## The modular curve $X_{73}$

Curve name $X_{73}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 5 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{26}$
Meaning/Special name
Chosen covering $X_{26}$
Curves that $X_{73}$ minimally covers $X_{26}$, $X_{45}$, $X_{50}$
Curves that minimally cover $X_{73}$ $X_{253}$, $X_{267}$, $X_{276}$, $X_{277}$, $X_{348}$, $X_{349}$, $X_{350}$, $X_{351}$
Curves that minimally cover $X_{73}$ and have infinitely many rational points. $X_{349}$, $X_{350}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{73}) = \mathbb{Q}(f_{73}), f_{26} = \frac{f_{73}^{2} - 8}{f_{73}^{2} + 8f_{73} + 8}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + 787952x + 97113753$, with conductor $16940$
Generic density of odd order reductions $1427/5376$