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# THE EXCEPTIONAL POINTS OF A CUBIC CURVE WHICH IS SYMMETRIC IN THE HOMOGENEOUS VARIABLES

ERNST S. SELMER
Mathematica Scandinavica
Vol. 2, No. 2 (February 24, 1955), pp. 227-236
Stable URL: http://www.jstor.org/stable/24489035
Page Count: 10
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## Abstract

It is shown that the symmetric ternary cubic equation a (x+y+z)3 + b (xy+xz+yz)(x+y+z) + c xyz = 0 will represent (be equivalent to) any cubic curve with three rational inflections. The exceptional points, in number n = 3n1, 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 then given for a and c when 9 | n. The case a = 0 covers the general curve with 6 | n, and corresponding representations for b and c are then given when there is a cyclic or a non-cyclic exceptional subgroup of order 12. It is assumed throughout that the coefficients and variables are absolutely rational. Because of the choice of normal form, the parametric representations are much simpler than those previously obtained for the Weierstrass curve y2 = x3 - Ax - B for n = 6, 9 and 12.

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