On Oct 22, 10:00 am, andrew ewart <aewartma...@googlemail.com> wrote: > What is the best way of writing out code to calculate the symmetry > group of the Klein Quartic > (The klein quatric is the set of the solutions to f=0, where f is the > polynomial x^3*y+y^3*z+z^3+x in QQ[x,y,z]) > I know of 2 symmetry types > one is the rotation of terms of order 3 ( (x,y,z), (z,x,y), (y,z,x)) > and one is the mapping of x->v*x, y->v^4*y, z->v^2*z, where v is a 7th > root of unity (v=(1/7)*(142)) > this is a symmetry of order 7 > thoughts?
Since the model is canonical, any automorphism arises from a projective linear transformation. A general approach is to find a set of geometrically defined points (the flexes for instance) and determine the linear transformations that act by permutation on those points. Since the flexes span the ambient space, the action of the automorphism group on the flexes is faithful. You can find the flexes by intersecting with the Hessian. You should find 24 flexes. The tangent line in a flex point intersects the curve in another flex point. This splits the flexes in 8 triangles. This structure should be preserved as well, cutting back considerably on the number of permutations you have to try, There is a further order 2 transformation by the way. MSRI's "The eightfold way" has a lot of nice introductory material on Klein's quartic: http://library.msri.org/books/Book35/index.html -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
