## The modular curve $X_{78e}$

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
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
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 $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} - 3348t^{10} - 27540t^{8} - 83808t^{6} - 110160t^{4} - 53568t^{2} - 1728$ $B(t) = -54t^{18} + 13284t^{16} + 305856t^{14} + 2274048t^{12} + 8278848t^{10} + 16557696t^{8} + 18192384t^{6} + 9787392t^{4} + 1700352t^{2} - 27648$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 7837923x + 8148740222$, with conductor $8280$
Generic density of odd order reductions $635/5376$