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

Curve name $X_{122e}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$ $8$ $24$ $X_{36f}$
Meaning/Special name
Chosen covering $X_{122}$
Curves that $X_{122e}$ minimally covers
Curves that minimally cover $X_{122e}$
Curves that minimally cover $X_{122e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -6912t^{12} + 27648t^{10} - 43200t^{8} + 32832t^{6} - 12123t^{4} + 1782t^{2} - 27$ $B(t) = -221184t^{18} + 1327104t^{16} - 3400704t^{14} + 4838400t^{12} - 4153680t^{10} + 2182464t^{8} - 674730t^{6} + 108702t^{4} - 6318t^{2} - 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 112899x - 14601022$, with conductor $1008$
Generic density of odd order reductions $25/224$