Abstract :
Suppose that E1 and E2 are elliptic curves over the rational field, , such that ords=1 L(E1/K,s)≡ords=1 L(E2/K,s) (mod 2) for all quadratic fields . We prove that their conductors N(E1), and N(E2) are equal up to squares. If for all quadratic fields , then the same conclusion holds, provided the 2-parts of their Tate–Shafarevich groups are finite.
