Explicit identities for invariants of elliptic curves

New explicit formulas are given for the supersingular polynomial ssp(t) and the Hasse invariant of an elliptic curve E in characteristic p. These formulas are used to derive identities for the Hasse invariants of elliptic curves En in Tate normal form with distinguished points of order n. This yields a proof that and are projective invariants (mod p) for the octahedral group and the icosahedral group, respectively; and that the set of fourth roots λ1/4 of supersingular parameters of the Legendre normal form Y2=X(X−1)(X−λ) in characteristic p has octahedral symmetry. For general n⩾4, the field of definition of a supersingular En is determined, along with the field of definition of the points of order n on En.