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The exceptional points of a cubic curve which is symmetric in the homogeneous variables. (English) Zbl 0057.03602

The author studies the cubic equation
\[ \begin{aligned} a(x+y+z)^3 + b(xy+xz+yz) (x+y+z) + cxyz = 0,\\ c(27a+9b+c) (b^3+b^2c-a c^2) \neq 0,\tag{1} \end{aligned} \]
in which case (1) represents a non-degenerate cubic curve of genus one. It is shown that (1) will be equivalent to any cubic curve with 3 rational inflections. The exceptional points, in number \(n=3n_1\) are studied in homogeneous coordinates, by a method based on the forming of tangentials. The case \(b=0\) will cover all curves with \(n > 6\), and a simple parametric representation is given for \(a\) and \(c\) when \(9\mid n\).
The case \(a=0\) covers the general curve with \(6\mid n\), and corresponding representations for \(b\) and \(c\) are given when there is a cyclic or a non-cyclic exceptional subgroup of order 12. It is assumed that the coefficients and variables are rational integers. Because of the choice of his normal form the author’s parametric representations are much simpler than those previously obtained for the Weierstrass curve \(y^2 = x^3-Ax-B\) for \(n = 6, 9\) and \(12\).
Reviewer: W. Ljunggren

MSC:

11D25 Cubic and quartic Diophantine equations
14H25 Arithmetic ground fields for curves
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