## The modular curve $X_{243}$

Curve name $X_{243}$
Index $48$
Level $32$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 16 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 16 & 1 \end{matrix}\right]$
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 $X_{243}$ minimally covers $X_{118}$
Curves that minimally cover $X_{243}$ $X_{490}$, $X_{492}$, $X_{494}$, $X_{495}$, $X_{498}$, $X_{499}$, $X_{243a}$, $X_{243b}$, $X_{243c}$, $X_{243d}$, $X_{243e}$, $X_{243f}$, $X_{243g}$, $X_{243h}$, $X_{243i}$, $X_{243j}$, $X_{243k}$, $X_{243l}$, $X_{243m}$, $X_{243n}$, $X_{243o}$, $X_{243p}$
Curves that minimally cover $X_{243}$ and have infinitely many rational points. $X_{243a}$, $X_{243b}$, $X_{243c}$, $X_{243d}$, $X_{243e}$, $X_{243f}$, $X_{243g}$, $X_{243h}$, $X_{243i}$, $X_{243j}$, $X_{243k}$, $X_{243l}$, $X_{243m}$, $X_{243n}$, $X_{243o}$, $X_{243p}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{243}) = \mathbb{Q}(f_{243}), f_{118} = \frac{2}{f_{243}^{2}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 720x - 8960$, with conductor $1530$
Generic density of odd order reductions $25/224$