We consider the space of binary cubic forms, equipped with the natural action of the group GL2 of invertible linear transformations of We describe explicitly the category of GL2-equivariant coherent -modules as the category of representations of a quiver with relations. We show moreover that this quiver is of tame representation type and we classify its indecomposable representations. We also give a construction of the simple equivariant -modules (of which there are 14), and give formulas for the characters of their underlying GL2-representations. We conclude the article with an explicit calculation of (iterated) local cohomology groups with supports given by orbit closures.