## The modular curve $X_{230}$

Curve name $X_{230}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{102}$
Meaning/Special name
Chosen covering $X_{102}$
Curves that $X_{230}$ minimally covers $X_{102}$, $X_{117}$, $X_{122}$
Curves that minimally cover $X_{230}$ $X_{467}$, $X_{473}$, $X_{475}$, $X_{487}$, $X_{230a}$, $X_{230b}$, $X_{230c}$, $X_{230d}$, $X_{230e}$, $X_{230f}$, $X_{230g}$, $X_{230h}$
Curves that minimally cover $X_{230}$ and have infinitely many rational points. $X_{230a}$, $X_{230b}$, $X_{230c}$, $X_{230d}$, $X_{230e}$, $X_{230f}$, $X_{230g}$, $X_{230h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{230}) = \mathbb{Q}(f_{230}), f_{102} = -2f_{230}^{2} + 1$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 36x - 176$, with conductor $126$
Generic density of odd order reductions $193/1792$