## The modular curve $X_{116}$

Curve name $X_{116}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 9 \\ 12 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{32}$
Meaning/Special name
Chosen covering $X_{32}$
Curves that $X_{116}$ minimally covers $X_{32}$
Curves that minimally cover $X_{116}$ $X_{241}$, $X_{242}$, $X_{316}$, $X_{333}$, $X_{407}$, $X_{408}$, $X_{116a}$, $X_{116b}$, $X_{116c}$, $X_{116d}$, $X_{116e}$, $X_{116f}$, $X_{116g}$, $X_{116h}$, $X_{116i}$, $X_{116j}$, $X_{116k}$, $X_{116l}$
Curves that minimally cover $X_{116}$ and have infinitely many rational points. $X_{241}$, $X_{242}$, $X_{116a}$, $X_{116b}$, $X_{116c}$, $X_{116d}$, $X_{116e}$, $X_{116f}$, $X_{116g}$, $X_{116h}$, $X_{116i}$, $X_{116j}$, $X_{116k}$, $X_{116l}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{116}) = \mathbb{Q}(f_{116}), f_{32} = -f_{116}^{2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 4101x + 100723$, with conductor $7275$
Generic density of odd order reductions $289/1792$