## The modular curve $X_{122p}$

Curve name $X_{122p}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36s}$
Meaning/Special name
Chosen covering $X_{122}$
Curves that $X_{122p}$ minimally covers
Curves that minimally cover $X_{122p}$
Curves that minimally cover $X_{122p}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -110592t^{16} + 552960t^{14} - 1161216t^{12} + 1327104t^{10} - 892080t^{8} + 353808t^{6} - 77436t^{4} + 7560t^{2} - 108$ $B(t) = -14155776t^{24} + 106168320t^{22} - 355663872t^{20} + 701595648t^{18} - 904172544t^{16} + 797879808t^{14} - 490783104t^{12} + 209718720t^{10} - 60686496t^{8} + 11218608t^{6} - 1167696t^{4} + 47952t^{2} + 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 22128204x - 40065204368$, with conductor $28224$
Generic density of odd order reductions $635/5376$