## The modular curve $X_{65}$

Curve name $X_{65}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 5 & 5 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right]$
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 $X_{65}$ minimally covers $X_{23}$, $X_{35}$, $X_{50}$
Curves that minimally cover $X_{65}$ $X_{251}$, $X_{261}$, $X_{317}$, $X_{318}$
Curves that minimally cover $X_{65}$ and have infinitely many rational points. $X_{318}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{65}) = \mathbb{Q}(f_{65}), f_{23} = \frac{f_{65}^{2} - 2}{f_{65}^{2} + 2}$
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 23x + 20$, with conductor $867$
Generic density of odd order reductions $401/1792$