| Curve name |
$X_{78e}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{78}$ |
| Curves that $X_{78e}$ minimally covers |
|
| Curves that minimally cover $X_{78e}$ |
|
| Curves that minimally cover $X_{78e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{12} - 3348t^{10} - 27540t^{8} - 83808t^{6} - 110160t^{4} -
53568t^{2} - 1728\]
\[B(t) = -54t^{18} + 13284t^{16} + 305856t^{14} + 2274048t^{12} + 8278848t^{10}
+ 16557696t^{8} + 18192384t^{6} + 9787392t^{4} + 1700352t^{2} - 27648\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 7837923x + 8148740222$, with conductor $8280$ |
| Generic density of odd order reductions |
$635/5376$ |