Cyclotomic Unit Calculator
Definitions
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εn := e2πi/n is an nth root of unity.
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For n=pα a prime power:
v(n,a) := (1 - εna)/ (1 - εn)
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For all other n:
v(n,a) := 1 - εna
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)
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Cyclotomic Units - Relations and Computations (online),
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