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Rational points on elliptic curves John Tate, Joseph H. Silverman
Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K

Let E / ℚ E ℚ E/\mathbb{Q} be an elliptic curve and let { P 1 , … , P r } subscript P 1 normal-… subscript P r \{P_{1},\ldots,P_{r}\} be a set of generators of the free part of E ⁢ ( ℚ ) E ℚ E(\mathbb{Q}) , i.e. Smyth, Minimal polynomials of algebraic numbers with rational parameters. Theorem 5 (on page vi) of Diem's thesis states that the discrete logarithm problem in the group of rational points of an elliptic curves E( F_{p^n} ) can be solved in an expected time of \tilde{O}( q^{2 – 2/n} ) bit operations. Be the group of rational points on the curve and let. Be a set of generators of the free part of. Be the Néron-Tate pairing: where. Silverman, Lehmer's Conjecture and points on elliptic curves that are congruent to torsion points. Rational points on elliptic curves. Rational.points.on.elliptic.curves.pdf. These finite étale coverings admit various symmetry properties arising from the additive and multiplicative structures on the ring Fl = Z/lZ acting on the l-torsion points of the elliptic curve. Is the canonical height on the elliptic curve. The points P i subscript P i P_{i} generate E .