The modular curve $X_{84}$

Curve name $X_{84}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{36}$
Curves that $X_{84}$ minimally covers $X_{36}$
Curves that minimally cover $X_{84}$ $X_{195}$, $X_{199}$, $X_{201}$, $X_{206}$, $X_{227}$, $X_{228}$, $X_{336}$, $X_{346}$, $X_{84a}$, $X_{84b}$, $X_{84c}$, $X_{84d}$, $X_{84e}$, $X_{84f}$, $X_{84g}$, $X_{84h}$, $X_{84i}$, $X_{84j}$, $X_{84k}$, $X_{84l}$, $X_{84m}$, $X_{84n}$, $X_{84o}$, $X_{84p}$
Curves that minimally cover $X_{84}$ and have infinitely many rational points. $X_{195}$, $X_{199}$, $X_{227}$, $X_{228}$, $X_{84a}$, $X_{84b}$, $X_{84c}$, $X_{84d}$, $X_{84e}$, $X_{84f}$, $X_{84g}$, $X_{84h}$, $X_{84i}$, $X_{84j}$, $X_{84k}$, $X_{84l}$, $X_{84m}$, $X_{84n}$, $X_{84o}$, $X_{84p}$
Model \[\mathbb{P}^{1}, \mathbb{Q}(X_{84}) = \mathbb{Q}(f_{84}), f_{36} = \frac{-4}{f_{84}^{2} + 1}\]
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 1989x - 458150$, with conductor $1989$
Generic density of odd order reductions $83/672$

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