## The modular curve $X_{28}$

Curve name $X_{28}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 6 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$
Meaning/Special name
Chosen covering $X_{11}$
Curves that $X_{28}$ minimally covers $X_{11}$
Curves that minimally cover $X_{28}$ $X_{67}$, $X_{68}$, $X_{81}$, $X_{90}$, $X_{128}$, $X_{146}$
Curves that minimally cover $X_{28}$ and have infinitely many rational points. $X_{67}$, $X_{68}$, $X_{81}$, $X_{90}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{28}) = \mathbb{Q}(f_{28}), f_{11} = -2f_{28}^{2} - 8$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 2703x - 54090$, with conductor $468$
Generic density of odd order reductions $89/336$