## The modular curve $X_{23}$

Curve name $X_{23}$
Index $12$
Level $4$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 2 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$
Meaning/Special name $X_{s}^{+}(4)$
Chosen covering $X_{11}$
Curves that $X_{23}$ minimally covers $X_{11}$
Curves that minimally cover $X_{23}$ $X_{58}$, $X_{60}$, $X_{64}$, $X_{65}$, $X_{68}$, $X_{69}$, $X_{76}$, $X_{80}$, $X_{127}$, $X_{136}$, $X_{146}$, $X_{148}$
Curves that minimally cover $X_{23}$ and have infinitely many rational points. $X_{58}$, $X_{60}$, $X_{64}$, $X_{65}$, $X_{68}$, $X_{69}$, $X_{76}$, $X_{80}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{23}) = \mathbb{Q}(f_{23}), f_{11} = \frac{8}{f_{23}^{2} - 1}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 128x + 540$, with conductor $1176$
Generic density of odd order reductions $25/112$