Infinitely many elliptic curves with torsion ℤ/2ℤ × ℤ/6ℤ and rank at least four

We show that there are infinitely many elliptic curves over ℚ with torsion subgroup ℤ/2ℤ × ℤ/6ℤ and rank at least 4, one more than the largest rank previously known to occur infinitely often with this torsion. The universal elliptic curve with this torsion acquires four independent points over the function field of an explicit elliptic curve of rank one, and Silverman's specialisation theorem gives the family. Three of the points come from quadratic twists by polynomials that become squares on that curve, as in the rank-3 constructions of Dujella and Peral and of Dujella, Kazalicki and Peral. The fourth comes from the twist by the product of two of them; it is pulled back from a quadratic twist of a rational elliptic surface, of which the universal curve is a base change along a double cover of the line. The family is sparse, but it contradicts the prediction of Park, Poonen, Voight and Wood that only finitely many curves with this torsion have rank greater than 3.