| Geometry of Totally Real Cubic Fields (2002/03/23) |
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Dan (02/03/15):
Now I have tuned the VORONOI algorithm for Signature (3,0) to run with 4th generation languages (4GL: Java, C++, Delphi). As a first trial, I have analyzed the totally real cubic field with discriminant d = 5768321, |
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Dan (02/03/16):
Johannes Buchmann has chosen this field to demonstrate the power of the algorithms presented in his Habilitations-Schrift 1987 at the University of Düsseldorf. 
The discriminant lies far outside the range d < 5*10^5 of Ennola and Turunen's tables from 1985, which contain the most recent computer calculated regulators and fundamental systems of units. (Llorente and Quer have computed a larger table in 1988. It covers the range d < 10^7, but it doesn't give regulators and units.) |
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Dan (02/03/23):
1. Buchmann starts with the generating polynomial P(X) = X^3 - 113 X - 11 P(X) has 3 real zeros: xi = xi_1 = 10.678488112627206... xi_2 = -10.581134814557196... xi_3 = -0.09735329807001065... I determined an integral basis with the aid of Voronoi's master's thesis: (om_1, om_2, om_3) = ( 1, xi, xi^2 - 113 ) Hence, the maximal order of the totally real cubic L = Q(xi) is O_L = Z*om_1 + Z*om_2 + Z*om_3 By LLL-reduction of the power basis, Buchman gets another integral basis (om_1, om_2, om_3) = ( 1, xi, xi^2 - 75 ) The polynomial index is i(P) = 1, whence polynomial discriminant and field discriminant coincide: d(P) = d_L = 5768321 = 13*17*43*607 The discriminant d_L being squarefree, we get the conductor f = 1, i. e., the normal field N of L is an unramified Hilbert class field over its quadratic subfield K with discriminant d = d_L. Indeed, counting cycles of reduced indefinite quadratic forms yields the class number h = 48 = 2^4*3 and thus the 3-class rank rho = 1 of K. The fundamental unit of K is rather big eta = 19245990076771983019414359282821719387667305+ +8013376486711217763609247832156537986212*d^1/2 the regulator is r = 100.3590238176445... and the period length l = 84, as the continued fraction algorithm shows. Of course, the Principal Factorization Type of L is DELTA1, i. e., we have a 1-dimensional capitulation of the class of K of order 3 in N and eta arises as the norm of a unit in N; 2. Using VORONOI's algorithm for signature (3,0) of his doctoral dissertation, I determined the geometry of lattice minima in the discrete geometric image of O_L, the maximal order of Buchmann's totally real cubic: Chain in X-direction: no pre period, chain length l_X = 18, N(n_X^18(1)) = 1 X-Branch in Y-direction: branch length l_XY = 7, association on X-chain at pos = 3, N(n_Y^7(1)) = N(n_X^3(1)) = 19 X-Branch in Z-direction: branch length l_XZ = 9, association on X-chain at pos = 7, N(n_Z^9(1)) = N(n_X^7(1)) = 161 Chain in Y-direction: no pre period, chain length l_Y = 68, N(n_Y^68(1)) = 1 Y-Branch in Z-direction: branch length l_YZ = 2, association on Y-chain at pos = 59, N(n_Z^2(1)) = N(n_Y^59(1)) = 63 Y-Branch in X-direction: branch length l_YX = 3, association on Y-chain at pos = 7, N(n_X^3(1)) = N(n_Y^7(1)) = 19 Chain in Z-direction: no pre period, chain length l_Z = 17, N(n_Z^17(1)) = 1 Z-Branch in X-direction: branch length l_ZX = 7, association on Z-chain at pos = 9, N(n_X^7(1)) = N(n_Z^9(1)) = 161 Z-Branch in Y-direction: branch length l_ZY = 14, association on Z-chain at pos = 11, N(n_Y^14(1)) = N(n_Z^11(1)) = 171 
Regulator R = 506.889707164989... None of the DIRICHLET systems of units is fundamental: XY-system: index ( U : < e_X,e_Y > ) = 8 YZ-system: index ( U : < e_Y,e_Z > ) = 6 ZX-system: index ( U : < e_Z,e_X > ) = 2 Of course, I got 6 fundamental pairs of units by VORONOI's algorithm: (e_X,e_XY),(e_X,e_XZ),(e_Y,e_YZ),(e_Y,e_YX),(e_Z,e_ZX),(e_Z,e_ZY) and I list the one with most similarity to Buchmann's result: e_X = 31 + 37xi + 43xi^2 1/e_XY = e_YX = -286276 - 2913780xi + 275375xi^2 = u_1 of Buchmann Buchmann gives the following fundamental pair of units w. r. to his LLL-basis: u_1 = 20366849om_1 - 2913780om_2 + 275375om_3 u_2 = 11501336om_1 - 346225om_2 + 153820om_3 3. Buchmann was able to determine the generalized 2-dimensional period length of Min(O_L), i. e., the number of orbits of U in Min(O_L): #(Min(O_L) / U) = 435 and he also gives the class number of L: h_L = 1 |