| Curve name |
$X_{199g}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{199}$ |
| Curves that $X_{199g}$ minimally covers |
|
| Curves that minimally cover $X_{199g}$ |
|
| Curves that minimally cover $X_{199g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} + 50112t^{14} - 316224t^{12} - 1029888t^{10} - 1226880t^{8}
- 4119552t^{6} - 5059584t^{4} + 3207168t^{2} - 27648\]
\[B(t) = -432t^{24} - 445824t^{22} + 19958400t^{20} - 8805888t^{18} -
37926144t^{16} + 962482176t^{14} + 3232051200t^{12} + 3849928704t^{10} -
606818304t^{8} - 563576832t^{6} + 5109350400t^{4} - 456523776t^{2} - 1769472\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 105217x + 15737087$, with conductor $3264$ |
| Generic density of odd order reductions |
$271/2688$ |