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

Ratpoints (C library): Michael Stoll's highly optimized C program for searching for certain rational points on hyperelliptic curves (i.e. 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. This brings the total Construct an elliptic curve from a plane curve of genus one (Lloyd Kilford, John Cremona ) — New function EllipticCurve_from_plane_curve() in the module sage/schemes/elliptic_curves/constructor.py to allow the construction of an elliptic curve from a smooth plane cubic with a rational point. The concrete example he described, which had been the original question of Masser, was the following: consider the Legendre family of elliptic curves. Silverman, Joseph H., Tate, John, Rational Points on Elliptic Curves, 1992 63. Rational Points on Elliptic Curves - Silverman, Tate.pdf. It also has It has no dependencies (instead of PARI), because Mark didn't want to have to license sympow under the GPL. P_t=(2,p_t),quad Q_t=(3,q_t These techniques are quite novel in this area, and rely ultimately (and quite strikingly) on the circle of ideas that started with the 1989 work of Bombieri and Pila on the number of rational (or integral) points on transcendental curves (in the plane, say). Devlin, Keith, The Joy of Sets – Fundamentals of Contemporary Set Theory, 1993 64. These new spkg's are mpmath for multiprecision floating-point arithmetic, and Ratpoints for computing rational points on hyperelliptic curves. Kinsey, L.Christine, Topology of Surfaces, 1993 65.