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A family of cyclic quartic fields arising from modular curves


Author: Lawrence C. Washington
Journal: Math. Comp. 57 (1991), 763-775
MSC: Primary 11R16; Secondary 11G05
DOI: https://doi.org/10.1090/S0025-5718-1991-1094964-6
MathSciNet review: 1094964
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Abstract: We study a family of cyclic quartic fields arising from the covering of modular curves $ {X_1}(16) \to {X_0}(16)$. An integral basis and a fundamental system of units are found. It is shown that a root of the quartic polynomial we construct is a translate of a cyclotomic period by an integer of the quadratic subfield of the quartic field.


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  • [1] A.-M. Bergé and J. Martinet, Notions relatives de régulateurs et de hauteurs, Acta Arith. (to appear). MR 1024424 (90m:11167)
  • [2] J. Buchmann and J. v. Schmettow, On the computation of unit groups and class groups of totally real quartic fields, Math. Comp. 53 (1989), 387-397. MR 970698 (90e:11192)
  • [3] G. Cornell and L. C. Washington, Class numbers of cyclotomic fields, J. Number Theory 21 (1985), 260-274. MR 814005 (87d:11079)
  • [4] H. Darmon, Note on a polynomial of Emma Lehmer, Math. Comp. 56 (1991), 795-800. MR 1068821 (91i:11149)
  • [5] M.-N. Gras, Table numérique du nombre de classes et des unités des extensions cycliques de degré 4 de $ \mathbb{Q}$, Publ. Math. Fac. Sci. Besançon, fasc. 2 (1977/78).
  • [6] -, Special units in real cyclic sextic fields, Math. Comp. 48 (1987), 179-182. MR 866107 (88m:11092)
  • [7] D. Kubert and S. Lang, Modular units, Springer-Verlag, New York-Heidelberg-Berlin, 1981. MR 648603 (84h:12009)
  • [8] A. Lazarus, The class number and cyclotomy of simplest quartic fields, Thesis, Univ. of California, Berkeley, 1989.
  • [9] O. Lecacheux, Unités d'une famille de corps cycliques réels de degré 6 liés à la courbe modulaire $ {X_1}(13)$, J. Number Theory 31 (1989), 54-63. MR 978099 (90i:11062)
  • [10] -, Unités d'une famille de corps liés à la courbe $ {X_1}(25)$, Ann. Inst. Fourier (Grenoble) 40 (1990), 237-253. MR 1070827 (91i:11065)
  • [11] E. Lehmer, Connection between Gaussian periods and cyclic units, Math. Comp. 50 (1988), 535-541. MR 929551 (89h:11067a)
  • [12] B. Levi, Saggio per una teoria aritmetica delle forme cubiche ternarie, Atti Accad. Reale Sci. Torino 43 (1908), 99-120.
  • [13] L. J. Mordell, Diophantine equations, Academic Press, London-New York, 1969. MR 0249355 (40:2600)
  • [14] A. Ogg, Rational points on certain elliptic modular curves, Proc. Sympos. Pure Math., vol. 24, Amer. Math. Soc., Providence, R.I., 1973, pp. 221-231. MR 0337974 (49:2743)
  • [15] R. Schoof and L. C. Washington, Quintic polynomials and real cyclotomic fields with large class numbers, Math. Comp. 50 (1988), 543-556. MR 929552 (89h:11067b)
  • [16] E. Seah, H. C. Williams, and L. C. Washington, The calculation of a large cubic class number with an application to real cyclotomic fields, Math. Comp. 41 (1983), 303-305. MR 701641 (84m:12008)
  • [17] D. Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137-1152. MR 0352049 (50:4537)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1991-1094964-6
Article copyright: © Copyright 1991 American Mathematical Society

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