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Letk/Fq(x) be a quadratic extension that is ramified over the unique pole ofx, and letAbe the integral closure of Fq[x] ink. Thenkis the function field analogue of an imaginary quadratic number field. A rank-one DrinfeldA-moduleφof generic characteristic is the analogue of an elliptic curve with complex multiplications by the full ring of integers of an imaginary quadratic number field. In this paper, we compute the degrees of the trace and norm down to Fq[x] ofj(φ), thej-invariant ofφ. Our results generalize previous ones wherekwas assumed to have genusg=1.