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

Curve name $X_{243m}$
Index $96$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 16 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36a}$ $16$ $48$ $X_{118m}$
Meaning/Special name
Chosen covering $X_{243}$
Curves that $X_{243m}$ minimally covers
Curves that minimally cover $X_{243m}$
Curves that minimally cover $X_{243m}$ 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 $y^2 + xy = x^3 - 23126x + 1661220$, with conductor $8670$
Generic density of odd order reductions $271/2688$