## The modular curve $X_{229}$

Curve name $X_{229}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{86}$
Meaning/Special name
Chosen covering $X_{86}$
Curves that $X_{229}$ minimally covers $X_{86}$, $X_{117}$, $X_{120}$
Curves that minimally cover $X_{229}$ $X_{467}$, $X_{476}$, $X_{229a}$, $X_{229b}$, $X_{229c}$, $X_{229d}$, $X_{229e}$, $X_{229f}$, $X_{229g}$, $X_{229h}$
Curves that minimally cover $X_{229}$ and have infinitely many rational points. $X_{229a}$, $X_{229b}$, $X_{229c}$, $X_{229d}$, $X_{229e}$, $X_{229f}$, $X_{229g}$, $X_{229h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{229}) = \mathbb{Q}(f_{229}), f_{86} = \frac{f_{229}^{2} - \frac{1}{2}}{f_{229}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 2034x + 60264$, with conductor $306$
Generic density of odd order reductions $635/5376$