## The modular curve $X_{233c}$

Curve name $X_{233c}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$ $8$ $48$ $X_{78h}$
Meaning/Special name
Chosen covering $X_{233}$
Curves that $X_{233c}$ minimally covers
Curves that minimally cover $X_{233c}$
Curves that minimally cover $X_{233c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -1811939328t^{16} - 56170119168t^{14} - 102261325824t^{12} - 56877907968t^{10} - 11355586560t^{8} - 888717312t^{6} - 24966144t^{4} - 214272t^{2} - 108$ $B(t) = 29686813949952t^{24} - 1825739057922048t^{22} - 10033215402147840t^{20} - 15574212840259584t^{18} - 10655833994035200t^{16} - 3557047085826048t^{14} - 611852501385216t^{12} - 55578860716032t^{10} - 2601521971200t^{8} - 59410907136t^{6} - 598026240t^{4} - 1700352t^{2} + 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 1775617x - 910101503$, with conductor $3264$
Generic density of odd order reductions $109/896$