## The modular curve $X_{36h}$

Curve name $X_{36h}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$
Meaning/Special name
Chosen covering $X_{36}$
Curves that $X_{36h}$ minimally covers
Curves that minimally cover $X_{36h}$
Curves that minimally cover $X_{36h}$ 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 $y^2 + xy = x^3 - x^2 - 27081x + 1667790$, with conductor $1287$
Generic density of odd order reductions $19/168$