## The modular curve $X_{210}$

Curve name $X_{210}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 14 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 10 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 14 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{24}$ $8$ $24$ $X_{61}$
Meaning/Special name
Chosen covering $X_{61}$
Curves that $X_{210}$ minimally covers $X_{61}$, $X_{111}$, $X_{112}$
Curves that minimally cover $X_{210}$ $X_{210a}$, $X_{210b}$, $X_{210c}$, $X_{210d}$
Curves that minimally cover $X_{210}$ and have infinitely many rational points. $X_{210a}$, $X_{210b}$, $X_{210c}$, $X_{210d}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{210}) = \mathbb{Q}(f_{210}), f_{61} = \frac{2}{f_{210}^{2}}$
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 51x + 152$, with conductor $153$
Generic density of odd order reductions $9249/57344$