## The modular curve $X_{215}$

Curve name $X_{215}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 14 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$ $8$ $24$ $X_{96}$
Meaning/Special name
Chosen covering $X_{96}$
Curves that $X_{215}$ minimally covers $X_{96}$, $X_{119}$, $X_{120}$
Curves that minimally cover $X_{215}$ $X_{469}$, $X_{470}$, $X_{483}$, $X_{485}$, $X_{215a}$, $X_{215b}$, $X_{215c}$, $X_{215d}$, $X_{215e}$, $X_{215f}$, $X_{215g}$, $X_{215h}$, $X_{215i}$, $X_{215j}$, $X_{215k}$, $X_{215l}$
Curves that minimally cover $X_{215}$ and have infinitely many rational points. $X_{215a}$, $X_{215b}$, $X_{215c}$, $X_{215d}$, $X_{215e}$, $X_{215f}$, $X_{215g}$, $X_{215h}$, $X_{215i}$, $X_{215j}$, $X_{215k}$, $X_{215l}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{215}) = \mathbb{Q}(f_{215}), f_{96} = \frac{2}{f_{215}^{2}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - 246x - 1485$, with conductor $735$
Generic density of odd order reductions $25/224$