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

Curve name $X_{122c}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$ $8$ $24$ $X_{36l}$
Meaning/Special name
Chosen covering $X_{122}$
Curves that $X_{122c}$ minimally covers
Curves that minimally cover $X_{122c}$
Curves that minimally cover $X_{122c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -6912t^{8} + 13824t^{6} - 8640t^{4} + 1728t^{2} - 27$ $B(t) = -221184t^{12} + 663552t^{10} - 746496t^{8} + 387072t^{6} - 89424t^{4} + 6480t^{2} + 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - 784x - 8515$, with conductor $21$
Generic density of odd order reductions $5/84$