| Curve name |
$X_{190g}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 2 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{190}$ |
| Curves that $X_{190g}$ minimally covers |
|
| Curves that minimally cover $X_{190g}$ |
|
| Curves that minimally cover $X_{190g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1404t^{16} - 41472t^{15} - 628992t^{14} - 6635520t^{13} -
54825984t^{12} - 366280704t^{11} - 1955708928t^{10} - 8174960640t^{9} -
26378403840t^{8} - 65399685120t^{7} - 125165371392t^{6} - 187535720448t^{5} -
224567230464t^{4} - 217432719360t^{3} - 164886478848t^{2} - 86973087744t -
23555211264\]
\[B(t) = -15120t^{24} - 497664t^{23} - 5681664t^{22} + 17915904t^{21} +
1520031744t^{20} + 25751126016t^{19} + 273235673088t^{18} + 2129396760576t^{17}
+ 12999169474560t^{16} + 64512287834112t^{15} + 267170114174976t^{14} +
941641313550336t^{13} + 2861569960574976t^{12} + 7533130508402688t^{11} +
17098887307198464t^{10} + 33030291371065344t^{9} + 53244598167797760t^{8} +
69776073050554368t^{7} + 71627092285980672t^{6} + 54004025426706432t^{5} +
25501900895944704t^{4} + 2404631929946112t^{3} - 6100640266715136t^{2} -
4274901208793088t - 1039038488248320\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 17x - 15$, with conductor $192$ |
| Generic density of odd order reductions |
$109/896$ |