The modular curve $X_{509}$

Curve name $X_{509}$
Index $96$
Level $16$
Genus $2$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 7 & 0 \\ 0 & 15 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 15 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 10 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $6$ $X_{8}$
$4$ $12$ $X_{25}$
$8$ $48$ $X_{193}$
Meaning/Special name
Chosen covering $X_{193}$
Curves that $X_{509}$ minimally covers $X_{193}$
Curves that minimally cover $X_{509}$
Curves that minimally cover $X_{509}$ and have infinitely many rational points.
Model \[y^2 = 2x^5 - 4x^4 - 4x^2 - 2x\]
Info about rational points
Rational pointImage on the $j$-line
$(1 : 0 : 0)$ \[ \infty \]
$(0 : 0 : 1)$ \[ \infty \]
Comments on finding rational points The rank of the Jacobian is 0. We use the method of Chabauty.
Elliptic curve whose $2$-adic image is the subgroup None
Generic density of odd order reductions N/A

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