## The modular curve $X_{236e}$

Curve name $X_{236e}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $48$ $X_{85o}$
Meaning/Special name
Chosen covering $X_{236}$
Curves that $X_{236e}$ minimally covers
Curves that minimally cover $X_{236e}$
Curves that minimally cover $X_{236e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -442368t^{24} + 4423680t^{22} + 10395648t^{20} - 187342848t^{18} + 485305344t^{16} - 123310080t^{14} - 527053824t^{12} - 30827520t^{10} + 30331584t^{8} - 2927232t^{6} + 40608t^{4} + 4320t^{2} - 108$ $B(t) = 113246208t^{36} - 1698693120t^{34} + 24715984896t^{32} - 219018166272t^{30} + 893087907840t^{28} - 1158763511808t^{26} - 2157644611584t^{24} + 6587744256000t^{22} - 1604147576832t^{20} - 4765666738176t^{18} - 401036894208t^{16} + 411734016000t^{14} - 33713197056t^{12} - 4526419968t^{10} + 872156160t^{8} - 53471232t^{6} + 1508544t^{4} - 25920t^{2} + 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 4214849x - 3329185215$, with conductor $9408$
Generic density of odd order reductions $109/896$