A new record for the canonical height on an elliptic curve over C(t)

We exhibit an elliptic curve E/C(t) of discriminant degree 84 with a nontorsion point P of canonical height 2987/120120 (a new record). We also prove that if (E, P) has Szpiro ratio sigma <= 4, then (h) over cap (P) must exceed this value, providing some evidence that our example may yield the smallest height possible over C(t). Using the same strategy, we find other E/C(t) with nontorsion points of small canonical height, including Elkies' previous record.