Common values of Dickson polynomials D_P(1,-m): a genus trichotomy and the coincidence 6301

For an odd prime P let f_P(m) = β^P − (β−1)^P, where β is the positive root of x²−x−m. This is the Dickson polynomial of the first kind D_P(1,−m), equivalently the Lucas companion value V_P(1,−m), a classical family. We study which integers are represented by two distinct members f_{P1} and f_{P2}. The common-value curve exhibits a genus trichotomy: it is rational when P1 = 3, so f_3 shares infinitely many values with every f_P (explicitly parametrized); it is elliptic in exactly one case, (P1,P2) = (5,7); and it has genus ≥ 2 otherwise. In the elliptic case we determine all common values: the only integers represented by both 5m²+5m+1 and 7m³+14m²+7m+1 are 1 and 6301. The proof reduces to an elliptic curve of conductor 28175 and rank 2, whose integral points are computed in SageMath. A longer, earlier manuscript by the author on the same polynomial family is available at DOI 10.5281/zenodo.23011124; the present note is a separate, self-contained paper.