## The modular curve $X_{233}$

Curve name $X_{233}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right]$
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 $X_{233}$ minimally covers $X_{78}$, $X_{119}$, $X_{122}$
Curves that minimally cover $X_{233}$ $X_{472}$, $X_{473}$, $X_{233a}$, $X_{233b}$, $X_{233c}$, $X_{233d}$, $X_{233e}$, $X_{233f}$, $X_{233g}$, $X_{233h}$
Curves that minimally cover $X_{233}$ and have infinitely many rational points. $X_{233a}$, $X_{233b}$, $X_{233c}$, $X_{233d}$, $X_{233e}$, $X_{233f}$, $X_{233g}$, $X_{233h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{233}) = \mathbb{Q}(f_{233}), f_{78} = \frac{f_{233}}{f_{233}^{2} + \frac{1}{8}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 249696x + 48087270$, with conductor $306$
Generic density of odd order reductions $635/5376$