## The modular curve $X_{240}$

Curve name $X_{240}$
Index $48$
Level $32$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 16 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 16 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{36}$ $16$ $24$ $X_{118}$
Meaning/Special name
Chosen covering $X_{118}$
Curves that $X_{240}$ minimally covers $X_{118}$
Curves that minimally cover $X_{240}$ $X_{488}$, $X_{489}$, $X_{491}$, $X_{493}$, $X_{496}$, $X_{497}$, $X_{240a}$, $X_{240b}$, $X_{240c}$, $X_{240d}$, $X_{240e}$, $X_{240f}$, $X_{240g}$, $X_{240h}$, $X_{240i}$, $X_{240j}$, $X_{240k}$, $X_{240l}$, $X_{240m}$, $X_{240n}$, $X_{240o}$, $X_{240p}$
Curves that minimally cover $X_{240}$ and have infinitely many rational points. $X_{240a}$, $X_{240b}$, $X_{240c}$, $X_{240d}$, $X_{240e}$, $X_{240f}$, $X_{240g}$, $X_{240h}$, $X_{240i}$, $X_{240j}$, $X_{240k}$, $X_{240l}$, $X_{240m}$, $X_{240n}$, $X_{240o}$, $X_{240p}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{240}) = \mathbb{Q}(f_{240}), f_{118} = f_{240}^{2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x - 64$, with conductor $735$
Generic density of odd order reductions $25/224$