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

Curve name $X_{243k}$
Index $96$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 16 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36s}$ $16$ $48$ $X_{118w}$
Meaning/Special name
Chosen covering $X_{243}$
Curves that $X_{243k}$ minimally covers
Curves that minimally cover $X_{243k}$
Curves that minimally cover $X_{243k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{24} - 54t^{20} + 405t^{16} + 864t^{12} - 864t^{4} - 432$ $B(t) = -54t^{36} - 162t^{32} - 1782t^{28} - 4914t^{24} + 324t^{20} + 13932t^{16} + 12096t^{12} - 5184t^{8} - 10368t^{4} - 3456$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 370016x - 106318080$, with conductor $69360$
Generic density of odd order reductions $109/896$