Quadratic Twists of an Elliptic Curve and Maps from a Hyperelliptic Curve

For an elliptic curve E over a number field k, we look for a polynomial f(t) such that rankE^f(t)(k(t)) is at least 3. To do so, we construct a family of hyperelliptic curves C : s² = f(t) over k of genus 3 such that J(C) is isogenous to E₁ × E₂ × E₃, and we give an example of C and E such that J(C) is isogenous to E × E × E over Q(√−3).