## The modular curve $X_{242}$

Curve name $X_{242}$
Index $48$
Level $32$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 15 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{32}$ $16$ $24$ $X_{116}$
Meaning/Special name
Chosen covering $X_{116}$
Curves that $X_{242}$ minimally covers $X_{116}$
Curves that minimally cover $X_{242}$ $X_{242a}$, $X_{242b}$, $X_{242c}$, $X_{242d}$, $X_{242e}$, $X_{242f}$, $X_{242g}$, $X_{242h}$
Curves that minimally cover $X_{242}$ and have infinitely many rational points. $X_{242a}$, $X_{242b}$, $X_{242c}$, $X_{242d}$, $X_{242e}$, $X_{242f}$, $X_{242g}$, $X_{242h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{242}) = \mathbb{Q}(f_{242}), f_{116} = -f_{242}^{2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 6x - 1$, with conductor $153$
Generic density of odd order reductions $9249/57344$