The modular curve $X_{71}$

Curve name $X_{71}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{11}$
Meaning/Special name
Chosen covering $X_{43}$
Curves that $X_{71}$ minimally covers $X_{43}$, $X_{49}$, $X_{50}$
Curves that minimally cover $X_{71}$ $X_{258}$, $X_{259}$, $X_{327}$, $X_{328}$
Curves that minimally cover $X_{71}$ and have infinitely many rational points. $X_{328}$
Model \[\mathbb{P}^{1}, \mathbb{Q}(X_{71}) = \mathbb{Q}(f_{71}), f_{43} = \frac{2f_{71}^{2} - 4}{f_{71}^{2} + 2}\]
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 + 62x + 224$, with conductor $867$
Generic density of odd order reductions $401/1792$

Back to the 2-adic image homepage.