Yes, it is a moon of Pluto, which is a Kuiper belt object.
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Yes. PLuto is the largest known object in the Kuiper belt.
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The Kuiper belt is not an object; it is a region of the solar
system. Therefore it does not have an axis of rotation
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No. Eris is not a Kuiper Belt object, but is instead part of the
scattered disk, beyond the Kuiper Belt. Eris is slightly smaller
than Pluto, though by an insignificant amount but has a greater
mass, meaning it is denser.
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Some known Kuiper-belt objects are hundreds of kilometers
across.