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

Curve name $X_{122h}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36c}$
Meaning/Special name
Chosen covering $X_{122}$
Curves that $X_{122h}$ minimally covers
Curves that minimally cover $X_{122h}$
Curves that minimally cover $X_{122h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27648t^{12} + 110592t^{10} - 172800t^{8} + 131328t^{6} - 48492t^{4} + 7128t^{2} - 108$ $B(t) = -1769472t^{18} + 10616832t^{16} - 27205632t^{14} + 38707200t^{12} - 33229440t^{10} + 17459712t^{8} - 5397840t^{6} + 869616t^{4} - 50544t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 79110x - 27294894$, with conductor $2070$
Generic density of odd order reductions $193/1792$