## The modular curve $X_{124}$

Curve name $X_{124}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 5 & 15 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 13 & 13 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 15 & 13 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $12$ $X_{39}$
Meaning/Special name
Chosen covering $X_{39}$
Curves that $X_{124}$ minimally covers $X_{39}$
Curves that minimally cover $X_{124}$ $X_{222}$, $X_{232}$, $X_{289}$, $X_{303}$, $X_{358}$, $X_{365}$
Curves that minimally cover $X_{124}$ and have infinitely many rational points. $X_{222}$, $X_{232}$, $X_{289}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{124}) = \mathbb{Q}(f_{124}), f_{39} = -2f_{124}^{2} - 4$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 1375x - 56250$, with conductor $700$
Generic density of odd order reductions $85091/344064$