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

Curve name $X_{122b}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36i}$
Meaning/Special name
Chosen covering $X_{122}$
Curves that $X_{122b}$ minimally covers
Curves that minimally cover $X_{122b}$
Curves that minimally cover $X_{122b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27648t^{8} + 55296t^{6} - 34560t^{4} + 6912t^{2} - 108$ $B(t) = -1769472t^{12} + 5308416t^{10} - 5971968t^{8} + 3096576t^{6} - 715392t^{4} + 51840t^{2} + 432$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 50177x - 4309503$, with conductor $1344$
Generic density of odd order reductions $635/5376$