  We formulate effective necessary and sufficient conditions to identify the
symmetry class of an elasticity tensor, a fourth-order tensor which is the
cornerstone of the theory of elasticity and a toy model for linear constitutive
laws in physics. The novelty is that these conditions are written using
polynomial covariants. As a corollary, we deduce that the symmetry classes are
affine algebraic sets, a result which seems to be new. Meanwhile, we have been
lead to produce a minimal set of 70 generators for the covariant algebra of a
fourth-order harmonic tensor and introduce an original generalized cross
product on totally symmetric tensors. Finally, using these tensorial
covariants, we produce a new minimal set of 294 generators for the invariant
algebra of the elasticity tensor.
