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

Curve name $X_{229c}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$ $8$ $48$ $X_{86j}$
Meaning/Special name
Chosen covering $X_{229}$
Curves that $X_{229c}$ minimally covers
Curves that minimally cover $X_{229c}$
Curves that minimally cover $X_{229c}$ 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 $y^2 = x^3 - x^2 + 3616x + 142848$, with conductor $816$
Generic density of odd order reductions $215/2688$