# 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 (relative mode)

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

v(420,1), v(420,61), v(420,73), v(420,193), v(420,253), v(420,313), v(420,361), v(420,373)

The rank (number of elements in basis) is 8

## Cyclotomic Units Base Representations for n = 420 = 2^2*3*5*7 (relative mode)

 v(420,1) v(420,61) v(420,73) v(420,193) v(420,253) v(420,313) v(420,361) v(420,373) v(420,11) 1 v(420,13) -1 -1 -1 -1 -1 v(420,17) -1 v(420,19) 1 1 1 v(420,23) 1 v(420,29) 1 -1 -1 -1 -1 v(420,31) 1 1 1 v(420,37) 1 v(420,41) 1 -1 -1 -1 -1 v(420,43) -1 v(420,47) 1 v(420,53) -1 v(420,59) 1 v(420,67) -1 v(420,71) 1 v(420,79) -1 v(420,83) -1 -1 -1 -1 -1 v(420,89) 1 1 1 v(420,97) 1 v(420,101) 1 1 1 v(420,103) -1 v(420,107) 1 v(420,109) -1 -1 -1 v(420,113) -1 v(420,121) -1 -1 -1 v(420,127) 1 1 1 1 1 v(420,131) 1 v(420,137) -1 v(420,139) -1 v(420,143) 1 v(420,149) -1 v(420,151) -1 v(420,157) 1 v(420,163) -1 v(420,167) 1 v(420,169) -1 1 1 1 1 v(420,173) -1 v(420,179) -1 -1 -1 v(420,181) -1 1 1 1 1 v(420,187) -1 v(420,191) -1 -1 -1 v(420,197) 1 1 1 1 1 v(420,199) -1 v(420,209) -1 v(420,211) -1 v(420,221) -1 v(420,223) 1 1 1 1 1 v(420,227) 1 v(420,229) -1 -1 -1 v(420,233) -1 v(420,239) -1 1 1 1 1 v(420,241) -1 -1 -1 v(420,247) -1 v(420,251) -1 1 1 1 1 v(420,257) -1 v(420,263) 1 v(420,269) -1 v(420,271) -1 v(420,277) 1 v(420,281) -1 v(420,283) -1 v(420,289) 1 v(420,293) 1 1 1 1 1 v(420,299) -1 -1 -1 v(420,307) -1 v(420,311) -1 -1 -1 v(420,317) -1 v(420,319) 1 1 1 v(420,323) 1 v(420,331) 1 1 1 v(420,337) -1 -1 -1 -1 -1 v(420,341) -1 v(420,347) 1 v(420,349) 1 v(420,353) -1 v(420,359) 1 v(420,367) -1 v(420,377) -1 v(420,379) 1 -1 -1 -1 -1 v(420,383) 1 v(420,389) 1 1 1 v(420,391) 1 -1 -1 -1 -1 v(420,397) 1 v(420,401) 1 1 1 v(420,403) -1 v(420,407) -1 -1 -1 -1 -1 v(420,409) 1 v(420,419) 1

## Cyclotomic Units Base Representations for n = 420 = 2^2*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(420,1) is in basis
11: v(420,11) = v(420,361)
13: v(420,13) = v(420,73)^-1 v(420,193)^-1 v(420,253)^-1 v(420,313)^-1 v(420,373)^-1
17: v(420,17) = v(420,193)^-1
19: v(420,19) = v(420,61) v(420,313) v(420,373)
23: v(420,23) = v(420,373)
29: v(420,29) = v(420,1) v(420,73)^-1 v(420,193)^-1 v(420,313)^-1 v(420,373)^-1
31: v(420,31) = v(420,73) v(420,193) v(420,361)
37: v(420,37) = v(420,313)
41: v(420,41) = v(420,1) v(420,73)^-1 v(420,193)^-1 v(420,313)^-1 v(420,373)^-1
43: v(420,43) = v(420,253)^-1
47: v(420,47) = v(420,373)
53: v(420,53) = v(420,193)^-1
59: v(420,59) = v(420,361)
61: v(420,61) is in basis
67: v(420,67) = v(420,73)^-1
71: v(420,71) = v(420,1)
73: v(420,73) is in basis
79: v(420,79) = v(420,61)^-1
83: v(420,83) = v(420,73)^-1 v(420,193)^-1 v(420,253)^-1 v(420,313)^-1 v(420,373)^-1
89: v(420,89) = v(420,61) v(420,313) v(420,373)
97: v(420,97) = v(420,253)
101: v(420,101) = v(420,73) v(420,193) v(420,361)
103: v(420,103) = v(420,313)^-1
107: v(420,107) = v(420,313)
109: v(420,109) = v(420,73)^-1 v(420,193)^-1 v(420,361)^-1
113: v(420,113) = v(420,253)^-1
121: v(420,121) = v(420,61)^-1 v(420,313)^-1 v(420,373)^-1
127: v(420,127) = v(420,73) v(420,193) v(420,253) v(420,313) v(420,373)
131: v(420,131) = v(420,61)
137: v(420,137) = v(420,73)^-1
139: v(420,139) = v(420,1)^-1
143: v(420,143) = v(420,73)
149: v(420,149) = v(420,61)^-1
151: v(420,151) = v(420,361)^-1
157: v(420,157) = v(420,193)
163: v(420,163) = v(420,373)^-1
167: v(420,167) = v(420,253)
169: v(420,169) = v(420,1)^-1 v(420,73) v(420,193) v(420,313) v(420,373)
173: v(420,173) = v(420,313)^-1
179: v(420,179) = v(420,73)^-1 v(420,193)^-1 v(420,361)^-1
181: v(420,181) = v(420,1)^-1 v(420,73) v(420,193) v(420,313) v(420,373)
187: v(420,187) = v(420,373)^-1
191: v(420,191) = v(420,61)^-1 v(420,313)^-1 v(420,373)^-1
193: v(420,193) is in basis
197: v(420,197) = v(420,73) v(420,193) v(420,253) v(420,313) v(420,373)
199: v(420,199) = v(420,361)^-1
209: v(420,209) = v(420,1)^-1
211: v(420,211) = v(420,1)^-1
221: v(420,221) = v(420,361)^-1
223: v(420,223) = v(420,73) v(420,193) v(420,253) v(420,313) v(420,373)
227: v(420,227) = v(420,193)
229: v(420,229) = v(420,61)^-1 v(420,313)^-1 v(420,373)^-1
233: v(420,233) = v(420,373)^-1
239: v(420,239) = v(420,1)^-1 v(420,73) v(420,193) v(420,313) v(420,373)
241: v(420,241) = v(420,73)^-1 v(420,193)^-1 v(420,361)^-1
247: v(420,247) = v(420,313)^-1
251: v(420,251) = v(420,1)^-1 v(420,73) v(420,193) v(420,313) v(420,373)
253: v(420,253) is in basis
257: v(420,257) = v(420,373)^-1
263: v(420,263) = v(420,193)
269: v(420,269) = v(420,361)^-1
271: v(420,271) = v(420,61)^-1
277: v(420,277) = v(420,73)
281: v(420,281) = v(420,1)^-1
283: v(420,283) = v(420,73)^-1
289: v(420,289) = v(420,61)
293: v(420,293) = v(420,73) v(420,193) v(420,253) v(420,313) v(420,373)
299: v(420,299) = v(420,61)^-1 v(420,313)^-1 v(420,373)^-1
307: v(420,307) = v(420,253)^-1
311: v(420,311) = v(420,73)^-1 v(420,193)^-1 v(420,361)^-1
313: v(420,313) is in basis
317: v(420,317) = v(420,313)^-1
319: v(420,319) = v(420,73) v(420,193) v(420,361)
323: v(420,323) = v(420,253)
331: v(420,331) = v(420,61) v(420,313) v(420,373)
337: v(420,337) = v(420,73)^-1 v(420,193)^-1 v(420,253)^-1 v(420,313)^-1 v(420,373)^-1
341: v(420,341) = v(420,61)^-1
347: v(420,347) = v(420,73)
349: v(420,349) = v(420,1)
353: v(420,353) = v(420,73)^-1
359: v(420,359) = v(420,61)
361: v(420,361) is in basis
367: v(420,367) = v(420,193)^-1
373: v(420,373) is in basis
377: v(420,377) = v(420,253)^-1
379: v(420,379) = v(420,1) v(420,73)^-1 v(420,193)^-1 v(420,313)^-1 v(420,373)^-1
383: v(420,383) = v(420,313)
389: v(420,389) = v(420,73) v(420,193) v(420,361)
391: v(420,391) = v(420,1) v(420,73)^-1 v(420,193)^-1 v(420,313)^-1 v(420,373)^-1
397: v(420,397) = v(420,373)
401: v(420,401) = v(420,61) v(420,313) v(420,373)
403: v(420,403) = v(420,193)^-1
407: v(420,407) = v(420,73)^-1 v(420,193)^-1 v(420,253)^-1 v(420,313)^-1 v(420,373)^-1
409: v(420,409) = v(420,361)
419: v(420,419) = v(420,1)
The rank is: 8

## Cyclotomic Units - Methods for First Development for n = 420 = 2^2*3*5*7

Compare with Algorithm 1.2 and 2.2

1: v(420,1) - method B (Case V,i - A12)
11: v(420,11) - method Z-2 (Case V,iv - A12)
13: v(420,13) - method Z-7 (Case V,iv - A12)
17: v(420,17) - method Z-3 (Case V,iv - A12)
19: v(420,19) - method Z-2 (Case V,iv - A12)
23: v(420,23) - method Z-2 (Case V,iv - A12)
29: v(420,29) - method Z-3 (Case V,iv - A12)
31: v(420,31) - method Z-2 (Case V,iv - A12)
37: v(420,37) - method S (Case V,ii - A12 for p=5)
41: v(420,41) - method Z-3 (Case V,iv - A12)
43: v(420,43) - method Z-2 (Case V,iv - A12)
47: v(420,47) - method Z-2 (Case V,iv - A12)
53: v(420,53) - method Z-3 (Case V,iv - A12)
59: v(420,59) - method Z-2 (Case V,iv - A12)
61: v(420,61) - method B (Case V,iii - A12)
67: v(420,67) - method Z-2 (Case V,iv - A12)
71: v(420,71) - method Z-2 (Case V,iv - A12)
73: v(420,73) - method B (Case V,iii - A12)
79: v(420,79) - method Z-2 (Case V,iv - A12)
83: v(420,83) - method Z-2 (Case V,iv - A12)
89: v(420,89) - method Z-3 (Case V,iv - A12)
97: v(420,97) - method S (Case V,ii - A12 for p=5)
101: v(420,101) - method Z-3 (Case V,iv - A12)
103: v(420,103) - method Z-2 (Case V,iv - A12)
107: v(420,107) - method Z-2 (Case V,iv - A12)
109: v(420,109) - method Z-5 (Case V,iv - A12)
113: v(420,113) - method Z-3 (Case V,iv - A12)
121: v(420,121) - method S (Case V,ii - A12 for p=7)
127: v(420,127) - method Z-2 (Case V,iv - A12)
131: v(420,131) - method Z-2 (Case V,iv - A12)
137: v(420,137) - method Z-3 (Case V,iv - A12)
139: v(420,139) - method Z-2 (Case V,iv - A12)
143: v(420,143) - method Z-2 (Case V,iv - A12)
149: v(420,149) - method Z-3 (Case V,iv - A12)
151: v(420,151) - method Z-2 (Case V,iv - A12)
157: v(420,157) - method S (Case V,ii - A12 for p=5)
163: v(420,163) - method Z-2 (Case V,iv - A12)
167: v(420,167) - method Z-2 (Case V,iv - A12)
169: v(420,169) - method Z-5 (Case V,iv - A12)
173: v(420,173) - method Z-3 (Case V,iv - A12)
179: v(420,179) - method Z-2 (Case V,iv - A12)
181: v(420,181) - method Z-7 (Case V,iv - A12)
187: v(420,187) - method Z-2 (Case V,iv - A12)
191: v(420,191) - method Z-2 (Case V,iv - A12)
193: v(420,193) - method B (Case V,iii - A12)
197: v(420,197) - method Z-3 (Case V,iv - A12)
199: v(420,199) - method Z-2 (Case V,iv - A12)
209: v(420,209) - method Z-3 (Case V,iv - A12)
211: v(420,211) - method Z-2 (Case V,iv - A12)
221: v(420,221) - method Z-3 (Case V,iv - A12)
223: v(420,223) - method Z-2 (Case V,iv - A12)
227: v(420,227) - method Z-2 (Case V,iv - A12)
229: v(420,229) - method Z-5 (Case V,iv - A12)
233: v(420,233) - method Z-3 (Case V,iv - A12)
239: v(420,239) - method Z-2 (Case V,iv - A12)
241: v(420,241) - method S (Case V,ii - A12 for p=7)
247: v(420,247) - method Z-2 (Case V,iv - A12)
251: v(420,251) - method Z-2 (Case V,iv - A12)
253: v(420,253) - method B (Case V,iii - A12)
257: v(420,257) - method Z-3 (Case V,iv - A12)
263: v(420,263) - method Z-2 (Case V,iv - A12)
269: v(420,269) - method Z-3 (Case V,iv - A12)
271: v(420,271) - method Z-2 (Case V,iv - A12)
277: v(420,277) - method S (Case V,ii - A12 for p=5)
281: v(420,281) - method Z-3 (Case V,iv - A12)
283: v(420,283) - method Z-2 (Case V,iv - A12)
289: v(420,289) - method Z-5 (Case V,iv - A12)
293: v(420,293) - method Z-3 (Case V,iv - A12)
299: v(420,299) - method Z-2 (Case V,iv - A12)
307: v(420,307) - method Z-2 (Case V,iv - A12)
311: v(420,311) - method Z-2 (Case V,iv - A12)
313: v(420,313) - method B (Case V,iii - A12)
317: v(420,317) - method Z-3 (Case V,iv - A12)
319: v(420,319) - method Z-2 (Case V,iv - A12)
323: v(420,323) - method Z-2 (Case V,iv - A12)
331: v(420,331) - method Z-2 (Case V,iv - A12)
337: v(420,337) - method S (Case V,ii - A12 for p=5)
341: v(420,341) - method Z-3 (Case V,iv - A12)
347: v(420,347) - method Z-2 (Case V,iv - A12)
349: v(420,349) - method Z-5 (Case V,iv - A12)
353: v(420,353) - method Z-3 (Case V,iv - A12)
359: v(420,359) - method Z-2 (Case V,iv - A12)
361: v(420,361) - method B (Case V,iii - A12)
367: v(420,367) - method Z-2 (Case V,iv - A12)
373: v(420,373) - method B (Case V,iii - A12)
377: v(420,377) - method Z-3 (Case V,iv - A12)
379: v(420,379) - method Z-2 (Case V,iv - A12)
383: v(420,383) - method Z-2 (Case V,iv - A12)
389: v(420,389) - method Z-3 (Case V,iv - A12)
391: v(420,391) - method Z-2 (Case V,iv - A12)
397: v(420,397) - method S (Case V,ii - A12 for p=5)
401: v(420,401) - method Z-3 (Case V,iv - A12)
403: v(420,403) - method Z-2 (Case V,iv - A12)
407: v(420,407) - method Z-2 (Case V,iv - A12)
409: v(420,409) - method Z-5 (Case V,iv - A12)
419: v(420,419) - method Z-2 (Case V,iv - A12)

 © 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).