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

Curve name $X_{78g}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 5 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$
Meaning/Special name
Chosen covering $X_{78}$
Curves that $X_{78g}$ minimally covers
Curves that minimally cover $X_{78g}$
Curves that minimally cover $X_{78g}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{8} - 12960t^{6} - 57888t^{4} - 51840t^{2} - 1728$ $B(t) = -432t^{12} + 108864t^{10} + 1798848t^{8} + 6096384t^{6} + 7195392t^{4} + 1741824t^{2} - 27648$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 189057x + 31547007$, with conductor $14784$
Generic density of odd order reductions $307/2688$