Elliptic curves with torsion group ℤ/6ℤ and generic rank four

We give an elliptic curve over ℚ(t) with non-constant j-invariant, torsion subgroup ℤ/6ℤ and Mordell–Weil rank exactly 4. The largest rank previously known for such a curve was 3, due to Lecacheux and reached by several later families. The curve is the pullback of the universal elliptic curve with a point of order 6 along an explicit rational function of degree 16 that is invariant under t ↦ −t and t ↦ 1/t. The upper bound for the rank comes from the specialization criterion of Gusić and Tadić and a 2-descent of a single fiber. The rank splits along the characters of the symmetry group: the curve descends to ℚ((t + 1/t)²), where it and three of its quadratic twists have rank exactly 1. All but finitely many rational fibers have torsion subgroup ℤ/6ℤ and rank at least 4. We also give an explicit family over an elliptic curve of rank two which shows that infinitely many elliptic curves over ℚ have torsion subgroup ℤ/6ℤ and rank at least 5, a bound that Dujella's tables credit to Eroshkin (2009, personal communication). One member of this family has rank exactly 6. Over ℚ(√2) the same kind of construction gives infinitely many curves with torsion subgroup ℤ/6ℤ and rank at least 6; whether the same holds over ℚ is open. All results are unconditional.