## The modular curve $X_{120}$

Curve name $X_{120}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
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 $X_{120}$ minimally covers $X_{36}$
Curves that minimally cover $X_{120}$ $X_{207}$, $X_{215}$, $X_{217}$, $X_{227}$, $X_{229}$, $X_{234}$, $X_{235}$, $X_{236}$, $X_{336}$, $X_{340}$, $X_{341}$, $X_{345}$, $X_{120a}$, $X_{120b}$, $X_{120c}$, $X_{120d}$, $X_{120e}$, $X_{120f}$, $X_{120g}$, $X_{120h}$, $X_{120i}$, $X_{120j}$, $X_{120k}$, $X_{120l}$, $X_{120m}$, $X_{120n}$, $X_{120o}$, $X_{120p}$
Curves that minimally cover $X_{120}$ and have infinitely many rational points. $X_{207}$, $X_{215}$, $X_{217}$, $X_{227}$, $X_{229}$, $X_{234}$, $X_{235}$, $X_{236}$, $X_{120a}$, $X_{120b}$, $X_{120c}$, $X_{120d}$, $X_{120e}$, $X_{120f}$, $X_{120g}$, $X_{120h}$, $X_{120i}$, $X_{120j}$, $X_{120k}$, $X_{120l}$, $X_{120m}$, $X_{120n}$, $X_{120o}$, $X_{120p}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{120}) = \mathbb{Q}(f_{120}), f_{36} = f_{120}^{2} - 4$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 + x^2 + 25x$, with conductor $525$
Generic density of odd order reductions $19/168$