TITLE: Hypersurfaces whose equation is not known
QUESTION [6 upvotes]: I would like to find some well-known/interesting hypersurfaces which arise as parametrizations where implicitization is computationally too difficult.
I have software which computes the Newton polytope of such hypersurfaces and would like to use it on an interesting example! 
What are some great parametrized hypersurfaces that would be of interest? I am currently working on the Luroth invariant, but would love more examples.
For what it's worth, the Luroth invariant is degree 54 in 15 variables and it is proving to be just within reach of the software. Something around this size or slightly smaller would be perfect for me.

REPLY [4 votes]: The projective dual of a variety is usually a hypersurface. So, take your favorite variety and try to compute its projective dual.