# Cyclotomic Unit Calculator

## Definitions

• εn := ei/n is an nth root of unity.
• For n=pα a prime power: v(n,a) := (1 - εna)/ (1 - εn)
• For all other n: v(n,a) := 1 - εna
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 -1 1 -1 1 -1 1 -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 -1 1 -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 -1 1 -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 -1 1 -1 1 -1 1 -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)

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