52 
TABLES OF THE BINARY CUBIC FORMS 
[496 
D. 
Reducirte 
Formen mit Charakteristik. 
Klassen. 
7 
(0, 1, 1, - 1) 
(2, 1, 4) 
1 
■ (0, -1,-1, 1) 
8 
(0, 1, 0, - 2) 
(2, 0, 4) 
1 
■ (0, - 1, o, 2) 
11 
(0, 1, 1, - 2) 
(2, 1, 6) 
| 
■ (0, -1,-1, 2) 
12 
(0, 1, 0, - 3) 
(2, 0, 6) 
! 
• (0, - 1, 0, 3) 
15 
(0, 1, 1, - 3) 
(2, 1, 8) 
i 
- (0, -1,-1, 3) 
44 
(0, 1, 0, -11)(1, -1, -2, 0) 1 
(2, 0, 22) (6, 2, 8) ) 
\ (0, - 1, 0, 11), (0, - 2, ± 1, 1) 
112 
(0, 1, 0, - 28) (0, 2, 2, — 2) (1, 2, 0,-4) ) 
| 
(2, 0, 56) 
(8, 4, 16) (8, 4, 16) 
(0, - 1, 0, 28), (0, 2, 2, - 2), 
(1, 
- 1, - 3, - 1) 
(8, 4, 16) J 
j (1, + 1, _ 3, + 1) 
144 
(0, 1, 0, - 36) (0, 2, 2, - 3) 
(2, 0, 72) (8, 4, 20) 
| (0, - 1, 0, 36), (0, -2,-2, 3) 
156 
(0, 1, 0, - 39) (1, -1, -3,1) 
(2, 0, 78) (8, 2, 20) 
| (0, - 1, 0, 39), (1, +1, -3, + 1) 
216 
(0, 1, 0, — 54) (1, -2,-3, 0) (2, 0,-3, 0) 
| (0, - 1, 0, 54), (0, + 3, 0, + 2), 
(2, 0, 108) 
(14, 6, 18) (12, 0, 18) 
j (0, + 3, 2, ± 1) 
The first column contains the value of the determinant, the second column contains 
the reduced forms, omitting the contrary and opposite forms; viz. for the cubic form 
(a, b, c, d), the contrary form (equal, that is, properly equivalent to the given form) is 
(— a, —b, —c, — d); and the opposite form (improperly equivalent to the given form) 
is (a, —b,c,— d) or (— a, b, — c, d)\ this second column contains also the characteristic 
of each cubic form, viz. the cubic form (a, b, c, d) has for its characteristic the 
quadratic form 
{2 (6 2 — ac), be — ad, 2 (c 2 - 6c£)}, 
(so that the cubic form and its characteristic have the same determinant 
— D — {be — adf — 4 (6 2 — ac) (c 2 — bd), = 1 or 0 (mod. 4)), 
and a cubic form which corresponds to a reduced characteristic is itself a reduced 
form. The third column contains for each determinant the entire series of unequal 
cubic forms (that is of the forms whereof no two are properly equivalent to each 
other), the representatives of the classes for this determinant. M. Arndt has included 
in his table the non-primitive classes (for example Det. = — 112, the form (0, 2, 2, — 2)), 
for which the terms (a, b, c, d) have a common divisor p, but as these are at once