The modular curve $X_{2}$

Curve name $X_{2}$
Index $2$
Level $2$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 0 & 1 \\ 1 & 1 \end{matrix}\right]$
Images in lower levels
Meaning/Special name Elliptic curves whose discriminant is a square
Chosen covering $X_{1}$
Curves that $X_{2}$ minimally covers $X_{1}$
Curves that minimally cover $X_{2}$ $X_{8}$, $X_{21}$, $X_{2a}$, $X_{2b}$
Curves that minimally cover $X_{2}$ and have infinitely many rational points. $X_{8}$, $X_{2a}$, $X_{2b}$
Model \[\mathbb{P}^{1}, \mathbb{Q}(X_{2}) = \mathbb{Q}(f_{2}), f_{1} = f_{2}^{2} + 1728\]
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 2x + 1$, with conductor $196$
Generic density of odd order reductions $5/7$

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