## The modular curve $X_{55}$

Curve name $X_{55}$
Index $16$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 5 \\ 7 & 2 \end{matrix}\right], \left[ \begin{matrix} 6 & 7 \\ 5 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $1$ $X_{1}$ $4$ $4$ $X_{7}$
Meaning/Special name $X_{ns}^{+}(8)$
Chosen covering $X_{7}$
Curves that $X_{55}$ minimally covers $X_{7}$
Curves that minimally cover $X_{55}$ $X_{179}$, $X_{180}$, $X_{253}$, $X_{441}$
Curves that minimally cover $X_{55}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{55}) = \mathbb{Q}(f_{55}), f_{7} = \frac{\frac{1}{8}f_{55}^{4} - \frac{1}{8}f_{55}^{2} + \frac{1}{32}}{f_{55}^{4} - f_{55}^{2} + f_{55} - \frac{1}{4}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 + 1995x + 58727$, with conductor $3388$
Generic density of odd order reductions $2867/5376$