349822 |
\( -\frac{6637246}{3375} a^{3} + \frac{11349478}{3375} a^{2} + \frac{20884829}{3375} a - \frac{16775914}{3375} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - a^{2} - 3 a\) , \( a^{3} - 4 a - 3\) , \( a^{3} - 4 a - 6\) , \( -a^{3} + 4 a^{2} - a - 10\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-4a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(a^{3}-4a-6\right){x}-a^{3}+4a^{2}-a-10$ |
| 45.1-a4 |
45.1-a |
$6$ |
$8$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{18} \cdot 5^{12} \) |
$13.86931$ |
$(-a), (-a^3+a^2+4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3Ns |
$4$ |
\( 2^{3} \) |
$2.264687853$ |
$18.17500335$ |
3.414349822 |
\( -\frac{5608173175704549785027}{129746337890625} a^{3} + \frac{15053308221079393467461}{129746337890625} a^{2} + \frac{2624391126667291079623}{129746337890625} a - \frac{9911446235169938034143}{129746337890625} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - a^{2} - 3 a\) , \( a^{3} - 4 a - 3\) , \( 46 a^{3} - 130 a^{2} + 11 a + 39\) , \( 446 a^{3} - 1239 a^{2} - 58 a + 674\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-4a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(46a^{3}-130a^{2}+11a+39\right){x}+446a^{3}-1239a^{2}-58a+674$ |
| 45.1-a5 |
45.1-a |
$6$ |
$8$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{12} \cdot 5^{24} \) |
$13.86931$ |
$(-a), (-a^3+a^2+4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$16$ |
\( 2^{2} \) |
$4.529375707$ |
$1.135937709$ |
3.414349822 |
\( -\frac{1838899427339088360320392720528}{43451786041259765625} a^{3} + \frac{239083282688250681080233210279}{43451786041259765625} a^{2} + \frac{9402497034463545997974102358322}{43451786041259765625} a + \frac{6341137901789312627182533751898}{43451786041259765625} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - a^{2} - 3 a\) , \( a^{3} - 4 a - 3\) , \( 101 a^{3} - 195 a^{2} - 224 a + 99\) , \( 416 a^{3} - 1025 a^{2} - 83 a - 256\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-4a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(101a^{3}-195a^{2}-224a+99\right){x}+416a^{3}-1025a^{2}-83a-256$ |
| 45.1-a6 |
45.1-a |
$6$ |
$8$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{30} \cdot 5^{6} \) |
$13.86931$ |
$(-a), (-a^3+a^2+4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$4$ |
\( 2^{3} \) |
$4.529375707$ |
$2.271875419$ |
3.414349822 |
\( -\frac{81577402532451076166045520603248}{4412961507515625} a^{3} + \frac{219331599661591677554721948716489}{4412961507515625} a^{2} + \frac{37516695901889429460362076037102}{4412961507515625} a - \frac{144929288746881365623568260159082}{4412961507515625} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - a^{2} - 3 a\) , \( a^{3} - 4 a - 3\) , \( 871 a^{3} - 2385 a^{2} - 234 a + 1419\) , \( 33788 a^{3} - 90569 a^{2} - 16465 a + 60596\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-4a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(871a^{3}-2385a^{2}-234a+1419\right){x}+33788a^{3}-90569a^{2}-16465a+60596$ |
| 49.1-a1 |
49.1-a |
$1$ |
$1$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{4} \) |
$14.01774$ |
$(a^2-2), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$27.18297454$ |
1.127436647 |
\( -\frac{29548068716}{49} a^{3} + \frac{52727787323}{49} a^{2} + \frac{106376614263}{49} a - \frac{112997933488}{49} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + 2 a^{2} + 4 a - 2\) , \( a^{3} - 4 a - 2\) , \( -2 a^{3} - 2 a^{2} + 2 a\) , \( -12 a^{3} - 20 a^{2} + 8 a + 12\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-4a-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-2\right){x}^{2}+\left(-2a^{3}-2a^{2}+2a\right){x}-12a^{3}-20a^{2}+8a+12$ |
| 49.1-b1 |
49.1-b |
$2$ |
$3$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{12} \) |
$14.01774$ |
$(a^2-2), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$9$ |
\( 2^{2} \) |
$1$ |
$1.431442919$ |
0.534332651 |
\( -\frac{12015094967435172140}{117649} a^{3} - \frac{19261658719636898941}{117649} a^{2} + \frac{9935012056352032719}{117649} a + \frac{13846932808971152480}{117649} \) |
\( \bigl[a^{2} - 3\) , \( a^{2} - a - 2\) , \( a^{3} - a^{2} - 3 a\) , \( -1117 a^{3} + 1919 a^{2} + 3961 a - 4173\) , \( 30960 a^{3} - 56092 a^{2} - 112142 a + 119623\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-a^{2}-3a\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-1117a^{3}+1919a^{2}+3961a-4173\right){x}+30960a^{3}-56092a^{2}-112142a+119623$ |
| 49.1-b2 |
49.1-b |
$2$ |
$3$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{20} \) |
$14.01774$ |
$(a^2-2), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$1$ |
$12.88298627$ |
0.534332651 |
\( -\frac{8979655417425289794}{1628413597910449} a^{3} - \frac{13671232780709261689}{1628413597910449} a^{2} + \frac{7405042981933616365}{1628413597910449} a + \frac{9817948894868981627}{1628413597910449} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - a^{2} - 3 a\) , \( a^{2} - 2\) , \( -a^{3} - 3 a^{2} - 3 a - 2\) , \( 348 a^{3} - 60 a^{2} - 1813 a - 1215\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(-a^{3}-3a^{2}-3a-2\right){x}+348a^{3}-60a^{2}-1813a-1215$ |
| 49.1-c1 |
49.1-c |
$1$ |
$1$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{11} \) |
$14.01774$ |
$(a^2-2), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$227.8833067$ |
4.725825552 |
\( -\frac{662790144}{823543} a^{3} + \frac{78151680}{823543} a^{2} + \frac{3414200320}{823543} a + \frac{2273931264}{823543} \) |
\( \bigl[0\) , \( a^{2} - 2 a - 3\) , \( a^{3} - 5 a - 2\) , \( 6 a^{3} - 32 a - 22\) , \( 8 a^{3} - a^{2} - 42 a - 29\bigr] \) |
${y}^2+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(6a^{3}-32a-22\right){x}+8a^{3}-a^{2}-42a-29$ |
| 49.1-d1 |
49.1-d |
$1$ |
$1$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{4} \) |
$14.01774$ |
$(a^2-2), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.010099872$ |
$1587.932347$ |
2.660743836 |
\( \frac{12738300}{49} a^{3} - \frac{3296037}{49} a^{2} - \frac{62183592}{49} a - \frac{40530220}{49} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{2} - 2 a - 3\) , \( a + 1\) , \( 2 a^{3} - 5 a^{2} - 9 a + 15\) , \( -3 a^{3} + 6 a^{2} + 10 a - 13\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(2a^{3}-5a^{2}-9a+15\right){x}-3a^{3}+6a^{2}+10a-13$ |
| 49.1-e1 |
49.1-e |
$1$ |
$1$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{3} \) |
$14.01774$ |
$(a^2-2), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.150890084$ |
$175.8018457$ |
2.200438810 |
\( -\frac{1183744}{49} a^{3} - \frac{454656}{49} a^{2} + \frac{761856}{7} a + \frac{3948544}{49} \) |
\( \bigl[0\) , \( a^{2} - a - 4\) , \( a^{2} - 3\) , \( 8 a^{3} - 2 a^{2} - 40 a - 24\) , \( -36 a^{3} + 5 a^{2} + 183 a + 121\bigr] \) |
${y}^2+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(8a^{3}-2a^{2}-40a-24\right){x}-36a^{3}+5a^{2}+183a+121$ |
| 49.3-a1 |
49.3-a |
$1$ |
$1$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
49.3 |
\( 7^{2} \) |
\( 7^{9} \) |
$14.01774$ |
$(a^2-2)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
|
\( 2 \) |
$1$ |
$32.74954243$ |
4.060937682 |
\( -299172 a^{3} + 534168 a^{2} + 1076463 a - 1144643 \) |
\( \bigl[a^{3} - 4 a - 2\) , \( -a^{2} + 3\) , \( a^{3} - a^{2} - 4 a\) , \( -2 a^{3} + 9 a + 6\) , \( -5 a^{3} - 3 a^{2} + 14 a + 8\bigr] \) |
${y}^2+\left(a^{3}-4a-2\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-2a^{3}+9a+6\right){x}-5a^{3}-3a^{2}+14a+8$ |
| 49.3-b1 |
49.3-b |
$1$ |
$1$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
49.3 |
\( 7^{2} \) |
\( 7^{3} \) |
$14.01774$ |
$(a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.056107611$ |
$793.7467402$ |
3.694272248 |
\( -299172 a^{3} + 534168 a^{2} + 1076463 a - 1144643 \) |
\( \bigl[a + 1\) , \( a^{3} - 6 a - 3\) , \( a^{3} - a^{2} - 4 a + 1\) , \( -6 a^{3} + 7 a^{2} + 24 a - 5\) , \( 5 a^{3} - 6 a^{2} - 21 a + 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(a^{3}-6a-3\right){x}^{2}+\left(-6a^{3}+7a^{2}+24a-5\right){x}+5a^{3}-6a^{2}-21a+7$ |
| 49.4-a1 |
49.4-a |
$1$ |
$1$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( - 7^{8} \) |
$14.01774$ |
$(a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$1$ |
$48.91697812$ |
1.521654493 |
\( 99930 a^{3} - 14353 a^{2} - 512410 a - 343169 \) |
\( \bigl[a^{3} - 4 a - 2\) , \( -a^{3} + a^{2} + 5 a + 1\) , \( 0\) , \( -5 a^{3} + a^{2} + 26 a + 17\) , \( -9 a^{3} - 8 a^{2} + 24 a + 20\bigr] \) |
${y}^2+\left(a^{3}-4a-2\right){x}{y}={x}^{3}+\left(-a^{3}+a^{2}+5a+1\right){x}^{2}+\left(-5a^{3}+a^{2}+26a+17\right){x}-9a^{3}-8a^{2}+24a+20$ |
| 49.4-b1 |
49.4-b |
$1$ |
$1$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( - 7^{7} \) |
$14.01774$ |
$(a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.114409679$ |
$267.6829191$ |
2.540437965 |
\( -\frac{73728}{7} a^{3} + \frac{12288}{7} a^{2} + \frac{364544}{7} a + \frac{225280}{7} \) |
\( \bigl[0\) , \( -a^{2} + 2 a + 2\) , \( a^{3} - a^{2} - 4 a + 1\) , \( 11 a^{3} - 59 a - 41\) , \( -31 a^{3} + 4 a^{2} + 159 a + 105\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(11a^{3}-59a-41\right){x}-31a^{3}+4a^{2}+159a+105$ |
| 49.4-c1 |
49.4-c |
$1$ |
$1$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( - 7^{2} \) |
$14.01774$ |
$(a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.062058786$ |
$638.6128616$ |
1.643751302 |
\( 99930 a^{3} - 14353 a^{2} - 512410 a - 343169 \) |
\( \bigl[a^{3} - 5 a - 3\) , \( a^{2} - a - 4\) , \( a^{2} - a - 2\) , \( -a^{3} + 5 a + 3\) , \( -a^{2} + a + 3\bigr] \) |
${y}^2+\left(a^{3}-5a-3\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-a^{3}+5a+3\right){x}-a^{2}+a+3$ |
| 51.1-a1 |
51.1-a |
$1$ |
$1$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
51.1 |
\( 3 \cdot 17 \) |
\( 3^{9} \cdot 17^{3} \) |
$14.08801$ |
$(-a), (a^3-a^2-4a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$83.70327993$ |
0.867915918 |
\( \frac{1981914615578}{96702579} a^{3} - \frac{281085011171}{96702579} a^{2} - \frac{10112310013918}{96702579} a - \frac{6734464736998}{96702579} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{3} + 6 a + 2\) , \( a^{3} - 4 a - 3\) , \( -3 a^{3} + 14 a + 11\) , \( -5 a^{3} - a^{2} + 19 a + 13\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{3}-4a-3\right){y}={x}^{3}+\left(-a^{3}+6a+2\right){x}^{2}+\left(-3a^{3}+14a+11\right){x}-5a^{3}-a^{2}+19a+13$ |
| 51.1-b1 |
51.1-b |
$1$ |
$1$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
51.1 |
\( 3 \cdot 17 \) |
\( - 3^{9} \cdot 17^{3} \) |
$14.08801$ |
$(-a), (a^3-a^2-4a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 3 \) |
$0.177175819$ |
$121.2239919$ |
2.672449183 |
\( -\frac{315432782989}{96702579} a^{3} - \frac{401383981952}{96702579} a^{2} + \frac{316713988436}{96702579} a + \frac{345749619758}{96702579} \) |
\( \bigl[a^{3} - 4 a - 3\) , \( -a^{3} + 6 a + 2\) , \( a^{3} - 5 a - 3\) , \( -5 a^{3} - 6 a^{2} + 38 a + 33\) , \( -18 a^{3} + 5 a^{2} + 88 a + 57\bigr] \) |
${y}^2+\left(a^{3}-4a-3\right){x}{y}+\left(a^{3}-5a-3\right){y}={x}^{3}+\left(-a^{3}+6a+2\right){x}^{2}+\left(-5a^{3}-6a^{2}+38a+33\right){x}-18a^{3}+5a^{2}+88a+57$ |
| 51.1-c1 |
51.1-c |
$4$ |
$6$ |
4.4.9301.1 |
$4$ |
$[4, 0]$ |
51.1 |
\( 3 \cdot 17 \) |
\( 3^{4} \cdot 17^{3} \) |
$14.08801$ |
$(-a), (a^3-a^2-4a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$4.765982476$ |
$2.314361278$ |
4.117383068 |
\( \frac{3145980571677213592387}{397953} a^{3} - \frac{5613926945615477005246}{397953} a^{2} - \frac{11325912207298148829797}{397953} a + \frac{12030886882774010107801}{397953} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -23 a^{2} - 21 a - 17\) , \( -39 a^{3} - 155 a^{2} - 59 a + 12\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(-23a^{2}-21a-17\right){x}-39a^{3}-155a^{2}-59a+12$ |
*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.
