Infinitely many elliptic curves with torsion ℤ/2ℤ × ℤ/4ℤ and rank at least six

We show that there are infinitely many elliptic curves over ℚ with torsion subgroup ℤ/2ℤ × ℤ/4ℤ and rank at least 6. This answers the case ℤ/2ℤ × ℤ/4ℤ of Problem 2.6 in Dujella's list of open problems. His tables give 5, due to Eroshkin, as the largest rank known to occur infinitely often with this torsion. The curves are fibres of the one-parameter family that Dujella and Peral obtained from their two-parameter model by a square condition and τ = 1/7, and we show that this family has rank exactly 3 over the rational function field. Over the function field of an elliptic curve of rank one the family acquires three further independent points, coming from quadratic twists of an elliptic K3 surface of which it is a base change. These six points stay independent at infinitely many rational points of that curve, by a computation at one such point and congruences modulo six primes; at these points the torsion subgroup is ℤ/2ℤ × ℤ/4ℤ, by Mazur's theorem and a congruence modulo 13². The result is unconditional and exceeds the bound 5 for this torsion group in the table of rank bounds suggested by the heuristic of Park, Poonen, Voight and Wood; the curves form a thin family, of a kind that they anticipate. A second, differently built family, over an elliptic curve of rank two, also gives infinitely many curves with torsion ℤ/2ℤ × ℤ/4ℤ and rank at least 6.