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Let k be a field with char(k) not 2 or 3. Let C_f be the projective curve ofa binary cubic form f, and k(C_f) the function field of C_f. In this paper weexplicitly describe the relative Brauer group Br(k(C_f)/k) of k(C_f) over k.When f is diagonalizable we show that every algebra in Br(k(C_f)/k) is a cyclicalgebra obtainable using the y-coordinate of a k-rational point on the JacobianE of C_f. But when f is not diagonalizable, the algebras in Br(k(C_f)/k) arepresented as cup products of cohomology classes, but not as cyclic algebras. Inparticular, we provide several specific examples of relative Brauer groups fork=Q, the rationals, and for k=Q(omega) where omega is a primitive third root ofunity. The approach is to realize C_f as a cyclic twist of its Jacobian E, anelliptic curve, and then apply a recent theorem of Ciperiani and Krashen.

Curves C that are Cyclic Twists of Y^2 = X^3+c and the Relative Brauer Groups Br(k(C)/k