| Curve name |
$X_{78c}$ |
| 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 \\ 8 & 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 $X_{78c}$ minimally covers |
|
| Curves that minimally cover $X_{78c}$ |
|
| Curves that minimally cover $X_{78c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} - 12960t^{14} - 57024t^{12} + 51840t^{10} + 459648t^{8} +
207360t^{6} - 912384t^{4} - 829440t^{2} - 27648\]
\[B(t) = 432t^{24} - 108864t^{22} - 1804032t^{20} - 4790016t^{18} +
14411520t^{16} + 66189312t^{14} - 264757248t^{10} - 230584320t^{8} +
306561024t^{6} + 461832192t^{4} + 111476736t^{2} - 1769472\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 1120920929x - 14431393673985$, with conductor
$1138368$ |
| Generic density of odd order reductions |
$335/2688$ |