The complete enumeration of extreme senary forms

Abstract
Let f(x 1 ..., x 6 )be a positive definite senary quadratic form of determinant D . Let M be its minimum value for integers x 1 ..., x 6 , not all zero. The form is said to be extreme if, for all infinitesimal variations of the coefficients, M 6 / D is maximum. It is proved here for the first time that there are exactly six classes of extreme senary forms, namely, the classes containing the six forms denoted by 0 O , 0 4 and 0 6 . (Another form 0 5 is shown to be only ‘perfect’, not extreme.) The forms 0 0 0 1 0 2 0 4 are equivalent to A 6 D 6 E 6 E 3/6 , in the notation of Coxeter (1951, p. 394); 0 3 was discovered simultaneously by M. Kneser and the author (1955); 0 6 is new. Although the analogous forms in fewer variables have been known since 1877, the only previous enumeration of extreme forms in six variables was by Hofreiter (1933), who missed 0 3 0 4 0 6 and proposed instead an incorrect form which he called F 4 .

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