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

Abstract : This paper provides a method for picking a rational point on elliptic curves over the finite field of characteristic 2. Is a smooth projective curve of genus 1 (i.e., topologically a torus) defined over {K} with a {K} -rational point {0} . Then there is a constant B(d) depending only on d such that, if E/K is an elliptic curve with a K -rational torsion point of order N , then N < B(d) . Ratpoints (C library): Michael Stoll's highly optimized C program for searching for certain rational points on hyperelliptic curves (i.e. 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. Rational Points on Elliptic Curves (Undergraduate Texts in Mathematics) By Joseph H. We prove that the presentation of a general elliptic curve E with two rational points and a zero point is the generic Calabi-Yau onefold in dP_2. Rational Points on Modular Elliptic Curves (Cbms Regional Conference Series in Mathematics) book download Download Rational Points on Modular Elliptic Curves (Cbms Regional Conference Series in Mathematics) . Theorem (Uniform Boundedness Theorem).Let K be a number field of degree d . The set of all rational points in an elliptic curve $C$ over $ℚ$ is denoted by $C(ℚ)$ and called the Mordell-Weil group, i.e.,$C(ℚ)=\{\text{points on } $C$ \text{ with coordinates in } ℚ\}∪\{∞\}$.. Hmmm… The “parametrize by slopes of lines through the origin” is a standard trick to get rational or integral points on an elliptic curve. We discuss its resolved elliptic fibrations over a general base B. This library is very, very good and fast for doing computations of many functions relevant to number theory, of "class groups of number fields", and for certain computations with elliptic curves. It also has It has no dependencies (instead of PARI), because Mark didn't want to have to license sympow under the GPL. An upper bound is established for certain exponential sums on the rational points of an elliptic curve over a residue class ring ZN , N=pq for two distinct odd primes p and q.