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

Curve name $X_{122k}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36q}$
Meaning/Special name
Chosen covering $X_{122}$
Curves that $X_{122k}$ minimally covers
Curves that minimally cover $X_{122k}$
Curves that minimally cover $X_{122k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27648t^{12} + 82944t^{10} - 96768t^{8} + 55296t^{6} - 15660t^{4} + 1836t^{2} - 27$ $B(t) = 1769472t^{18} - 7962624t^{16} + 15261696t^{14} - 16257024t^{12} + 10502784t^{10} - 4193856t^{8} + 1000944t^{6} - 127656t^{4} + 6156t^{2} + 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 38417x + 2882228$, with conductor $147$
Generic density of odd order reductions $25/224$