| Curve name |
$X_{193n}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
$X_1(2,8)$ |
| Chosen covering |
$X_{193}$ |
| Curves that $X_{193n}$ minimally covers |
|
| Curves that minimally cover $X_{193n}$ |
|
| Curves that minimally cover $X_{193n}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1769472t^{16} + 3538944t^{14} - 1327104t^{12} - 221184t^{10} -
1589760t^{8} - 13824t^{6} - 5184t^{4} + 864t^{2} - 27\]
\[B(t) = 905969664t^{24} - 2717908992t^{22} + 2378170368t^{20} - 396361728t^{18}
- 1985347584t^{16} + 1847328768t^{14} + 229146624t^{12} + 115458048t^{10} -
7755264t^{8} - 96768t^{6} + 36288t^{4} - 2592t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - 1070x + 7812$, with conductor $210$ |
| Generic density of odd order reductions |
$1/28$ |