| 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 |
| 4.1-a1 |
4.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$9.19558$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.845874701$ |
$156.5570771$ |
1.530367813 |
\( -818626500 a^{3} + 1637253000 a^{2} + 2455879500 a + 599278500 \) |
\( \bigl[a^{3} - a^{2} - 6 a\) , \( a^{2} - 2 a - 2\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 110 a^{3} - 141 a^{2} - 549 a - 163\) , \( 1102 a^{3} - 1434 a^{2} - 5414 a - 1580\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(110a^{3}-141a^{2}-549a-163\right){x}+1102a^{3}-1434a^{2}-5414a-1580$ |
| 4.1-a2 |
4.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$9.19558$ |
$(a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.211468675$ |
$2504.913233$ |
1.530367813 |
\( 818626500 a^{3} - 1637253000 a^{2} - 2455879500 a + 2236531500 \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{3} + a^{2} + 7 a + 2\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( -21 a^{3} + 40 a^{2} + 105 a - 66\) , \( 71 a^{3} - 147 a^{2} - 296 a + 273\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+7a+2\right){x}^{2}+\left(-21a^{3}+40a^{2}+105a-66\right){x}+71a^{3}-147a^{2}-296a+273$ |
| 4.1-a3 |
4.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$9.19558$ |
$(a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$0.845874701$ |
$626.2283084$ |
1.530367813 |
\( 0 \) |
\( \bigl[0\) , \( a^{3} - a^{2} - 7 a\) , \( a^{3} - a^{2} - 6 a\) , \( -2 a^{3} + 4 a^{2} + 6 a + 3\) , \( 3 a^{3} - 4 a^{2} - 15 a - 3\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a\right){y}={x}^{3}+\left(a^{3}-a^{2}-7a\right){x}^{2}+\left(-2a^{3}+4a^{2}+6a+3\right){x}+3a^{3}-4a^{2}-15a-3$ |
| 4.1-a4 |
4.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$9.19558$ |
$(a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 3 \) |
$0.281958233$ |
$626.2283084$ |
1.530367813 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} + a^{2} + 7 a\) , \( a + 1\) , \( -2 a^{3} + 4 a^{2} + 6 a + 3\) , \( -2 a^{3} + 2 a^{2} + 11 a + 1\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+7a\right){x}^{2}+\left(-2a^{3}+4a^{2}+6a+3\right){x}-2a^{3}+2a^{2}+11a+1$ |
| 4.1-a5 |
4.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$9.19558$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 3 \) |
$0.281958233$ |
$156.5570771$ |
1.530367813 |
\( 818626500 a^{3} - 1637253000 a^{2} - 2455879500 a + 2236531500 \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( a^{3} - a^{2} - 5 a - 1\) , \( a^{3} - a^{2} - 6 a\) , \( -18 a^{3} + 35 a^{2} + 87 a - 74\) , \( -93 a^{3} + 201 a^{2} + 362 a - 325\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-a^{2}-6a\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a-1\right){x}^{2}+\left(-18a^{3}+35a^{2}+87a-74\right){x}-93a^{3}+201a^{2}+362a-325$ |
| 4.1-a6 |
4.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$9.19558$ |
$(a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 3 \) |
$0.070489558$ |
$2504.913233$ |
1.530367813 |
\( -818626500 a^{3} + 1637253000 a^{2} + 2455879500 a + 599278500 \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( a^{3} - a^{2} - 5 a - 1\) , \( a + 1\) , \( 51 a^{3} - 179 a^{2} + 63 a + 29\) , \( -484 a^{3} + 1660 a^{2} - 440 a - 336\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a-1\right){x}^{2}+\left(51a^{3}-179a^{2}+63a+29\right){x}-484a^{3}+1660a^{2}-440a-336$ |
| 4.1-a7 |
4.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$9.19558$ |
$(a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$0.422937350$ |
$1252.456616$ |
1.530367813 |
\( 54000 \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( -a^{3} + 2 a^{2} + 3 a - 2\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( -1\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-2\right){x}^{2}+\left(-a^{3}+2a^{2}+3a-4\right){x}-1$ |
| 4.1-a8 |
4.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$9.19558$ |
$(a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 3 \) |
$0.140979116$ |
$1252.456616$ |
1.530367813 |
\( 54000 \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( -1\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( -a^{3} + 2 a^{2} + 3 a - 2\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}-{x}^{2}+\left(-a^{3}+2a^{2}+3a-4\right){x}-a^{3}+2a^{2}+3a-2$ |
| 9.1-a1 |
9.1-a |
$6$ |
$18$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1.457067043$ |
$5.494316968$ |
1.665263001 |
\( 4577527672747874049472 a^{3} - \frac{31954464968147703823552}{3} a^{2} - \frac{44484143963085684580640}{3} a + \frac{42007413456669872157944}{3} \) |
\( \bigl[a^{2} - a - 2\) , \( a\) , \( a^{3} - 2 a^{2} - 3 a + 1\) , \( -144 a^{3} - 74 a^{2} + 109 a - 114\) , \( -1618 a^{3} - 1326 a^{2} + 1233 a - 262\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-2a^{2}-3a+1\right){y}={x}^{3}+a{x}^{2}+\left(-144a^{3}-74a^{2}+109a-114\right){x}-1618a^{3}-1326a^{2}+1233a-262$ |
| 9.1-a2 |
9.1-a |
$6$ |
$18$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{12} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.485689014$ |
$445.0396744$ |
1.665263001 |
\( \frac{67512848}{9} a^{3} - \frac{135025696}{9} a^{2} - \frac{67512848}{3} a + \frac{553427224}{27} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( -a^{3} + 3 a^{2} + 2 a - 2\) , \( a^{2} - 2 a - 2\) , \( 142 a^{3} - 185 a^{2} - 699 a - 202\) , \( -869 a^{3} + 1130 a^{2} + 4269 a + 1244\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{2}-2a-2\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+2a-2\right){x}^{2}+\left(142a^{3}-185a^{2}-699a-202\right){x}-869a^{3}+1130a^{2}+4269a+1244$ |
| 9.1-a3 |
9.1-a |
$6$ |
$18$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1.457067043$ |
$445.0396744$ |
1.665263001 |
\( \frac{1467590246567884959584}{3} a^{3} + 518039479508229869184 a^{2} - \frac{1116375831348836743360}{3} a - \frac{479767813323440326600}{3} \) |
\( \bigl[a + 1\) , \( a^{3} - 2 a^{2} - 5 a\) , \( a^{3} - 2 a^{2} - 4 a + 2\) , \( -1196 a^{3} + 2755 a^{2} + 3913 a - 3668\) , \( 10698 a^{3} - 24764 a^{2} - 34801 a + 32764\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-2a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-5a\right){x}^{2}+\left(-1196a^{3}+2755a^{2}+3913a-3668\right){x}+10698a^{3}-24764a^{2}-34801a+32764$ |
| 9.1-a4 |
9.1-a |
$6$ |
$18$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$2.914134087$ |
$445.0396744$ |
1.665263001 |
\( -\frac{63260152520620576768}{3} a^{3} + \frac{82310502753736586240}{3} a^{2} + 103521294495504877568 a + \frac{90519447988781264960}{3} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( -a^{3} + a^{2} + 6 a + 1\) , \( 1\) , \( 359 a^{3} - 479 a^{2} - 1734 a - 503\) , \( -5241 a^{3} + 6851 a^{2} + 25646 a + 7469\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+6a+1\right){x}^{2}+\left(359a^{3}-479a^{2}-1734a-503\right){x}-5241a^{3}+6851a^{2}+25646a+7469$ |
| 9.1-a5 |
9.1-a |
$6$ |
$18$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 3 \) |
$0.971378029$ |
$445.0396744$ |
1.665263001 |
\( -\frac{12413440}{9} a^{3} + \frac{24826880}{9} a^{2} + \frac{12413440}{3} a + \frac{9047744}{9} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( -a^{3} + a^{2} + 6 a + 1\) , \( 1\) , \( 4 a^{3} - 9 a^{2} - 9 a - 3\) , \( a^{3} - 6 a^{2} + 8 a + 3\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+6a+1\right){x}^{2}+\left(4a^{3}-9a^{2}-9a-3\right){x}+a^{3}-6a^{2}+8a+3$ |
| 9.1-a6 |
9.1-a |
$6$ |
$18$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$2.914134087$ |
$5.494316968$ |
1.665263001 |
\( -\frac{30896331170976808960}{3} a^{3} + \frac{106002464629458185216}{3} a^{2} - 9364810803907491840 a - \frac{21592118461949468608}{3} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( -a^{3} + a^{2} + 6 a + 1\) , \( 1\) , \( 179 a^{3} - 599 a^{2} + 126 a + 97\) , \( 2760 a^{3} - 9391 a^{2} + 2303 a + 1829\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+6a+1\right){x}^{2}+\left(179a^{3}-599a^{2}+126a+97\right){x}+2760a^{3}-9391a^{2}+2303a+1829$ |
| 9.1-b1 |
9.1-b |
$4$ |
$10$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$4$ |
\( 1 \) |
$1$ |
$1737.297665$ |
0.803066130 |
\( -\frac{1024}{3} a^{3} + \frac{2048}{3} a^{2} + 1024 a + \frac{3392}{3} \) |
\( \bigl[a^{2} - 2 a - 1\) , \( a^{3} - 2 a^{2} - 5 a + 1\) , \( a\) , \( -a + 3\) , \( a^{3} - 3 a^{2} - 2 a + 2\bigr] \) |
${y}^2+\left(a^{2}-2a-1\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-2a^{2}-5a+1\right){x}^{2}+\left(-a+3\right){x}+a^{3}-3a^{2}-2a+2$ |
| 9.1-b2 |
9.1-b |
$4$ |
$10$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{10} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$100$ |
\( 1 \) |
$1$ |
$2.779676265$ |
0.803066130 |
\( -\frac{12326555106304}{27} a^{3} + \frac{24653110212608}{27} a^{2} + \frac{12326555106304}{9} a + \frac{9023655698240}{27} \) |
\( \bigl[a^{2} - 2 a - 1\) , \( -a^{3} + a^{2} + 7 a + 2\) , \( a^{3} - a^{2} - 6 a - 1\) , \( 409 a^{3} - 539 a^{2} - 2021 a - 597\) , \( 8516 a^{3} - 11129 a^{2} - 41898 a - 12219\bigr] \) |
${y}^2+\left(a^{2}-2a-1\right){x}{y}+\left(a^{3}-a^{2}-6a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+7a+2\right){x}^{2}+\left(409a^{3}-539a^{2}-2021a-597\right){x}+8516a^{3}-11129a^{2}-41898a-12219$ |
| 9.1-b3 |
9.1-b |
$4$ |
$10$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{20} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 2 \) |
$1$ |
$5.559352530$ |
0.803066130 |
\( \frac{7693908914896}{243} a^{3} - \frac{15387817829792}{243} a^{2} - \frac{7693908914896}{81} a + \frac{21020455836104}{243} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( -a^{3} + a^{2} + 5 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( -116 a^{3} + 312 a^{2} + 320 a - 582\) , \( -1303 a^{3} + 3449 a^{2} + 3581 a - 5939\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-a^{2}-5a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+2\right){x}^{2}+\left(-116a^{3}+312a^{2}+320a-582\right){x}-1303a^{3}+3449a^{2}+3581a-5939$ |
| 9.1-b4 |
9.1-b |
$4$ |
$10$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$3474.595331$ |
0.803066130 |
\( -\frac{1097776}{3} a^{3} + \frac{2195552}{3} a^{2} + 1097776 a + 271064 \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( -a^{3} + a^{2} + 5 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( -a^{3} + 2 a^{2} + 5 a - 2\) , \( -a - 1\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-a^{2}-5a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+2\right){x}^{2}+\left(-a^{3}+2a^{2}+5a-2\right){x}-a-1$ |
| 9.1-c1 |
9.1-c |
$4$ |
$10$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{10} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$909.3680994$ |
2.627222204 |
\( -\frac{12326555106304}{27} a^{3} + \frac{24653110212608}{27} a^{2} + \frac{12326555106304}{9} a + \frac{9023655698240}{27} \) |
\( \bigl[a^{2} - 2 a - 1\) , \( -a^{3} + 2 a^{2} + 4 a - 2\) , \( a\) , \( 411 a^{3} - 541 a^{2} - 2037 a - 599\) , \( -7094 a^{3} + 9257 a^{2} + 34889 a + 10185\bigr] \) |
${y}^2+\left(a^{2}-2a-1\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-2\right){x}^{2}+\left(411a^{3}-541a^{2}-2037a-599\right){x}-7094a^{3}+9257a^{2}+34889a+10185$ |
| 9.1-c2 |
9.1-c |
$4$ |
$10$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$909.3680994$ |
2.627222204 |
\( -\frac{1024}{3} a^{3} + \frac{2048}{3} a^{2} + 1024 a + \frac{3392}{3} \) |
\( \bigl[a^{2} - 2 a - 1\) , \( -a^{2} + a + 2\) , \( a^{3} - 2 a^{2} - 4 a + 2\) , \( a^{3} - 3 a^{2} + 3\) , \( -a^{2} + 3 a - 1\bigr] \) |
${y}^2+\left(a^{2}-2a-1\right){x}{y}+\left(a^{3}-2a^{2}-4a+2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(a^{3}-3a^{2}+3\right){x}-a^{2}+3a-1$ |
| 9.1-c3 |
9.1-c |
$4$ |
$10$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{20} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$454.6840497$ |
2.627222204 |
\( \frac{7693908914896}{243} a^{3} - \frac{15387817829792}{243} a^{2} - \frac{7693908914896}{81} a + \frac{21020455836104}{243} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - a^{2} - 6 a - 1\) , \( a\) , \( -115 a^{3} + 311 a^{2} + 313 a - 583\) , \( 1188 a^{3} - 3139 a^{2} - 3266 a + 5356\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-6a-1\right){x}^{2}+\left(-115a^{3}+311a^{2}+313a-583\right){x}+1188a^{3}-3139a^{2}-3266a+5356$ |
| 9.1-c4 |
9.1-c |
$4$ |
$10$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$454.6840497$ |
2.627222204 |
\( -\frac{1097776}{3} a^{3} + \frac{2195552}{3} a^{2} + 1097776 a + 271064 \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - a^{2} - 6 a - 1\) , \( a\) , \( a^{2} - 2 a - 3\) , \( a - 2\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-6a-1\right){x}^{2}+\left(a^{2}-2a-3\right){x}+a-2$ |
| 9.1-d1 |
9.1-d |
$6$ |
$18$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1.457067043$ |
$5.494316968$ |
1.665263001 |
\( \frac{1467590246567884959584}{3} a^{3} + 518039479508229869184 a^{2} - \frac{1116375831348836743360}{3} a - \frac{479767813323440326600}{3} \) |
\( \bigl[a^{3} - a^{2} - 6 a\) , \( a + 1\) , \( 1\) , \( -1198 a^{3} + 2758 a^{2} + 3923 a - 3665\) , \( -11895 a^{3} + 27521 a^{2} + 38718 a - 36433\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1198a^{3}+2758a^{2}+3923a-3665\right){x}-11895a^{3}+27521a^{2}+38718a-36433$ |
| 9.1-d2 |
9.1-d |
$6$ |
$18$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1.457067043$ |
$445.0396744$ |
1.665263001 |
\( 4577527672747874049472 a^{3} - \frac{31954464968147703823552}{3} a^{2} - \frac{44484143963085684580640}{3} a + \frac{42007413456669872157944}{3} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( a + 1\) , \( a^{3} - 2 a^{2} - 3 a + 1\) , \( -145 a^{3} - 73 a^{2} + 113 a - 115\) , \( 1110 a^{3} + 785 a^{2} - 946 a + 291\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-2a^{2}-3a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-145a^{3}-73a^{2}+113a-115\right){x}+1110a^{3}+785a^{2}-946a+291$ |
| 9.1-d3 |
9.1-d |
$6$ |
$18$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{12} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.485689014$ |
$445.0396744$ |
1.665263001 |
\( \frac{67512848}{9} a^{3} - \frac{135025696}{9} a^{2} - \frac{67512848}{3} a + \frac{553427224}{27} \) |
\( \bigl[a + 1\) , \( -a^{2} + 3 a + 1\) , \( 1\) , \( 65 a^{3} - 226 a^{2} + 73 a + 37\) , \( -587 a^{3} + 2021 a^{2} - 557 a - 401\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+3a+1\right){x}^{2}+\left(65a^{3}-226a^{2}+73a+37\right){x}-587a^{3}+2021a^{2}-557a-401$ |
| 9.1-d4 |
9.1-d |
$6$ |
$18$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$2.914134087$ |
$5.494316968$ |
1.665263001 |
\( -\frac{63260152520620576768}{3} a^{3} + \frac{82310502753736586240}{3} a^{2} + 103521294495504877568 a + \frac{90519447988781264960}{3} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{2} - 3 a - 1\) , \( 1\) , \( 360 a^{3} - 479 a^{2} - 1743 a - 505\) , \( 5601 a^{3} - 7331 a^{2} - 27386 a - 7973\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+{y}={x}^{3}+\left(a^{2}-3a-1\right){x}^{2}+\left(360a^{3}-479a^{2}-1743a-505\right){x}+5601a^{3}-7331a^{2}-27386a-7973$ |
| 9.1-d5 |
9.1-d |
$6$ |
$18$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 3 \) |
$0.971378029$ |
$445.0396744$ |
1.665263001 |
\( -\frac{12413440}{9} a^{3} + \frac{24826880}{9} a^{2} + \frac{12413440}{3} a + \frac{9047744}{9} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{2} - 3 a - 1\) , \( 1\) , \( 5 a^{3} - 9 a^{2} - 18 a - 5\) , \( 4 a^{3} - 4 a^{2} - 23 a - 7\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+{y}={x}^{3}+\left(a^{2}-3a-1\right){x}^{2}+\left(5a^{3}-9a^{2}-18a-5\right){x}+4a^{3}-4a^{2}-23a-7$ |
| 9.1-d6 |
9.1-d |
$6$ |
$18$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$10.17658$ |
$(a^3-2a^2-3a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$2.914134087$ |
$445.0396744$ |
1.665263001 |
\( -\frac{30896331170976808960}{3} a^{3} + \frac{106002464629458185216}{3} a^{2} - 9364810803907491840 a - \frac{21592118461949468608}{3} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{2} - 3 a - 1\) , \( 1\) , \( 180 a^{3} - 599 a^{2} + 117 a + 95\) , \( -2580 a^{3} + 8791 a^{2} - 2183 a - 1733\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+{y}={x}^{3}+\left(a^{2}-3a-1\right){x}^{2}+\left(180a^{3}-599a^{2}+117a+95\right){x}-2580a^{3}+8791a^{2}-2183a-1733$ |
| 11.1-a1 |
11.1-a |
$1$ |
$1$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
11.1 |
\( 11 \) |
\( - 11^{5} \) |
$10.43507$ |
$(-a^3+2a^2+4a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$109.1134905$ |
1.260943221 |
\( -\frac{149006623}{161051} a^{3} - \frac{676285268}{161051} a^{2} - \frac{10886640}{14641} a + \frac{507579014}{161051} \) |
\( \bigl[a^{2} - 2 a - 2\) , \( -a^{3} + 2 a^{2} + 4 a\) , \( 0\) , \( -5 a^{3} + 9 a^{2} + 19 a - 2\) , \( -2 a^{2} + a + 7\bigr] \) |
${y}^2+\left(a^{2}-2a-2\right){x}{y}={x}^{3}+\left(-a^{3}+2a^{2}+4a\right){x}^{2}+\left(-5a^{3}+9a^{2}+19a-2\right){x}-2a^{2}+a+7$ |
| 11.1-b1 |
11.1-b |
$1$ |
$1$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
11.1 |
\( 11 \) |
\( - 11^{5} \) |
$10.43507$ |
$(-a^3+2a^2+4a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 5 \) |
$0.005203758$ |
$1091.654626$ |
1.312953821 |
\( -\frac{149006623}{161051} a^{3} - \frac{676285268}{161051} a^{2} - \frac{10886640}{14641} a + \frac{507579014}{161051} \) |
\( \bigl[a^{3} - a^{2} - 5 a\) , \( -a^{3} + 2 a^{2} + 4 a - 2\) , \( a^{2} - a - 1\) , \( -3 a^{3} + 4 a^{2} + 13 a - 1\) , \( -3 a^{3} + 3 a^{2} + 15 a + 5\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-2\right){x}^{2}+\left(-3a^{3}+4a^{2}+13a-1\right){x}-3a^{3}+3a^{2}+15a+5$ |
| 11.2-a1 |
11.2-a |
$1$ |
$1$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
11.2 |
\( 11 \) |
\( - 11^{5} \) |
$10.43507$ |
$(-a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$109.1134905$ |
1.260943221 |
\( -\frac{2664376560}{161051} a^{3} + \frac{573004694}{14641} a^{2} + \frac{8559902589}{161051} a - \frac{8420354916}{161051} \) |
\( \bigl[a^{2} - 2 a - 2\) , \( -a - 1\) , \( a^{2} - 2 a - 2\) , \( -a^{3} + 3 a^{2} + a - 6\) , \( a^{2} + 2 a - 4\bigr] \) |
${y}^2+\left(a^{2}-2a-2\right){x}{y}+\left(a^{2}-2a-2\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a^{3}+3a^{2}+a-6\right){x}+a^{2}+2a-4$ |
| 11.2-b1 |
11.2-b |
$1$ |
$1$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
11.2 |
\( 11 \) |
\( - 11^{5} \) |
$10.43507$ |
$(-a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 5 \) |
$0.005203758$ |
$1091.654626$ |
1.312953821 |
\( -\frac{2664376560}{161051} a^{3} + \frac{573004694}{14641} a^{2} + \frac{8559902589}{161051} a - \frac{8420354916}{161051} \) |
\( \bigl[a^{3} - a^{2} - 5 a\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 0\) , \( -a^{3} + 3 a^{2} + 3 a - 1\) , \( -a^{3} + 2 a^{2}\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a\right){x}{y}={x}^{3}+\left(a^{3}-2a^{2}-3a+2\right){x}^{2}+\left(-a^{3}+3a^{2}+3a-1\right){x}-a^{3}+2a^{2}$ |
| 13.1-a1 |
13.1-a |
$1$ |
$1$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
13.1 |
\( 13 \) |
\( 13^{5} \) |
$10.65527$ |
$(-a^3+3a^2+2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$159.3371930$ |
1.841341088 |
\( \frac{6434431994832}{371293} a^{3} - \frac{9609694163356}{371293} a^{2} - \frac{28207203562225}{371293} a - \frac{7970337758378}{371293} \) |
\( \bigl[a^{3} - a^{2} - 6 a - 1\) , \( a^{3} - 2 a^{2} - 5 a + 1\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( -3 a^{3} + 9 a^{2} + 12 a - 13\) , \( 4 a^{3} - 18 a^{2} - 14 a + 18\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-1\right){x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-5a+1\right){x}^{2}+\left(-3a^{3}+9a^{2}+12a-13\right){x}+4a^{3}-18a^{2}-14a+18$ |
| 13.1-b1 |
13.1-b |
$1$ |
$1$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
13.1 |
\( 13 \) |
\( 13^{5} \) |
$10.65527$ |
$(-a^3+3a^2+2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$78.48400852$ |
0.906981144 |
\( \frac{6434431994832}{371293} a^{3} - \frac{9609694163356}{371293} a^{2} - \frac{28207203562225}{371293} a - \frac{7970337758378}{371293} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 2\) , \( a^{2} - 3 a - 2\) , \( a^{3} - a^{2} - 5 a - 1\) , \( -4 a^{3} + 11 a^{2} + 14 a - 14\) , \( -9 a^{3} + 26 a^{2} + 30 a - 37\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+2\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(a^{2}-3a-2\right){x}^{2}+\left(-4a^{3}+11a^{2}+14a-14\right){x}-9a^{3}+26a^{2}+30a-37$ |
| 13.2-a1 |
13.2-a |
$1$ |
$1$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
13.2 |
\( 13 \) |
\( 13^{5} \) |
$10.65527$ |
$(-a^3+a^2+3a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$159.3371930$ |
1.841341088 |
\( \frac{4048864069719}{371293} a^{3} - \frac{11356897965746}{371293} a^{2} - \frac{3242684631428}{371293} a + \frac{295205696628}{371293} \) |
\( \bigl[a^{2} - a - 1\) , \( a^{3} - 2 a^{2} - 3 a\) , \( 1\) , \( 5 a^{3} - 11 a^{2} - 17 a + 12\) , \( -12 a^{3} + 29 a^{2} + 39 a - 39\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-2a^{2}-3a\right){x}^{2}+\left(5a^{3}-11a^{2}-17a+12\right){x}-12a^{3}+29a^{2}+39a-39$ |
| 13.2-b1 |
13.2-b |
$1$ |
$1$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
13.2 |
\( 13 \) |
\( 13^{5} \) |
$10.65527$ |
$(-a^3+a^2+3a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$78.48400852$ |
0.906981144 |
\( \frac{4048864069719}{371293} a^{3} - \frac{11356897965746}{371293} a^{2} - \frac{3242684631428}{371293} a + \frac{295205696628}{371293} \) |
\( \bigl[a\) , \( -a^{3} + a^{2} + 6 a\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 3 a^{3} - 9 a^{2} - 12 a + 14\) , \( 11 a^{3} - 29 a^{2} - 40 a + 33\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a\right){x}^{2}+\left(3a^{3}-9a^{2}-12a+14\right){x}+11a^{3}-29a^{2}-40a+33$ |
| 16.1-a1 |
16.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$10.93545$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$626.2283084$ |
1.809213362 |
\( -818626500 a^{3} + 1637253000 a^{2} + 2455879500 a + 599278500 \) |
\( \bigl[a^{2} - a - 2\) , \( a^{2} - a - 2\) , \( a^{3} - a^{2} - 5 a - 1\) , \( a^{3} - 4 a^{2} - 5 a - 1\) , \( -3 a^{3} - 9 a^{2} + 1\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(a^{3}-4a^{2}-5a-1\right){x}-3a^{3}-9a^{2}+1$ |
| 16.1-a2 |
16.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$10.93545$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$626.2283084$ |
1.809213362 |
\( 818626500 a^{3} - 1637253000 a^{2} - 2455879500 a + 2236531500 \) |
\( \bigl[a^{2} - a - 2\) , \( a^{2} - a - 2\) , \( 0\) , \( -10 a^{3} - 7 a^{2} + 10 a\) , \( 41 a^{3} + 45 a^{2} - 30 a - 15\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-10a^{3}-7a^{2}+10a\right){x}+41a^{3}+45a^{2}-30a-15$ |
| 16.1-a3 |
16.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$10.93545$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$1252.456616$ |
1.809213362 |
\( 54000 \) |
\( \bigl[a^{2} - 2 a - 1\) , \( -a^{3} + 2 a^{2} + 5 a - 2\) , \( a^{2} - 2 a - 1\) , \( 54 a^{3} - 184 a^{2} + 46 a + 40\) , \( -438 a^{3} + 1503 a^{2} - 398 a - 307\bigr] \) |
${y}^2+\left(a^{2}-2a-1\right){x}{y}+\left(a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+5a-2\right){x}^{2}+\left(54a^{3}-184a^{2}+46a+40\right){x}-438a^{3}+1503a^{2}-398a-307$ |
| 16.1-a4 |
16.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$10.93545$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$1252.456616$ |
1.809213362 |
\( 54000 \) |
\( \bigl[a^{2} - 2 a - 1\) , \( a^{3} - 3 a^{2} - 3 a + 4\) , \( a^{2} - 2 a - 1\) , \( 110 a^{3} - 146 a^{2} - 534 a - 148\) , \( -896 a^{3} + 1164 a^{2} + 4402 a + 1287\bigr] \) |
${y}^2+\left(a^{2}-2a-1\right){x}{y}+\left(a^{2}-2a-1\right){y}={x}^{3}+\left(a^{3}-3a^{2}-3a+4\right){x}^{2}+\left(110a^{3}-146a^{2}-534a-148\right){x}-896a^{3}+1164a^{2}+4402a+1287$ |
| 16.1-a5 |
16.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$10.93545$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$626.2283084$ |
1.809213362 |
\( -818626500 a^{3} + 1637253000 a^{2} + 2455879500 a + 599278500 \) |
\( \bigl[a^{3} - a^{2} - 6 a\) , \( -a^{2} + 3 a + 3\) , \( a^{2} - 2 a - 1\) , \( -14 a^{3} + 28 a^{2} + 55 a - 28\) , \( -27 a^{3} + 61 a^{2} + 91 a - 77\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a\right){x}{y}+\left(a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{2}+3a+3\right){x}^{2}+\left(-14a^{3}+28a^{2}+55a-28\right){x}-27a^{3}+61a^{2}+91a-77$ |
| 16.1-a6 |
16.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$10.93545$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$626.2283084$ |
1.809213362 |
\( 0 \) |
\( \bigl[0\) , \( a^{2} - a - 1\) , \( a^{3} - a^{2} - 6 a\) , \( a^{2}\) , \( a^{3} - a^{2} - 4 a - 2\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a\right){y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}+a^{2}{x}+a^{3}-a^{2}-4a-2$ |
| 16.1-a7 |
16.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$10.93545$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$626.2283084$ |
1.809213362 |
\( 0 \) |
\( \bigl[0\) , \( -a^{2} + a + 1\) , \( a + 1\) , \( a^{2}\) , \( -a^{2}\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+a^{2}{x}-a^{2}$ |
| 16.1-a8 |
16.1-a |
$8$ |
$12$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$10.93545$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$626.2283084$ |
1.809213362 |
\( 818626500 a^{3} - 1637253000 a^{2} - 2455879500 a + 2236531500 \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( -a^{2} + 3 a + 3\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( -13 a^{3} - 4 a^{2} + 21 a + 3\) , \( -82 a^{3} - 82 a^{2} + 70 a + 27\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{2}+3a+3\right){x}^{2}+\left(-13a^{3}-4a^{2}+21a+3\right){x}-82a^{3}-82a^{2}+70a+27$ |
| 22.1-a1 |
22.1-a |
$4$ |
$15$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{15} \cdot 11 \) |
$11.37953$ |
$(a+1), (-a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$116.1774220$ |
1.342575808 |
\( -\frac{206523}{88} a^{3} - \frac{143937}{16} a^{2} + \frac{592191}{176} a + \frac{21762}{11} \) |
\( \bigl[a^{2} - 2 a - 2\) , \( a^{3} - 3 a^{2} - 2 a + 3\) , \( a + 1\) , \( -2 a^{2} + 5\) , \( a^{3} - 2 a\bigr] \) |
${y}^2+\left(a^{2}-2a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-3a^{2}-2a+3\right){x}^{2}+\left(-2a^{2}+5\right){x}+a^{3}-2a$ |
| 22.1-a2 |
22.1-a |
$4$ |
$15$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{3} \cdot 11^{5} \) |
$11.37953$ |
$(a+1), (-a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$116.1774220$ |
1.342575808 |
\( -\frac{239911034371671678746479117074}{161051} a^{3} - \frac{46192006744148769339275912673}{29282} a^{2} + \frac{364994086161069889411346160597}{322102} a + \frac{78428970636351301079852030865}{161051} \) |
\( \bigl[a^{2} - 2 a - 2\) , \( a^{3} - 3 a^{2} - 2 a + 3\) , \( a + 1\) , \( -235 a^{3} - 207 a^{2} + 205 a + 30\) , \( 7870 a^{3} + 8247 a^{2} - 6035 a - 2463\bigr] \) |
${y}^2+\left(a^{2}-2a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-3a^{2}-2a+3\right){x}^{2}+\left(-235a^{3}-207a^{2}+205a+30\right){x}+7870a^{3}+8247a^{2}-6035a-2463$ |
| 22.1-a3 |
22.1-a |
$4$ |
$15$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{5} \cdot 11^{3} \) |
$11.37953$ |
$(a+1), (-a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$116.1774220$ |
1.342575808 |
\( -\frac{29175220575}{5324} a^{3} + \frac{6171704199}{484} a^{2} + \frac{94505480973}{5324} a - \frac{89241217425}{5324} \) |
\( \bigl[a^{3} - a^{2} - 5 a\) , \( -a - 1\) , \( a^{2} - 2 a - 2\) , \( -a^{3} + 3 a^{2} + 3 a - 7\) , \( -2 a^{3} + 5 a^{2} + 7 a - 8\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a\right){x}{y}+\left(a^{2}-2a-2\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a^{3}+3a^{2}+3a-7\right){x}-2a^{3}+5a^{2}+7a-8$ |
| 22.1-a4 |
22.1-a |
$4$ |
$15$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2 \cdot 11^{15} \) |
$11.37953$ |
$(a+1), (-a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$116.1774220$ |
1.342575808 |
\( \frac{945471202761628384833933}{8354496338831302} a^{3} - \frac{131390442129560266984851}{759499667166482} a^{2} - \frac{4983393061162814171576643}{8354496338831302} a - \frac{1441611487960520416115607}{8354496338831302} \) |
\( \bigl[a^{3} - a^{2} - 5 a\) , \( -a - 1\) , \( a^{2} - 2 a - 2\) , \( 19 a^{3} - 97 a^{2} - 172 a - 2\) , \( 145 a^{3} + 494 a^{2} + 458 a + 71\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a\right){x}{y}+\left(a^{2}-2a-2\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(19a^{3}-97a^{2}-172a-2\right){x}+145a^{3}+494a^{2}+458a+71$ |
| 22.1-b1 |
22.1-b |
$4$ |
$15$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{5} \cdot 11^{3} \) |
$11.37953$ |
$(a+1), (-a+2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 3 \cdot 5 \) |
$1$ |
$392.4528985$ |
2.721171248 |
\( -\frac{29175220575}{5324} a^{3} + \frac{6171704199}{484} a^{2} + \frac{94505480973}{5324} a - \frac{89241217425}{5324} \) |
\( \bigl[a^{2} - 2 a - 2\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( -a^{3} + 3 a^{2} + 3 a - 3\) , \( 0\bigr] \) |
${y}^2+\left(a^{2}-2a-2\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+2\right){x}^{2}+\left(-a^{3}+3a^{2}+3a-3\right){x}$ |
| 22.1-b2 |
22.1-b |
$4$ |
$15$ |
4.4.7488.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2 \cdot 11^{15} \) |
$11.37953$ |
$(a+1), (-a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.2 |
$25$ |
\( 3 \cdot 5 \) |
$1$ |
$0.627924637$ |
2.721171248 |
\( \frac{945471202761628384833933}{8354496338831302} a^{3} - \frac{131390442129560266984851}{759499667166482} a^{2} - \frac{4983393061162814171576643}{8354496338831302} a - \frac{1441611487960520416115607}{8354496338831302} \) |
\( \bigl[a^{2} - 2 a - 2\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 19 a^{3} - 97 a^{2} - 172 a + 2\) , \( -127 a^{3} - 589 a^{2} - 626 a - 74\bigr] \) |
${y}^2+\left(a^{2}-2a-2\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+2\right){x}^{2}+\left(19a^{3}-97a^{2}-172a+2\right){x}-127a^{3}-589a^{2}-626a-74$ |
*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.
