List of Positive Definite Primitive Ternary Quadratic Forms over Z

Let Q(x,y,z)=axx+byy+czz+2dxy+2exz+2fyz be a positive definite quadratic form over Z.
We say that Q is primitive when a,b,c,d,e,f are integers and GCD(a,b,c,d,e,f)=1.
By taking a suitable linear transformation of variables, we can assume that Q is reduced, that is, the coefficinets a,b,c,d,e,f are satisfies one of the following:

(I) 0 < a <= b <= c, 0 <= d <= a/2, 0 <= e <= a/2, 0 <= f <= b/2,
   f <= e if a = b,   e <= d if b = c,
   e <= 2f if a=2d,   d <= 2f if a = 2e,   d <= 2e if b = 2f.
(II) 0 < a <= b <= c, 0 < -d < a/2, 0 < -e < a/2, 0 < -f < b/2,
   f <= e if a = b,   e <= d if b = c,
   a + b + 2d + 2e + 2f >= 0,
   a + d + 2e <= 0 if a + b + 2d + 2e + 2f = 0.

We note that the reduced form is determined uniquely in each class.
The discriminant D of Q is defined as D = abc + 2def - aff - bee - cdd.

We explain the notation of genus which is used in the following list of ternary quadratic forms.

the factorization of discriminant : D = (p^e)·(q^f)·...

the genus invariants : {X/Y/Z/...}

X for p^e (D = p^e · u)

p = 2 : (Q0 = binary hyperbolic, Q1 = binary non-hyperbolic, e = 2f)

   [H] for Q = Q0 + <7D>, [K] for Q = Q1 + <3D>, [E] for [H]=[K] (e = 1)

   [h] for Q = <7u> + (2^s)Q0, [k] for Q = <3u> + (2^s)Q1, [o] for [h]=[k] (f = 1)

   [sij] for Q = <i> + <(2^s)j> + <(2^t)k> (0 <= s <= t, s + t = e, ijk = u)

p : odd (Q0 = binary hyperbolic, Q1 = binary non-hyperbolic, e = 2f)

   [H] for Q = Q0 + <-D>, [K] for Q = Q1 + <-zD> (z:non-square unit)

   [h] for Q = <-u> + (p^s)Q0, [k] for Q = <-zu> + (p^s)Q1 (f > 0)

   [smn] for Q = <i> + <(p^s)j> + <(p^t)k> (0 < s < t, e = s + t, ijk = u, i = 1,z (m = +,-), j = 1,z (n = +,-))