| Curve name |
$X_{85h}$ |
| Index |
$48$ |
| Level |
$16$ |
| 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} 7 & 0 \\ 8 & 7 \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 $X_{85h}$ minimally covers |
|
| Curves that minimally cover $X_{85h}$ |
|
| Curves that minimally cover $X_{85h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} + 2160t^{14} - 11664t^{12} - 20736t^{10} + 423360t^{8} -
1596672t^{6} + 2509056t^{4} - 1382400t^{2} - 110592\]
\[B(t) = -432t^{24} + 12960t^{22} - 228096t^{20} + 2785536t^{18} -
22726656t^{16} + 120434688t^{14} - 406038528t^{12} + 827117568t^{10} -
856313856t^{8} + 58613760t^{6} + 729907200t^{4} - 498991104t^{2} + 14155776\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 15003x - 1979636$, with conductor $441$ |
| Generic density of odd order reductions |
$25/224$ |