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

Curve name $X_{32h}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 9 \\ 12 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{32}$
Meaning/Special name
Chosen covering $X_{32}$
Curves that $X_{32h}$ minimally covers
Curves that minimally cover $X_{32h}$
Curves that minimally cover $X_{32h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{6} - 432t^{4} - 432t^{2}$ $B(t) = 54t^{9} + 1296t^{7} + 6480t^{5} - 3456t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 542x + 4991$, with conductor $1025$
Generic density of odd order reductions $289/1792$