Simon, Denis On the parametrization of solutions of quadratic equations. (Sur la paramétrisation des solutions des équations quadratiques.) (French) Zbl 1116.11024 J. Théor. Nombres Bordx. 18, No. 1, 265-283 (2006). Let \(Q(X,Y)\) be a binary integral quadratic form, then if \(Q=1\) has a solution it easily leads to an integral parametrization by quadratic forms \(q_i(s,t)\), \((i=1,2,3)\) such that \(Q(q_1(s,t),q_2(s,t))=q^2_3 (s,t)\). This also serves to make the class of \(Q\) the square of the class of \(q_3(s,t)\). As a consequence \(q_3\) is independent of the solution, and there are also applications to the 2-descent for elliptic curves [see the author, LMS J. Comput. Math. 5, 7–17 (2002; Zbl 1067.11015)]. The exposition also discusses the 2-class number and is remarkably self-contained. Reviewer: Harvey Cohn (Laguna Woods) Cited in 1 Document MSC: 11E16 General binary quadratic forms Keywords:quadratic form; elliptic curves; 2-class number Citations:Zbl 1067.11015 PDFBibTeX XMLCite \textit{D. Simon}, J. Théor. Nombres Bordx. 18, No. 1, 265--283 (2006; Zbl 1116.11024) Full Text: DOI Numdam EuDML References: [1] W. Bosma, P. Stevenhagen, On the computation of quadratic \(2\)-class groups. J. Théor. Nombres Bordeaux 8 (1996), no. 2, 283-313. · Zbl 0870.11080 [2] J.W.S. Cassels, Rational Quadratic Forms. L.M.S. Monographs, Academic Press (1978). · Zbl 0395.10029 [3] H. Cohen, A Course in Computational Algebraic Number Theory. Graduate Texts in Math. 138, Third corrected printing, Springer-Verlag (1996). · Zbl 0786.11071 [4] D. Cox, Primes of the form \(x^2+ny^2\), Fermat, class field theory, and complex multiplication. Wiley-Interscience (1989). · Zbl 0956.11500 [5] J. Cremona, D. Rusin, Efficient solution of rational conics. Math. Comp. 72 (2003), 1417-1441. · Zbl 1022.11031 [6] K.F. Gauss, Recherches Arithmétiques. Poullet-Delisle, A.C.M. (trad.), A. Blanchard, 1953. · Zbl 0051.03003 [7] K. Hardy, K. Williams, The squareroot of an ambiguous form in the principal genus. Proc. Edinburgh Math. Soc. (2) 36 (1993), no. 1, 145-150. · Zbl 0790.11030 [8] J.C. Lagarias, Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. J. Ameri. Math. Soc. 2 (1989), no 4, 143-186. · Zbl 0473.68030 [9] D. Shanks, Gauss’s ternary form reduction and the \(2\)-Sylow subgroup. Math. Comp. 25, no 116 (1971), 837-853 ; Erratum : Math. Comp. 32 (1978), 1328-1329. · Zbl 0227.12002 [10] D. Simon, Computing the rank of elliptic curves over number fields. London Math. Soc. Journal of Computation and Mathematics, vol 5 (2002) 7-17. · Zbl 1067.11015 [11] D. Simon, Solving quadratic equations using reduced unimodular quadratic forms. Math. Comp. 74, no 251 (2005), 1531-1543. · Zbl 1078.11072 [12] N.P. Smart, The algorithmic resolution of Diophantine equations. London Math. Soc. Student Texts 41, Cambridge University Press, 1998. · Zbl 0907.11001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.