## The modular curve $X_{78}$

Curve name $X_{78}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{33}$
Curves that $X_{78}$ minimally covers $X_{33}$, $X_{36}$, $X_{44}$
Curves that minimally cover $X_{78}$ $X_{201}$, $X_{202}$, $X_{233}$, $X_{234}$, $X_{331}$, $X_{332}$, $X_{78a}$, $X_{78b}$, $X_{78c}$, $X_{78d}$, $X_{78e}$, $X_{78f}$, $X_{78g}$, $X_{78h}$, $X_{78i}$, $X_{78j}$
Curves that minimally cover $X_{78}$ and have infinitely many rational points. $X_{202}$, $X_{233}$, $X_{234}$, $X_{78a}$, $X_{78b}$, $X_{78c}$, $X_{78d}$, $X_{78e}$, $X_{78f}$, $X_{78g}$, $X_{78h}$, $X_{78i}$, $X_{78j}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{78}) = \mathbb{Q}(f_{78}), f_{33} = \frac{8f_{78}}{f_{78}^{2} - 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 425379x + 106683838$, with conductor $5544$
Generic density of odd order reductions $643/5376$