If f(x,y,z) = ax² + by² + cz² + ryz + sxz + txy is a positive definite, primitive, ternary quadratic form, (a ternary form, for short) then its discriminant is given by d = 4abc + rst - ar² - bs² - ct², and its level is N = 4d/m where m is the greatest common divisor of 4bc-r², 4ac-s², 4ab-t², 2st-4ar, 2rt-4bs, and 2rs-4ct. (Since f is positive definite, each of a, b, c, 4bc-r², 4ac-s², 4ab-t², and d is positive.)

Every ternary form is equivalent to exactly one reduced form, which, by definition, satisfies the following conditions:

• a is less than or equal to b, which is less than or equal to c.
• r, s, and t are all positive or all non-positive.
• |t| is less than or equal to a.
• |s| is less than or equal to a.
• |r| is less than or equal to b.
• If a = b, then |r| is less than or equal to |s|.
• If b = c, then |s| is less than or equal to |t|.
• a + b + r + s + t is non-negative.
• If a + b + r + s + t = 0, then 2a + 2s + t is non-positive.
• If a = -t, then s = 0.
• If a = -s, then t = 0.
• If b = -r, then t = 0.
• If a = t, then s is less than or equal to 2r.
• If a = s, then t is less than or equal to 2r.
• If b = r, then t is less than or equal to 2s.

It can be shown that if f is reduced, then abc is greater than or equal to d/4 and less than or equal to d/2. Furthermore, the following facts are easy to establish:

• If m is even, then each of r, s, and t must be even.
• If mu = 4N/m is odd, then each of a, b, and c has the property of being congruent to 0 or to -mu modulo 4.
• ab is greater than or equal to m/4.

Using these facts, we have the following algorithm for finding all reduced ternary forms having discriminant d and level N.

Algorithm:

1. Let a vary from 1 to the greatest integer less than or equal to the cube root of d/2, subject to the condition that if mu is odd, then a is congruent to 0 or -mu modulo 4. For each a, use the Euclidean Algorithm to find the greatest common divisor, g, of 4a and m, and integers u and v such that g = 4au + mv.
2. Given a, let t vary from 0 to a, subject to the conditions that if m is even, then t is even, and that g divides t².
3. Given a and t, let b vary over all integers congruent to ut²/g modulo m/g, such that b is greater than or equal to the larger of a and m/4a, b is less than or equal to the square root of d/2a, and subject to the condition that if mu is odd, then b is congruent to 0 or -mu modulo 4. (Note: In this step and the previous one, we are using the fact that 4ab-t² must be divisible by m. Thus b is a solution of the linear congruence 4ax = t² (mod m). This congruence has a unique solution modulo m/g, namely ut²/g, if and only if g divides t².)
4. Given a, t, and b, let s vary from 0 to a, subject to the conditions that if m is even, then s is even, and that g divides 2st.
5. Given a, t, b, and s, let r vary over all integers that are congruent to 2stu/g modulo m/g, that are less than or equal to b in absolute value, and such that if m is even, then r is even, and if st = 0, then r is less than or equal to 0. (In this step and the previous one, we use the fact that 2st-4ar is divisible by m.)
6. Given a, t, b, s, and r, test whether c = (d - rst + ar² + bs²)/(4ab - t²) is an integer. If so, test whether a, b, c, r, s, and t satisfy all the other properties of a reduced ternary form of level N and discriminant d. (If r is less than or equal to 0, replace s and t by their negatives. It is easy to see that any two of r, s, and t can be changed in sign without affecting the invariants N and d.)