| Curve name |
$X_{78d}$ |
| 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 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{78}$ |
| Curves that $X_{78d}$ minimally covers |
|
| Curves that minimally cover $X_{78d}$ |
|
| Curves that minimally cover $X_{78d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} - 3240t^{14} - 14256t^{12} + 12960t^{10} + 114912t^{8} +
51840t^{6} - 228096t^{4} - 207360t^{2} - 6912\]
\[B(t) = 54t^{24} - 13608t^{22} - 225504t^{20} - 598752t^{18} + 1801440t^{16} +
8273664t^{14} - 33094656t^{10} - 28823040t^{8} + 38320128t^{6} + 57729024t^{4} +
13934592t^{2} - 221184\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 280230232x - 1803784094132$, with conductor $142296$ |
| Generic density of odd order reductions |
$307/2688$ |