## The modular curve $X_{24}$

Curve name $X_{24}$
Index $12$
Level $4$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 2 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$
Meaning/Special name
Chosen covering $X_{8}$
Curves that $X_{24}$ minimally covers $X_{8}$, $X_{9}$, $X_{11}$
Curves that minimally cover $X_{24}$ $X_{58}$, $X_{59}$, $X_{61}$, $X_{66}$, $X_{67}$, $X_{132}$, $X_{141}$, $X_{24a}$, $X_{24b}$, $X_{24c}$, $X_{24d}$, $X_{24e}$, $X_{24f}$, $X_{24g}$, $X_{24h}$
Curves that minimally cover $X_{24}$ and have infinitely many rational points. $X_{58}$, $X_{61}$, $X_{66}$, $X_{67}$, $X_{24a}$, $X_{24b}$, $X_{24c}$, $X_{24d}$, $X_{24e}$, $X_{24f}$, $X_{24g}$, $X_{24h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{24}) = \mathbb{Q}(f_{24}), f_{8} = -2f_{24}^{2} - 1$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 240x - 469$, with conductor $1845$
Generic density of odd order reductions $9/56$