The modular curve $X_{64}$

Curve name $X_{64}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{23}$
Meaning/Special name
Chosen covering $X_{23}$
Curves that $X_{64}$ minimally covers $X_{23}$, $X_{43}$, $X_{47}$
Curves that minimally cover $X_{64}$ $X_{191}$, $X_{196}$, $X_{255}$, $X_{257}$, $X_{263}$, $X_{278}$
Curves that minimally cover $X_{64}$ and have infinitely many rational points. $X_{191}$, $X_{196}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{64}) = \mathbb{Q}(f_{64}), f_{23} = \frac{f_{64}^{2} + \frac{1}{4}}{f_{64}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 204083x + 35462018$, with conductor $424830$
Generic density of odd order reductions $271/1344$