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In Magma there is a function integralQuarticPoints that can be used to get all the integer solutions to $$y^2 = ax^4+bx^3+cx^2+dx+e$$ when $e$ is a square. There is a function .integral_points() for ElipticCurves, but this is a quartic not an eliptic curve.

Is there a way to do this in Sage?

PS: Would have added link to the docs of the Magma function, but not allowed to due insufficient karma.