| Curve name |
$X_{85m}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{85}$ |
| Curves that $X_{85m}$ minimally covers |
|
| Curves that minimally cover $X_{85m}$ |
|
| Curves that minimally cover $X_{85m}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{12} + 432t^{10} - 1080t^{8} - 11232t^{6} + 65232t^{4} -
93312t^{2} - 6912\]
\[B(t) = 54t^{18} - 1296t^{16} + 20088t^{14} - 211680t^{12} + 1319328t^{10} -
4437504t^{8} + 6604416t^{6} + 41472t^{4} - 7464960t^{2} + 221184\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 306x + 5859$, with conductor $63$ |
| Generic density of odd order reductions |
$17/168$ |