| Curve name |
$X_{192g}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{192}$ |
| Curves that $X_{192g}$ minimally covers |
|
| Curves that minimally cover $X_{192g}$ |
|
| Curves that minimally cover $X_{192g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} - 25056t^{14} - 316224t^{12} + 2059776t^{10} - 4907520t^{8} +
32956416t^{6} - 80953344t^{4} - 102629376t^{2} - 1769472\]
\[B(t) = -54t^{24} + 111456t^{22} + 9979200t^{20} + 8805888t^{18} -
75852288t^{16} - 3849928704t^{14} + 25856409600t^{12} - 61598859264t^{10} -
19418185728t^{8} + 36068917248t^{6} + 653996851200t^{4} + 116870086656t^{2} -
905969664\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 1920800x + 1025280000$, with conductor $1680$ |
| Generic density of odd order reductions |
$19/336$ |