## The modular curve $X_{63}$

Curve name $X_{63}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 6 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$
Meaning/Special name
Chosen covering $X_{35}$
Curves that $X_{63}$ minimally covers $X_{35}$, $X_{39}$, $X_{47}$
Curves that minimally cover $X_{63}$ $X_{254}$, $X_{263}$, $X_{302}$, $X_{307}$
Curves that minimally cover $X_{63}$ and have infinitely many rational points. $X_{302}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{63}) = \mathbb{Q}(f_{63}), f_{35} = \frac{8f_{63}}{f_{63}^{2} - 2}$
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 538x - 4628$, with conductor $2400$
Generic density of odd order reductions $1343/5376$