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computations of selfpoints

let X_0(p) be the modular curve of prime level p. suppose that E/Q is an elliptic curve which is a quotient of the jacobian of X_0(p). choose any isogeny phi on E of degree p. it is defined over a field K which in most cases is a degreee p+1 extension of Q. hence the couple (E,phi) represents a point on the modular curve defined over K. we call its image in E a selfpoint on E. there are p+1 selfpoints, which we believe to be of infinite order on E. we have computed the first examples, namely for p = 11, p = 17 and p = 19. (Note the points are not written in the shortest possible form, at all.)


 
* a picture where i tried to make an illustration for the self-points on the curve X_0(11). this was presented at the annual meeting of the swiss academy of sciences (sep 07).