The modular curve $X_{6}$

Curve name	$X_{6}$
Index	$3$
Level	$2$
Genus	$0$
Does the subgroup contain $-I$?	Yes
Generating matrices	$\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
Meaning/Special name	Elliptic curves with a rational $2$-torsion point
Chosen covering	$X_{1}$
Curves that $X_{6}$ minimally covers	$X_{1}$
Curves that minimally cover $X_{6}$	$X_{8}$, $X_{9}$, $X_{10}$, $X_{11}$, $X_{12}$, $X_{13}$, $X_{14}$, $X_{15}$, $X_{16}$, $X_{17}$, $X_{18}$, $X_{19}$
Curves that minimally cover $X_{6}$ and have infinitely many rational points.	$X_{8}$, $X_{9}$, $X_{10}$, $X_{11}$, $X_{12}$, $X_{13}$, $X_{14}$, $X_{15}$, $X_{16}$, $X_{17}$, $X_{18}$, $X_{19}$
Model	$$\mathbb{P}^{1}, \mathbb{Q}(X_{6}) = \mathbb{Q}(f_{6}), f_{1} = \frac{f_{6}^{3}}{f_{6} + 16}$$
Info about rational points	None
Comments on finding rational points	None
Elliptic curve whose $2$-adic image is the subgroup	$y^2 + xy + y = x^3 - x - 1$, with conductor $69$
Generic density of odd order reductions	$5/21$

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