## The modular curve $X_{115}$

Curve name $X_{115}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 9 \\ 12 & 7 \end{matrix}\right]$
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 $X_{115}$ minimally covers $X_{32}$
Curves that minimally cover $X_{115}$ $X_{314}$, $X_{334}$, $X_{115a}$, $X_{115b}$, $X_{115c}$, $X_{115d}$, $X_{115e}$, $X_{115f}$, $X_{115g}$, $X_{115h}$
Curves that minimally cover $X_{115}$ and have infinitely many rational points. $X_{115a}$, $X_{115b}$, $X_{115c}$, $X_{115d}$, $X_{115e}$, $X_{115f}$, $X_{115g}$, $X_{115h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{115}) = \mathbb{Q}(f_{115}), f_{32} = \frac{2}{f_{115}^{2}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 18x - 27$, with conductor $360$
Generic density of odd order reductions $13/84$