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

Curve name $X_{25d}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 0 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{25}$
Curves that $X_{25d}$ minimally covers
Curves that minimally cover $X_{25d}$
Curves that minimally cover $X_{25d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{10} + 324t^{8} - 432t^{6} + 324t^{4} - 108t^{2}$ $B(t) = 432t^{15} - 1944t^{13} + 2592t^{11} - 2592t^{7} + 1944t^{5} - 432t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 2042442x + 1039319716$, with conductor $54450$
Generic density of odd order reductions $643/5376$