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

Curve name $X_{122j}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36a}$
Meaning/Special name
Chosen covering $X_{122}$
Curves that $X_{122j}$ minimally covers
Curves that minimally cover $X_{122j}$
Curves that minimally cover $X_{122j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -110592t^{12} + 331776t^{10} - 387072t^{8} + 221184t^{6} - 62640t^{4} + 7344t^{2} - 108$ $B(t) = 14155776t^{18} - 63700992t^{16} + 122093568t^{14} - 130056192t^{12} + 84022272t^{10} - 33550848t^{8} + 8007552t^{6} - 1021248t^{4} + 49248t^{2} + 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 + 489x + 5865$, with conductor $1470$
Generic density of odd order reductions $193/1792$