| Curve name |
$X_{190h}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \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 $X_{190h}$ minimally covers |
|
| Curves that minimally cover $X_{190h}$ |
|
| Curves that minimally cover $X_{190h}$ 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 |
$271/2688$ |