## The modular curve $X_{120c}$

Curve name $X_{120c}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36k}$
Meaning/Special name
Chosen covering $X_{120}$
Curves that $X_{120c}$ minimally covers
Curves that minimally cover $X_{120c}$
Curves that minimally cover $X_{120c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{8} + 1728t^{6} - 8640t^{4} + 13824t^{2} - 1728$ $B(t) = 432t^{12} - 10368t^{10} + 93312t^{8} - 387072t^{6} + 715392t^{4} - 414720t^{2} - 27648$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 + 63x - 63$, with conductor $1344$
Generic density of odd order reductions $635/5376$