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Values of Dedekind zeta functions of totally real cubic fields at negative odd integers
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Numbers
$D$
$k$
$s$ 
$\zeta_K(s)$
49
1
-1:
-1/21
comment: $x^3-x^2-2x+1=0$; $C_3$, conductor $7$; $h_K=1$; $K=\mathbb{Q}(\zeta_7)^+$
49
1
-3:
79/210
comment: $x^3-x^2-2x+1=0$; $C_3$, conductor $7$; $h_K=1$; $K=\mathbb{Q}(\zeta_7)^+$
49
1
-5:
-7393/63
comment: $x^3-x^2-2x+1=0$; $C_3$, conductor $7$; $h_K=1$; $K=\mathbb{Q}(\zeta_7)^+$
81
1
-1:
-1/9
comment: $x^3-3x-1=0$; $C_3$, conductor $9$; $h_K=1$; $K=\mathbb{Q}(\zeta_9)^+$
81
1
-3:
199/90
comment: $x^3-3x-1=0$; $C_3$, conductor $9$; $h_K=1$; $K=\mathbb{Q}(\zeta_9)^+$
81
1
-5:
-50353/27
comment: $x^3-3x-1=0$; $C_3$, conductor $9$; $h_K=1$; $K=\mathbb{Q}(\zeta_9)^+$
148
1
-1:
-1/3
comment: $x^3-x^2-3x+1=0$; $S_3$; $h_K=1$
148
1
-3:
577/30
comment: $x^3-x^2-3x+1=0$; $S_3$; $h_K=1$
148
1
-5:
-3281281/63
comment: $x^3-x^2-3x+1=0$; $S_3$; $h_K=1$
169
1
-1:
-1/3
comment: $x^3-x^2-4x-1=0$; $C_3$, conductor $13$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{13})$
169
1
-3:
11227/390
comment: $x^3-x^2-4x-1=0$; $C_3$, conductor $13$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{13})$
169
1
-5:
-6701911/63
comment: $x^3-x^2-4x-1=0$; $C_3$, conductor $13$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{13})$
229
1
-1:
-2/3
comment: $x^3-4x-1=0$; $S_3$; $h_K=1$
229
1
-3:
1333/15
comment: $x^3-4x-1=0$; $S_3$; $h_K=1$
229
1
-5:
-36206042/63
comment: $x^3-4x-1=0$; $S_3$; $h_K=1$
257
1
-1:
-2/3
comment: $x^3-x^2-4x+3=0$; $S_3$; $h_K=1$
257
1
-3:
1891/15
comment: $x^3-x^2-4x+3=0$; $S_3$; $h_K=1$
257
1
-5:
-67297502/63
comment: $x^3-x^2-4x+3=0$; $S_3$; $h_K=1$
316
1
-1:
-4/3
comment: $x^3-x^2-4x+2=0$; $S_3$; $h_K=1$
316
1
-3:
874/3
comment: $x^3-x^2-4x+2=0$; $S_3$; $h_K=1$
316
1
-5:
-216119884/63
comment: $x^3-x^2-4x+2=0$; $S_3$; $h_K=1$
321
1
-1:
-1
comment: $x^3-x^2-4x+1=0$; $S_3$; $h_K=1$
321
1
-3:
555/2
comment: $x^3-x^2-4x+1=0$; $S_3$; $h_K=1$
321
1
-5:
-76311671/21
comment: $x^3-x^2-4x+1=0$; $S_3$; $h_K=1$
361
1
-1:
-1
comment: $x^3-x^2-6x+7=0$; $C_3$, conductor $19$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{19})$
361
1
-3:
4087/10
comment: $x^3-x^2-6x+7=0$; $C_3$, conductor $19$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{19})$
361
1
-5:
-2758494229/399
comment: $x^3-x^2-6x+7=0$; $C_3$, conductor $19$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{19})$
404
1
-1:
-5/3
comment: $x^3-x^2-5x-1=0$; $S_3$; $h_K=1$
404
1
-3:
19613/30
comment: $x^3-x^2-5x-1=0$; $S_3$; $h_K=1$
404
1
-5:
-822762485/63
comment: $x^3-x^2-5x-1=0$; $S_3$; $h_K=1$
469
1
-1:
-2
comment: $x^3-x^2-5x+4=0$; $S_3$; $h_K=1$
469
1
-3:
1093
comment: $x^3-x^2-5x+4=0$; $S_3$; $h_K=1$
469
1
-5:
-207442574/7
comment: $x^3-x^2-5x+4=0$; $S_3$; $h_K=1$
473
1
-1:
-5/3
comment: $x^3-5x-1=0$; $S_3$; $h_K=1$
473
1
-3:
31979/30
comment: $x^3-5x-1=0$; $S_3$; $h_K=1$
473
1
-5:
-1927968515/63
comment: $x^3-5x-1=0$; $S_3$; $h_K=1$
564
1
-1:
-3
comment: $x^3-x^2-5x+3=0$; $S_3$; $h_K=1$
564
1
-3:
21267/10
comment: $x^3-x^2-5x+3=0$; $S_3$; $h_K=1$
564
1
-5:
-81933783
comment: $x^3-x^2-5x+3=0$; $S_3$; $h_K=1$
568
1
-1:
-10/3
comment: $x^3-x^2-6x-2=0$; $S_3$; $h_K=1$
568
1
-3:
34079/15
comment: $x^3-x^2-6x-2=0$; $S_3$; $h_K=1$
568
1
-5:
-776714410/9
comment: $x^3-x^2-6x-2=0$; $S_3$; $h_K=1$
621
1
-1:
-10/3
comment: $x^3-6x-3=0$; $S_3$; $h_K=1$
621
1
-3:
44321/15
comment: $x^3-6x-3=0$; $S_3$; $h_K=1$
621
1
-5:
-8755597090/63
comment: $x^3-6x-3=0$; $S_3$; $h_K=1$
697
1
-1:
-8/3
comment: $x^3-7x-5=0$; $S_3$; $h_K=1$
697
1
-3:
61336/15
comment: $x^3-7x-5=0$; $S_3$; $h_K=1$
697
1
-5:
-16238550248/63
comment: $x^3-7x-5=0$; $S_3$; $h_K=1$
733
1
-1:
-4
comment: $x^3-x^2-7x+8=0$; $S_3$; $h_K=1$
733
1
-3:
26114/5
comment: $x^3-x^2-7x+8=0$; $S_3$; $h_K=1$
733
1
-5:
-7255453204/21
comment: $x^3-x^2-7x+8=0$; $S_3$; $h_K=1$
756
1
-1:
-13/3
comment: $x^3-6x-2=0$; $S_3$; $h_K=1$
756
1
-3:
175861/30
comment: $x^3-6x-2=0$; $S_3$; $h_K=1$
756
1
-5:
-25825564573/63
comment: $x^3-6x-2=0$; $S_3$; $h_K=1$
761
1
-1:
-10/3
comment: $x^3-x^2-6x-1=0$; $S_3$; $h_K=1$
761
1
-3:
84359/15
comment: $x^3-x^2-6x-1=0$; $S_3$; $h_K=1$
761
1
-5:
-26360578870/63
comment: $x^3-x^2-6x-1=0$; $S_3$; $h_K=1$
785
1
-1:
-11/3
comment: $x^3-x^2-6x+5=0$; $S_3$; $h_K=1$
785
1
-3:
188597/30
comment: $x^3-x^2-6x+5=0$; $S_3$; $h_K=1$
785
1
-5:
-31273327181/63
comment: $x^3-x^2-6x+5=0$; $S_3$; $h_K=1$
788
1
-1:
-14/3
comment: $x^3-x^2-7x-3=0$; $S_3$; $h_K=1$
788
1
-3:
20351/3
comment: $x^3-x^2-7x-3=0$; $S_3$; $h_K=1$
788
1
-5:
-32441085854/63
comment: $x^3-x^2-7x-3=0$; $S_3$; $h_K=1$
837
1
-1:
-16/3
comment: $x^3-6x-1=0$; $S_3$; $h_K=1$
837
1
-3:
25228/3
comment: $x^3-6x-1=0$; $S_3$; $h_K=1$
837
1
-5:
-45216419296/63
comment: $x^3-6x-1=0$; $S_3$; $h_K=1$
892
1
-1:
-20/3
comment: $x^3-x^2-8x+10=0$; $S_3$; $h_K=1$
892
1
-3:
165466/15
comment: $x^3-x^2-8x+10=0$; $S_3$; $h_K=1$
892
1
-5:
-65080934780/63
comment: $x^3-x^2-8x+10=0$; $S_3$; $h_K=1$
940
1
-1:
-22/3
comment: $x^3-7x-4=0$; $S_3$; $h_K=1$
940
1
-3:
199013/15
comment: $x^3-7x-4=0$; $S_3$; $h_K=1$
940
1
-5:
-86830900042/63
comment: $x^3-7x-4=0$; $S_3$; $h_K=1$
961
1
-1:
-28/3
comment: $x^3-x^2-10x+8=0$; $C_3$, conductor $31$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{31})$
961
1
-3:
228614/15
comment: $x^3-x^2-10x+8=0$; $C_3$, conductor $31$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{31})$
961
1
-5:
-99594088828/63
comment: $x^3-x^2-10x+8=0$; $C_3$, conductor $31$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{31})$
985
1
-1:
-14/3
comment: $x^3-x^2-6x+1=0$; $S_3$; $h_K=1$
985
1
-3:
206173/15
comment: $x^3-x^2-6x+1=0$; $S_3$; $h_K=1$
985
1
-5:
-108817102514/63
comment: $x^3-x^2-6x+1=0$; $S_3$; $h_K=1$
993
1
-1:
-17/3
comment: $x^3-x^2-6x+3=0$; $S_3$; $h_K=1$
993
1
-3:
86827/6
comment: $x^3-x^2-6x+3=0$; $S_3$; $h_K=1$
993
1
-5:
-16296150401/9
comment: $x^3-x^2-6x+3=0$; $S_3$; $h_K=1$
1016
1
-1:
-26/3
comment: $x^3-x^2-6x+2=0$; $S_3$; $h_K=1$
1016
1
-3:
263719/15
comment: $x^3-x^2-6x+2=0$; $S_3$; $h_K=1$
1016
1
-5:
-133330020566/63
comment: $x^3-x^2-6x+2=0$; $S_3$; $h_K=1$
1076
1
-1:
-22/3
comment: $x^3-8x-6=0$; $S_3$; $h_K=1$
1076
1
-3:
60475/3
comment: $x^3-8x-6=0$; $S_3$; $h_K=1$
1076
1
-5:
-25706708746/9
comment: $x^3-8x-6=0$; $S_3$; $h_K=1$
1101
1
-1:
-26/3
comment: $x^3-x^2-9x+12=0$; $S_3$; $h_K=1$
1101
1
-3:
332857/15
comment: $x^3-x^2-9x+12=0$; $S_3$; $h_K=1$
1101
1
-5:
-204505351346/63
comment: $x^3-x^2-9x+12=0$; $S_3$; $h_K=1$
1129
1
-1:
-22/3
comment: $x^3-7x-3=0$; $S_3$; $h_K=1$
1129
1
-3:
343757/15
comment: $x^3-7x-3=0$; $S_3$; $h_K=1$
1129
1
-5:
-231389577082/63
comment: $x^3-7x-3=0$; $S_3$; $h_K=1$
1229
1
-1:
-28/3
comment: $x^3-x^2-7x+6=0$; $S_3$; $h_K=1$
1229
1
-3:
96646/3
comment: $x^3-x^2-7x+6=0$; $S_3$; $h_K=1$
1229
1
-5:
-373951086028/63
comment: $x^3-x^2-7x+6=0$; $S_3$; $h_K=1$
1257
1
-1:
-8
comment: $x^3-x^2-8x+9=0$; $S_3$; $h_K=1$
1257
1
-3:
165072/5
comment: $x^3-x^2-8x+9=0$; $S_3$; $h_K=1$
1257
1
-5:
-139053348808/21
comment: $x^3-x^2-8x+9=0$; $S_3$; $h_K=1$
1300
1
-1:
-9
comment: $x^3-10x-10=0$; $S_3$; $h_K=1$
1300
1
-3:
386521/10
comment: $x^3-10x-10=0$; $S_3$; $h_K=1$
1300
1
-5:
-169502156969/21
comment: $x^3-10x-10=0$; $S_3$; $h_K=1$
1304
1
-1:
-38/3
comment: $x^3-11x-2=0$; $S_3$; $h_K=1$
1304
1
-3:
631693/15
comment: $x^3-11x-2=0$; $S_3$; $h_K=1$
1304
1
-5:
-526065092858/63
comment: $x^3-11x-2=0$; $S_3$; $h_K=1$
1345
1
-1:
-23/3
comment: $x^3-7x-1=0$; $S_3$; $h_K=1$
1345
1
-3:
1227713/30
comment: $x^3-7x-1=0$; $S_3$; $h_K=1$
1345
1
-5:
-86234211239/9
comment: $x^3-7x-1=0$; $S_3$; $h_K=1$
1369
1
-1:
-7
comment: $x^3-x^2-12x-11=0$; $C_3$, conductor $37$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{37})$
1369
1
-3:
433513/10
comment: $x^3-x^2-12x-11=0$; $C_3$, conductor $37$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{37})$
1369
1
-5:
-221736281617/21
comment: $x^3-x^2-12x-11=0$; $C_3$, conductor $37$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{37})$
1373
1
-1:
-34/3
comment: $x^3-8x-5=0$; $S_3$; $h_K=1$
1373
1
-3:
142645/3
comment: $x^3-8x-5=0$; $S_3$; $h_K=1$
1373
1
-5:
-98264552422/9
comment: $x^3-8x-5=0$; $S_3$; $h_K=1$
1384
1
-1:
-38/3
comment: $x^3-x^2-10x+14=0$; $S_3$; $h_K=1$
1384
1
-3:
768781/15
comment: $x^3-x^2-10x+14=0$; $S_3$; $h_K=1$
1384
1
-5:
-728900966138/63
comment: $x^3-x^2-10x+14=0$; $S_3$; $h_K=1$
1396
1
-1:
-32/3
comment: $x^3-x^2-7x+5=0$; $S_3$; $h_K=1$
1396
1
-3:
746308/15
comment: $x^3-x^2-7x+5=0$; $S_3$; $h_K=1$
1396
1
-5:
-752546403632/63
comment: $x^3-x^2-7x+5=0$; $S_3$; $h_K=1$
1425
1
-1:
-29/3
comment: $x^3-x^2-8x-3=0$; $S_3$; $h_K=1$
1425
1
-3:
1536419/30
comment: $x^3-x^2-8x-3=0$; $S_3$; $h_K=1$
1425
1
-5:
-831649869899/63
comment: $x^3-x^2-8x-3=0$; $S_3$; $h_K=1$
1436
1
-1:
-44/3
comment: $x^3-11x-12=0$; $S_3$; $h_K=1$
1436
1
-3:
885286/15
comment: $x^3-11x-12=0$; $S_3$; $h_K=1$
1436
1
-5:
-894050845124/63
comment: $x^3-11x-12=0$; $S_3$; $h_K=1$
1489
1
-1:
-8
comment: $x^3-x^2-10x-7=0$; $S_3$; $h_K=1$
1489
1
-3:
290944/5
comment: $x^3-x^2-10x-7=0$; $S_3$; $h_K=1$
1489
1
-5:
-351999233288/21
comment: $x^3-x^2-10x-7=0$; $S_3$; $h_K=1$
1492
1
-1:
-34/3
comment: $x^3-x^2-9x-5=0$; $S_3$; $h_K=1$
1492
1
-3:
187933/3
comment: $x^3-x^2-9x-5=0$; $S_3$; $h_K=1$
1492
1
-5:
-1084785143074/63
comment: $x^3-x^2-9x-5=0$; $S_3$; $h_K=1$
1509
1
-1:
-14
comment: $x^3-x^2-7x+4=0$; $S_3$; $h_K=1$
1509
1
-3:
334443/5
comment: $x^3-x^2-7x+4=0$; $S_3$; $h_K=1$
1509
1
-5:
-55137388282/3
comment: $x^3-x^2-7x+4=0$; $S_3$; $h_K=1$
1524
1
-1:
-41/3
comment: $x^3-x^2-7x+1=0$; $S_3$; $h_K=1$
1524
1
-3:
2070137/30
comment: $x^3-x^2-7x+1=0$; $S_3$; $h_K=1$
1524
1
-5:
-1222333368521/63
comment: $x^3-x^2-7x+1=0$; $S_3$; $h_K=1$
1556
1
-1:
-38/3
comment: $x^3-x^2-9x+11=0$; $S_3$; $h_K=1$
1556
1
-3:
1099279/15
comment: $x^3-x^2-9x+11=0$; $S_3$; $h_K=1$
1556
1
-5:
-1368442609238/63
comment: $x^3-x^2-9x+11=0$; $S_3$; $h_K=1$
1573
1
-1:
-38/3
comment: $x^3-x^2-7x+2=0$; $S_3$; $h_K=1$
1573
1
-3:
1134223/15
comment: $x^3-x^2-7x+2=0$; $S_3$; $h_K=1$
1573
1
-5:
-207311775074/9
comment: $x^3-x^2-7x+2=0$; $S_3$; $h_K=1$
1593
1
-1:
-32/3
comment: $x^3-9x-7=0$; $S_3$; $h_K=1$
1593
1
-3:
1122004/15
comment: $x^3-9x-7=0$; $S_3$; $h_K=1$
1593
1
-5:
-1533063691952/63
comment: $x^3-9x-7=0$; $S_3$; $h_K=1$
1620
1
-1:
-43/3
comment: $x^3-12x-14=0$; $S_3$; $h_K=1$
1620
1
-3:
2539963/30
comment: $x^3-12x-14=0$; $S_3$; $h_K=1$
1620
1
-5:
-244042885789/9
comment: $x^3-12x-14=0$; $S_3$; $h_K=1$
1708
1
-1:
-18
comment: $x^3-x^2-8x-2=0$; $S_3$; $h_K=1$
1708
1
-3:
536039/5
comment: $x^3-x^2-8x-2=0$; $S_3$; $h_K=1$
1708
1
-5:
-772703316878/21
comment: $x^3-x^2-8x-2=0$; $S_3$; $h_K=1$
1765
1
-1:
-46/3
comment: $x^3-x^2-11x+16=0$; $S_3$; $h_K=1$
1765
1
-3:
1699247/15
comment: $x^3-x^2-11x+16=0$; $S_3$; $h_K=1$
1765
1
-5:
-2734222785286/63
comment: $x^3-x^2-11x+16=0$; $S_3$; $h_K=1$
1772
1
-1:
-62/3
comment: $x^3-x^2-12x+8=0$; $S_3$; $h_K=1$
1772
1
-3:
1850617/15
comment: $x^3-x^2-12x+8=0$; $S_3$; $h_K=1$
1772
1
-5:
-2841781644482/63
comment: $x^3-x^2-12x+8=0$; $S_3$; $h_K=1$
1825
1
-1:
-34/3
comment: $x^3-x^2-8x+7=0$; $S_3$; $h_K=1$
1825
1
-3:
1782119/15
comment: $x^3-x^2-8x+7=0$; $S_3$; $h_K=1$
1825
1
-5:
-461974379842/9
comment: $x^3-x^2-8x+7=0$; $S_3$; $h_K=1$
1849
1
-1:
-76/3
comment: $x^3-x^2-14x-8=0$; $C_3$, conductor $43$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{43})$
1849
1
-3:
2259086/15
comment: $x^3-x^2-14x-8=0$; $C_3$, conductor $43$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{43})$
1849
1
-5:
-3642576372076/63
comment: $x^3-x^2-14x-8=0$; $C_3$, conductor $43$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{43})$
1901
1
-1:
-18
comment: $x^3-x^2-9x-4=0$; $S_3$; $h_K=1$
1901
1
-3:
741373/5
comment: $x^3-x^2-9x-4=0$; $S_3$; $h_K=1$
1901
1
-5:
-1372649819818/21
comment: $x^3-x^2-9x-4=0$; $S_3$; $h_K=1$
1929
1
-1:
-46/3
comment: $x^3-x^2-10x+13=0$; $S_3$; $h_K=1$
1929
1
-3:
2216153/15
comment: $x^3-x^2-10x+13=0$; $S_3$; $h_K=1$
1929
1
-5:
-4398139787266/63
comment: $x^3-x^2-10x+13=0$; $S_3$; $h_K=1$
1937
1
-1:
-14
comment: $x^3-x^2-8x-1=0$; $S_3$; $h_K=1$
1937
1
-3:
740953/5
comment: $x^3-x^2-8x-1=0$; $S_3$; $h_K=1$
1937
1
-5:
-1497816493154/21
comment: $x^3-x^2-8x-1=0$; $S_3$; $h_K=1$
1940
1
-1:
-56/3
comment: $x^3-8x-2=0$; $S_3$; $h_K=1$
1940
1
-3:
2386072/15
comment: $x^3-8x-2=0$; $S_3$; $h_K=1$
1940
1
-5:
-4604030639336/63
comment: $x^3-8x-2=0$; $S_3$; $h_K=1$
1944
1
-1:
-70/3
comment: $x^3-9x-6=0$; $S_3$; $h_K=1$
1944
1
-3:
2556221/15
comment: $x^3-9x-6=0$; $S_3$; $h_K=1$
1944
1
-5:
-4729825529050/63
comment: $x^3-9x-6=0$; $S_3$; $h_K=1$
1957
1
-1:
-52/3
comment: $x^3-x^2-9x+10=0$; $S_3$; $h_K=2$
1957
1
-3:
2435222/15
comment: $x^3-x^2-9x+10=0$; $S_3$; $h_K=2$
1957
1
-5:
-4824583850932/63
comment: $x^3-x^2-9x+10=0$; $S_3$; $h_K=2$
2021
1
-1:
-20
comment: $x^3-8x-1=0$; $S_3$; $h_K=1$
2021
1
-3:
918874/5
comment: $x^3-8x-1=0$; $S_3$; $h_K=1$
2021
1
-5:
-1922102647460/21
comment: $x^3-8x-1=0$; $S_3$; $h_K=1$
2024
1
-1:
-74/3
comment: $x^3-x^2-10x-6=0$; $S_3$; $h_K=1$
2024
1
-3:
588611/3
comment: $x^3-x^2-10x-6=0$; $S_3$; $h_K=1$
2024
1
-5:
-843475980842/9
comment: $x^3-x^2-10x-6=0$; $S_3$; $h_K=1$
2057
1
-1:
-46/3
comment: $x^3-11x-11=0$; $S_3$; $h_K=1$
2057
1
-3:
2743373/15
comment: $x^3-11x-11=0$; $S_3$; $h_K=1$
2057
1
-5:
-6253967874706/63
comment: $x^3-11x-11=0$; $S_3$; $h_K=1$
2089
1
-1:
-92/3
comment: $x^3-13x-4=0$; $S_3$; $h_K=1$
2089
1
-3:
3463894/15
comment: $x^3-13x-4=0$; $S_3$; $h_K=1$
2089
1
-5:
-7127222029052/63
comment: $x^3-13x-4=0$; $S_3$; $h_K=1$
2101
1
-1:
-56/3
comment: $x^3-x^2-11x-8=0$; $S_3$; $h_K=1$
2101
1
-3:
623480/3
comment: $x^3-x^2-11x-8=0$; $S_3$; $h_K=1$
2101
1
-5:
-7128957661256/63
comment: $x^3-x^2-11x-8=0$; $S_3$; $h_K=1$
2177
1
-1:
-17
comment: $x^3-x^2-8x+5=0$; $S_3$; $h_K=1$
2177
1
-3:
2231327/10
comment: $x^3-x^2-8x+5=0$; $S_3$; $h_K=1$
2177
1
-5:
-406792419761/3
comment: $x^3-x^2-8x+5=0$; $S_3$; $h_K=1$
2213
1
-1:
-70/3
comment: $x^3-x^2-13x-12=0$; $S_3$; $h_K=1$
2213
1
-3:
3791711/15
comment: $x^3-x^2-13x-12=0$; $S_3$; $h_K=1$
2213
1
-5:
-9499591224910/63
comment: $x^3-x^2-13x-12=0$; $S_3$; $h_K=1$
2228
1
-1:
-67/3
comment: $x^3-x^2-13x+9=0$; $S_3$; $h_K=1$
2228
1
-3:
7734883/30
comment: $x^3-x^2-13x+9=0$; $S_3$; $h_K=1$
2228
1
-5:
-1408110330661/9
comment: $x^3-x^2-13x+9=0$; $S_3$; $h_K=1$
2233
1
-1:
-47/3
comment: $x^3-x^2-8x+1=0$; $S_3$; $h_K=1$
2233
1
-3:
7225241/30
comment: $x^3-x^2-8x+1=0$; $S_3$; $h_K=1$
2233
1
-5:
-9809841235097/63
comment: $x^3-x^2-8x+1=0$; $S_3$; $h_K=1$
2241
1
-1:
-55/3
comment: $x^3-9x-5=0$; $S_3$; $h_K=1$
2241
1
-3:
7425001/30
comment: $x^3-9x-5=0$; $S_3$; $h_K=1$
2241
1
-5:
-10019554553665/63
comment: $x^3-9x-5=0$; $S_3$; $h_K=1$
2292
1
-1:
-26
comment: $x^3-x^2-13x+1=0$; $S_3$; $h_K=1$
2292
1
-3:
1441497/5
comment: $x^3-x^2-13x+1=0$; $S_3$; $h_K=1$
2292
1
-5:
-3844669433866/21
comment: $x^3-x^2-13x+1=0$; $S_3$; $h_K=1$
2296
1
-1:
-82/3
comment: $x^3-x^2-14x-14=0$; $S_3$; $h_K=1$
2296
1
-3:
4522259/15
comment: $x^3-x^2-14x-14=0$; $S_3$; $h_K=1$
2296
1
-5:
-11796919174462/63
comment: $x^3-x^2-14x-14=0$; $S_3$; $h_K=1$
2300
1
-1:
-92/3
comment: $x^3-x^2-8x+2=0$; $S_3$; $h_K=1$
2300
1
-3:
4610494/15
comment: $x^3-x^2-8x+2=0$; $S_3$; $h_K=1$
2300
1
-5:
-11927333915732/63
comment: $x^3-x^2-8x+2=0$; $S_3$; $h_K=1$
2349
1
-1:
-74/3
comment: $x^3-12x-13=0$; $S_3$; $h_K=1$
2349
1
-3:
4664197/15
comment: $x^3-12x-13=0$; $S_3$; $h_K=1$
2349
1
-5:
-13186677848114/63
comment: $x^3-12x-13=0$; $S_3$; $h_K=1$
2429
1
-1:
-80/3
comment: $x^3-x^2-14x-4=0$; $S_3$; $h_K=1$
2429
1
-3:
5248588/15
comment: $x^3-x^2-14x-4=0$; $S_3$; $h_K=1$
2429
1
-5:
-2264852089760/9
comment: $x^3-x^2-14x-4=0$; $S_3$; $h_K=1$
2505
1
-1:
-71/3
comment: $x^3-x^2-10x-5=0$; $S_3$; $h_K=1$
2505
1
-3:
11087633/30
comment: $x^3-x^2-10x-5=0$; $S_3$; $h_K=1$
2505
1
-5:
-18511118662241/63
comment: $x^3-x^2-10x-5=0$; $S_3$; $h_K=1$
2557
1
-1:
-26
comment: $x^3-x^2-9x-2=0$; $S_3$; $h_K=1$
2557
1
-3:
2069849/5
comment: $x^3-x^2-9x-2=0$; $S_3$; $h_K=1$
2557
1
-5:
-2333389111142/7
comment: $x^3-x^2-9x-2=0$; $S_3$; $h_K=1$
2589
1
-1:
-32
comment: $x^3-x^2-14x+12=0$; $S_3$; $h_K=1$
2589
1
-3:
2213292/5
comment: $x^3-x^2-14x+12=0$; $S_3$; $h_K=1$
2589
1
-5:
-7515938740432/21
comment: $x^3-x^2-14x+12=0$; $S_3$; $h_K=1$
2597
1
-1:
-30
comment: $x^3-x^2-9x+8=0$; $S_3$; $h_K=3$
2597
1
-3:
2213503/5
comment: $x^3-x^2-9x+8=0$; $S_3$; $h_K=3$
2597
1
-5:
-7634578180630/21
comment: $x^3-x^2-9x+8=0$; $S_3$; $h_K=3$
2636
1
-1:
-110/3
comment: $x^3-14x-4=0$; $S_3$; $h_K=1$
2636
1
-3:
7419169/15
comment: $x^3-14x-4=0$; $S_3$; $h_K=1$
2636
1
-5:
-25247158360850/63
comment: $x^3-14x-4=0$; $S_3$; $h_K=1$
2673
1
-1:
-68/3
comment: $x^3-9x-3=0$; $S_3$; $h_K=1$
2673
1
-3:
6861718/15
comment: $x^3-9x-3=0$; $S_3$; $h_K=1$
2673
1
-5:
-26415619856828/63
comment: $x^3-9x-3=0$; $S_3$; $h_K=1$
2677
1
-1:
-28
comment: $x^3-10x-7=0$; $S_3$; $h_K=1$
2677
1
-3:
486170
comment: $x^3-10x-7=0$; $S_3$; $h_K=1$
2677
1
-5:
-3002874730036/7
comment: $x^3-10x-7=0$; $S_3$; $h_K=1$
2700
1
-1:
-118/3
comment: $x^3-15x-20=0$; $S_3$; $h_K=1$
2700
1
-3:
8083301/15
comment: $x^3-15x-20=0$; $S_3$; $h_K=1$
2700
1
-5:
-28810039979818/63
comment: $x^3-15x-20=0$; $S_3$; $h_K=1$
2708
1
-1:
-91/3
comment: $x^3-x^2-11x-7=0$; $S_3$; $h_K=1$
2708
1
-3:
15317419/30
comment: $x^3-x^2-11x-7=0$; $S_3$; $h_K=1$
2708
1
-5:
-28825206727291/63
comment: $x^3-x^2-11x-7=0$; $S_3$; $h_K=1$
2713
1
-1:
-85/3
comment: $x^3-13x-15=0$; $S_3$; $h_K=1$
2713
1
-3:
2962367/6
comment: $x^3-13x-15=0$; $S_3$; $h_K=1$
2713
1
-5:
-28742774430835/63
comment: $x^3-13x-15=0$; $S_3$; $h_K=1$
2777
1
-1:
-24
comment: $x^3-x^2-14x+23=0$; $S_3$; $h_K=2$
2777
1
-3:
2614228/5
comment: $x^3-x^2-14x+23=0$; $S_3$; $h_K=2$
2777
1
-5:
-10862079137704/21
comment: $x^3-x^2-14x+23=0$; $S_3$; $h_K=2$
2804
1
-1:
-101/3
comment: $x^3-x^2-9x-1=0$; $S_3$; $h_K=1$
2804
1
-3:
3470305/6
comment: $x^3-x^2-9x-1=0$; $S_3$; $h_K=1$
2804
1
-5:
-34916903412581/63
comment: $x^3-x^2-9x-1=0$; $S_3$; $h_K=1$
2808
1
-1:
-124/3
comment: $x^3-9x-2=0$; $S_3$; $h_K=1$
2808
1
-3:
9269474/15
comment: $x^3-9x-2=0$; $S_3$; $h_K=1$
2808
1
-5:
-35745692598724/63
comment: $x^3-9x-2=0$; $S_3$; $h_K=1$
2836
1
-1:
-86/3
comment: $x^3-x^2-9x+7=0$; $S_3$; $h_K=1$
2836
1
-3:
8882887/15
comment: $x^3-x^2-9x+7=0$; $S_3$; $h_K=1$
2836
1
-5:
-5301191459018/9
comment: $x^3-x^2-9x+7=0$; $S_3$; $h_K=1$
2857
1
-1:
-65/3
comment: $x^3-x^2-10x+11=0$; $S_3$; $h_K=1$
2857
1
-3:
17101679/30
comment: $x^3-x^2-10x+11=0$; $S_3$; $h_K=1$
2857
1
-5:
-38042819668775/63
comment: $x^3-x^2-10x+11=0$; $S_3$; $h_K=1$
2917
1
-1:
-32
comment: $x^3-x^2-13x+20=0$; $S_3$; $h_K=1$
2917
1
-3:
3283192/5
comment: $x^3-x^2-13x+20=0$; $S_3$; $h_K=1$
2917
1
-5:
-14445849065792/21
comment: $x^3-x^2-13x+20=0$; $S_3$; $h_K=1$
2920
1
-1:
-122/3
comment: $x^3-x^2-15x-5=0$; $S_3$; $h_K=1$
2920
1
-3:
10514827/15
comment: $x^3-x^2-15x-5=0$; $S_3$; $h_K=1$
2920
1
-5:
-44266677385862/63
comment: $x^3-x^2-15x-5=0$; $S_3$; $h_K=1$
2941
1
-1:
-34
comment: $x^3-x^2-14x+4=0$; $S_3$; $h_K=1$
2941
1
-3:
3387821/5
comment: $x^3-x^2-14x+4=0$; $S_3$; $h_K=1$
2941
1
-5:
-15113565363994/21
comment: $x^3-x^2-14x+4=0$; $S_3$; $h_K=1$
2981
1
-1:
-36
comment: $x^3-x^2-11x+14=0$; $S_3$; $h_K=1$
2981
1
-3:
3581458/5
comment: $x^3-x^2-11x+14=0$; $S_3$; $h_K=1$
2981
1
-5:
-16298697270196/21
comment: $x^3-x^2-11x+14=0$; $S_3$; $h_K=1$
2993
1
-1:
-79/3
comment: $x^3-x^2-12x+17=0$; $S_3$; $h_K=1$
2993
1
-3:
20378209/30
comment: $x^3-x^2-12x+17=0$; $S_3$; $h_K=1$
2993
1
-5:
-49198297404169/63
comment: $x^3-x^2-12x+17=0$; $S_3$; $h_K=1$
3021
1
-1:
-40
comment: $x^3-x^2-9x+6=0$; $S_3$; $h_K=1$
3021
1
-3:
759432
comment: $x^3-x^2-9x+6=0$; $S_3$; $h_K=1$
3021
1
-5:
-17562345423320/21
comment: $x^3-x^2-9x+6=0$; $S_3$; $h_K=1$
3028
1
-1:
-130/3
comment: $x^3-10x-6=0$; $S_3$; $h_K=1$
3028
1
-3:
11599409/15
comment: $x^3-10x-6=0$; $S_3$; $h_K=1$
3028
1
-5:
-53425624488130/63
comment: $x^3-10x-6=0$; $S_3$; $h_K=1$
3124
1
-1:
-133/3
comment: $x^3-16x-12=0$; $S_3$; $h_K=1$
3124
1
-3:
25838941/30
comment: $x^3-16x-12=0$; $S_3$; $h_K=1$
3124
1
-5:
-9061031039779/9
comment: $x^3-16x-12=0$; $S_3$; $h_K=1$
3132
1
-1:
-148/3
comment: $x^3-15x-6=0$; $S_3$; $h_K=1$
3132
1
-3:
13589834/15
comment: $x^3-15x-6=0$; $S_3$; $h_K=1$
3132
1
-5:
-65172296055868/63
comment: $x^3-15x-6=0$; $S_3$; $h_K=1$
3137
1
-1:
-85/3
comment: $x^3-11x-9=0$; $S_3$; $h_K=1$
3137
1
-3:
24022147/30
comment: $x^3-11x-9=0$; $S_3$; $h_K=1$
3137
1
-5:
-63707884432195/63
comment: $x^3-11x-9=0$; $S_3$; $h_K=1$
3144
1
-1:
-158/3
comment: $x^3-x^2-16x-8=0$; $S_3$; $h_K=1$
3144
1
-3:
13916809/15
comment: $x^3-x^2-16x-8=0$; $S_3$; $h_K=1$
3144
1
-5:
-66644124311138/63
comment: $x^3-x^2-16x-8=0$; $S_3$; $h_K=1$
3173
1
-1:
-40
comment: $x^3-14x-17=0$; $S_3$; $h_K=1$
3173
1
-3:
4460692/5
comment: $x^3-14x-17=0$; $S_3$; $h_K=1$
3173
1
-5:
-7658607321240/7
comment: $x^3-14x-17=0$; $S_3$; $h_K=1$
3221
1
-1:
-124/3
comment: $x^3-x^2-9x+2=0$; $S_3$; $h_K=1$
3221
1
-3:
14098766/15
comment: $x^3-x^2-9x+2=0$; $S_3$; $h_K=1$
3221
1
-5:
-74858159972524/63
comment: $x^3-x^2-9x+2=0$; $S_3$; $h_K=1$
3229
1
-1:
-118/3
comment: $x^3-x^2-9x+4=0$; $S_3$; $h_K=1$
3229
1
-3:
14095487/15
comment: $x^3-x^2-9x+4=0$; $S_3$; $h_K=1$
3229
1
-5:
-75794918817118/63
comment: $x^3-x^2-9x+4=0$; $S_3$; $h_K=1$
3252
1
-1:
-131/3
comment: $x^3-x^2-9x+3=0$; $S_3$; $h_K=1$
3252
1
-3:
29417411/30
comment: $x^3-x^2-9x+3=0$; $S_3$; $h_K=1$
3252
1
-5:
-78999525360131/63
comment: $x^3-x^2-9x+3=0$; $S_3$; $h_K=1$
3261
1
-1:
-136/3
comment: $x^3-x^2-11x-6=0$; $S_3$; $h_K=1$
3261
1
-3:
14891168/15
comment: $x^3-x^2-11x-6=0$; $S_3$; $h_K=1$
3261
1
-5:
-80224609438936/63
comment: $x^3-x^2-11x-6=0$; $S_3$; $h_K=1$
3281
1
-1:
-89/3
comment: $x^3-x^2-14x-13=0$; $S_3$; $h_K=1$
3281
1
-3:
28074311/30
comment: $x^3-x^2-14x-13=0$; $S_3$; $h_K=1$
3281
1
-5:
-11648642722457/9
comment: $x^3-x^2-14x-13=0$; $S_3$; $h_K=1$
3305
1
-1:
-32
comment: $x^3-x^2-10x-3=0$; $S_3$; $h_K=1$
3305
1
-3:
4813588/5
comment: $x^3-x^2-10x-3=0$; $S_3$; $h_K=1$
3305
1
-5:
-28295212029872/21
comment: $x^3-x^2-10x-3=0$; $S_3$; $h_K=1$
3316
1
-1:
-106/3
comment: $x^3-16x-22=0$; $S_3$; $h_K=1$
3316
1
-3:
15343949/15
comment: $x^3-16x-22=0$; $S_3$; $h_K=1$
3316
1
-5:
-87692083135306/63
comment: $x^3-16x-22=0$; $S_3$; $h_K=1$
3325
1
-1:
-118/3
comment: $x^3-10x-5=0$; $S_3$; $h_K=1$
3325
1
-3:
15579431/15
comment: $x^3-10x-5=0$; $S_3$; $h_K=1$
3325
1
-5:
-89037182680318/63
comment: $x^3-10x-5=0$; $S_3$; $h_K=1$
3356
1
-1:
-160/3
comment: $x^3-x^2-15x-1=0$; $S_3$; $h_K=1$
3356
1
-3:
17281424/15
comment: $x^3-x^2-15x-1=0$; $S_3$; $h_K=1$
3356
1
-5:
-13612533728800/9
comment: $x^3-x^2-15x-1=0$; $S_3$; $h_K=1$
3368
1
-1:
-164/3
comment: $x^3-x^2-15x+11=0$; $S_3$; $h_K=1$
3368
1
-3:
17519734/15
comment: $x^3-x^2-15x+11=0$; $S_3$; $h_K=1$
3368
1
-5:
-97182252932924/63
comment: $x^3-x^2-15x+11=0$; $S_3$; $h_K=1$
3496
1
-1:
-152/3
comment: $x^3-13x-14=0$; $S_3$; $h_K=1$
3496
1
-3:
3938288/3
comment: $x^3-13x-14=0$; $S_3$; $h_K=1$
3496
1
-5:
-119141298931352/63
comment: $x^3-13x-14=0$; $S_3$; $h_K=1$
3508
1
-1:
-39
comment: $x^3-x^2-11x+13=0$; $S_3$; $h_K=1$
3508
1
-3:
12470407/10
comment: $x^3-x^2-11x+13=0$; $S_3$; $h_K=1$
3508
1
-5:
-39839208551399/21
comment: $x^3-x^2-11x+13=0$; $S_3$; $h_K=1$
3540
1
-1:
-155/3
comment: $x^3-x^2-15x-15=0$; $S_3$; $h_K=1$
3540
1
-3:
39665387/30
comment: $x^3-x^2-15x-15=0$; $S_3$; $h_K=1$
3540
1
-5:
-125992761568955/63
comment: $x^3-x^2-15x-15=0$; $S_3$; $h_K=1$
3569
1
-1:
-101/3
comment: $x^3-x^2-10x+9=0$; $S_3$; $h_K=1$
3569
1
-3:
37687859/30
comment: $x^3-x^2-10x+9=0$; $S_3$; $h_K=1$
3569
1
-5:
-18503210043893/9
comment: $x^3-x^2-10x+9=0$; $S_3$; $h_K=1$
3576
1
-1:
-194/3
comment: $x^3-x^2-15x+3=0$; $S_3$; $h_K=1$
3576
1
-3:
21847627/15
comment: $x^3-x^2-15x+3=0$; $S_3$; $h_K=1$
3576
1
-5:
-135300004214414/63
comment: $x^3-x^2-15x+3=0$; $S_3$; $h_K=1$
3580
1
-1:
-56
comment: $x^3-x^2-15x+7=0$; $S_3$; $h_K=1$
3580
1
-3:
7154876/5
comment: $x^3-x^2-15x+7=0$; $S_3$; $h_K=1$
3580
1
-5:
-15086518690232/7
comment: $x^3-x^2-15x+7=0$; $S_3$; $h_K=1$
3592
1
-1:
-54
comment: $x^3-x^2-18x+34=0$; $S_3$; $h_K=1$
3592
1
-3:
7225541/5
comment: $x^3-x^2-18x+34=0$; $S_3$; $h_K=1$
3592
1
-5:
-46096969704554/21
comment: $x^3-x^2-18x+34=0$; $S_3$; $h_K=1$
3596
1
-1:
-178/3
comment: $x^3-11x-8=0$; $S_3$; $h_K=1$
3596
1
-3:
22008611/15
comment: $x^3-11x-8=0$; $S_3$; $h_K=1$
3596
1
-5:
-19903374076594/9
comment: $x^3-11x-8=0$; $S_3$; $h_K=1$
3604
1
-1:
-119/3
comment: $x^3-x^2-17x+31=0$; $S_3$; $h_K=1$
3604
1
-3:
41069039/30
comment: $x^3-x^2-17x+31=0$; $S_3$; $h_K=1$
3604
1
-5:
-138642495096359/63
comment: $x^3-x^2-17x+31=0$; $S_3$; $h_K=1$
3624
1
-1:
-66
comment: $x^3-x^2-10x-2=0$; $S_3$; $h_K=1$
3624
1
-3:
7630479/5
comment: $x^3-x^2-10x-2=0$; $S_3$; $h_K=1$
3624
1
-5:
-16177225539402/7
comment: $x^3-x^2-10x-2=0$; $S_3$; $h_K=1$
3721
1
-1:
-133/3
comment: $x^3-x^2-20x+9=0$; $C_3$, conductor $61$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{61})$
3721
1
-3:
44689099/30
comment: $x^3-x^2-20x+9=0$; $C_3$, conductor $61$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{61})$
3721
1
-5:
-163363743993283/63
comment: $x^3-x^2-20x+9=0$; $C_3$, conductor $61$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{61})$
3732
1
-1:
-54
comment: $x^3-x^2-13x+19=0$; $S_3$; $h_K=1$
3732
1
-3:
7938891/5
comment: $x^3-x^2-13x+19=0$; $S_3$; $h_K=1$
3732
1
-5:
-18716877293778/7
comment: $x^3-x^2-13x+19=0$; $S_3$; $h_K=1$
3736
1
-1:
-56
comment: $x^3-x^2-14x+22=0$; $S_3$; $h_K=1$
3736
1
-3:
8281064/5
comment: $x^3-x^2-14x+22=0$; $S_3$; $h_K=1$
3736
1
-5:
-2724688203576
comment: $x^3-x^2-14x+22=0$; $S_3$; $h_K=1$
3753
1
-1:
-110/3
comment: $x^3-15x-19=0$; $S_3$; $h_K=1$
3753
1
-3:
22492621/15
comment: $x^3-15x-19=0$; $S_3$; $h_K=1$
3753
1
-5:
-170782991970770/63
comment: $x^3-15x-19=0$; $S_3$; $h_K=1$
3873
1
-1:
-131/3
comment: $x^3-x^2-16x-17=0$; $S_3$; $h_K=1$
3873
1
-3:
50858381/30
comment: $x^3-x^2-16x-17=0$; $S_3$; $h_K=1$
3873
1
-5:
-203338860822581/63
comment: $x^3-x^2-16x-17=0$; $S_3$; $h_K=1$
3877
1
-1:
-146/3
comment: $x^3-x^2-13x-10=0$; $S_3$; $h_K=1$
3877
1
-3:
26658481/15
comment: $x^3-x^2-13x-10=0$; $S_3$; $h_K=1$
3877
1
-5:
-207224234733626/63
comment: $x^3-x^2-13x-10=0$; $S_3$; $h_K=1$
3889
1
-1:
-106/3
comment: $x^3-x^2-10x-1=0$; $S_3$; $h_K=1$
3889
1
-3:
25154987/15
comment: $x^3-x^2-10x-1=0$; $S_3$; $h_K=1$
3889
1
-5:
-29631963017578/9
comment: $x^3-x^2-10x-1=0$; $S_3$; $h_K=1$
3892
1
-1:
-143/3
comment: $x^3-10x-2=0$; $S_3$; $h_K=1$
3892
1
-3:
53864951/30
comment: $x^3-10x-2=0$; $S_3$; $h_K=1$
3892
1
-5:
-211622895914783/63
comment: $x^3-10x-2=0$; $S_3$; $h_K=1$
3941
1
-1:
-166/3
comment: $x^3-17x-12=0$; $S_3$; $h_K=1$
3941
1
-3:
5710915/3
comment: $x^3-17x-12=0$; $S_3$; $h_K=1$
3941
1
-5:
-227050242293806/63
comment: $x^3-17x-12=0$; $S_3$; $h_K=1$
3957
1
-1:
-62
comment: $x^3-x^2-18x-12=0$; $S_3$; $h_K=1$
3957
1
-3:
9781311/5
comment: $x^3-x^2-18x-12=0$; $S_3$; $h_K=1$
3957
1
-5:
-77498725997302/21
comment: $x^3-x^2-18x-12=0$; $S_3$; $h_K=1$
3969
1
-1:
-133/3
comment: $x^3-21x-35=0$; $C_3$, conductor $63$; $h_K=3$; a cyclic cubic field of conductor $63$
3969
1
-3:
54922771/30
comment: $x^3-21x-35=0$; $C_3$, conductor $63$; $h_K=3$; a cyclic cubic field of conductor $63$
3969
1
-5:
-232363717924243/63
comment: $x^3-21x-35=0$; $C_3$, conductor $63$; $h_K=3$; a cyclic cubic field of conductor $63$
3969
2
-1:
-268/3
comment: $x^3-21x-28=0$; $C_3$, conductor $63$; $h_K=3$; a cyclic cubic field of conductor $63$
3969
2
-3:
33153134/15
comment: $x^3-21x-28=0$; $C_3$, conductor $63$; $h_K=3$; a cyclic cubic field of conductor $63$
3969
2
-5:
-243557105345068/63
comment: $x^3-21x-28=0$; $C_3$, conductor $63$; $h_K=3$; a cyclic cubic field of conductor $63$
3973
1
-1:
-50
comment: $x^3-10x-1=0$; $S_3$; $h_K=1$
3973
1
-3:
1935461
comment: $x^3-10x-1=0$; $S_3$; $h_K=1$
3973
1
-5:
-79020746993210/21
comment: $x^3-10x-1=0$; $S_3$; $h_K=1$
3981
1
-1:
-184/3
comment: $x^3-x^2-11x+12=0$; $S_3$; $h_K=2$
3981
1
-3:
5987128/3
comment: $x^3-x^2-11x+12=0$; $S_3$; $h_K=2$
3981
1
-5:
-240345143251144/63
comment: $x^3-x^2-11x+12=0$; $S_3$; $h_K=2$
3988
1
-1:
-48
comment: $x^3-16x-4=0$; $S_3$; $h_K=1$
3988
1
-3:
9771316/5
comment: $x^3-16x-4=0$; $S_3$; $h_K=1$
3988
1
-5:
-80656601806048/21
comment: $x^3-16x-4=0$; $S_3$; $h_K=1$
4001
1
-1:
-42
comment: $x^3-11x-7=0$; $S_3$; $h_K=1$
4001
1
-3:
9378599/5
comment: $x^3-11x-7=0$; $S_3$; $h_K=1$
4001
1
-5:
-80938034850902/21
comment: $x^3-11x-7=0$; $S_3$; $h_K=1$
4065
1
-1:
-49
comment: $x^3-x^2-10x+7=0$; $S_3$; $h_K=1$
4065
1
-3:
20118783/10
comment: $x^3-x^2-10x+7=0$; $S_3$; $h_K=1$
4065
1
-5:
-88450829640119/21
comment: $x^3-x^2-10x+7=0$; $S_3$; $h_K=1$
4104
1
-1:
-214/3
comment: $x^3-18x-16=0$; $S_3$; $h_K=1$
4104
1
-3:
34934837/15
comment: $x^3-18x-16=0$; $S_3$; $h_K=1$
4104
1
-5:
-288172712869834/63
comment: $x^3-18x-16=0$; $S_3$; $h_K=1$
4193
1
-1:
-137/3
comment: $x^3-x^2-12x-7=0$; $S_3$; $h_K=1$
4193
1
-3:
66378167/30
comment: $x^3-x^2-12x-7=0$; $S_3$; $h_K=1$
4193
1
-5:
-314237171467967/63
comment: $x^3-x^2-12x-7=0$; $S_3$; $h_K=1$
4212
1
-1:
-176/3
comment: $x^3-12x-10=0$; $S_3$; $h_K=3$
4212
1
-3:
35937964/15
comment: $x^3-12x-10=0$; $S_3$; $h_K=3$
4212
1
-5:
-327257898969536/63
comment: $x^3-12x-10=0$; $S_3$; $h_K=3$
4281
1
-1:
-54
comment: $x^3-x^2-12x+15=0$; $S_3$; $h_K=1$
4281
1
-3:
2414421
comment: $x^3-x^2-12x+15=0$; $S_3$; $h_K=1$
4281
1
-5:
-39198594448878/7
comment: $x^3-x^2-12x+15=0$; $S_3$; $h_K=1$
4312
1
-1:
-72
comment: $x^3-x^2-16x+8=0$; $S_3$; $h_K=3$
4312
1
-3:
13700528/5
comment: $x^3-x^2-16x+8=0$; $S_3$; $h_K=3$
4312
1
-5:
-125909757330632/21
comment: $x^3-x^2-16x+8=0$; $S_3$; $h_K=3$
4344
1
-1:
-86
comment: $x^3-x^2-16x+4=0$; $S_3$; $h_K=1$
4344
1
-3:
14383641/5
comment: $x^3-x^2-16x+4=0$; $S_3$; $h_K=1$
4344
1
-5:
-131486412659866/21
comment: $x^3-x^2-16x+4=0$; $S_3$; $h_K=1$
4345
1
-1:
-44
comment: $x^3-x^2-10x+5=0$; $S_3$; $h_K=1$
4345
1
-3:
2478242
comment: $x^3-x^2-10x+5=0$; $S_3$; $h_K=1$
4345
1
-5:
-127239195659444/21
comment: $x^3-x^2-10x+5=0$; $S_3$; $h_K=1$
4360
1
-1:
-226/3
comment: $x^3-x^2-10x+2=0$; $S_3$; $h_K=1$
4360
1
-3:
42789359/15
comment: $x^3-x^2-10x+2=0$; $S_3$; $h_K=1$
4360
1
-5:
-401467678752766/63
comment: $x^3-x^2-10x+2=0$; $S_3$; $h_K=1$
4364
1
-1:
-238/3
comment: $x^3-x^2-19x+27=0$; $S_3$; $h_K=1$
4364
1
-3:
8666509/3
comment: $x^3-x^2-19x+27=0$; $S_3$; $h_K=1$
4364
1
-5:
-404001063374098/63
comment: $x^3-x^2-19x+27=0$; $S_3$; $h_K=1$
4409
1
-1:
-47
comment: $x^3-x^2-10x+3=0$; $S_3$; $h_K=1$
4409
1
-3:
26329097/10
comment: $x^3-x^2-10x+3=0$; $S_3$; $h_K=1$
4409
1
-5:
-138066747264377/21
comment: $x^3-x^2-10x+3=0$; $S_3$; $h_K=1$
4481
1
-1:
-320/3
comment: $x^3-17x-8=0$; $S_3$; $h_K=1$
4481
1
-3:
50680936/15
comment: $x^3-17x-8=0$; $S_3$; $h_K=1$
4481
1
-5:
-474694388465120/63
comment: $x^3-17x-8=0$; $S_3$; $h_K=1$
4489
1
-1:
-193/3
comment: $x^3-x^2-22x-5=0$; $C_3$, conductor $67$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{67})$
4489
1
-3:
86578159/30
comment: $x^3-x^2-22x-5=0$; $C_3$, conductor $67$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{67})$
4489
1
-5:
-458598853953703/63
comment: $x^3-x^2-22x-5=0$; $C_3$, conductor $67$; $h_K=1$; the cubic subfield of $\mathbb{Q}(\zeta_{67})$
4493
1
-1:
-206/3
comment: $x^3-x^2-13x+18=0$; $S_3$; $h_K=1$
4493
1
-3:
45231727/15
comment: $x^3-x^2-13x+18=0$; $S_3$; $h_K=1$
4493
1
-5:
-466939890731366/63
comment: $x^3-x^2-13x+18=0$; $S_3$; $h_K=1$
4596
1
-1:
-74
comment: $x^3-x^2-11x-3=0$; $S_3$; $h_K=1$
4596
1
-3:
16446369/5
comment: $x^3-x^2-11x-3=0$; $S_3$; $h_K=1$
4596
1
-5:
-176501503133434/21
comment: $x^3-x^2-11x-3=0$; $S_3$; $h_K=1$
4597
1
-1:
-190/3
comment: $x^3-x^2-15x-14=0$; $S_3$; $h_K=1$
4597
1
-3:
9679003/3
comment: $x^3-x^2-15x-14=0$; $S_3$; $h_K=1$
4597
1
-5:
-528839319522070/63
comment: $x^3-x^2-15x-14=0$; $S_3$; $h_K=1$
4628
1
-1:
-67
comment: $x^3-x^2-13x-9=0$; $S_3$; $h_K=1$
4628
1
-3:
33305291/10
comment: $x^3-x^2-13x-9=0$; $S_3$; $h_K=1$
4628
1
-5:
-61041075977009/7
comment: $x^3-x^2-13x-9=0$; $S_3$; $h_K=1$
4641
1
-1:
-57
comment: $x^3-x^2-14x+21=0$; $S_3$; $h_K=1$
4641
1
-3:
6380955/2
comment: $x^3-x^2-14x+21=0$; $S_3$; $h_K=1$
4641
1
-5:
-61102581757829/7
comment: $x^3-x^2-14x+21=0$; $S_3$; $h_K=1$
4649
1
-1:
-166/3
comment: $x^3-11x-5=0$; $S_3$; $h_K=1$
4649
1
-3:
47746289/15
comment: $x^3-11x-5=0$; $S_3$; $h_K=1$
4649
1
-5:
-554493558816346/63
comment: $x^3-11x-5=0$; $S_3$; $h_K=1$
4684
1
-1:
-232/3
comment: $x^3-x^2-19x-13=0$; $S_3$; $h_K=1$
4684
1
-3:
54797672/15
comment: $x^3-x^2-19x-13=0$; $S_3$; $h_K=1$
4684
1
-5:
-595399831417672/63
comment: $x^3-x^2-19x-13=0$; $S_3$; $h_K=1$
4692
1
-1:
-227/3
comment: $x^3-x^2-17x-3=0$; $S_3$; $h_K=1$
4692
1
-3:
106126211/30
comment: $x^3-x^2-17x-3=0$; $S_3$; $h_K=1$
4692
1
-5:
-84755412176261/9
comment: $x^3-x^2-17x-3=0$; $S_3$; $h_K=1$
4729
1
-1:
-44
comment: $x^3-19x-29=0$; $S_3$; $h_K=1$
4729
1
-3:
16603246/5
comment: $x^3-19x-29=0$; $S_3$; $h_K=1$
4729
1
-5:
-28956795690332/3
comment: $x^3-19x-29=0$; $S_3$; $h_K=1$
4749
1
-1:
-80
comment: $x^3-x^2-15x+24=0$; $S_3$; $h_K=1$
4749
1
-3:
18501348/5
comment: $x^3-x^2-15x+24=0$; $S_3$; $h_K=1$
4749
1
-5:
-30197501684800/3
comment: $x^3-x^2-15x+24=0$; $S_3$; $h_K=1$
4764
1
-1:
-296/3
comment: $x^3-x^2-12x-6=0$; $S_3$; $h_K=1$
4764
1
-3:
59601988/15
comment: $x^3-x^2-12x-6=0$; $S_3$; $h_K=1$
4764
1
-5:
-93616962560648/9
comment: $x^3-x^2-12x-6=0$; $S_3$; $h_K=1$
4765
1
-1:
-208/3
comment: $x^3-x^2-11x+10=0$; $S_3$; $h_K=1$
4765
1
-3:
10992148/3
comment: $x^3-x^2-11x+10=0$; $S_3$; $h_K=1$
4765
1
-5:
-644299412350528/63
comment: $x^3-x^2-11x+10=0$; $S_3$; $h_K=1$
4825
1
-1:
-332/3
comment: $x^3-x^2-18x-8=0$; $S_3$; $h_K=1$
4825
1
-3:
64946278/15
comment: $x^3-x^2-18x-8=0$; $S_3$; $h_K=1$
4825
1
-5:
-712062845170892/63
comment: $x^3-x^2-18x-8=0$; $S_3$; $h_K=1$
4841
1
-1:
-53
comment: $x^3-x^2-16x+27=0$; $S_3$; $h_K=1$
4841
1
-3:
36503651/10
comment: $x^3-x^2-16x+27=0$; $S_3$; $h_K=1$
4841
1
-5:
-32980692153029/3
comment: $x^3-x^2-16x+27=0$; $S_3$; $h_K=1$
4844
1
-1:
-94
comment: $x^3-x^2-12x+14=0$; $S_3$; $h_K=1$
4844
1
-3:
20819533/5
comment: $x^3-x^2-12x+14=0$; $S_3$; $h_K=1$
4844
1
-5:
-79689329891238/7
comment: $x^3-x^2-12x+14=0$; $S_3$; $h_K=1$
4852
1
-1:
-67
comment: $x^3-x^2-17x+13=0$; $S_3$; $h_K=1$
4852
1
-3:
38853483/10
comment: $x^3-x^2-17x+13=0$; $S_3$; $h_K=1$
4852
1
-5:
-237169225427027/21
comment: $x^3-x^2-17x+13=0$; $S_3$; $h_K=1$
4853
1
-1:
-76
comment: $x^3-x^2-18x+20=0$; $S_3$; $h_K=1$
4853
1
-3:
3947714
comment: $x^3-x^2-18x+20=0$; $S_3$; $h_K=1$
4853
1
-5:
-79272258025412/7
comment: $x^3-x^2-18x+20=0$; $S_3$; $h_K=1$
4857
1
-1:
-62
comment: $x^3-x^2-18x-21=0$; $S_3$; $h_K=1$
4857
1
-3:
3745437
comment: $x^3-x^2-18x-21=0$; $S_3$; $h_K=1$
4857
1
-5:
-235432148626162/21
comment: $x^3-x^2-18x-21=0$; $S_3$; $h_K=1$
4860
1
-1:
-292/3
comment: $x^3-18x-12=0$; $S_3$; $h_K=1$
4860
1
-3:
12664810/3
comment: $x^3-18x-12=0$; $S_3$; $h_K=1$
4860
1
-5:
-730410181396972/63
comment: $x^3-18x-12=0$; $S_3$; $h_K=1$
4892
1
-1:
-292/3
comment: $x^3-11x-4=0$; $S_3$; $h_K=1$
4892
1
-3:
64730018/15
comment: $x^3-11x-4=0$; $S_3$; $h_K=1$
4892
1
-5:
-757215787088812/63
comment: $x^3-11x-4=0$; $S_3$; $h_K=1$
4933
1
-1:
-70
comment: $x^3-x^2-11x-2=0$; $S_3$; $h_K=1$
4933
1
-3:
4129611
comment: $x^3-x^2-11x-2=0$; $S_3$; $h_K=1$
4933
1
-5:
-259838561755310/21
comment: $x^3-x^2-11x-2=0$; $S_3$; $h_K=1$
Definition
Let $K$ be a totally real [8] cubic field [7] of discriminant $D$ and $\zeta_K(s)=\sum_{\mathfrak{a}}N(\mathfrak{a})^{-s}$ its Dedekind zeta function [6], summed over the nonzero ideals of $\mathcal{O}_K$. Listed is $\zeta_K(s)$ at negative odd integers $s$.
Parameters
$D$
—   discriminant of $K$ ($D$ the positive discriminant of a totally real cubic field)
$k$
—   index of $K$ among the totally real cubic fields of discriminant $D$ ($k\geq 1$)
$s$
—   argument ($s$ a negative odd integer)
Formulas
(1)
For $m=1,2,3$, $\zeta_K(1-2m)=8\sum_{\ell=1}^{r_m}b_\ell(6m)S_\ell^K(2m)$ [3], where $r_1=1$ and $r_2=r_3=2$. Here $S_\ell^K(2m)=\sum_\nu\sum_{\mathfrak{b}\mid(\nu)\mathfrak{D}_{K/\mathbb{Q}}} N(\mathfrak{b})^{2m-1}$, with $\nu$ ranging over the totally positive elements of the codifferent $\mathfrak{D}_{K/\mathbb{Q}}^{-1}$ whose trace is $\ell$, and $\mathfrak{b}$ ranging over the integral ideals dividing $(\nu)\mathfrak{D}_{K/\mathbb{Q}}$. The coefficients used here are $b_1(6)=-1/504$, $(b_1(12),b_2(12))=(1/8190,1/196560)$, and $(b_1(18),b_2(18))=(-22/3591,-1/86184)$.
(2)
$\zeta_K(2m)= \frac{(-1)^m2^{6m}m^3\pi^{6m}}{((2m)!)^3D^{2m-1/2}}\, \zeta_K(1-2m)$ for $m\geq 1$, from $\Lambda_K(s)=D^{s/2}(\pi^{-s/2}\Gamma(s/2))^3\zeta_K(s)= \Lambda_K(1-s)$. In particular $\zeta_K(2)=-8\pi^6\zeta_K(-1)/D^{3/2}$.
(3)
If $K$ is cyclic and $\chi$ is either primitive cubic Dirichlet character belonging to $K$, then $\zeta_K(1-2m)=\zeta(1-2m)L(1-2m,\chi)L(1-2m,\bar\chi) =-B_{2m}B_{2m,\chi}B_{2m,\bar\chi}/(2m)^3$. The factor $B_{2m}$ is in the table of Bernoulli numbers, and the generalized Bernoulli numbers $B_{2m,\chi}$ are the quantities indexed in the table of generalized Bernoulli numbers.
Comments
(4)
Non-isomorphic cubic fields can share a discriminant, so the discriminant $D$ is not always enough to identify $K$. The fields of discriminant $D$ are numbered $k=1,2,\ldots$ in lexicographic order of the coefficient vectors $(a_2,a_1,a_0)$ of their reduced defining polynomials $x^3+a_2x^2+a_1x+a_0$, the polynomial PARI's polredabs returns for the field and the LMFDB displays as its defining polynomial. This is the index used by the table of regulators of cubic fields and the table of residues of Dedekind zeta functions of cubic fields. In the range computed here, $D=3969$ is the only discriminant carrying two fields; every other $D$ has $k=1$.
(5)
The Siegel-Klingen theorem [1] [2] says that $\zeta_K(1-2m)$ is rational for every totally real field $K$ and every $m\geq 1$. For a totally real cubic field, the values at negative even integers vanish, while $\zeta_K(1-2m)$ has sign $(-1)^m$ by the functional equation (2); so the entries at $s=-1$ and $s=-5$ are negative, and the entries at $s=-3$ are positive.
(6)
The comment on each entry gives the reduced polynomial $f$, so that $K=\mathbb{Q}(x)/(f)$; the Galois group of the Galois closure, $C_3$ or $S_3$; and the class number $h_K$. The regulator $R_K$ and the residue at $s=1$ of $\zeta_K$ are held in the cubic regulator table and the cubic residue table for the fields with $D\leq 3000$, under the same $D$ and $k$ convention.
(7)
A totally real cubic field is cyclic exactly when $D$ is a square. Then $\sqrt{D}$ is the conductor, and $\zeta_K(s)=\zeta(s)L(s,\chi)L(s,\bar\chi)$ for either primitive cubic Dirichlet character $\chi$ belonging to the field. In this range the square discriminants are $49,81,169,361,961,1369,1849,3721,3969$ and $4489$; $D=3969$ has two cyclic cubic fields.
(8)
The formula (1) gives fixed denominator bounds for this rectangle: the denominator divides $63$ at $s=-1$, $24570$ at $s=-3$, and $10773$ at $s=-5$. These bounds are consequences of the coefficients $b_\ell(6m)$ and are much smaller than the $10^{20}$ denominator bound used only for the PARI lfun agreement check.
(9)
The real quadratic case is the degree-two table. There the values are a product $\zeta(1-2m)L(1-2m,\chi_D)$ for a quadratic Dirichlet character. The cyclic rows of this table have an analogous factorisation with two conjugate cubic characters; the non-cyclic rows are values of a two-dimensional Artin $L$-function.
Programs
(P1)
PARI/GP
default(realprecision, 80);
bestappr(lfun(lfuncreate(x^3 - x^2 - 2*x + 1), -1), 10^6)    \\ -1/21, for D = 49
References
[1]
Helmut Klingen, "Uber die Werte der Dedekindschen Zetafunktion", Mathematische Annalen 145 (1962), 265-272.
[2]
Carl Ludwig Siegel, "Berechnung von Zetafunktionen an ganzzahligen Stellen", Nachrichten der Akademie der Wissenschaften in Gottingen, Math.-Phys. Klasse 1969, 87-102.
[3]
Stephane R. Louboutin, "Numerical Evaluation at Negative Integers of the Dedekind Zeta Functions of Totally Real Cubic Number Fields", in Algorithmic Number Theory, ANTS-VI, Lecture Notes in Computer Science 3076, Springer, 2004, 318-326. (doi)
[4]
U. Halbritter and M. Pohst, "On the computation of the values of zeta functions of totally real cubic fields", Journal of Number Theory 36 (1990), 266-288. (doi)
[5]
H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138, Springer, 1993; Theorem 6.4.2 (Hunter).
Links
Similar tables
Values of Dedekind zeta functions of real quadratic fields at negative odd integers —   the degree-two table at the same negative odd arguments
Residues of Dedekind zeta functions of cubic fields —   the same cubic fields for $D\leq 3000$, at the pole $s=1$
Regulators of cubic fields —   the same cubic fields for $D\leq 3000$, with the same $D$ and $k$ convention
Bernoulli numbers —   the factor $B_{2m}$ in the cyclic formula
Generalized Bernoulli numbers —   the generalized Bernoulli factors in the cyclic formula
Values of the Riemann zeta function at rational numbers —   the factor $\zeta(1-2m)=-B_{2m}/(2m)$
Data properties
Entries are of type: rational number
How well the digits are known: exact
Table is complete: no (it holds every totally real cubic field with $D\leq 5000$, at $s=-1,-3,-5$)
How they were obtained:

Each value is computed in exact rational arithmetic from Siegel's formula (1). The generator enumerates the 173 totally real cubic fields with $D\leq 5000$ by Hunter's theorem [5], using the same polredabs ordering as the cubic regulator and residue tables, and refuses the run unless the count is 173.

more

The Siegel coefficients are checked in exact arithmetic against the cyclic formulas for $\mathbb{Q}(\zeta_7)^+$ and $\mathbb{Q}(\zeta_9)^+$, using generalized Bernoulli numbers for cubic Dirichlet characters. Every returned value is also compared with PARI's lfun value at 120 digits, recognised as a rational with denominator at most $10^{20}$, and the generator refuses a value if the two computations disagree or if the sign is not $(-1)^m$ at $s=1-2m$. The class number in each entry comment is computed by PARI's bnfinit and certified by bnfcertify.



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