Generalisation of algebraic point families of the Fermat curve quotient C1,2(7)

Summary: We explicitly determine the algebraic point families of a given degree over Q of the curve C1,2(7) with affine equation: y7 = x(x − 1)2 This curve is a special case of the family of quotients of Fermat curves Cr,s(p) described in [11] of affine equation: yp = xr(x − 1)s with 1 ≤ r, s, r + s ≤ p; for r = 1, s = 2 and p = 7 such a curve was considered in [10]. This curve has been studied by O. Sall in [14], where the author gives a parametrisation of the cubic points. It should be noted, however, that the method used by O. Sall does not allow us to determine the set of points of degree greater than 3. We have therefore used a geometric method to extend this work and determine the quartic points [4]. In this note, we describe all the families of algebraic points of given degree, geometrically specifying the contact lines and the curve containing them, by applying the fundamental Abel-Jacobi theorem [1, 9], before using these results with a Q-basis of the linear systems L(m∞) and combining the contact order of the curve and specific points to obtain analytical expressions for these families of points.