## The modular curve $X_{243h}$

Curve name $X_{243h}$
Index $96$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 16 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36i}$ $16$ $48$ $X_{118g}$
Meaning/Special name
Chosen covering $X_{243}$
Curves that $X_{243h}$ minimally covers
Curves that minimally cover $X_{243h}$
Curves that minimally cover $X_{243h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{16} + 1728t^{8} - 1728$ $B(t) = -432t^{24} - 12960t^{16} + 41472t^{8} - 27648$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 5121x - 171615$, with conductor $16320$
Generic density of odd order reductions $299/2688$