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             LEHMER'S NUMBER - Coxeter group approach

                                      2      3      4
Twisting qs by p(a), p = 5 + 3 x - 7 x  - 2 x  + 2 x
2 x^4-2 x^3-7 x^2+3 x+5
                                    2    3    4
Twisting qc by p(a), p = 1 + x + 2 x  - x  - x
-x^4-x^3+2 x^2+x+1

		Picard group 
Glue group {11, 23, 23}, Signature {11, 1}, lambda 1.17628

Positivity verified.  Crossing: 4 Cyclic: 22 Fixed: 2
                                      3    4    5    6    7    9    10
Char poly  (-1 + x) (1 + x) (1 + x - x  - x  - x  - x  - x  + x  + x  )
Localization at 11: 1 + x
Localization at 23: (4 + x) (6 + x)

		Transcendental cycles
Glue group {11, 23, 23}, Signature {8, 2}, lambda = 1.
                    2    3    4    5    6    7    8    9    10
Char poly  1 - x + x  - x  + x  - x  + x  - x  + x  - x  + x    Period 1
Localization at 11: 1 + x
Localization at 23: (4 + x) (6 + x)
Rotation of T(X): 1/22

		Periodic factor of Pic
Glue group {11}, Signature {2, 0}

Positivity verified.  Cyclic: 22 Fixed: 2
Char poly  (-1 + x) (1 + x)  Period 2
Localization at 11: 1 + x

		Salem factor of Pic
Glue group {23, 23}, Signature {9, 1}, lambda 1.17628

Positivity verified.  Crossing: 4 Cyclic: Infinity Fixed: Infinity
                    3    4    5    6    7    9    10
Char poly  1 + x - x  - x  - x  - x  - x  + x  + x
Localization at 23: (4 + x) (6 + x)

		Field Information
A/Z[x], h(K), B/Z[y], h(k), Ramif: {1, 1, 1, 1, 1}
                        2      3      4
Twisted by 5 + 3 x - 7 x  - 2 x  + 2 x
Prime 23^1  <-  (10 + x), splits in K
Checking Coxeter obstruction: {}

		Primes and Twists

Feasible Primes: {3, 5, 13, 23, 29}
                         2    3    4
Prime 3^4  <-  (1 + x + x  + x  + x ), inert in K
Prime 3^1  <-  (x), inert in K
                             2    3
Prime 5^3  <-  (2 + 2 x + 3 x  + x ), inert in K
                           2
Prime 5^2  <-  (4 + 3 x + x ), inert in K
                               2      3    4
Prime 13^4  <-  (6 + 9 x + 11 x  + 7 x  + x ), inert in K
Prime 13^1  <-  (7 + x), inert in K
                                 2       3    4
Prime 23^4  <-  (21 + 19 x + 16 x  + 14 x  + x ), splits in K
Prime 23^1  <-  (10 + x), splits in K
                                 2      3    4
Prime 29^4  <-  (21 + 26 x + 15 x  + 5 x  + x ), inert in K
Prime 29^1  <-  (25 + x), inert in K
All twists:  those with dim'n small enough: 16
All twists:  those with period small enough: 10
All twists:  those with right signature: 4
Twists found: 4
Ideal 23         Periods {22}           Min 4  
Obstruction found after 567 trials
Ideal 3^2        Periods {4, 12}        Min 2  OBSTRUCTED
Ideal 3 29       Periods {4, 15}        Min 4  
Ideal 3 13       Periods {4, 14}        Min 4  

	Strategy Details
Period 23^4: 25440 requires dim 74
Period 23^1: 22 requires dim 10
                                                             2       3    4
Factorization of S(x) mod 23: (4 + x) (6 + x) (22 + 4 x + 6 x  + 18 x  + x ) 
 
                     2       3    4
>    (22 + 5 x + 17 x  + 19 x  + x )
(x+4) (x+6) \left(x^4+18 x^3+6 x^2+4 x+22\right) \left(x^4+19 x^3+17 x^2+5\
 
>   x+22\right)

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