## The modular curve $X_{25i}$

Curve name $X_{25i}$
Index $24$
Level $4$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 2 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$
Meaning/Special name
Chosen covering $X_{25}$
Curves that $X_{25i}$ minimally covers
Curves that minimally cover $X_{25i}$
Curves that minimally cover $X_{25i}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{8} + 81t^{6} - 108t^{4} + 81t^{2} - 27$ $B(t) = 54t^{12} - 243t^{10} + 324t^{8} - 324t^{4} + 243t^{2} - 54$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 4266x - 70713$, with conductor $1287$
Generic density of odd order reductions $19/168$