Cyclotomic Unit Calculator

Definitions

Cyclotomic Unit Calculation

Basis for Cyclotomic Units for n = 105 = 3*5*7 (relative mode)

Note that the union of all bases over all n extends to a universal basis

v(105,43), v(105,46), v(105,58), v(105,61), v(105,73), v(105,88), v(105,103)

The rank (number of elements in basis) is 7

Cyclotomic Units Base Representations for n = 105 = 3*5*7 (relative mode)

 
v(105,43)
v(105,46)
v(105,58)
v(105,61)
v(105,73)
v(105,88)
v(105,103)
v(105,1)-1-1 -1 -1-1
v(105,2)      1
v(105,4) -1  1-1 
v(105,8)-1      
v(105,11) -1     
v(105,13)-1 -1 -1-1-1
v(105,16)  -11  1
v(105,17)     1 
v(105,19)  1-1  -1
v(105,22)1 1 111
v(105,23)  -1    
v(105,26)   -1   
v(105,29)1-11-11  
v(105,31) 1  -11 
v(105,32)    1  
v(105,34)11 1 11
v(105,37)      -1
v(105,38)    -1  
v(105,41)-11-11-1  
v(105,44)   1   
v(105,47)  1    
v(105,52)     -1 
v(105,53)     -1 
v(105,59) 1     
v(105,62)1      
v(105,64)-11-11-1  
v(105,67)    -1  
v(105,68)      -1
v(105,71)11 1 11
v(105,74) 1  -11 
v(105,76)1-11-11  
v(105,79)   -1   
v(105,82)  -1    
v(105,83)1 1 111
v(105,86)  1-1  -1
v(105,89)  -11  1
v(105,92)-1 -1 -1-1-1
v(105,94) -1     
v(105,97)-1      
v(105,101) -1  1-1 
v(105,104)-1-1 -1 -1-1

Cyclotomic Units Base Representations for n = 105 = 3*5*7 (relative mode)

See Algorithm 2.4 for details

Note that equality is modulo multiplication by an nth unit root and modulo dth cyclotomic units where d is a proper divisor of n.

1: v(105,1) = v(105,43)^-1 v(105,46)^-1 v(105,61)^-1 v(105,88)^-1 v(105,103)^-1
2: v(105,2) = v(105,103)
4: v(105,4) = v(105,46)^-1 v(105,73) v(105,88)^-1
8: v(105,8) = v(105,43)^-1
11: v(105,11) = v(105,46)^-1
13: v(105,13) = v(105,43)^-1 v(105,58)^-1 v(105,73)^-1 v(105,88)^-1 v(105,103)^-1
16: v(105,16) = v(105,58)^-1 v(105,61) v(105,103)
17: v(105,17) = v(105,88)
19: v(105,19) = v(105,58) v(105,61)^-1 v(105,103)^-1
22: v(105,22) = v(105,43) v(105,58) v(105,73) v(105,88) v(105,103)
23: v(105,23) = v(105,58)^-1
26: v(105,26) = v(105,61)^-1
29: v(105,29) = v(105,43) v(105,46)^-1 v(105,58) v(105,61)^-1 v(105,73)
31: v(105,31) = v(105,46) v(105,73)^-1 v(105,88)
32: v(105,32) = v(105,73)
34: v(105,34) = v(105,43) v(105,46) v(105,61) v(105,88) v(105,103)
37: v(105,37) = v(105,103)^-1
38: v(105,38) = v(105,73)^-1
41: v(105,41) = v(105,43)^-1 v(105,46) v(105,58)^-1 v(105,61) v(105,73)^-1
43: v(105,43) is in basis
44: v(105,44) = v(105,61)
46: v(105,46) is in basis
47: v(105,47) = v(105,58)
52: v(105,52) = v(105,88)^-1
53: v(105,53) = v(105,88)^-1
58: v(105,58) is in basis
59: v(105,59) = v(105,46)
61: v(105,61) is in basis
62: v(105,62) = v(105,43)
64: v(105,64) = v(105,43)^-1 v(105,46) v(105,58)^-1 v(105,61) v(105,73)^-1
67: v(105,67) = v(105,73)^-1
68: v(105,68) = v(105,103)^-1
71: v(105,71) = v(105,43) v(105,46) v(105,61) v(105,88) v(105,103)
73: v(105,73) is in basis
74: v(105,74) = v(105,46) v(105,73)^-1 v(105,88)
76: v(105,76) = v(105,43) v(105,46)^-1 v(105,58) v(105,61)^-1 v(105,73)
79: v(105,79) = v(105,61)^-1
82: v(105,82) = v(105,58)^-1
83: v(105,83) = v(105,43) v(105,58) v(105,73) v(105,88) v(105,103)
86: v(105,86) = v(105,58) v(105,61)^-1 v(105,103)^-1
88: v(105,88) is in basis
89: v(105,89) = v(105,58)^-1 v(105,61) v(105,103)
92: v(105,92) = v(105,43)^-1 v(105,58)^-1 v(105,73)^-1 v(105,88)^-1 v(105,103)^-1
94: v(105,94) = v(105,46)^-1
97: v(105,97) = v(105,43)^-1
101: v(105,101) = v(105,46)^-1 v(105,73) v(105,88)^-1
103: v(105,103) is in basis
104: v(105,104) = v(105,43)^-1 v(105,46)^-1 v(105,61)^-1 v(105,88)^-1 v(105,103)^-1
The rank is: 7

Cyclotomic Units - Methods for First Development for n = 105 = 3*5*7

Compare with Algorithm 1.2 and 2.2

1: v(105,1) - method E (Case V,i - A12)
2: v(105,2) - method Z-3 (Case V,iv - A12)
4: v(105,4) - method Z-5 (Case V,iv - A12)
8: v(105,8) - method Z-3 (Case V,iv - A12)
11: v(105,11) - method Z-3 (Case V,iv - A12)
13: v(105,13) - method Z-7 (Case V,iv - A12)
16: v(105,16) - method S (Case V,ii - A12 for p=7)
17: v(105,17) - method Z-3 (Case V,iv - A12)
19: v(105,19) - method Z-5 (Case V,iv - A12)
22: v(105,22) - method S (Case V,ii - A12 for p=5)
23: v(105,23) - method Z-3 (Case V,iv - A12)
26: v(105,26) - method Z-3 (Case V,iv - A12)
29: v(105,29) - method Z-3 (Case V,iv - A12)
31: v(105,31) - method S (Case V,ii - A12 for p=7)
32: v(105,32) - method Z-3 (Case V,iv - A12)
34: v(105,34) - method Z-5 (Case V,iv - A12)
37: v(105,37) - method S (Case V,ii - A12 for p=5)
38: v(105,38) - method Z-3 (Case V,iv - A12)
41: v(105,41) - method Z-3 (Case V,iv - A12)
43: v(105,43) - method B (Case V,iii - A12)
44: v(105,44) - method Z-3 (Case V,iv - A12)
46: v(105,46) - method B (Case V,iii - A12)
47: v(105,47) - method Z-3 (Case V,iv - A12)
52: v(105,52) - method S (Case V,ii - A12 for p=5)
53: v(105,53) - method Z-3 (Case V,iv - A12)
58: v(105,58) - method B (Case V,iii - A12)
59: v(105,59) - method Z-3 (Case V,iv - A12)
61: v(105,61) - method B (Case V,iii - A12)
62: v(105,62) - method Z-3 (Case V,iv - A12)
64: v(105,64) - method Z-5 (Case V,iv - A12)
67: v(105,67) - method S (Case V,ii - A12 for p=5)
68: v(105,68) - method Z-3 (Case V,iv - A12)
71: v(105,71) - method Z-3 (Case V,iv - A12)
73: v(105,73) - method B (Case V,iii - A12)
74: v(105,74) - method Z-3 (Case V,iv - A12)
76: v(105,76) - method Z-7 (Case V,iv - A12)
79: v(105,79) - method Z-5 (Case V,iv - A12)
82: v(105,82) - method S (Case V,ii - A12 for p=5)
83: v(105,83) - method Z-3 (Case V,iv - A12)
86: v(105,86) - method Z-3 (Case V,iv - A12)
88: v(105,88) - method B (Case V,iii - A12)
89: v(105,89) - method Z-3 (Case V,iv - A12)
92: v(105,92) - method Z-3 (Case V,iv - A12)
94: v(105,94) - method Z-5 (Case V,iv - A12)
97: v(105,97) - method S (Case V,ii - A12 for p=5)
101: v(105,101) - method Z-3 (Case V,iv - A12)
103: v(105,103) - method B (Case V,iii - A12)
104: v(105,104) - method Z-3 (Case V,iv - A12)

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