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

Curve name $X_{32c}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{32}$
Curves that $X_{32c}$ minimally covers
Curves that minimally cover $X_{32c}$
Curves that minimally cover $X_{32c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{8} - 5184t^{6} - 84672t^{4} - 497664t^{2} - 442368$ $B(t) = 432t^{12} + 31104t^{10} + 881280t^{8} + 12192768t^{6} + 80953344t^{4} + 191102976t^{2} - 113246208$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 2008x + 25012$, with conductor $2400$
Generic density of odd order reductions $289/1792$