## The modular curve $X_{32f}$

Curve name $X_{32f}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$
Meaning/Special name
Chosen covering $X_{32}$
Curves that $X_{32f}$ minimally covers
Curves that minimally cover $X_{32f}$
Curves that minimally cover $X_{32f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{8} - 1296t^{6} - 21168t^{4} - 124416t^{2} - 110592$ $B(t) = -54t^{12} - 3888t^{10} - 110160t^{8} - 1524096t^{6} - 10119168t^{4} - 23887872t^{2} + 14155776$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 188675x - 1623250$, with conductor $2200$
Generic density of odd order reductions $13/84$