Cube-sum curve x^3 + y^3 = 111690 (k = 2*3^2*5*17*73; Weierstrass model y^2 = x^3 - 432k^2, minimal model shown). Found by an exhaustive scan of cube-free k <= 7*10^5: minimal-model naive height computed for every k, a Mestre-Nagao a_p sieve over primes p = 1 mod 3 to rank candidates, then PARI ellrank 2-descent. k divisible by 9 gains a factor 3^6 in the minimal model, so k = 111690 has smaller naive height than the classical smallest rank-4 k = 21691; it is the minimum naive height over all rank-4 k in the scanned range (any larger k has naive height > 57).
David Renshaw ·