## The modular curve $X_{229h}$

Curve name $X_{229h}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $48$ $X_{86k}$
Meaning/Special name
Chosen covering $X_{229}$
Curves that $X_{229h}$ minimally covers
Curves that minimally cover $X_{229h}$
Curves that minimally cover $X_{229h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27648t^{16} - 110592t^{14} + 1575936t^{12} + 3345408t^{10} - 4544640t^{8} + 836352t^{6} + 98496t^{4} - 1728t^{2} - 108$ $B(t) = -1769472t^{24} - 10616832t^{22} - 241532928t^{20} - 898007040t^{18} + 1176809472t^{16} + 3905667072t^{14} - 4125413376t^{12} + 976416768t^{10} + 73550592t^{8} - 14031360t^{6} - 943488t^{4} - 10368t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 + 14463x + 1157247$, with conductor $3264$
Generic density of odd order reductions $271/2688$