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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