Canonical form for primes in polynomial progressions

The primes of the form $4n^2-8n-9$ are just those of the form $n^2-13.$ Various substitutions can change this into many other forms. Is there a canonical choice for the polynomial, such that $C(p_1)=C(p_2)$ iff the primes of the form $p_1$ are those of the form $p_2$? (I suppose such form must exist; I guess the real question is whether there is a canonical canonical form.

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OK, but I think one usually concentrates on quadratic forms with squarefree discriminant, and both of those polynomials have discriminant $-3$ once you chuck out the square factors. –  Gerry Myerson Jan 2 '12 at 15:03