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(/usr/share/texmf/tex/latex/amsfonts/umsb.fd File: umsb.fd 1995/01/05 v2.2e AMS font definitions ) Overfull \hbox (1.49698pt too wide) in paragraph at lines 50--50 []\OT1/cmtt/m/n/10 ? f = x^18 - x^17 + x^16 - x^15 - x^12 + x^11 - x^10 + x^9 - x^8 \[] [] [1 ] Overfull \hbox (64.49643pt too wide) in paragraph at lines 99--99 []\OT1/cmtt/m/n/10 x^9 - 3*x^8 - 23*x^7 + 35*x^6 + 81*x^5 - 116*x^4 - 67*x^3 + 104*x^2 + 8*x - 19[] [] Overfull \hbox (2027.97931pt too wide) in paragraph at lines 99--99 []\OT1/cmtt/m/n/10 [ <-0.468111906454000689322313234631336807368973901776, 1>, <0.558484342299467301848260817414060396183155100382, 1>, <1.2313073740889668988 677367163865053658, 1>, <1.3706568287964283199650620889897443169, 1>, <1.147493 0126118010257418142242557119149, 1>, <5.6479185634867740095959132409297992529, 1>, <-3.6958681103558892478674250772828809120, 1>, <-1.784325060209241050083476 7489211165859, 1>, <-1.00755504426430656874557202714048694160132114234, 1> ][] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 99--99 []\OT1/cmtt/m/n/10 [47385, 30164, 87513, 43915, 73086, 88147, 40429, 20990, -13 3906],[] [] [2] Overfull \hbox (53.99652pt too wide) in paragraph at lines 110--110 []\OT1/cmtt/m/n/10 9*x^8 - 24*x^7 - 161*x^6 + 210*x^5 + 405*x^4 - 464*x^3 - 201 *x^2 + 208*x + 8[] [] Overfull \hbox (64.49643pt too wide) in paragraph at lines 123--123 []\OT1/cmtt/m/n/10 > R := [r[1] : r in Roots(RR!g)]; // these correspond to th e real embeddings.[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 163--163 [] \OT1/cmtt/m/n/10 if EmbeddingSigns(phi(U.r))[c] eq -1 then A[r,c] := 1; end if;[] [] [3] Overfull \hbox (206.2452pt too wide) in paragraph at lines 183--183 []\OT1/cmtt/m/n/10 $.1^9 - 37*$.1^8 + 334*$.1^7 - 1251*$.1^6 + 2322*$.1^5 - 225 0*$.1^4 + 1125*$.1^3 - 272*$.1^2 + 28*$.1 - 1[] [] Overfull \hbox (111.74602pt too wide) in paragraph at lines 183--183 []\OT1/cmtt/m/n/10 x^9 - 37*x^8 + 334*x^7 - 1251*x^6 + 2322*x^5 - 2250*x^4 + 11 25*x^3 - 272*x^2 + 28*x - 1[] [] [4] Overfull \hbox (1.49698pt too wide) in paragraph at lines 239--239 []\OT1/cmtt/m/n/10 I should clarify that I had checked that the totally positiv e unit[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 239--239 []\OT1/cmtt/m/n/10 I think by "strict class number" you mean the "narrow class number",[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 239--239 []\OT1/cmtt/m/n/10 strict class group of a quadratic number field" says. Using PARI, I[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 239--239 []\OT1/cmtt/m/n/10 ? f = x^18 - x^17 + x^16 - x^15 - x^12 + x^11 - x^10 + x^9 - x^8 \[] [] [5] Overfull \hbox (27.74675pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 In my paper, the field k is defined by the 9th degree polyno mial, which[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 y^9 - y^8 - 8*y^7 + 7*y^6 + 20*y^5 - 14*y^4 - 17*y^3 + 8*y^2 + 4*y - 1[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 and the field K is defined by the 18th degree polynomial I s ent you.[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 The strict class number of k is 2, and the strict class numb er of K[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 You are right that there is a relation on the signs of units in k, so[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 there is a totally positive unit which is not a square. The relation[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 complex in K is equal to 1; these correspond to the 8 roots of the[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 g'(y) is positive at 4 of these roots, and negative at 4 of them, by[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 what I used to teach in math 1a. So there should be a unit u which[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 positive unit you sent me, and that gives the two lattices L 1 and L2[] [] Overfull \hbox (80.24629pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 Let g(y) = y^9 - y^8 - 8*y^7 + 7*y^6 + 20*y^5 - 14*y^4 - 17* y^3 + 8*y^2 + 4*y - 1[] [] Overfull \hbox (59.24648pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 be the polynomial for the degree 9 field k, and index the re al places of k by[] [] Overfull \hbox (59.24648pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 the roots of g(y), in increasing order, from -1.995.. to 2.0 29.. Just fooling[] [] [6] Overfull \hbox (48.74657pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 Anyhow, this fits my prediction that the only relation on si gns is that the[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 first 8 entries (the real places ramified in K/k) have an ev en number of[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 - signs. Since g'(y) has the signs + - + - + - + - +, we sho uld be able to[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 303--303 []\OT1/cmtt/m/n/10 Since g(y) + 1 = (y-2)*(y-1)^2*(y)*(y+1)*(y+2)*(y^3-3*y-1), the elements[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 480--480 []\OT1/cmtt/m/n/10 > f := x^18 - x^17 + x^16 - x^15 - x^12 + x^11 - x^10 + x^9 - x^8[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 480--480 []\OT1/cmtt/m/n/10 x^9 - x^8 - 8*x^7 + 7*x^6 + 20*x^5 - 14*x^4 - 17*x^3 + 8*x^2 + 4*x - 1[] [] Overfull \hbox (347.99396pt too wide) in paragraph at lines 480--480 []\OT1/cmtt/m/n/10 Number Field with defining polynomial x^9 - x^8 - 8*x^7 + 7* x^6 + 20*x^5 - 14*x^4 - 17*x^3 + 8*x^2 + 4*x - 1 over the Rational Field[] [] [7] Overfull \hbox (22.4968pt too wide) in paragraph at lines 480--480 []\OT1/cmtt/m/n/10 9*x^8 - 8*x^7 - 56*x^6 + 42*x^5 + 100*x^4 - 56*x^3 - 51*x^2 + 16*x + 4[] [] Overfull \hbox (64.49643pt too wide) in paragraph at lines 480--480 []\OT1/cmtt/m/n/10 > R := [r[1] : r in Roots(RR!g)]; // these correspond to th e real embeddings.[] [] Overfull \hbox (64.49643pt too wide) in paragraph at lines 480--480 []\OT1/cmtt/m/n/10 > for|for> if EmbeddingSigns(phi(U.r))[c] eq -1 then A[ r,c] := 1; end if;[] [] [8] Overfull \hbox (536.99231pt too wide) in paragraph at lines 480--480 []\OT1/cmtt/m/n/10 1/53995179961*(8462152596*b^8 - 6852528273*b^7 - 64893174399 *b^6 + 44327911581*b^5 + 148290056856*b^4 - 74228933970*b^3 - 99225969383*b^2 + 23785291597*b + 10873550202)[] [] Overfull \hbox (510.74254pt too wide) in paragraph at lines 480--480 []\OT1/cmtt/m/n/10 1/53995179961*(-3426147347*b^8 + 817946757*b^7 + 29470572047 *b^6 - 3984702109*b^5 - 80455461231*b^4 + 2178527185*b^3 + 70870946633*b^2 + 34 09710051*b - 2902988849)[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 480--480 []\OT1/cmtt/m/n/10 > // d is the totally positive generator for the inverse dif ferent![] [] [9] Overfull \hbox (311.24428pt too wide) in paragraph at lines 480--480 []\OT1/cmtt/m/n/10 Maximal Equation Order with defining polynomial x^9 - x^8 - 8*x^7 + 7*x^6 + 20*x^5 - 14*x^4 - 17*x^3 + 8*x^2 + 4*x - 1 over Z[] [] Overfull \hbox (80.24629pt too wide) in paragraph at lines 480--480 []\OT1/cmtt/m/n/10 > for r in [1..9] do for c in [1..9] do B[r,c] := bin(O.r,O. c); end for; end for;[] [] [10] Overfull \hbox (22.4968pt too wide) in paragraph at lines 521--521 []\OT1/cmtt/m/n/10 x^9 - x^8 + 21031903322/53995179961*x^7 - 4130791902/5399517 9961*x^6 +[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 521--521 [] \OT1/cmtt/m/n/10 1015898/53995179961*x^3 - 21138/53995179961*x^2 + 230/53 995179961*x[] [] [11] Overfull \hbox (510.74254pt too wide) in paragraph at lines 573--573 []\OT1/cmtt/m/n/10 1/53995179961*(-3426147347*b^8 + 817946757*b^7 + 29470572047 *b^6 - 3984702109*b^5 - 80455461231*b^4 + 2178527185*b^3 + 70870946633*b^2 + 34 09710051*b - 2902988849)[] [] Overfull \hbox (90.7462pt too wide) in paragraph at lines 573--573 []\OT1/cmtt/m/n/10 > for r in [1..9] do for c in [1..9] do B2[r,c] := bin2(O.r, O.c); end for; end for;[] [] [12] Overfull \hbox (27.74675pt too wide) in paragraph at lines 610--610 []\OT1/cmtt/m/n/10 > Great. Could you now multiply the totally positive generat or you used[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 610--610 []\OT1/cmtt/m/n/10 > by the totally positive unit (x+2) and calculate the new G ram matrix and[] [] [13] (lattices-aug9.aux) ) Here is how much of TeX's memory you used: 1529 strings out of 20898 14611 string characters out of 196686 64244 words of memory out of 350001 4478 multiletter control sequences out of 10000+15000 12645 words of font info for 49 fonts, out of 400000 for 1000 14 hyphenation exceptions out of 1000 31i,6n,24p,472b,323s stack positions out of 1500i,100n,500p,50000b,4000s Output written on lattices-aug9.dvi (13 pages, 25792 bytes).