## The modular curve $X_{79}$

Curve name $X_{79}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{32}$
Curves that $X_{79}$ minimally covers $X_{32}$, $X_{34}$, $X_{44}$
Curves that minimally cover $X_{79}$ $X_{226}$, $X_{333}$, $X_{334}$, $X_{368}$, $X_{369}$, $X_{79a}$, $X_{79b}$, $X_{79c}$, $X_{79d}$, $X_{79e}$, $X_{79f}$, $X_{79g}$, $X_{79h}$, $X_{79i}$, $X_{79j}$
Curves that minimally cover $X_{79}$ and have infinitely many rational points. $X_{226}$, $X_{79a}$, $X_{79b}$, $X_{79c}$, $X_{79d}$, $X_{79e}$, $X_{79f}$, $X_{79g}$, $X_{79h}$, $X_{79i}$, $X_{79j}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{79}) = \mathbb{Q}(f_{79}), f_{32} = \frac{4f_{79}^{2} - 8}{f_{79}^{2} + 4f_{79} + 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 117603x - 15523002$, with conductor $10080$
Generic density of odd order reductions $289/1792$