The modular curve $X_{533}$

Curve name $X_{533}$
Index $96$
Level $32$
Genus $2$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 3 \\ 2 & 19 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 10 & 31 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 10 & 31 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{11}$
$8$ $24$ $X_{77}$
$16$ $48$ $X_{216}$
Meaning/Special name
Chosen covering $X_{216}$
Curves that $X_{533}$ minimally covers $X_{216}$
Curves that minimally cover $X_{533}$
Curves that minimally cover $X_{533}$ and have infinitely many rational points.
Model \[y^2 = -x^6 + 8x^5 - 6x^4 - 12x^2 - 32x - 8\]
Info about rational points No non-singular rational points
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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