## The modular curve $X_{122}$

Curve name $X_{122}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{36}$
Meaning/Special name
Chosen covering $X_{36}$
Curves that $X_{122}$ minimally covers $X_{36}$
Curves that minimally cover $X_{122}$ $X_{207}$, $X_{208}$, $X_{211}$, $X_{212}$, $X_{217}$, $X_{228}$, $X_{230}$, $X_{233}$, $X_{335}$, $X_{337}$, $X_{346}$, $X_{347}$, $X_{122a}$, $X_{122b}$, $X_{122c}$, $X_{122d}$, $X_{122e}$, $X_{122f}$, $X_{122g}$, $X_{122h}$, $X_{122i}$, $X_{122j}$, $X_{122k}$, $X_{122l}$, $X_{122m}$, $X_{122n}$, $X_{122o}$, $X_{122p}$
Curves that minimally cover $X_{122}$ and have infinitely many rational points. $X_{207}$, $X_{208}$, $X_{211}$, $X_{212}$, $X_{217}$, $X_{228}$, $X_{230}$, $X_{233}$, $X_{122a}$, $X_{122b}$, $X_{122c}$, $X_{122d}$, $X_{122e}$, $X_{122f}$, $X_{122g}$, $X_{122h}$, $X_{122i}$, $X_{122j}$, $X_{122k}$, $X_{122l}$, $X_{122m}$, $X_{122n}$, $X_{122o}$, $X_{122p}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{122}) = \mathbb{Q}(f_{122}), f_{36} = 8f_{122}^{2} - 4$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 + x^2 - 19600x - 1064375$, with conductor $525$
Generic density of odd order reductions $19/168$