## The modular curve $X_{69}$

Curve name $X_{69}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 6 & 7 \end{matrix}\right]$
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 $X_{69}$ minimally covers $X_{23}$, $X_{45}$, $X_{50}$
Curves that minimally cover $X_{69}$ $X_{250}$, $X_{251}$, $X_{252}$, $X_{258}$, $X_{262}$, $X_{277}$, $X_{319}$, $X_{320}$, $X_{321}$, $X_{322}$, $X_{323}$, $X_{324}$
Curves that minimally cover $X_{69}$ and have infinitely many rational points. $X_{320}$, $X_{323}$, $X_{324}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{69}) = \mathbb{Q}(f_{69}), f_{23} = \frac{f_{69}^{2} + 2}{f_{69}^{2} - 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + 529x - 20491$, with conductor $11109$
Generic density of odd order reductions $401/1792$