| 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 |
| 7.1-a1 |
7.1-a |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( 7^{4} \) |
$10.67151$ |
$(-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$155.3164161$ |
0.829348559 |
\( -\frac{3330378294119680}{2401} a^{3} + \frac{12188115690766272}{2401} a^{2} - \frac{3576468865195392}{2401} a - \frac{14046479459413248}{2401} \) |
\( \bigl[a^{3} - 4 a - 3\) , \( 0\) , \( a\) , \( 31 a^{3} - 117 a^{2} + 42 a + 140\) , \( -243 a^{3} + 890 a^{2} - 260 a - 1025\bigr] \) |
${y}^2+\left(a^{3}-4a-3\right){x}{y}+a{y}={x}^{3}+\left(31a^{3}-117a^{2}+42a+140\right){x}-243a^{3}+890a^{2}-260a-1025$ |
| 7.1-a2 |
7.1-a |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( 7^{8} \) |
$10.67151$ |
$(-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$155.3164161$ |
0.829348559 |
\( \frac{57165234770944}{5764801} a^{3} + \frac{29324092490688}{5764801} a^{2} - \frac{177134335713792}{5764801} a - \frac{137148082835392}{5764801} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{2} - a - 4\) , \( a^{2} - 2\) , \( 8 a^{3} - 30 a^{2} + 8 a + 34\) , \( -29 a^{3} + 103 a^{2} - 28 a - 118\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(8a^{3}-30a^{2}+8a+34\right){x}-29a^{3}+103a^{2}-28a-118$ |
| 7.1-b1 |
7.1-b |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( 7^{2} \) |
$10.67151$ |
$(-a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.043271130$ |
$1477.732640$ |
1.365758242 |
\( \frac{30306816}{49} a^{3} - \frac{86052416}{49} a^{2} - \frac{79682944}{49} a + \frac{250463744}{49} \) |
\( \bigl[a^{2} - a - 3\) , \( -a^{3} + 5 a + 3\) , \( a^{3} - a^{2} - 4 a\) , \( -2 a^{3} + 11 a + 9\) , \( 2 a^{3} - 11 a - 9\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{3}+5a+3\right){x}^{2}+\left(-2a^{3}+11a+9\right){x}+2a^{3}-11a-9$ |
| 7.1-b2 |
7.1-b |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( 7^{4} \) |
$10.67151$ |
$(-a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.021635565$ |
$1477.732640$ |
1.365758242 |
\( -\frac{3744000}{2401} a^{3} + \frac{3354304}{2401} a^{2} + \frac{22264576}{2401} a + \frac{10547776}{2401} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a^{2} - 2\) , \( -2 a^{2} + a + 7\) , \( -a^{3} + 2 a^{2} - 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(-2a^{2}+a+7\right){x}-a^{3}+2a^{2}-1$ |
| 7.2-a1 |
7.2-a |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
7.2 |
\( 7 \) |
\( 7^{7} \) |
$10.67151$ |
$(a^3-2a^2-3a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 7 \) |
$0.057415531$ |
$109.9898435$ |
1.888381478 |
\( -\frac{1423105976}{823543} a^{3} + \frac{4066274288}{823543} a^{2} + \frac{3628847529}{823543} a - \frac{1661776752}{117649} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + a^{2} + 4 a + 1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( 3 a + 6\) , \( -18 a^{3} - 10 a^{2} + 60 a + 45\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a+1\right){x}^{2}+\left(3a+6\right){x}-18a^{3}-10a^{2}+60a+45$ |
| 7.3-a1 |
7.3-a |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
7.3 |
\( 7 \) |
\( 7^{7} \) |
$10.67151$ |
$(-a^2+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 7 \) |
$0.057415531$ |
$109.9898435$ |
1.888381478 |
\( \frac{1423105976}{823543} a^{3} - \frac{203043640}{823543} a^{2} - \frac{7492078177}{823543} a - \frac{765774489}{117649} \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} + 2 a + 2\) , \( a^{2} - a - 3\) , \( -a^{3} + 6 a^{2} - 2 a - 7\) , \( 21 a^{3} - 73 a^{2} + 18 a + 84\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(-a^{3}+6a^{2}-2a-7\right){x}+21a^{3}-73a^{2}+18a+84$ |
| 7.4-a1 |
7.4-a |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
7.4 |
\( 7 \) |
\( 7^{8} \) |
$10.67151$ |
$(-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$155.3164161$ |
0.829348559 |
\( -\frac{57165234770944}{5764801} a^{3} + \frac{200819796803520}{5764801} a^{2} - \frac{53009553580416}{5764801} a - \frac{32541870183936}{823543} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{2} - a - 4\) , \( a^{2} - 3\) , \( -9 a^{3} - 4 a^{2} + 30 a + 17\) , \( 28 a^{3} + 18 a^{2} - 90 a - 74\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-9a^{3}-4a^{2}+30a+17\right){x}+28a^{3}+18a^{2}-90a-74$ |
| 7.4-a2 |
7.4-a |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
7.4 |
\( 7 \) |
\( 7^{4} \) |
$10.67151$ |
$(-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$155.3164161$ |
0.829348559 |
\( \frac{3330378294119680}{2401} a^{3} + \frac{2196980808407232}{2401} a^{2} - \frac{10808627633978112}{2401} a - \frac{1252172989708864}{343} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( a - 1\) , \( a^{2} - 2\) , \( -32 a^{3} - 23 a^{2} + 105 a + 92\) , \( 189 a^{3} + 127 a^{2} - 614 a - 505\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-32a^{3}-23a^{2}+105a+92\right){x}+189a^{3}+127a^{2}-614a-505$ |
| 7.4-b1 |
7.4-b |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
7.4 |
\( 7 \) |
\( 7^{4} \) |
$10.67151$ |
$(-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.021635565$ |
$1477.732640$ |
1.365758242 |
\( \frac{3744000}{2401} a^{3} - \frac{7877696}{2401} a^{2} - \frac{17741184}{2401} a + \frac{4631808}{343} \) |
\( \bigl[a^{3} - 4 a - 3\) , \( -a^{3} + a^{2} + 4 a + 1\) , \( a^{2} - 3\) , \( -8 a^{3} + 13 a^{2} + 31 a - 20\) , \( -11 a^{3} + 22 a^{2} + 39 a - 46\bigr] \) |
${y}^2+\left(a^{3}-4a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a+1\right){x}^{2}+\left(-8a^{3}+13a^{2}+31a-20\right){x}-11a^{3}+22a^{2}+39a-46$ |
| 7.4-b2 |
7.4-b |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
7.4 |
\( 7 \) |
\( 7^{2} \) |
$10.67151$ |
$(-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.043271130$ |
$1477.732640$ |
1.365758242 |
\( -\frac{30306816}{49} a^{3} + \frac{4868032}{49} a^{2} + \frac{160867328}{49} a + \frac{16433600}{7} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - 5 a - 2\) , \( a^{3} - 5 a - 2\) , \( a^{3} - 3 a^{2} - 2 a + 10\) , \( -3 a^{3} + 11 a^{2} + 5 a - 36\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(a^{3}-3a^{2}-2a+10\right){x}-3a^{3}+11a^{2}+5a-36$ |
| 28.1-a1 |
28.1-a |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( - 2^{2} \cdot 7 \) |
$12.69064$ |
$(-a), (a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$253.6739628$ |
2.709103659 |
\( -\frac{2391172730382}{7} a^{3} - \frac{3154823624777}{14} a^{2} + \frac{15520946769649}{14} a + \frac{12586656735363}{14} \) |
\( \bigl[a^{3} - 5 a - 3\) , \( -a^{3} + 2 a^{2} + 4 a - 4\) , \( a^{3} - a^{2} - 4 a\) , \( -2 a^{3} - 11 a^{2} + 33 a + 29\) , \( 6 a^{3} - 49 a^{2} + 44 a + 117\bigr] \) |
${y}^2+\left(a^{3}-5a-3\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-4\right){x}^{2}+\left(-2a^{3}-11a^{2}+33a+29\right){x}+6a^{3}-49a^{2}+44a+117$ |
| 28.1-b1 |
28.1-b |
$2$ |
$3$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( - 2^{2} \cdot 7 \) |
$12.69064$ |
$(-a), (a^2-a-3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$1076.820954$ |
1.277764205 |
\( \frac{7462577}{14} a^{3} - \frac{13233471}{14} a^{2} - \frac{27170221}{14} a + \frac{12737230}{7} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{2} + a + 3\) , \( a^{3} - a^{2} - 3 a + 1\) , \( -a^{3} + 2 a^{2} - 1\) , \( -27 a^{3} + 97 a^{2} - 27 a - 112\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-a^{3}+2a^{2}-1\right){x}-27a^{3}+97a^{2}-27a-112$ |
| 28.1-b2 |
28.1-b |
$2$ |
$3$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( - 2^{6} \cdot 7^{3} \) |
$12.69064$ |
$(-a), (a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$13.29408585$ |
1.277764205 |
\( \frac{661426621493919350957}{343} a^{3} - \frac{7553259107304969906021}{1372} a^{2} - \frac{6771160831123005516347}{1372} a + \frac{21662985339131396964743}{1372} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{2} + a + 3\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 4 a^{3} - 18 a^{2} + 10 a + 19\) , \( 714 a^{3} - 2617 a^{2} + 775 a + 3011\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(4a^{3}-18a^{2}+10a+19\right){x}+714a^{3}-2617a^{2}+775a+3011$ |
| 28.1-c1 |
28.1-c |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( - 2^{6} \cdot 7^{5} \) |
$12.69064$ |
$(-a), (a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$110.5322042$ |
1.180425438 |
\( \frac{2035510669}{67228} a^{3} - \frac{7032400899}{67228} a^{2} + \frac{409886903}{67228} a + \frac{2618144985}{16807} \) |
\( \bigl[a^{3} - 5 a - 2\) , \( a^{3} - 5 a - 2\) , \( 1\) , \( 2 a^{3} - 5 a^{2} + 4 a\) , \( 6 a^{3} - 20 a^{2} + 11 a + 10\bigr] \) |
${y}^2+\left(a^{3}-5a-2\right){x}{y}+{y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(2a^{3}-5a^{2}+4a\right){x}+6a^{3}-20a^{2}+11a+10$ |
| 28.1-d1 |
28.1-d |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{28} \cdot 7 \) |
$12.69064$ |
$(-a), (a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$17.69036254$ |
2.644931923 |
\( -\frac{1050799201833}{56} a^{3} - \frac{5547216337951}{448} a^{2} + \frac{54556742363853}{896} a + \frac{691322109897}{14} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{3} + a^{2} + 5 a - 1\) , \( a^{3} - 5 a - 3\) , \( 6 a^{3} - 5 a^{2} - 32 a - 19\) , \( -38 a^{3} + 104 a^{2} + 6 a - 95\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{3}-5a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-1\right){x}^{2}+\left(6a^{3}-5a^{2}-32a-19\right){x}-38a^{3}+104a^{2}+6a-95$ |
| 28.1-e1 |
28.1-e |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{7} \) |
$12.69064$ |
$(-a), (a^2-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$0.050473301$ |
$93.50613839$ |
2.822536639 |
\( \frac{817855417}{1647086} a^{3} - \frac{194156684}{823543} a^{2} - \frac{3288385623}{1647086} a - \frac{540085410}{823543} \) |
\( \bigl[a\) , \( a^{3} - 2 a^{2} - 4 a + 2\) , \( a^{3} - 4 a - 2\) , \( 2 a^{3} - 11 a^{2} - a + 38\) , \( 13 a^{3} - 38 a^{2} - 35 a + 106\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-4a-2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+2\right){x}^{2}+\left(2a^{3}-11a^{2}-a+38\right){x}+13a^{3}-38a^{2}-35a+106$ |
| 28.2-a1 |
28.2-a |
$2$ |
$3$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{8} \cdot 7 \) |
$12.69064$ |
$(a^3-2a^2-3a+3), (a^2-a-3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$391.5662567$ |
1.858542388 |
\( \frac{1681633}{28} a^{3} - \frac{66257}{7} a^{2} - \frac{8823963}{28} a - \frac{449645}{2} \) |
\( \bigl[a + 1\) , \( -a^{3} + 2 a^{2} + 4 a - 4\) , \( a^{3} - 5 a - 2\) , \( -4 a^{2} + 5 a + 18\) , \( 2 a^{2} - a - 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-4\right){x}^{2}+\left(-4a^{2}+5a+18\right){x}+2a^{2}-a-9$ |
| 28.2-a2 |
28.2-a |
$2$ |
$3$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{24} \cdot 7^{3} \) |
$12.69064$ |
$(a^3-2a^2-3a+3), (a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$4.834151318$ |
1.858542388 |
\( \frac{53992007028089}{21952} a^{3} - \frac{24708914377163}{2744} a^{2} + \frac{3628719958247}{1372} a + \frac{8137310632245}{784} \) |
\( \bigl[a + 1\) , \( -a^{3} + 2 a^{2} + 4 a - 4\) , \( a^{3} - 5 a - 2\) , \( 10 a^{3} - 49 a^{2} + 35 a + 68\) , \( 57 a^{3} - 233 a^{2} + 119 a + 261\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-4\right){x}^{2}+\left(10a^{3}-49a^{2}+35a+68\right){x}+57a^{3}-233a^{2}+119a+261$ |
| 28.3-a1 |
28.3-a |
$2$ |
$3$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.3 |
\( 2^{2} \cdot 7 \) |
\( 2^{8} \cdot 7 \) |
$12.69064$ |
$(-a^2+2), (a^2-a-3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$391.5662567$ |
1.858542388 |
\( -\frac{1681633}{28} a^{3} + \frac{4779871}{28} a^{2} + \frac{1077280}{7} a - 489371 \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( a^{3} - 6 a - 4\) , \( a^{3} - 5 a - 2\) , \( -5 a^{3} - 3 a^{2} + 14 a + 10\) , \( -20 a^{3} - 14 a^{2} + 63 a + 51\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(a^{3}-6a-4\right){x}^{2}+\left(-5a^{3}-3a^{2}+14a+10\right){x}-20a^{3}-14a^{2}+63a+51$ |
| 28.3-a2 |
28.3-a |
$2$ |
$3$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.3 |
\( 2^{2} \cdot 7 \) |
\( 2^{24} \cdot 7^{3} \) |
$12.69064$ |
$(-a^2+2), (a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$4.834151318$ |
1.858542388 |
\( -\frac{53992007028089}{21952} a^{3} - \frac{35695293933037}{21952} a^{2} + \frac{175307089618389}{21952} a + \frac{20317844149371}{3136} \) |
\( \bigl[a\) , \( a^{3} - a^{2} - 3 a - 1\) , \( a^{3} - 5 a - 2\) , \( -12 a^{3} - 12 a^{2} + 37 a + 43\) , \( -81 a^{3} - 71 a^{2} + 250 a + 239\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a-1\right){x}^{2}+\left(-12a^{3}-12a^{2}+37a+43\right){x}-81a^{3}-71a^{2}+250a+239$ |
| 28.4-a1 |
28.4-a |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.4 |
\( 2^{2} \cdot 7 \) |
\( - 2^{2} \cdot 7 \) |
$12.69064$ |
$(-a+1), (a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$253.6739628$ |
2.709103659 |
\( \frac{2391172730382}{7} a^{3} - \frac{17501860007069}{14} a^{2} + \frac{5135736862197}{14} a + \frac{2881490631353}{2} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( -a^{3} + 6 a + 3\) , \( 0\) , \( -7 a^{3} - 2 a^{2} + 29 a + 22\) , \( -14 a^{3} - 7 a^{2} + 49 a + 38\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}={x}^{3}+\left(-a^{3}+6a+3\right){x}^{2}+\left(-7a^{3}-2a^{2}+29a+22\right){x}-14a^{3}-7a^{2}+49a+38$ |
| 28.4-b1 |
28.4-b |
$2$ |
$3$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.4 |
\( 2^{2} \cdot 7 \) |
\( - 2^{2} \cdot 7 \) |
$12.69064$ |
$(-a+1), (a^2-a-3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$1076.820954$ |
1.277764205 |
\( -\frac{7462577}{14} a^{3} + \frac{4577130}{7} a^{2} + \frac{15624716}{7} a - \frac{1066665}{2} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{2} + a + 3\) , \( a^{3} - 4 a - 2\) , \( 0\) , \( 26 a^{3} + 17 a^{2} - 85 a - 69\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-4a-2\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+26a^{3}+17a^{2}-85a-69$ |
| 28.4-b2 |
28.4-b |
$2$ |
$3$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.4 |
\( 2^{2} \cdot 7 \) |
\( - 2^{6} \cdot 7^{3} \) |
$12.69064$ |
$(-a+1), (a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$13.29408585$ |
1.277764205 |
\( -\frac{661426621493919350957}{343} a^{3} + \frac{383860350622062305463}{1372} a^{2} + \frac{13940559587805913116905}{1372} a + \frac{1426324555239871278029}{196} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{2} + a + 3\) , \( a^{3} - 4 a - 2\) , \( -5 a^{3} - 5 a^{2} + 15 a + 15\) , \( -715 a^{3} - 474 a^{2} + 2318 a + 1883\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-4a-2\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-5a^{3}-5a^{2}+15a+15\right){x}-715a^{3}-474a^{2}+2318a+1883$ |
| 28.4-c1 |
28.4-c |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.4 |
\( 2^{2} \cdot 7 \) |
\( - 2^{6} \cdot 7^{5} \) |
$12.69064$ |
$(-a+1), (a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$110.5322042$ |
1.180425438 |
\( -\frac{2035510669}{67228} a^{3} - \frac{231467223}{16807} a^{2} + \frac{1887095722}{16807} a + \frac{840796659}{9604} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{2} + 2\) , \( a + 1\) , \( -2 a^{3} - 2 a^{2} + 7 a + 8\) , \( a^{3} - 2 a^{2} - 9 a - 5\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-2a^{3}-2a^{2}+7a+8\right){x}+a^{3}-2a^{2}-9a-5$ |
| 28.4-d1 |
28.4-d |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.4 |
\( 2^{2} \cdot 7 \) |
\( 2^{28} \cdot 7 \) |
$12.69064$ |
$(-a+1), (a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$17.69036254$ |
2.644931923 |
\( \frac{1050799201833}{56} a^{3} - \frac{30766397181943}{448} a^{2} + \frac{18070484675935}{896} a + \frac{10127733927433}{128} \) |
\( \bigl[a^{3} - 5 a - 2\) , \( -a^{3} + 2 a^{2} + 4 a - 2\) , \( a + 1\) , \( -15 a^{3} + 23 a^{2} + 64 a - 49\) , \( 71 a^{3} - 129 a^{2} - 161 a + 345\bigr] \) |
${y}^2+\left(a^{3}-5a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-2\right){x}^{2}+\left(-15a^{3}+23a^{2}+64a-49\right){x}+71a^{3}-129a^{2}-161a+345$ |
| 28.4-e1 |
28.4-e |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
28.4 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{7} \) |
$12.69064$ |
$(-a+1), (a^2-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$0.050473301$ |
$93.50613839$ |
2.822536639 |
\( -\frac{817855417}{1647086} a^{3} + \frac{2065252883}{1647086} a^{2} + \frac{805723054}{823543} a - \frac{281358171}{117649} \) |
\( \bigl[a + 1\) , \( -a^{3} + a^{2} + 4 a\) , \( a^{3} - a^{2} - 4 a\) , \( -6 a^{3} + a^{2} + 31 a + 23\) , \( -17 a^{3} + 3 a^{2} + 89 a + 62\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a\right){x}^{2}+\left(-6a^{3}+a^{2}+31a+23\right){x}-17a^{3}+3a^{2}+89a+62$ |
| 47.1-a1 |
47.1-a |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( -47 \) |
$13.53945$ |
$(a^3-3a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.115226424$ |
$494.1936863$ |
2.432534457 |
\( \frac{2551808}{47} a^{3} - \frac{7278592}{47} a^{2} - 139264 a + \frac{20893696}{47} \) |
\( \bigl[0\) , \( a^{2} - 2\) , \( a^{3} - a^{2} - 4 a\) , \( a^{3} + a^{2} - 5 a - 4\) , \( 3 a^{3} - 15 a - 12\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(a^{3}+a^{2}-5a-4\right){x}+3a^{3}-15a-12$ |
| 47.1-b1 |
47.1-b |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( -47 \) |
$13.53945$ |
$(a^3-3a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.019745470$ |
$2245.999079$ |
1.894465812 |
\( -\frac{72957952}{47} a^{3} - \frac{48173056}{47} a^{2} + 5038080 a + \frac{192151552}{47} \) |
\( \bigl[0\) , \( -a^{3} + 2 a^{2} + 3 a - 2\) , \( a^{3} - 5 a - 3\) , \( -2 a^{3} + 4 a^{2} + 6 a - 7\) , \( a^{3} - a^{2} - 5 a\bigr] \) |
${y}^2+\left(a^{3}-5a-3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-2\right){x}^{2}+\left(-2a^{3}+4a^{2}+6a-7\right){x}+a^{3}-a^{2}-5a$ |
| 47.2-a1 |
47.2-a |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
47.2 |
\( 47 \) |
\( -47 \) |
$13.53945$ |
$(a^2-6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.115226424$ |
$494.1936863$ |
2.432534457 |
\( -\frac{2551808}{47} a^{3} + \frac{376832}{47} a^{2} + \frac{13447168}{47} a + \frac{9621504}{47} \) |
\( \bigl[0\) , \( a^{2} - 2 a - 4\) , \( a^{3} - 5 a - 2\) , \( -a^{3} + 2 a^{2} + 4 a - 2\) , \( -a^{3} + 4 a^{2} - 16\bigr] \) |
${y}^2+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(a^{2}-2a-4\right){x}^{2}+\left(-a^{3}+2a^{2}+4a-2\right){x}-a^{3}+4a^{2}-16$ |
| 47.2-b1 |
47.2-b |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
47.2 |
\( 47 \) |
\( -47 \) |
$13.53945$ |
$(a^2-6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.019745470$ |
$2245.999079$ |
1.894465812 |
\( \frac{72957952}{47} a^{3} - \frac{267046912}{47} a^{2} + \frac{78430208}{47} a + \frac{307810304}{47} \) |
\( \bigl[0\) , \( a^{3} - a^{2} - 4 a - 1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( 0\) , \( -a^{3} + a^{2} + 4 a\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a-1\right){x}^{2}-a^{3}+a^{2}+4a$ |
| 49.1-a1 |
49.1-a |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{12} \) |
$13.61016$ |
$(-a), (a^3-2a^2-3a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.114749467$ |
$517.3581679$ |
2.536014356 |
\( -\frac{44760930627344}{282475249} a^{3} + \frac{168976543383552}{282475249} a^{2} - \frac{73341307103017}{282475249} a - \frac{157744926916982}{282475249} \) |
\( \bigl[1\) , \( a^{3} - 5 a - 2\) , \( 1\) , \( 6 a^{3} - 30 a - 23\) , \( -22 a^{3} + 4 a^{2} + 118 a + 84\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(6a^{3}-30a-23\right){x}-22a^{3}+4a^{2}+118a+84$ |
| 49.1-a2 |
49.1-a |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{6} \) |
$13.61016$ |
$(-a), (a^3-2a^2-3a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.229498934$ |
$1034.716335$ |
2.536014356 |
\( -\frac{16354304}{16807} a^{3} + \frac{912599}{16807} a^{2} + \frac{99864196}{16807} a + \frac{73738428}{16807} \) |
\( \bigl[1\) , \( a^{3} - 5 a - 2\) , \( 1\) , \( a^{3} - 5 a - 3\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(a^{3}-5a-3\right){x}$ |
| 49.1-b1 |
49.1-b |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{5} \) |
$13.61016$ |
$(-a), (a^3-2a^2-3a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \cdot 3 \) |
$1$ |
$32.80866345$ |
2.102274964 |
\( -\frac{290358137165245440}{343} a^{3} + \frac{1062617589116427072}{343} a^{2} - \frac{311813455979452288}{343} a - \frac{1224638565232102912}{343} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - a^{2} - 5 a - 1\) , \( a\) , \( -11 a^{3} + 27 a^{2} + 31 a - 73\) , \( -26 a^{3} + 73 a^{2} + 67 a - 210\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-5a-1\right){x}^{2}+\left(-11a^{3}+27a^{2}+31a-73\right){x}-26a^{3}+73a^{2}+67a-210$ |
| 49.1-b2 |
49.1-b |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{10} \) |
$13.61016$ |
$(-a), (a^3-2a^2-3a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$32.80866345$ |
2.102274964 |
\( \frac{1293030515712}{117649} a^{3} - \frac{3376965535296}{117649} a^{2} + \frac{33867079424}{117649} a + \frac{3305156021568}{117649} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( -a^{3} + 2 a^{2} + 2 a - 4\) , \( a^{3} - a^{2} - 4 a + 1\) , \( 7 a^{3} - 23 a^{2} - 23 a + 63\) , \( -30 a^{3} + 70 a^{2} + 80 a - 195\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-4\right){x}^{2}+\left(7a^{3}-23a^{2}-23a+63\right){x}-30a^{3}+70a^{2}+80a-195$ |
| 49.1-c1 |
49.1-c |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{6} \) |
$13.61016$ |
$(-a), (a^3-2a^2-3a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.105980174$ |
$816.5448599$ |
3.696701625 |
\( -\frac{7138560}{2401} a^{3} - \frac{5082944}{2401} a^{2} + \frac{49843200}{2401} a + \frac{5979840}{343} \) |
\( \bigl[a^{3} - 4 a - 3\) , \( -1\) , \( a + 1\) , \( -4 a^{3} - 2 a^{2} + 15 a + 12\) , \( -6 a^{3} - 4 a^{2} + 19 a + 15\bigr] \) |
${y}^2+\left(a^{3}-4a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-4a^{3}-2a^{2}+15a+12\right){x}-6a^{3}-4a^{2}+19a+15$ |
| 49.1-c2 |
49.1-c |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{3} \) |
$13.61016$ |
$(-a), (a^3-2a^2-3a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.211960348$ |
$816.5448599$ |
3.696701625 |
\( -\frac{399201536}{49} a^{3} + \frac{1461376192}{49} a^{2} - \frac{430704768}{49} a - \frac{240186624}{7} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( -a^{2} + 2 a + 2\) , \( a^{3} - a^{2} - 4 a\) , \( 3 a^{3} + 4 a^{2} - 11 a - 14\) , \( 4 a^{3} + 3 a^{2} - 14 a - 14\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(3a^{3}+4a^{2}-11a-14\right){x}+4a^{3}+3a^{2}-14a-14$ |
| 49.1-d1 |
49.1-d |
$4$ |
$6$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{3} \) |
$13.61016$ |
$(-a), (a^3-2a^2-3a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2 \) |
$1$ |
$1384.524355$ |
3.285774959 |
\( \frac{765877760}{49} a^{3} - \frac{2202481472}{49} a^{2} - \frac{1944993920}{49} a + \frac{906237440}{7} \) |
\( \bigl[a^{3} - 4 a - 3\) , \( -a^{3} + 4 a + 4\) , \( a^{2} - 3\) , \( -6 a^{3} - a^{2} + 26 a + 19\) , \( a^{3} + 3 a^{2} + a - 2\bigr] \) |
${y}^2+\left(a^{3}-4a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+4a+4\right){x}^{2}+\left(-6a^{3}-a^{2}+26a+19\right){x}+a^{3}+3a^{2}+a-2$ |
| 49.1-d2 |
49.1-d |
$4$ |
$6$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{9} \) |
$13.61016$ |
$(-a), (a^3-2a^2-3a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$4$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$17.09289327$ |
3.285774959 |
\( -\frac{12483790472214352746752}{117649} a^{3} + \frac{1811248607191129664192}{117649} a^{2} + \frac{65778659077294775822208}{117649} a + \frac{6730125577940443591936}{16807} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - 2 a^{2} - 4 a + 3\) , \( a^{2} - 3\) , \( -203 a^{3} + 576 a^{2} + 532 a - 1670\) , \( -479 a^{3} + 1360 a^{2} + 1253 a - 3947\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+3\right){x}^{2}+\left(-203a^{3}+576a^{2}+532a-1670\right){x}-479a^{3}+1360a^{2}+1253a-3947$ |
| 49.1-d3 |
49.1-d |
$4$ |
$6$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{6} \) |
$13.61016$ |
$(-a), (a^3-2a^2-3a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \) |
$1$ |
$1384.524355$ |
3.285774959 |
\( \frac{906803712}{2401} a^{3} + \frac{609987520}{2401} a^{2} - \frac{2943977216}{2401} a - \frac{345815360}{343} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - 2 a^{2} - 2 a + 4\) , \( a^{3} - 5 a - 2\) , \( -7 a^{3} - 8 a^{2} + 22 a + 27\) , \( 24 a^{3} + 14 a^{2} - 79 a - 61\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-2a+4\right){x}^{2}+\left(-7a^{3}-8a^{2}+22a+27\right){x}+24a^{3}+14a^{2}-79a-61$ |
| 49.1-d4 |
49.1-d |
$4$ |
$6$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{18} \) |
$13.61016$ |
$(-a), (a^3-2a^2-3a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$17.09289327$ |
3.285774959 |
\( -\frac{1628587733026164992}{13841287201} a^{3} + \frac{236282966458738880}{13841287201} a^{2} + \frac{8581248695245708288}{13841287201} a + \frac{877992424829113536}{1977326743} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( -a^{3} + 4 a + 4\) , \( a + 1\) , \( 203 a^{3} - 588 a^{2} - 514 a + 1699\) , \( -44 a^{3} + 120 a^{2} + 125 a - 357\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+4a+4\right){x}^{2}+\left(203a^{3}-588a^{2}-514a+1699\right){x}-44a^{3}+120a^{2}+125a-357$ |
| 49.10-a1 |
49.10-a |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.10 |
\( 7^{2} \) |
\( 7^{10} \) |
$13.61016$ |
$(-a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.115530191$ |
$858.7849669$ |
4.238280019 |
\( -\frac{3744000}{2401} a^{3} + \frac{3354304}{2401} a^{2} + \frac{22264576}{2401} a + \frac{10547776}{2401} \) |
\( \bigl[a^{3} - 4 a - 3\) , \( -a + 1\) , \( a^{3} - 5 a - 2\) , \( a^{3} - 6 a - 4\) , \( 3 a^{2} - 5 a - 7\bigr] \) |
${y}^2+\left(a^{3}-4a-3\right){x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(a^{3}-6a-4\right){x}+3a^{2}-5a-7$ |
| 49.10-a2 |
49.10-a |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.10 |
\( 7^{2} \) |
\( 7^{8} \) |
$13.61016$ |
$(-a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.231060382$ |
$858.7849669$ |
4.238280019 |
\( \frac{30306816}{49} a^{3} - \frac{86052416}{49} a^{2} - \frac{79682944}{49} a + \frac{250463744}{49} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( -a^{2} + 3\) , \( a^{3} - a^{2} - 4 a\) , \( 2 a^{3} - 6 a^{2} + 3\) , \( -2 a^{3} + 9 a^{2} - 5 a - 13\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(2a^{3}-6a^{2}+3\right){x}-2a^{3}+9a^{2}-5a-13$ |
| 49.10-b1 |
49.10-b |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.10 |
\( 7^{2} \) |
\( 7^{10} \) |
$13.61016$ |
$(-a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.942095165$ |
$72.43223032$ |
3.004568365 |
\( -\frac{3330378294119680}{2401} a^{3} + \frac{12188115690766272}{2401} a^{2} - \frac{3576468865195392}{2401} a - \frac{14046479459413248}{2401} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - 5 a - 4\) , \( a\) , \( 13 a^{3} - 74 a^{2} + 102 a + 7\) , \( -358 a^{3} + 1250 a^{2} - 90 a - 1869\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-5a-4\right){x}^{2}+\left(13a^{3}-74a^{2}+102a+7\right){x}-358a^{3}+1250a^{2}-90a-1869$ |
| 49.10-b2 |
49.10-b |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.10 |
\( 7^{2} \) |
\( 7^{14} \) |
$13.61016$ |
$(-a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.971047582$ |
$72.43223032$ |
3.004568365 |
\( \frac{57165234770944}{5764801} a^{3} + \frac{29324092490688}{5764801} a^{2} - \frac{177134335713792}{5764801} a - \frac{137148082835392}{5764801} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( -a^{3} + a^{2} + 5 a - 1\) , \( a + 1\) , \( 6 a^{3} - 27 a^{2} + 19 a + 26\) , \( -54 a^{3} + 186 a^{2} + 3 a - 309\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-1\right){x}^{2}+\left(6a^{3}-27a^{2}+19a+26\right){x}-54a^{3}+186a^{2}+3a-309$ |
| 49.10-c1 |
49.10-c |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.10 |
\( 7^{2} \) |
\( 7^{3} \) |
$13.61016$ |
$(-a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.031833306$ |
$1490.667613$ |
4.054173041 |
\( -418578 a^{3} + 1008528 a^{2} + 1910559 a - 4342514 \) |
\( \bigl[a + 1\) , \( -a^{3} + 4 a + 2\) , \( a\) , \( -a^{3} + 5 a + 3\) , \( 45 a^{3} - 7 a^{2} - 237 a - 169\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+4a+2\right){x}^{2}+\left(-a^{3}+5a+3\right){x}+45a^{3}-7a^{2}-237a-169$ |
| 49.10-d1 |
49.10-d |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.10 |
\( 7^{2} \) |
\( - 7^{3} \) |
$13.61016$ |
$(-a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.418467123$ |
$399.7197143$ |
3.572700715 |
\( 3113216 a^{3} + 2053056 a^{2} - 10104064 a - 8192192 \) |
\( \bigl[a^{3} - 4 a - 3\) , \( -a^{2} + 4\) , \( a^{3} - a^{2} - 4 a + 1\) , \( -2 a^{3} - 3 a^{2} + 6 a + 10\) , \( 3 a^{3} + 2 a^{2} - 11 a - 10\bigr] \) |
${y}^2+\left(a^{3}-4a-3\right){x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-2a^{3}-3a^{2}+6a+10\right){x}+3a^{3}+2a^{2}-11a-10$ |
| 49.10-d2 |
49.10-d |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.10 |
\( 7^{2} \) |
\( - 7^{3} \) |
$13.61016$ |
$(-a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.209233561$ |
$799.4394287$ |
3.572700715 |
\( -471552 a^{3} + 74432 a^{2} + 2471552 a + 1766400 \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( -a\) , \( a^{3} - 5 a - 3\) , \( a^{3} - 9 a - 7\) , \( 3 a^{3} - 15 a - 11\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{3}-5a-3\right){y}={x}^{3}-a{x}^{2}+\left(a^{3}-9a-7\right){x}+3a^{3}-15a-11$ |
| 49.10-e1 |
49.10-e |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.10 |
\( 7^{2} \) |
\( - 7^{9} \) |
$13.61016$ |
$(-a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.143545861$ |
$725.9474792$ |
2.225746009 |
\( -471552 a^{3} + 74432 a^{2} + 2471552 a + 1766400 \) |
\( \bigl[a^{3} - 4 a - 3\) , \( -a^{3} + 5 a + 2\) , \( a^{3} - 5 a - 3\) , \( 2 a^{2} - 3 a - 10\) , \( 2 a^{3} - 3 a^{2} - 8 a + 4\bigr] \) |
${y}^2+\left(a^{3}-4a-3\right){x}{y}+\left(a^{3}-5a-3\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(2a^{2}-3a-10\right){x}+2a^{3}-3a^{2}-8a+4$ |
| 49.10-e2 |
49.10-e |
$2$ |
$2$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.10 |
\( 7^{2} \) |
\( - 7^{9} \) |
$13.61016$ |
$(-a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.287091723$ |
$362.9737396$ |
2.225746009 |
\( 3113216 a^{3} + 2053056 a^{2} - 10104064 a - 8192192 \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - 2 a^{2} - 3 a + 3\) , \( a^{2} - 2\) , \( a^{3} - 4 a^{2} - 3 a + 11\) , \( -5 a^{2} + 4 a + 20\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+3\right){x}^{2}+\left(a^{3}-4a^{2}-3a+11\right){x}-5a^{2}+4a+20$ |
| 49.10-f1 |
49.10-f |
$1$ |
$1$ |
4.4.8768.1 |
$4$ |
$[4, 0]$ |
49.10 |
\( 7^{2} \) |
\( 7^{9} \) |
$13.61016$ |
$(-a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.029911116$ |
$976.0538341$ |
2.494285191 |
\( -418578 a^{3} + 1008528 a^{2} + 1910559 a - 4342514 \) |
\( \bigl[a^{3} - 4 a - 2\) , \( -a^{3} + 5 a + 2\) , \( 1\) , \( -2 a^{3} + a^{2} + 13 a + 4\) , \( -2 a^{3} + a^{2} + 9 a + 13\bigr] \) |
${y}^2+\left(a^{3}-4a-2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-2a^{3}+a^{2}+13a+4\right){x}-2a^{3}+a^{2}+9a+13$ |
*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.
