  We calculate the mod (p, v_1, v_2) homotopy V(2)_* TC(BP<2>) of the
topological cyclic homology of the truncated Brown--Peterson spectrum BP<2>, at
all primes p\ge7, and show that it is a finitely generated and free
F_p[v_3]-module on 12p+4 generators in explicit degrees within the range -1 \le
* \le 2p^3+2p^2+2p-3. At these primes BP<2> is a form of elliptic cohomology,
and our result also determines the mod (p, v_1, v_2) homotopy of its algebraic
K-theory. Our computation is the first that exhibits chromatic redshift from
pure v_2-periodicity to pure v_3-periodicity in a precise quantitative manner.
