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

Curve name $X_{243e}$
Index $96$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 16 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 16 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 16 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$ $8$ $24$ $X_{36f}$ $16$ $48$ $X_{118c}$
Meaning/Special name
Chosen covering $X_{243}$
Curves that $X_{243e}$ minimally covers
Curves that minimally cover $X_{243e}$
Curves that minimally cover $X_{243e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{32} + 486t^{24} - 1323t^{16} + 1296t^{8} - 432$ $B(t) = 54t^{48} + 1458t^{40} - 9882t^{32} + 23814t^{24} - 27540t^{16} + 15552t^{8} - 3456$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 - 1414945355x - 20485667336853$, with conductor $130050$
Generic density of odd order reductions $299/2688$