## The modular curve $X_{236}$

Curve name $X_{236}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{85}$
Meaning/Special name
Chosen covering $X_{85}$
Curves that $X_{236}$ minimally covers $X_{85}$, $X_{117}$, $X_{120}$
Curves that minimally cover $X_{236}$ $X_{466}$, $X_{474}$, $X_{476}$, $X_{483}$, $X_{236a}$, $X_{236b}$, $X_{236c}$, $X_{236d}$, $X_{236e}$, $X_{236f}$, $X_{236g}$, $X_{236h}$
Curves that minimally cover $X_{236}$ and have infinitely many rational points. $X_{236a}$, $X_{236b}$, $X_{236c}$, $X_{236d}$, $X_{236e}$, $X_{236f}$, $X_{236g}$, $X_{236h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{236}) = \mathbb{Q}(f_{236}), f_{85} = \frac{f_{236}^{2} + \frac{1}{2}}{f_{236}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 12096x - 509036$, with conductor $126$
Generic density of odd order reductions $193/1792$