| Curve name |
$X_{192d}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 9 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 15 & 14 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 14 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{192}$ |
| Curves that $X_{192d}$ minimally covers |
|
| Curves that minimally cover $X_{192d}$ |
|
| Curves that minimally cover $X_{192d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{26} + 864t^{25} - 101952t^{24} + 805248t^{23} - 2878848t^{22} +
13312512t^{21} - 21648384t^{20} - 38320128t^{19} - 34919424t^{18} -
319832064t^{17} + 259227648t^{16} - 19464192t^{15} + 1686306816t^{14} +
77856768t^{13} + 4147642368t^{12} + 20469252096t^{11} - 8939372544t^{10} +
39239811072t^{9} - 88671780864t^{8} - 218112196608t^{7} - 188668182528t^{6} -
211090931712t^{5} - 106904420352t^{4} - 3623878656t^{3} - 1811939328t^{2}\]
\[B(t) = 432t^{39} - 5184t^{38} - 870912t^{37} + 10651392t^{36} -
122487552t^{35} + 1057701888t^{34} - 4202053632t^{33} + 10128457728t^{32} -
29601054720t^{31} + 32630833152t^{30} + 41044672512t^{29} - 281494683648t^{28} +
1915552530432t^{27} - 508928458752t^{26} + 2275909042176t^{25} +
5980758736896t^{24} - 47762324324352t^{23} - 28417776943104t^{22} -
113671107772416t^{20} + 764197189189632t^{19} + 382768559161344t^{18} -
582632714797056t^{17} - 521142741762048t^{16} - 7846103164649472t^{15} -
4612008896888832t^{14} - 2689903657746432t^{13} + 8553977125797888t^{12} +
31038955554078720t^{11} + 42481830762381312t^{10} + 70498761427648512t^{9} +
70981172154335232t^{8} + 32880001875443712t^{7} + 11436845074219008t^{6} +
3740538557693952t^{5} - 89060441849856t^{4} - 29686813949952t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 432180075x - 3458159099750$, with conductor $25200$ |
| Generic density of odd order reductions |
$139/1344$ |