# 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 = 420 = 2^2*3*5*7

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(420,1), v(35,1), v(28,1), v(21,1), v(20,1), v(15,1), v(12,1), v(35,3), v(420,61), v(420,73), v(28,5), v(21,4), v(60,13), v(35,8), v(140,33), v(7,2), v(84,25), v(35,11), v(140,53), v(5,2), v(105,43), v(7,3), v(140,61), v(105,46), v(420,193), v(35,18), v(140,73), v(105,58), v(140,81), v(105,61), v(420,253), v(20,13), v(35,23), v(140,93), v(105,73), v(84,61), v(35,26), v(420,313), v(140,113), v(105,88), v(420,361), v(15,13), v(420,373), v(28,25), v(21,19), v(35,33), v(105,103)

The rank (number of elements in basis) is 47

## Cyclotomic Units Base Representations for n = 420 = 2^2*3*5*7

 v(420,1) v(35,1) v(28,1) v(21,1) v(20,1) v(15,1) v(12,1) v(35,3) v(420,61) v(420,73) v(28,5) v(21,4) v(60,13) v(35,8) v(140,33) v(7,2) v(84,25) v(35,11) v(140,53) v(5,2) v(105,43) v(7,3) v(140,61) v(105,46) v(420,193) v(35,18) v(140,73) v(105,58) v(140,81) v(105,61) v(420,253) v(20,13) v(35,23) v(140,93) v(105,73) v(84,61) v(35,26) v(420,313) v(140,113) v(105,88) v(420,361) v(15,13) v(420,373) v(28,25) v(21,19) v(35,33) v(105,103) v(210,1) 1 1 -1 -1 -1 1 1 -1 v(140,1) 1 1 -1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 v(105,1) 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 -1 v(84,1) 1 1 1 -1 -1 -1 1 v(70,1) 1 -1 v(60,1) 1 1 -1 1 v(105,2) 1 v(140,3) 1 -1 -1 1 1 1 v(42,1) 1 1 -1 v(420,11) -1 1 -1 -1 -1 1 1 v(420,13) 1 -1 -1 -1 1 -1 1 -1 v(30,1) 1 -1 1 v(105,4) 1 -1 1 -1 1 -1 1 -1 1 v(420,17) -1 -1 1 v(70,3) 1 -1 1 1 1 v(420,19) 1 1 1 -1 -3 1 -1 -1 1 -1 1 2 -1 -1 1 1 -1 -2 -2 -1 1 1 -1 1 2 -1 -1 v(210,11) -1 1 -1 -1 v(420,23) 1 -2 -1 1 -1 1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 v(35,2) 1 v(84,5) 1 1 -1 -1 1 v(210,13) 1 1 -1 -1 -1 -1 -1 1 -1 v(140,9) 1 1 -1 -1 1 1 1 1 v(420,29) 1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 1 v(14,1) -1 v(420,31) -1 -1 1 1 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 v(105,8) 1 -1 -1 v(140,11) 1 -1 -1 v(210,17) -1 1 v(420,37) 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 1 -1 1 1 -2 v(210,19) 1 -1 -1 -1 1 -1 1 -1 -1 1 -1 v(140,13) -1 -1 -1 -1 1 -1 -1 v(21,2) 1 v(420,41) 1 -2 -1 1 -1 2 -1 -1 1 -1 1 -2 -1 -1 2 -1 1 1 1 1 2 -1 1 -1 1 1 1 -1 -1 -2 1 1 v(10,1) -1 v(420,43) 1 -1 1 -1 1 1 1 -1 -1 -1 1 -1 1 -1 v(105,11) -1 1 -1 v(28,3) 1 v(210,23) 1 -1 1 -1 1 1 1 -1 -1 1 1 -1 1 v(420,47) 1 v(35,4) -1 -1 -1 1 -1 v(60,7) 1 -1 1 1 v(42,5) 1 2 -1 -1 v(140,17) -1 1 -1 v(105,13) 1 1 -1 -1 -1 -1 1 -1 v(420,53) -1 1 1 -1 -1 1 -1 -1 1 1 v(70,9) 1 1 -1 1 1 1 1 v(84,11) 1 -1 -1 1 2 -1 1 v(15,2) 1 v(140,19) 1 1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -2 v(210,29) 1 -2 1 -1 -1 1 -1 1 -1 2 1 -1 1 v(420,59) 1 v(7,1) v(210,31) -1 1 -1 1 -1 1 1 -1 1 -1 1 v(20,3) 1 -1 v(105,16) -1 1 1 1 -1 -1 1 -1 -1 -1 1 v(84,13) -1 -1 1 -1 -1 -1 1 v(70,11) 1 -1 v(420,67) -1 1 -1 1 -1 1 1 -1 -1 -1 -1 2 1 -1 1 -1 1 1 v(105,17) 1 v(140,23) 1 -1 1 1 -1 -1 -1 v(6,1) v(420,71) 1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 1 1 -1 1 1 -1 -1 -1 1 v(35,6) -1 1 -1 -1 1 1 1 v(210,37) 1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 -2 v(105,19) 1 -1 -1 1 -1 1 -1 -1 1 -1 v(60,11) -1 1 -1 -1 1 v(70,13) -1 -1 -1 1 -1 -1 -1 v(420,79) -1 1 -1 v(140,27) 1 v(210,41) -1 2 -1 1 1 -1 -1 -1 1 1 -1 1 1 -2 -1 1 -1 v(420,83) -1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 2 1 1 -1 1 1 -1 1 1 v(5,1) v(84,17) 1 -1 -1 v(210,43) 1 -1 1 -1 1 1 1 -1 -1 1 -1 1 -1 v(140,29) 1 -1 1 -1 1 1 -1 -1 v(105,22) -1 -1 -1 -1 1 -1 1 1 1 1 -1 1 v(420,89) -1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 1 1 v(14,3) -1 1 v(105,23) 1 1 1 -1 1 -1 1 v(140,31) -1 -1 1 -1 1 -1 -1 -1 1 -1 v(210,47) 1 -1 1 1 -1 1 1 -1 -1 1 -1 v(84,19) -1 -1 1 v(420,97) -1 2 -1 2 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 -1 1 v(30,7) 1 2 -1 v(21,5) 1 -1 -1 v(420,101) 1 -1 1 1 1 -1 1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 v(70,17) 1 -1 v(420,103) -1 -1 -1 1 -1 1 -1 1 1 1 -1 1 -1 2 v(105,26) -1 -1 1 v(4,1) v(210,53) 1 1 1 1 -1 -1 -1 -1 v(420,107) 1 v(35,9) 1 v(420,109) 1 -1 -1 -1 1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 v(42,11) -1 -1 2 1 v(140,37) -1 -1 1 v(15,4) -1 -1 v(420,113) -1 -1 1 1 -1 -1 -1 -1 1 -1 v(70,19) -1 1 -1 -1 -1 v(84,23) 1 v(105,29) -1 1 -1 -1 1 -1 1 -1 1 1 -1 1 v(140,39) 1 -2 -1 1 1 -1 -1 1 1 1 -1 v(210,59) -1 -1 -1 1 1 -1 1 -1 v(60,17) -1 -1 1 v(420,121) -1 1 2 -1 1 1 -1 -1 1 1 -1 -1 1 2 1 -1 -1 -1 1 -1 -2 -1 1 v(210,61) 1 1 1 1 -1 1 -1 1 -1 -1 -1 1 -1 v(140,41) -1 -1 1 -1 -1 -1 1 1 -1 1 1 1 v(105,31) -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 v(10,3) 1 v(420,127) 1 1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 1 -2 1 -1 -1 1 -1 v(105,32) 1 v(140,43) 1 1 1 1 -1 1 -1 v(42,13) -1 -1 -1 v(420,131) -1 -1 1 -1 1 -1 1 1 -1 1 1 1 -1 1 v(60,19) -1 1 -1 -1 v(210,67) -1 -1 2 1 -1 1 -1 1 1 -1 -1 1 v(28,9) -1 1 v(105,34) -1 -1 1 -1 1 -1 1 1 1 -1 1 v(420,137) 1 -1 -1 -1 -1 1 1 -1 1 1 1 1 -1 v(70,23) 1 -1 1 1 -1 -1 v(420,139) -1 1 1 -1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 1 1 v(3,1) v(140,47) 1 v(210,71) -1 -1 1 -1 1 -1 1 1 -1 1 v(420,143) -1 1 1 -1 1 -1 1 1 1 -1 -1 -1 1 1 -1 1 -1 1 -1 -1 -1 1 v(35,12) 1 v(84,29) 1 -1 -1 -1 1 1 1 -1 v(210,73) 1 -1 -1 -1 1 1 -1 1 1 1 1 -1 v(20,7) 1 v(105,37) -1 1 -1 v(420,149) 1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 1 -1 v(14,5) 1 v(420,151) -1 -1 -1 1 1 -1 1 -1 -1 v(105,38) -1 1 -1 v(140,51) -1 -1 1 1 -1 -1 -1 -1 1 1 1 v(30,11) -1 -1 -1 v(84,31) -1 -1 1 v(35,13) -1 -1 1 -1 -1 -1 v(420,157) 1 1 -1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 v(210,79) -1 -1 -1 1 1 -1 1 1 1 -1 1 v(21,8) -1 -1 v(60,23) -1 1 1 1 -1 -1 v(70,27) 1 1 1 1 -1 1 v(420,163) 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 v(105,41) -1 2 -1 1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 -1 v(28,11) -1 1 -1 v(210,83) -1 -1 -1 -1 1 1 -1 1 1 1 1 -1 1 v(420,167) 1 v(420,169) -1 1 1 -1 1 1 1 1 -1 1 -1 1 -1 1 -1 -1 -2 -1 -1 -1 1 -1 -1 1 1 1 -1 -1 v(42,17) 1 -1 v(140,57) 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 v(420,173) 1 -1 1 1 -1 v(70,29) -1 -1 -1 1 1 1 v(12,5) -1 v(105,44) 1 v(140,59) 1 v(210,89) -1 1 -1 1 1 1 -1 -1 1 -1 -1 1 v(420,179) -1 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 1 1 1 -1 -1 1 v(420,181) -1 1 1 -2 1 1 -1 -1 1 -1 1 -2 1 -1 -1 -1 -1 1 -1 1 -1 -1 1 1 2 -1 v(30,13) -1 1 v(84,37) 1 -1 1 1 -1 -1 v(70,31) -1 -1 -1 1 -1 -1 v(420,187) 1 1 -1 1 -1 1 1 1 1 -1 -1 1 v(105,47) 1 v(20,9) -1 -1 v(42,19) -1 1 v(420,191) -1 -1 1 2 -1 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 1 -1 -1 1 v(35,16) 1 -1 -1 -1 v(210,97) -1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 v(28,13) -1 -1 v(15,7) 1 -1 v(420,197) 1 -1 1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1 v(70,33) -1 1 v(420,199) 1 -1 -1 v(21,10) -1 1 v(140,67) 1 v(210,101) 1 -1 1 -1 1 -1 1 -1 1 -1 v(60,29) -1 1 -1 1 -1 v(35,17) 1 v(84,41) 1 -1 -1 1 1 -1 v(210,103) -1 -1 -1 1 -1 1 -1 1 1 1 1 -1 2 v(140,69) 1 -1 -1 1 -1 -1 1 1 1 1 -2 1 -1 1 -1 -1 v(105,52) 1 -1 1 1 -1 -1 -1 v(420,209) -1 1 1 -1 -1 -1 1 1 -1 v(2,1) v(420,211) -1 1 1 -1 -1 -1 1 1 -1 v(105,53) 1 -1 1 1 -1 -1 -1 v(140,71) 1 -1 -1 1 -1 -1 1 1 1 1 -2 1 -1 1 -1 -1 v(210,107) -1 -1 -1 1 -1 1 -1 1 1 1 1 -1 2 v(84,43) 1 -1 -1 1 1 -1 v(60,31) -1 1 -1 1 -1 v(210,109) 1 -1 1 -1 1 -1 1 -1 1 -1 v(21,11) -1 1 v(420,221) 1 -1 -1 v(70,37) -1 1 v(420,223) 1 -1 1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1 v(15,8) 1 -1 v(28,15) -1 -1 v(210,113) -1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 v(420,227) 1 v(35,19) 1 -1 -1 -1 v(420,229) -1 -1 1 2 -1 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 1 -1 -1 1 v(42,23) -1 1 v(20,11) -1 -1 v(420,233) 1 1 -1 1 -1 1 1 1 1 -1 -1 1 v(70,39) -1 -1 -1 1 -1 -1 v(84,47) 1 -1 1 1 -1 -1 v(105,59) 1 v(140,79) 1 v(30,17) -1 1 v(420,239) -1 1 1 -2 1 1 -1 -1 1 -1 1 -2 1 -1 -1 -1 -1 1 -1 1 -1 -1 1 1 2 -1 v(7,4) 1 v(420,241) -1 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 1 1 1 -1 -1 1 v(210,121) -1 1 -1 1 1 1 -1 -1 1 -1 -1 1 v(12,7) -1 v(70,41) -1 -1 -1 1 1 1 v(420,247) 1 -1 1 1 -1 v(105,62) 1 v(140,83) 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 v(42,25) 1 -1 v(420,251) -1 1 1 -1 1 1 1 1 -1 1 -1 1 -1 1 -1 -1 -2 -1 -1 -1 1 -1 -1 1 1 1 -1 -1 v(5,3) 1 v(210,127) -1 -1 -1 -1 1 1 -1 1 1 1 1 -1 1 v(28,17) -1 1 -1 v(105,64) -1 2 -1 1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 -1 v(420,257) 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 v(70,43) 1 1 1 1 -1 1 v(60,37) -1 1 1 1 -1 -1 v(21,13) -1 -1 v(140,87) 1 v(210,131) -1 -1 -1 1 1 -1 1 1 1 -1 1 v(420,263) 1 1 -1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 v(35,22) -1 -1 1 -1 -1 -1 v(84,53) -1 -1 1 v(30,19) -1 -1 -1 v(140,89) -1 -1 1 1 -1 -1 -1 -1 1 1 1 v(105,67) -1 1 -1 v(420,269) -1 -1 -1 1 1 -1 1 -1 -1 v(14,9) 1 v(420,271) 1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 1 -1 v(105,68) -1 1 -1 v(210,137) 1 -1 -1 -1 1 1 -1 1 1 1 1 -1 v(84,55) 1 -1 -1 -1 1 1