## The modular curve $X_{225}$

Curve name $X_{225}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{85}$
Meaning/Special name
Chosen covering $X_{85}$
Curves that $X_{225}$ minimally covers $X_{85}$, $X_{118}$, $X_{121}$
Curves that minimally cover $X_{225}$ $X_{471}$, $X_{482}$, $X_{495}$, $X_{497}$, $X_{225a}$, $X_{225b}$, $X_{225c}$, $X_{225d}$, $X_{225e}$, $X_{225f}$, $X_{225g}$, $X_{225h}$, $X_{225i}$, $X_{225j}$, $X_{225k}$, $X_{225l}$
Curves that minimally cover $X_{225}$ and have infinitely many rational points. $X_{225a}$, $X_{225b}$, $X_{225c}$, $X_{225d}$, $X_{225e}$, $X_{225f}$, $X_{225g}$, $X_{225h}$, $X_{225i}$, $X_{225j}$, $X_{225k}$, $X_{225l}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{225}) = \mathbb{Q}(f_{225}), f_{85} = \frac{f_{225}}{f_{225}^{2} + \frac{1}{8}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - 105841x + 13244636$, with conductor $735$
Generic density of odd order reductions $25/224$