Cyclotomic Unit Calculator

Definitions

Cyclotomic Unit Calculation

Basis for Cyclotomic Units for n = 81 = 3^4

Note that this basis extends to a universal basis, i.e the basis for d is a subset of the basis for n iff d|n.

v(81,28), v(81,29), v(27,10), v(81,31), v(81,32), v(27,11), v(81,34), v(81,35), v(9,4), v(81,37), v(81,38), v(27,13), v(81,40), v(81,55), v(81,56), v(27,19), v(81,58), v(81,59), v(27,20), v(81,61), v(81,62), v(9,7), v(81,64), v(81,65), v(27,22), v(81,67)

The rank (number of elements in basis) is 26

Cyclotomic Units Base Representations for n = 81 = 3^4

 
v(81,28)
v(81,29)
v(27,10)
v(81,31)
v(81,32)
v(27,11)
v(81,34)
v(81,35)
v(9,4)
v(81,37)
v(81,38)
v(27,13)
v(81,40)
v(81,55)
v(81,56)
v(27,19)
v(81,58)
v(81,59)
v(27,20)
v(81,61)
v(81,62)
v(9,7)
v(81,64)
v(81,65)
v(27,22)
v(81,67)
v(81,1)                          
v(81,2)1-11  -1       1-11  -1  1    
v(27,1)                          
v(81,4)1 1-1    1  -1 1 1-1       -1 
v(81,5)1   -1        1   -1      1 
v(27,2)  1  -1         1  -1  1    
v(81,7)1     -1      1    1-1      
v(81,8)1      -1     1 1    -1     
v(9,1)                          
v(81,10)1 1      -1   1        -1   
v(81,11)1    1    -1  1         -1  
v(27,4)  1     1  -1   1        -1 
v(81,13)1          1-11           -1
v(81,14)                         1
v(27,5)                        1 
v(81,16)                       1  
v(81,17)                      1   
v(9,2)                     1    
v(81,19)                    1     
v(81,20)                   1      
v(27,7)                  1       
v(81,22)                 1        
v(81,23)                1         
v(27,8)               1          
v(81,25)              1           
v(81,26)             1            
v(3,1)                          
v(81,41)            1             
v(27,14)           1              
v(81,43)          1               
v(81,44)         1                
v(9,5)        1                 
v(81,46)       1                  
v(81,47)      1                   
v(27,16)     1                    
v(81,49)    1                     
v(81,50)   1                      
v(27,17)  1                       
v(81,52) 1                        
v(81,53)1                         
v(3,2)                          
v(81,68)1          1-11           -1
v(27,23)  1     1  -1   1        -1 
v(81,70)1    1    -1  1         -1  
v(81,71)1 1      -1   1        -1   
v(9,8)                          
v(81,73)1      -1     1 1    -1     
v(81,74)1     -1      1    1-1      
v(27,25)  1  -1         1  -1  1    
v(81,76)1   -1        1   -1      1 
v(81,77)1 1-1    1  -1 1 1-1       -1 
v(27,26)                          
v(81,79)1-11  -1       1-11  -1  1    
v(81,80)                          

Cyclotomic Units Base Representations for n = 81 = 3^4

See Algorithm 2.4 for details

Note that equality is modulo multiplication by an nth unit root.

1: v(81,1) = 1
2: v(81,2) = v(81,28) v(81,29)^-1 v(81,55) v(81,56)^-1 v(27,10) v(27,11)^-1 v(27,19) v(27,20)^-1 v(9,7)
3: v(27,1) = 1
4: v(81,4) = v(81,28) v(81,31)^-1 v(81,55) v(81,58)^-1 v(27,10) v(27,13)^-1 v(27,19) v(27,22)^-1 v(9,4)
5: v(81,5) = v(81,28) v(81,32)^-1 v(81,55) v(81,59)^-1 v(27,22)
6: v(27,2) = v(27,10) v(27,11)^-1 v(27,19) v(27,20)^-1 v(9,7)
7: v(81,7) = v(81,28) v(81,34)^-1 v(81,55) v(81,61)^-1 v(27,20)
8: v(81,8) = v(81,28) v(81,35)^-1 v(81,55) v(81,62)^-1 v(27,19)
9: v(9,1) = 1
10: v(81,10) = v(81,28) v(81,37)^-1 v(81,55) v(81,64)^-1 v(27,10)
11: v(81,11) = v(81,28) v(81,38)^-1 v(81,55) v(81,65)^-1 v(27,11)
12: v(27,4) = v(27,10) v(27,13)^-1 v(27,19) v(27,22)^-1 v(9,4)
13: v(81,13) = v(81,28) v(81,40)^-1 v(81,55) v(81,67)^-1 v(27,13)
14: v(81,14) = v(81,67)
15: v(27,5) = v(27,22)
16: v(81,16) = v(81,65)
17: v(81,17) = v(81,64)
18: v(9,2) = v(9,7)
19: v(81,19) = v(81,62)
20: v(81,20) = v(81,61)
21: v(27,7) = v(27,20)
22: v(81,22) = v(81,59)
23: v(81,23) = v(81,58)
24: v(27,8) = v(27,19)
25: v(81,25) = v(81,56)
26: v(81,26) = v(81,55)
27: v(3,1) = 1
28: v(81,28) is in basis
29: v(81,29) is in basis
30: v(27,10) is in basis
31: v(81,31) is in basis
32: v(81,32) is in basis
33: v(27,11) is in basis
34: v(81,34) is in basis
35: v(81,35) is in basis
36: v(9,4) is in basis
37: v(81,37) is in basis
38: v(81,38) is in basis
39: v(27,13) is in basis
40: v(81,40) is in basis
41: v(81,41) = v(81,40)
42: v(27,14) = v(27,13)
43: v(81,43) = v(81,38)
44: v(81,44) = v(81,37)
45: v(9,5) = v(9,4)
46: v(81,46) = v(81,35)
47: v(81,47) = v(81,34)
48: v(27,16) = v(27,11)
49: v(81,49) = v(81,32)
50: v(81,50) = v(81,31)
51: v(27,17) = v(27,10)
52: v(81,52) = v(81,29)
53: v(81,53) = v(81,28)
54: v(3,2) = 1
55: v(81,55) is in basis
56: v(81,56) is in basis
57: v(27,19) is in basis
58: v(81,58) is in basis
59: v(81,59) is in basis
60: v(27,20) is in basis
61: v(81,61) is in basis
62: v(81,62) is in basis
63: v(9,7) is in basis
64: v(81,64) is in basis
65: v(81,65) is in basis
66: v(27,22) is in basis
67: v(81,67) is in basis
68: v(81,68) = v(81,28) v(81,40)^-1 v(81,55) v(81,67)^-1 v(27,13)
69: v(27,23) = v(27,10) v(27,13)^-1 v(27,19) v(27,22)^-1 v(9,4)
70: v(81,70) = v(81,28) v(81,38)^-1 v(81,55) v(81,65)^-1 v(27,11)
71: v(81,71) = v(81,28) v(81,37)^-1 v(81,55) v(81,64)^-1 v(27,10)
72: v(9,8) = 1
73: v(81,73) = v(81,28) v(81,35)^-1 v(81,55) v(81,62)^-1 v(27,19)
74: v(81,74) = v(81,28) v(81,34)^-1 v(81,55) v(81,61)^-1 v(27,20)
75: v(27,25) = v(27,10) v(27,11)^-1 v(27,19) v(27,20)^-1 v(9,7)
76: v(81,76) = v(81,28) v(81,32)^-1 v(81,55) v(81,59)^-1 v(27,22)
77: v(81,77) = v(81,28) v(81,31)^-1 v(81,55) v(81,58)^-1 v(27,10) v(27,13)^-1 v(27,19) v(27,22)^-1 v(9,4)
78: v(27,26) = 1
79: v(81,79) = v(81,28) v(81,29)^-1 v(81,55) v(81,56)^-1 v(27,10) v(27,11)^-1 v(27,19) v(27,20)^-1 v(9,7)
80: v(81,80) = 1
The rank is: 26

Cyclotomic Units - Methods for First Development for n = 81 = 3^4

Compare with Algorithm 2.2

1: v(81,1) - method T (Case IV,iii - A22)
2: v(81,2) - method Y-3 (Case IV,iv - A22)
3: v(27,1) - method T (Case IV,iii - A22)
4: v(81,4) - method Y-3 (Case IV,iv - A22)
5: v(81,5) - method Y-3 (Case IV,iv - A22)
6: v(27,2) - method Y-3 (Case IV,iv - A22)
7: v(81,7) - method Y-3 (Case IV,iv - A22)
8: v(81,8) - method Y-3 (Case IV,iv - A22)
9: v(9,1) - method T (Case IV,iii - A22)
10: v(81,10) - method Y-3 (Case IV,iv - A22)
11: v(81,11) - method Y-3 (Case IV,iv - A22)
12: v(27,4) - method Y-3 (Case IV,iv - A22)
13: v(81,13) - method Y-3 (Case IV,iv - A22)
14: v(81,14) - method S (Case IV,i - A22)
15: v(27,5) - method S (Case IV,i - A22)
16: v(81,16) - method S (Case IV,i - A22)
17: v(81,17) - method S (Case IV,i - A22)
18: v(9,2) - method S (Case IV,i - A22)
19: v(81,19) - method S (Case IV,i - A22)
20: v(81,20) - method S (Case IV,i - A22)
21: v(27,7) - method S (Case IV,i - A22)
22: v(81,22) - method S (Case IV,i - A22)
23: v(81,23) - method S (Case IV,i - A22)
24: v(27,8) - method S (Case IV,i - A22)
25: v(81,25) - method S (Case IV,i - A22)
26: v(81,26) - method S (Case IV,i - A22)
27: v(3,1) - method T (Case III, A22)
28: v(81,28) - method B (Case IV,ii - A22)
29: v(81,29) - method B (Case IV,ii - A22)
30: v(27,10) - method B (Case IV,ii - A22)
31: v(81,31) - method B (Case IV,ii - A22)
32: v(81,32) - method B (Case IV,ii - A22)
33: v(27,11) - method B (Case IV,ii - A22)
34: v(81,34) - method B (Case IV,ii - A22)
35: v(81,35) - method B (Case IV,ii - A22)
36: v(9,4) - method B (Case IV,ii - A22)
37: v(81,37) - method B (Case IV,ii - A22)
38: v(81,38) - method B (Case IV,ii - A22)
39: v(27,13) - method B (Case IV,ii - A22)
40: v(81,40) - method B (Case IV,ii - A22)
41: v(81,41) - method S (Case IV,i - A22)
42: v(27,14) - method S (Case IV,i - A22)
43: v(81,43) - method S (Case IV,i - A22)
44: v(81,44) - method S (Case IV,i - A22)
45: v(9,5) - method S (Case IV,i - A22)
46: v(81,46) - method S (Case IV,i - A22)
47: v(81,47) - method S (Case IV,i - A22)
48: v(27,16) - method S (Case IV,i - A22)
49: v(81,49) - method S (Case IV,i - A22)
50: v(81,50) - method S (Case IV,i - A22)
51: v(27,17) - method S (Case IV,i - A22)
52: v(81,52) - method S (Case IV,i - A22)
53: v(81,53) - method S (Case IV,i - A22)
54: v(3,2) - method S (Case III, A22)
55: v(81,55) - method B (Case IV,ii - A22)
56: v(81,56) - method B (Case IV,ii - A22)
57: v(27,19) - method B (Case IV,ii - A22)
58: v(81,58) - method B (Case IV,ii - A22)
59: v(81,59) - method B (Case IV,ii - A22)
60: v(27,20) - method B (Case IV,ii - A22)
61: v(81,61) - method B (Case IV,ii - A22)
62: v(81,62) - method B (Case IV,ii - A22)
63: v(9,7) - method B (Case IV,ii - A22)
64: v(81,64) - method B (Case IV,ii - A22)
65: v(81,65) - method B (Case IV,ii - A22)
66: v(27,22) - method B (Case IV,ii - A22)
67: v(81,67) - method B (Case IV,ii - A22)
68: v(81,68) - method S (Case IV,i - A22)
69: v(27,23) - method S (Case IV,i - A22)
70: v(81,70) - method S (Case IV,i - A22)
71: v(81,71) - method S (Case IV,i - A22)
72: v(9,8) - method S (Case IV,i - A22)
73: v(81,73) - method S (Case IV,i - A22)
74: v(81,74) - method S (Case IV,i - A22)
75: v(27,25) - method S (Case IV,i - A22)
76: v(81,76) - method S (Case IV,i - A22)
77: v(81,77) - method S (Case IV,i - A22)
78: v(27,26) - method S (Case IV,i - A22)
79: v(81,79) - method S (Case IV,i - A22)
80: v(81,80) - method S (Case IV,i - A22)

Back to the Cyclotomic Unit Resources page


Olive Tree Icon © by: Marc Conrad, 2016. If you want me to come to your organisation and talk about cyclotomic units please contact me.
The material on this page is presented "as is". There is no warranty implied by presenting this stuff.
Feel free to use and modify the material for your own teaching. When doing so please link to this web site (http://perisic.com/cyclotomic). In acadmic publications cite this page as: Conrad, Marc (2016) Cyclotomic Units - Relations and Computations (online), available at: http://perisic.com/cyclotomic.
The webspace for this project is kindly provided by the Perisic Guesthouse (www.perisic.com).