Let \mathbb F be any field of characteristic p . It is well-known that there are exactly p inequivalent indecomposable representations V_1,V_2,\dots,V_p of C_p defined over \mathbb F . Thus if V is any finite dimensional C_p -representation there are non-negative integers 0\leq n_1,n_2,\dots, n_k \leq p-1 such that V \cong \oplus_{i=1}^k V_{n_i+1} . It is also well-known there is a unique (up to equivalence) d+1 dimensional irreducible complex representation of SL _2(\mathbb C) given by its action on the space R_d of d forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation of the ring of C_p -invariants \mathbb F[ \oplus_{i=1}^k V_{n_i+1}]^{C_p} to the computation of the classical ring of invariants (or covariants) \mathbb C[R_1 \oplus (\oplus_{i=1}^k R_{n_i})]^{\mathrm {SL}_2(\mathbb C)} . This shows that the problem of computing modular C_p invariants is equivalent to the problem of computing classical SL _2(\mathbb C) invariants. This allows us to compute for the first time the ring of invariants for many representations of C_p . In particular, we easily obtain from this generators for the rings of vector invariants \mathbb F[m\,V_2]^{C_p} , \mathbb F[m\,V_3]^{C_p} and \mathbb F[m\,V_4]^{C_p} for all m \in \mathbb N . This is the first computation of the latter two families of rings of invariants.