The modular curve $X_{289}$

Curve name $X_{289}$
Index $48$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 9 & 2 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 9 & 9 \\ 2 & 7 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{11}$
$8$ $24$ $X_{91}$
Meaning/Special name
Chosen covering $X_{91}$
Curves that $X_{289}$ minimally covers $X_{91}$, $X_{124}$, $X_{166}$
Curves that minimally cover $X_{289}$ $X_{582}$, $X_{585}$
Curves that minimally cover $X_{289}$ and have infinitely many rational points.
Model \[y^2 = x^3 + x^2 - 13x - 21\]
Info about rational points $X_{289}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 + 173490x + 156628973$, with conductor $4046$
Generic density of odd order reductions $12833/57344$

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