## The modular curve $X_{20}$

Curve name $X_{20}$
Index $8$
Level $4$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 3 & 0 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $1$ $X_{1}$
Meaning/Special name Elliptic curves whose discriminant is minus a square with $a_{p}(E) \equiv 0 \pmod{4}$ for $p \equiv 3 \pmod{4}$
Chosen covering $X_{7}$
Curves that $X_{20}$ minimally covers $X_{3}$, $X_{7}$
Curves that minimally cover $X_{20}$ $X_{60}$, $X_{180}$, $X_{20a}$, $X_{20b}$
Curves that minimally cover $X_{20}$ and have infinitely many rational points. $X_{60}$, $X_{20a}$, $X_{20b}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{20}) = \mathbb{Q}(f_{20}), f_{7} = \frac{f_{20} + 1}{f_{20}^{2} - 3}$
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 6x + 8$, with conductor $162$
Generic density of odd order reductions $37/84$