Small problem in six-dimensional space

[fivemack]

I'm considering the codimension-1-in-P5-object defined by

x^6+y^6+z^6 = u^6+v^6+w^6

with the obvious solutions-by-permutation removed and GCD(x,y,z,u,v,w)=1. I've searched for z,w<2000, and am finding that the number of these points with z,w<N seems to be slightly more than linear in N, so I suspect there are some families of points lurking. Suspiciously many of the points I've found have x+y+u=z+v+w, for example.

http://www.jstor.org/pss/2005335 gives a set of homogeneous quartics parameterising some solutions, but clearly not all solutions since the quartics happen also to satisfy x^2+y^2+z^2=u^2+v^2+w^2.

What are the right sort of questions to ask about the set of points on a high-dimensional variety?

[R.D. Silverman]

Originally Posted by fivemack

I'm considering the codimension-1-in-P5-object defined by

x^6+y^6+z^6 = u^6+v^6+w^6

<snip>

What are the right sort of questions to ask about the set of points on a high-dimensional variety?

Is there an associated L-function? Is there a motive?
What does the j-invariant look like? Is the variety even Abelian?
How many localizations yield singularities? (i.e. how many primes have
bad reduction)

[R.D. Silverman]

Originally Posted by R.D. Silverman

Is there an associated L-function? Is there a motive?
What does the j-invariant look like? Is the variety even Abelian?
How many localizations yield singularities? (i.e. how many primes have
bad reduction)

And I stupidly forgot perhaps the most obvious question: What is its
genus? i.e. if viewed as a Riemann surface over C, how many holes
does the surface have? Then one might study conformal maps of
that surface.