## The modular curve $X_{25}$

Curve name $X_{25}$
Index $12$
Level $4$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$
Meaning/Special name $\tilde{X}_{1}(2,4)$
Chosen covering $X_{8}$
Curves that $X_{25}$ minimally covers $X_{8}$, $X_{12}$, $X_{13}$
Curves that minimally cover $X_{25}$ $X_{58}$, $X_{62}$, $X_{87}$, $X_{96}$, $X_{98}$, $X_{99}$, $X_{100}$, $X_{101}$, $X_{129}$, $X_{142}$, $X_{25a}$, $X_{25b}$, $X_{25c}$, $X_{25d}$, $X_{25e}$, $X_{25f}$, $X_{25g}$, $X_{25h}$, $X_{25i}$, $X_{25j}$, $X_{25k}$, $X_{25l}$, $X_{25m}$, $X_{25n}$
Curves that minimally cover $X_{25}$ and have infinitely many rational points. $X_{58}$, $X_{62}$, $X_{87}$, $X_{96}$, $X_{98}$, $X_{99}$, $X_{100}$, $X_{101}$, $X_{25a}$, $X_{25b}$, $X_{25c}$, $X_{25d}$, $X_{25e}$, $X_{25f}$, $X_{25g}$, $X_{25h}$, $X_{25i}$, $X_{25j}$, $X_{25k}$, $X_{25l}$, $X_{25m}$, $X_{25n}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{25}) = \mathbb{Q}(f_{25}), f_{8} = -2f_{25}^{2} + 1$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 2430x - 42449$, with conductor $3465$
Generic density of odd order reductions $5/42$