## The modular curve $X_{105}$

Curve name $X_{105}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 13 & 10 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 13 & 13 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 15 & 13 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
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 $X_{105}$ minimally covers $X_{39}$
Curves that minimally cover $X_{105}$ $X_{214}$, $X_{221}$, $X_{224}$, $X_{232}$, $X_{282}$, $X_{288}$, $X_{301}$, $X_{302}$, $X_{388}$, $X_{389}$, $X_{390}$, $X_{394}$
Curves that minimally cover $X_{105}$ and have infinitely many rational points. $X_{214}$, $X_{221}$, $X_{224}$, $X_{232}$, $X_{288}$, $X_{302}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{105}) = \mathbb{Q}(f_{105}), f_{39} = -f_{105}^{2} + 4$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 9235x + 342453$, with conductor $4606$
Generic density of odd order reductions $85091/344064$