The modular curve $X_{323}$

Curve name $X_{323}$
Index $48$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 10 & 15 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{23}$
$8$ $24$ $X_{69}$
Meaning/Special name
Chosen covering $X_{69}$
Curves that $X_{323}$ minimally covers $X_{69}$, $X_{112}$, $X_{150}$
Curves that minimally cover $X_{323}$ $X_{568}$, $X_{578}$, $X_{649}$, $X_{651}$, $X_{691}$, $X_{692}$
Curves that minimally cover $X_{323}$ and have infinitely many rational points.
Model \[y^2 = x^3 + x^2 - 3x + 1\]
Info about rational points $X_{323}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 + x^2 + 174549339394871087x - 36189713187335153010390608$, with conductor $20167985106465$
Generic density of odd order reductions $12833/57344$

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