## The modular curve $X_{27}$

Curve name $X_{27}$
Index $12$
Level $4$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$
Meaning/Special name
Chosen covering $X_{10}$
Curves that $X_{27}$ minimally covers $X_{10}$, $X_{11}$, $X_{13}$
Curves that minimally cover $X_{27}$ $X_{60}$, $X_{92}$, $X_{94}$, $X_{95}$, $X_{134}$, $X_{135}$, $X_{27a}$, $X_{27b}$, $X_{27c}$, $X_{27d}$, $X_{27e}$, $X_{27f}$, $X_{27g}$, $X_{27h}$
Curves that minimally cover $X_{27}$ and have infinitely many rational points. $X_{60}$, $X_{92}$, $X_{94}$, $X_{95}$, $X_{27a}$, $X_{27b}$, $X_{27c}$, $X_{27d}$, $X_{27e}$, $X_{27f}$, $X_{27g}$, $X_{27h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{27}) = \mathbb{Q}(f_{27}), f_{10} = \frac{f_{27}}{f_{27}^{2} - \frac{1}{4}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 885x - 4294$, with conductor $1845$
Generic density of odd order reductions $9/56$