For every number field K and every 3-generic elliptic curve E/K with full rational 2-torsion, there are infinitely many quadratic twists of E with rank exactly 1.
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Elliptic curves of rank one over number fields
For every number field K and every 3-generic elliptic curve E/K with full rational 2-torsion, there are infinitely many quadratic twists of E with rank exactly 1.