## The modular curve $X_{117}$

Curve name $X_{117}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{36}$
Meaning/Special name
Chosen covering $X_{36}$
Curves that $X_{117}$ minimally covers $X_{36}$
Curves that minimally cover $X_{117}$ $X_{219}$, $X_{223}$, $X_{228}$, $X_{229}$, $X_{230}$, $X_{236}$, $X_{306}$, $X_{315}$, $X_{330}$, $X_{332}$, $X_{117a}$, $X_{117b}$, $X_{117c}$, $X_{117d}$, $X_{117e}$, $X_{117f}$, $X_{117g}$, $X_{117h}$, $X_{117i}$, $X_{117j}$, $X_{117k}$, $X_{117l}$, $X_{117m}$, $X_{117n}$, $X_{117o}$, $X_{117p}$
Curves that minimally cover $X_{117}$ and have infinitely many rational points. $X_{219}$, $X_{223}$, $X_{228}$, $X_{229}$, $X_{230}$, $X_{236}$, $X_{117a}$, $X_{117b}$, $X_{117c}$, $X_{117d}$, $X_{117e}$, $X_{117f}$, $X_{117g}$, $X_{117h}$, $X_{117i}$, $X_{117j}$, $X_{117k}$, $X_{117l}$, $X_{117m}$, $X_{117n}$, $X_{117o}$, $X_{117p}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{117}) = \mathbb{Q}(f_{117}), f_{36} = \frac{2}{f_{117}^{2}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 18714x - 985367$, with conductor $5544$
Generic density of odd order reductions $643/5376$