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

Curve name $X_{236b}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$ $8$ $48$ $X_{85k}$
Meaning/Special name
Chosen covering $X_{236}$
Curves that $X_{236b}$ minimally covers
Curves that minimally cover $X_{236b}$
Curves that minimally cover $X_{236b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -6912t^{16} + 27648t^{14} + 393984t^{12} - 836352t^{10} - 1136160t^{8} - 209088t^{6} + 24624t^{4} + 432t^{2} - 27$ $B(t) = 221184t^{24} - 1327104t^{22} + 30191616t^{20} - 112250880t^{18} - 147101184t^{16} + 488208384t^{14} + 515676672t^{12} + 122052096t^{10} - 9193824t^{8} - 1753920t^{6} + 117936t^{4} - 1296t^{2} + 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 1344x + 18405$, with conductor $42$
Generic density of odd order reductions $53/896$