## The modular curve $X_{234}$

Curve name $X_{234}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{78}$
Meaning/Special name
Chosen covering $X_{78}$
Curves that $X_{234}$ minimally covers $X_{78}$, $X_{120}$, $X_{121}$
Curves that minimally cover $X_{234}$ $X_{471}$, $X_{474}$, $X_{234a}$, $X_{234b}$, $X_{234c}$, $X_{234d}$, $X_{234e}$, $X_{234f}$, $X_{234g}$, $X_{234h}$
Curves that minimally cover $X_{234}$ and have infinitely many rational points. $X_{234a}$, $X_{234b}$, $X_{234c}$, $X_{234d}$, $X_{234e}$, $X_{234f}$, $X_{234g}$, $X_{234h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{234}) = \mathbb{Q}(f_{234}), f_{78} = \frac{2f_{234}^{2} + 8}{f_{234}^{2} + 4f_{234} - 4}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 9608x - 365712$, with conductor $600$
Generic density of odd order reductions $635/5376$