Leprévost, Franck Modular curves and 11-rank of quadratic fields. (Courbes modulaires et 11-rang de corps quadratiques.) (French) Zbl 0804.11062 Exp. Math. 2, No. 2, 137-146 (1993). The author exhibits 53 explicit complex quadratic number fields whose ideal class groups admit a subgroup isomorphic to \(\mathbb{Z}/11 \mathbb{Z} \times \mathbb{Z}/11 \mathbb{Z} \times \mathbb{Z}/11 \mathbb{Z}\). The field with the smallest discriminant is \(\mathbb{Q} (\sqrt {-107212102879})\). The author exploits the arithmetic of the modular curves \(X_ 0(23)\) and \(X_ 0(46)\) to obtain his examples. Reviewer: R.Schoof (Povo) Cited in 3 Documents MSC: 11R29 Class numbers, class groups, discriminants 11R11 Quadratic extensions 11Y40 Algebraic number theory computations Keywords:complex quadratic fields; ideal class groups; modular curves Software:PARI/GP × Cite Format Result Cite Review PDF Full Text: DOI EuDML EMIS References: [1] Batut C., User’s Guide to PARI-GP (1992) [2] Buell D. A., Math. Comp. 48 pp 85– (1987) · doi:10.1090/S0025-5718-1987-0866100-9 [3] Chevalley C., Comptes R. Acad. Set. Paris 195 pp 570– (1932) [4] Diaz F., Math. Comp. 32 pp 636– (1973) [5] Fermigier S., Comptes R. Acad. Sci. Paris 315 pp 719– (1992) [6] Fricke R., Lehrbuch der Algebra 3 (1928) · JFM 54.0187.20 [7] Gonzàlez Rovira J., Ann. Inst. Fourier 41 (4) pp 779– (1991) · Zbl 0758.14010 · doi:10.5802/aif.1273 [8] Ling S., Courbes modulaires et courbes de Shimura pp 171– (1991) [9] Llorente, P. and Quer, J. [Llorente et Quer], Tables (non publicé) [10] Mestre J.-F., Sémi-naire de Théorie des Nombres de Paris pp 189– (1982) [11] Mestre J.-F., J. reine angew. Math. 343 pp 23– (1983) [12] Mestre J.-F., Comptes R. Acad. Sci. Paris 315 pp 371– (1992) [13] Quer J., Comptes R. Acad. Sci. Paris 305 pp 215– (1987) [14] Schoof R., Math. Comp. 43 pp 295– (1983) · doi:10.1090/S0025-5718-1983-0701640-0 [15] Shanks, D. 1971.”Class number, a theory of factorisation and genera”415–440. Providence, RI: Amer. Math. Soc. [Shanks 1971], in 1969 Number Theory Institute, Proc. Sympos. Pure Math. 20 [16] Shanks D., Acta Arithm. 21 pp 71– (1972) [17] Solderitsch J. J., Thesis, in: ”Quadratic fields with special class groups” (1977) 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.