# 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 = 60 = 2^2*3*5 (relative mode)

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

v(60,13)

The rank (number of elements in basis) is 1

## Cyclotomic Units Base Representations for n = 60 = 2^2*3*5 (relative mode)

 v(60,13) v(60,1) -1 v(60,7) -1 v(60,11) -1 v(60,17) -1 v(60,19) 1 v(60,23) 1 v(60,29) 1 v(60,31) 1 v(60,37) 1 v(60,41) 1 v(60,43) -1 v(60,47) 1 v(60,49) -1 v(60,53) -1 v(60,59) -1

## Cyclotomic Units Base Representations for n = 60 = 2^2*3*5 (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(60,1) = v(60,13)^-1
7: v(60,7) = v(60,13)^-1
11: v(60,11) = v(60,13)^-1
13: v(60,13) is in basis
17: v(60,17) = v(60,13)^-1
19: v(60,19) = v(60,13)
23: v(60,23) = v(60,13)
29: v(60,29) = v(60,13)
31: v(60,31) = v(60,13)
37: v(60,37) = v(60,13)
41: v(60,41) = v(60,13)
43: v(60,43) = v(60,13)^-1
47: v(60,47) = v(60,13)
49: v(60,49) = v(60,13)^-1
53: v(60,53) = v(60,13)^-1
59: v(60,59) = v(60,13)^-1
The rank is: 1

## Cyclotomic Units - Methods for First Development for n = 60 = 2^2*3*5

Compare with Algorithm 1.2 and 2.2

1: v(60,1) - method E (Case V,i - A12)
7: v(60,7) - method Z-2 (Case V,iv - A12)
11: v(60,11) - method Z-2 (Case V,iv - A12)
13: v(60,13) - method B (Case V,iii - A12)
17: v(60,17) - method Z-3 (Case V,iv - A12)
19: v(60,19) - method Z-2 (Case V,iv - A12)
23: v(60,23) - method Z-2 (Case V,iv - A12)
29: v(60,29) - method Z-3 (Case V,iv - A12)
31: v(60,31) - method Z-2 (Case V,iv - A12)
37: v(60,37) - method S (Case V,ii - A12 for p=5)
41: v(60,41) - method Z-3 (Case V,iv - A12)
43: v(60,43) - method Z-2 (Case V,iv - A12)
47: v(60,47) - method Z-2 (Case V,iv - A12)
49: v(60,49) - method Z-5 (Case V,iv - A12)
53: v(60,53) - method Z-3 (Case V,iv - A12)
59: v(60,59) - method Z-2 (Case V,iv - A12)