| Curve name |
$X_{78b}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{78}$ |
| Curves that $X_{78b}$ minimally covers |
|
| Curves that minimally cover $X_{78b}$ |
|
| Curves that minimally cover $X_{78b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{12} - 3132t^{10} - 1620t^{8} + 31968t^{6} - 6480t^{4} -
50112t^{2} - 1728\]
\[B(t) = 54t^{18} - 13932t^{16} - 142560t^{14} + 423360t^{12} + 1083456t^{10} -
2166912t^{8} - 3386880t^{6} + 4561920t^{4} + 1783296t^{2} - 27648\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 2315952x - 1356051852$, with conductor $25872$ |
| Generic density of odd order reductions |
$193/1792$ |