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Values of Dedekind zeta functions of totally real quartic fields at negative odd integers
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Numbers
$D$
$k$
$s$ 
$\zeta_K(s)$
725
1
-1:
2/15
comment: $x^4-x^3-3x^2+x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.725.1.
725
1
-3:
541/15
comment: $x^4-x^3-3x^2+x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.725.1.
725
1
-5:
393276734/315
comment: $x^4-x^3-3x^2+x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.725.1.
1125
1
-1:
4/15
comment: $x^4-x^3-4x^2+4x+1=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.1125.1.
1125
1
-3:
2522/15
comment: $x^4-x^3-4x^2+4x+1=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.1125.1.
1125
1
-5:
4407640828/315
comment: $x^4-x^3-4x^2+4x+1=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.1125.1.
1600
1
-1:
7/15
comment: $x^4-6x^2+4=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.1600.1.
1600
1
-3:
17347/30
comment: $x^4-6x^2+4=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.1600.1.
1600
1
-5:
30591886909/315
comment: $x^4-6x^2+4=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.1600.1.
1957
1
-1:
2/3
comment: $x^4-4x^2-x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.1957.1.
1957
1
-3:
3541/3
comment: $x^4-4x^2-x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.1957.1.
1957
1
-5:
18544626782/63
comment: $x^4-4x^2-x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.1957.1.
2000
1
-1:
2/3
comment: $x^4-5x^2+5=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.2000.1.
2000
1
-3:
3793/3
comment: $x^4-5x^2+5=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.2000.1.
2000
1
-5:
20876972870/63
comment: $x^4-5x^2+5=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.2000.1.
2048
1
-1:
5/6
comment: $x^4-4x^2+2=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.2048.1.
2048
1
-3:
87439/60
comment: $x^4-4x^2+2=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.2048.1.
2048
1
-5:
48311765405/126
comment: $x^4-4x^2+2=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.2048.1.
2225
1
-1:
4/5
comment: $x^4-x^3-5x^2+2x+4=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.2225.1.
2225
1
-3:
9202/5
comment: $x^4-x^3-5x^2+2x+4=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.2225.1.
2225
1
-5:
20850830676/35
comment: $x^4-x^3-5x^2+2x+4=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.2225.1.
2304
1
-1:
1
comment: $x^4-4x^2+1=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.2304.1.
2304
1
-3:
22011/10
comment: $x^4-4x^2+1=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.2304.1.
2304
1
-5:
2198584943/3
comment: $x^4-4x^2+1=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.2304.1.
2525
1
-1:
14/15
comment: $x^4-2x^3-4x^2+5x+5=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.2525.1.
2525
1
-3:
42787/15
comment: $x^4-2x^3-4x^2+5x+5=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.2525.1.
2525
1
-5:
376125166418/315
comment: $x^4-2x^3-4x^2+5x+5=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.2525.1.
2624
1
-1:
1
comment: $x^4-2x^3-3x^2+2x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.2624.1.
2624
1
-3:
32681/10
comment: $x^4-2x^3-3x^2+2x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.2624.1.
2624
1
-5:
30985745851/21
comment: $x^4-2x^3-3x^2+2x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.2624.1.
2777
1
-1:
4/3
comment: $x^4-x^3-4x^2+x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.2777.1.
2777
1
-3:
63478/15
comment: $x^4-x^3-4x^2+x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.2777.1.
2777
1
-5:
128936936644/63
comment: $x^4-x^3-4x^2+x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.2777.1.
3600
1
-1:
8/5
comment: $x^4-2x^3-7x^2+8x+1=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.3600.1.
3600
1
-3:
49404/5
comment: $x^4-2x^3-7x^2+8x+1=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.3600.1.
3600
1
-5:
882041504056/105
comment: $x^4-2x^3-7x^2+8x+1=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.3600.1.
3981
1
-1:
2
comment: $x^4-x^3-4x^2+2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.3981.1.
3981
1
-3:
70953/5
comment: $x^4-x^3-4x^2+2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.3981.1.
3981
1
-5:
307134978142/21
comment: $x^4-x^3-4x^2+2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.3981.1.
4205
1
-1:
2
comment: $x^4-x^3-5x^2-x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.4205.1.
4205
1
-3:
84937/5
comment: $x^4-x^3-5x^2-x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.4205.1.
4205
1
-5:
138157529674/7
comment: $x^4-x^3-5x^2-x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.4205.1.
4225
1
-1:
32/15
comment: $x^4-9x^2+4=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.4225.1.
4225
1
-3:
260536/15
comment: $x^4-9x^2+4=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.4225.1.
4225
1
-5:
82993232144192/4095
comment: $x^4-9x^2+4=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.4225.1.
4352
1
-1:
8/3
comment: $x^4-6x^2-4x+2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.4352.1.
4352
1
-3:
61204/3
comment: $x^4-6x^2-4x+2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.4352.1.
4352
1
-5:
1525823800808/63
comment: $x^4-6x^2-4x+2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.4352.1.
4400
1
-1:
34/15
comment: $x^4-7x^2+11=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.4400.1.
4400
1
-3:
300017/15
comment: $x^4-7x^2+11=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.4400.1.
4400
1
-5:
7979768310118/315
comment: $x^4-7x^2+11=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.4400.1.
4525
1
-1:
34/15
comment: $x^4-x^3-7x^2+3x+9=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.4525.1.
4525
1
-3:
329717/15
comment: $x^4-x^3-7x^2+3x+9=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.4525.1.
4525
1
-5:
9306752421118/315
comment: $x^4-x^3-7x^2+3x+9=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.4525.1.
4752
1
-1:
8/3
comment: $x^4-2x^3-3x^2+4x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.4752.1.
4752
1
-3:
396428/15
comment: $x^4-2x^3-3x^2+4x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.4752.1.
4752
1
-5:
2440010378168/63
comment: $x^4-2x^3-3x^2+4x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.4752.1.
4913
1
-1:
8/3
comment: $x^4-x^3-6x^2+x+1=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.4913.1.
4913
1
-3:
1501748/51
comment: $x^4-x^3-6x^2+x+1=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.4913.1.
4913
1
-5:
2927442275768/63
comment: $x^4-x^3-6x^2+x+1=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.4913.1.
5125
1
-1:
8/3
comment: $x^4-2x^3-6x^2+7x+11=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.5125.1.
5125
1
-3:
101812/3
comment: $x^4-2x^3-6x^2+7x+11=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.5125.1.
5125
1
-5:
3691602450680/63
comment: $x^4-2x^3-6x^2+7x+11=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.5125.1.
5225
1
-1:
44/15
comment: $x^4-x^3-8x^2+x+11=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.5225.1.
5225
1
-3:
547942/15
comment: $x^4-x^3-8x^2+x+11=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.5225.1.
5225
1
-5:
20536485958868/315
comment: $x^4-x^3-8x^2+x+11=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.5225.1.
5725
1
-1:
46/15
comment: $x^4-x^3-8x^2+6x+11=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.5725.1.
5725
1
-3:
748883/15
comment: $x^4-x^3-8x^2+6x+11=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.5725.1.
5725
1
-5:
4847395841086/45
comment: $x^4-x^3-8x^2+6x+11=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.5725.1.
5744
1
-1:
10/3
comment: $x^4-5x^2-2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.5744.1.
5744
1
-3:
761737/15
comment: $x^4-5x^2-2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.5744.1.
5744
1
-5:
6913306068190/63
comment: $x^4-5x^2-2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.5744.1.
6125
1
-1:
52/15
comment: $x^4-x^3-9x^2+9x+11=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.6125.1.
6125
1
-3:
949826/15
comment: $x^4-x^3-9x^2+9x+11=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.6125.1.
6125
1
-5:
49198405871884/315
comment: $x^4-x^3-9x^2+9x+11=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.6125.1.
6224
1
-1:
14/3
comment: $x^4-6x^2-2x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.6224.1.
6224
1
-3:
214357/3
comment: $x^4-6x^2-2x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.6224.1.
6224
1
-5:
10917479887694/63
comment: $x^4-6x^2-2x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.6224.1.
6809
1
-1:
16/3
comment: $x^4-5x^2-x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.6809.1.
6809
1
-3:
1467568/15
comment: $x^4-5x^2-x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.6809.1.
6809
1
-5:
2556219350608/9
comment: $x^4-5x^2-x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.6809.1.
7053
1
-1:
14/3
comment: $x^4-2x^3-4x^2+3x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.7053.1.
7053
1
-3:
1573751/15
comment: $x^4-2x^3-4x^2+3x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.7053.1.
7053
1
-5:
21405527213234/63
comment: $x^4-2x^3-4x^2+3x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.7053.1.
7056
1
-1:
16/3
comment: $x^4-5x^2+1=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.7056.1.
7056
1
-3:
1600984/15
comment: $x^4-5x^2+1=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.7056.1.
7056
1
-5:
21490136139376/63
comment: $x^4-5x^2+1=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.7056.1.
7168
1
-1:
17/3
comment: $x^4-6x^2+7=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.7168.1.
7168
1
-3:
701855/6
comment: $x^4-6x^2+7=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.7168.1.
7168
1
-5:
23735722165817/63
comment: $x^4-6x^2+7=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.7168.1.
7225
1
-1:
24/5
comment: $x^4-11x^2+9=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.7225.1.
7225
1
-3:
567932/5
comment: $x^4-11x^2+9=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.7225.1.
7225
1
-5:
40694841858728/105
comment: $x^4-11x^2+9=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.7225.1.
7232
1
-1:
22/3
comment: $x^4-2x^3-5x^2+4x+4=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.7232.1.
7232
1
-3:
1929367/15
comment: $x^4-2x^3-5x^2+4x+4=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.7232.1.
7232
1
-5:
25320300946522/63
comment: $x^4-2x^3-5x^2+4x+4=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.7232.1.
7488
1
-1:
6
comment: $x^4-2x^3-4x^2+2x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.7488.1.
7488
1
-3:
681237/5
comment: $x^4-2x^3-4x^2+2x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.7488.1.
7488
1
-5:
3353352939362/7
comment: $x^4-2x^3-4x^2+2x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.7488.1.
7537
1
-1:
20/3
comment: $x^4-x^3-5x^2+4x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.7537.1.
7537
1
-3:
2116766/15
comment: $x^4-x^3-5x^2+4x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.7537.1.
7537
1
-5:
31325482066580/63
comment: $x^4-x^3-5x^2+4x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.7537.1.
7600
1
-1:
74/15
comment: $x^4-9x^2+19=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.7600.1.
7600
1
-3:
2026237/15
comment: $x^4-9x^2+19=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.7600.1.
7600
1
-5:
161221009169918/315
comment: $x^4-9x^2+19=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.7600.1.
7625
1
-1:
16/3
comment: $x^4-x^3-9x^2+4x+16=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.7625.1.
7625
1
-3:
412064/3
comment: $x^4-x^3-9x^2+4x+16=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.7625.1.
7625
1
-5:
32841895292560/63
comment: $x^4-x^3-9x^2+4x+16=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.7625.1.
8000
1
-1:
82/15
comment: $x^4-10x^2+20=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.8000.1.
8000
1
-3:
2428121/15
comment: $x^4-10x^2+20=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.8000.1.
8000
1
-5:
213780668106934/315
comment: $x^4-10x^2+20=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.8000.1.
8069
1
-1:
16/3
comment: $x^4-x^3-5x^2+5x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.8069.1.
8069
1
-3:
2494048/15
comment: $x^4-x^3-5x^2+5x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.8069.1.
8069
1
-5:
44813871114256/63
comment: $x^4-x^3-5x^2+5x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.8069.1.
8112
1
-1:
6
comment: $x^4-5x^2+3=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.8112.1.
8112
1
-3:
858951/5
comment: $x^4-5x^2+3=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.8112.1.
8112
1
-5:
66753866394926/91
comment: $x^4-5x^2+3=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.8112.1.
8468
1
-1:
28/3
comment: $x^4-x^3-5x^2+3x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.8468.1.
8468
1
-3:
3351718/15
comment: $x^4-x^3-5x^2+3x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.8468.1.
8468
1
-5:
60303492412228/63
comment: $x^4-x^3-5x^2+3x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.8468.1.
8525
1
-1:
88/15
comment: $x^4-2x^3-8x^2+9x+19=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.8525.1.
8525
1
-3:
3025964/15
comment: $x^4-2x^3-8x^2+9x+19=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.8525.1.
8525
1
-5:
303189519784936/315
comment: $x^4-2x^3-8x^2+9x+19=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.8525.1.
8725
1
-1:
86/15
comment: $x^4-x^3-10x^2+2x+19=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.8725.1.
8725
1
-3:
3272143/15
comment: $x^4-x^3-10x^2+2x+19=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.8725.1.
8725
1
-5:
344390186265002/315
comment: $x^4-x^3-10x^2+2x+19=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.8725.1.
8768
1
-1:
19/3
comment: $x^4-2x^3-5x^2+6x+7=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.8768.1.
8768
1
-3:
6691883/30
comment: $x^4-2x^3-5x^2+6x+7=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.8768.1.
8768
1
-5:
70785200542369/63
comment: $x^4-2x^3-5x^2+6x+7=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.8768.1.
8789
1
-1:
6
comment: $x^4-x^3-6x^2-2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.8789.1.
8789
1
-3:
1120867/5
comment: $x^4-x^3-6x^2-2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.8789.1.
8789
1
-5:
23902668538106/21
comment: $x^4-x^3-6x^2-2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.8789.1.
8957
1
-1:
22/3
comment: $x^4-x^3-5x^2+x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.8957.1.
8957
1
-3:
3676243/15
comment: $x^4-x^3-5x^2+x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.8957.1.
8957
1
-5:
1037288381006146/819
comment: $x^4-x^3-5x^2+x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.8957.1.
9225
1
-1:
104/15
comment: $x^4-x^3-10x^2+7x+19=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.9225.1.
9225
1
-3:
4007332/15
comment: $x^4-x^3-10x^2+7x+19=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.9225.1.
9225
1
-5:
66875819176424/45
comment: $x^4-x^3-10x^2+7x+19=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.9225.1.
9248
1
-1:
32/3
comment: $x^4-5x^2+2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.9248.1.
9248
1
-3:
912496/3
comment: $x^4-5x^2+2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.9248.1.
9248
1
-5:
97906322775392/63
comment: $x^4-5x^2+2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.9248.1.
9301
1
-1:
22/3
comment: $x^4-x^3-5x^2+x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.9301.1.
9301
1
-3:
4152019/15
comment: $x^4-x^3-5x^2+x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.9301.1.
9301
1
-5:
14006030113126/9
comment: $x^4-x^3-5x^2+x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.9301.1.
9792
1
-1:
22/3
comment: $x^4-2x^3-7x^2+2x+7=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.9792.1.
9792
1
-3:
4921939/15
comment: $x^4-2x^3-7x^2+2x+7=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.9792.1.
9792
1
-5:
18564260978206/9
comment: $x^4-2x^3-7x^2+2x+7=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.9792.1.
9909
1
-1:
8
comment: $x^4-6x^2-3x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.9909.1.
9909
1
-3:
1726836/5
comment: $x^4-6x^2-3x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.9909.1.
9909
1
-5:
46295576191288/21
comment: $x^4-6x^2-3x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.9909.1.
10025
1
-1:
124/15
comment: $x^4-x^3-11x^2+10x+20=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.10025.1.
10025
1
-3:
5376902/15
comment: $x^4-x^3-11x^2+10x+20=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.10025.1.
10025
1
-5:
739725298767268/315
comment: $x^4-x^3-11x^2+10x+20=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.10025.1.
10273
1
-1:
32/3
comment: $x^4-2x^3-5x^2+x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.10273.1.
10273
1
-3:
6258008/15
comment: $x^4-2x^3-5x^2+x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.10273.1.
10273
1
-5:
172044628777472/63
comment: $x^4-2x^3-5x^2+x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.10273.1.
10304
1
-1:
38/3
comment: $x^4-2x^3-7x^2+8x+8=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.10304.1.
10304
1
-3:
6663503/15
comment: $x^4-2x^3-7x^2+8x+8=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.10304.1.
10304
1
-5:
177453021743498/63
comment: $x^4-2x^3-7x^2+8x+8=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.10304.1.
10309
1
-1:
22/3
comment: $x^4-x^3-6x^2+8x-1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.10309.1.
10309
1
-3:
1173311/3
comment: $x^4-x^3-6x^2+8x-1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.10309.1.
10309
1
-5:
2241325247409826/819
comment: $x^4-x^3-6x^2+8x-1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.10309.1.
10512
1
-1:
8
comment: $x^4-7x^2-6x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.10512.1.
10512
1
-3:
420348
comment: $x^4-7x^2-6x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.10512.1.
10512
1
-5:
9141687262984/3
comment: $x^4-7x^2-6x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.10512.1.
10816
1
-1:
25/3
comment: $x^4-2x^3-9x^2+10x-1=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.10816.1.
10816
1
-3:
13933601/30
comment: $x^4-2x^3-9x^2+10x-1=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.10816.1.
10816
1
-5:
224565569919235/63
comment: $x^4-2x^3-9x^2+10x-1=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.10816.1.
10889
1
-1:
32/3
comment: $x^4-x^3-5x^2+2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.10889.1.
10889
1
-3:
7581944/15
comment: $x^4-x^3-5x^2+2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.10889.1.
10889
1
-5:
33810368948096/9
comment: $x^4-x^3-5x^2+2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.10889.1.
11025
1
-1:
48/5
comment: $x^4-13x^2+16=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.11025.1.
11025
1
-3:
2500344/5
comment: $x^4-13x^2+16=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.11025.1.
11025
1
-5:
138658645437552/35
comment: $x^4-13x^2+16=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.11025.1.
11197
1
-1:
28/3
comment: $x^4-2x^3-4x^2+3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.11197.1.
11197
1
-3:
7933414/15
comment: $x^4-2x^3-4x^2+3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.11197.1.
11197
1
-5:
271974614751748/63
comment: $x^4-2x^3-4x^2+3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.11197.1.
11324
1
-1:
12
comment: $x^4-x^3-5x^2+4x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.11324.1.
11324
1
-3:
2912506/5
comment: $x^4-x^3-5x^2+4x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.11324.1.
11324
1
-5:
97885877355452/21
comment: $x^4-x^3-5x^2+4x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.11324.1.
11344
1
-1:
38/3
comment: $x^4-2x^3-4x^2+4x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.11344.1.
11344
1
-3:
8866901/15
comment: $x^4-2x^3-4x^2+4x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.11344.1.
11344
1
-5:
296856451142918/63
comment: $x^4-2x^3-4x^2+4x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.11344.1.
11348
1
-1:
44/3
comment: $x^4-x^3-5x^2+x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.11348.1.
11348
1
-3:
9341246/15
comment: $x^4-x^3-5x^2+x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.11348.1.
11348
1
-5:
301722393522164/63
comment: $x^4-x^3-5x^2+x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.11348.1.
11525
1
-1:
46/5
comment: $x^4-x^3-11x^2+5x+25=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.11525.1.
11525
1
-3:
2897563/5
comment: $x^4-x^3-11x^2+5x+25=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.11525.1.
11525
1
-5:
176879480894854/35
comment: $x^4-x^3-11x^2+5x+25=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.11525.1.
11661
1
-1:
12
comment: $x^4-2x^3-4x^2+5x+3=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.11661.1.
11661
1
-3:
3123126/5
comment: $x^4-2x^3-4x^2+5x+3=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.11661.1.
11661
1
-5:
492498456805772/91
comment: $x^4-2x^3-4x^2+5x+3=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.11661.1.
12197
1
-1:
10
comment: $x^4-x^3-5x^2+3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.12197.1.
12197
1
-3:
3533149/5
comment: $x^4-x^3-5x^2+3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.12197.1.
12197
1
-5:
144942266782870/21
comment: $x^4-x^3-5x^2+3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.12197.1.
12357
1
-1:
12
comment: $x^4-x^3-5x^2+3x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.12357.1.
12357
1
-3:
3780342/5
comment: $x^4-x^3-5x^2+3x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.12357.1.
12357
1
-5:
52040829980604/7
comment: $x^4-x^3-5x^2+3x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.12357.1.
12400
1
-1:
164/15
comment: $x^4-12x^2+31=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.12400.1.
12400
1
-3:
11275762/15
comment: $x^4-12x^2+31=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.12400.1.
12400
1
-5:
2381339065986668/315
comment: $x^4-12x^2+31=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.12400.1.
12544
1
-1:
40/3
comment: $x^4-8x^2+9=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.12544.1.
12544
1
-3:
12444916/15
comment: $x^4-8x^2+9=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.12544.1.
12544
1
-5:
515365535956360/63
comment: $x^4-8x^2+9=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.12544.1.
12725
1
-1:
10
comment: $x^4-2x^3-10x^2+11x+29=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.12725.1.
12725
1
-3:
817121
comment: $x^4-2x^3-10x^2+11x+29=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.12725.1.
12725
1
-5:
26137753769434/3
comment: $x^4-2x^3-10x^2+11x+29=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.12725.1.
13025
1
-1:
184/15
comment: $x^4-x^3-12x^2+3x+29=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13025.1.
13025
1
-3:
13441892/15
comment: $x^4-x^3-12x^2+3x+29=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13025.1.
13025
1
-5:
3121647803507368/315
comment: $x^4-x^3-12x^2+3x+29=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13025.1.
13068
1
-1:
16
comment: $x^4-x^3-6x^2-x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13068.1.
13068
1
-3:
4860552/5
comment: $x^4-x^3-6x^2-x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13068.1.
13068
1
-5:
215496026629136/21
comment: $x^4-x^3-6x^2-x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13068.1.
13448
1
-1:
16
comment: $x^4-7x^2+2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13448.1.
13448
1
-3:
5324032/5
comment: $x^4-7x^2+2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13448.1.
13448
1
-5:
251979861916336/21
comment: $x^4-7x^2+2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13448.1.
13525
1
-1:
178/15
comment: $x^4-x^3-12x^2+8x+29=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13525.1.
13525
1
-3:
15223349/15
comment: $x^4-x^3-12x^2+8x+29=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13525.1.
13525
1
-5:
548344371255298/45
comment: $x^4-x^3-12x^2+8x+29=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13525.1.
13625
1
-1:
64/5
comment: $x^4-2x^3-11x^2+12x+31=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13625.1.
13625
1
-3:
5237912/5
comment: $x^4-2x^3-11x^2+12x+31=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13625.1.
13625
1
-5:
1332933938664928/105
comment: $x^4-2x^3-11x^2+12x+31=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13625.1.
13676
1
-1:
16
comment: $x^4-x^3-6x^2+7x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.13676.1.
13676
1
-3:
5638312/5
comment: $x^4-x^3-6x^2+7x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.13676.1.
13676
1
-5:
276373274156176/21
comment: $x^4-x^3-6x^2+7x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.13676.1.
13725
1
-1:
56/5
comment: $x^4-x^3-13x^2+x+31=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13725.1.
13725
1
-3:
5324268/5
comment: $x^4-x^3-13x^2+x+31=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13725.1.
13725
1
-5:
1386873156941512/105
comment: $x^4-x^3-13x^2+x+31=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13725.1.
13768
1
-1:
52/3
comment: $x^4-x^3-5x^2+2x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.13768.1.
13768
1
-3:
17503798/15
comment: $x^4-x^3-5x^2+2x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.13768.1.
13768
1
-5:
861390576341572/63
comment: $x^4-x^3-5x^2+2x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.13768.1.
13824
1
-1:
50/3
comment: $x^4-6x^2+6=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13824.1.
13824
1
-3:
17687759/15
comment: $x^4-6x^2+6=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13824.1.
13824
1
-5:
880623785856530/63
comment: $x^4-6x^2+6=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13824.1.
13888
1
-1:
38/3
comment: $x^4-2x^3-7x^2+6x+9=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13888.1.
13888
1
-3:
16732331/15
comment: $x^4-2x^3-7x^2+6x+9=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13888.1.
13888
1
-5:
888189891329858/63
comment: $x^4-2x^3-7x^2+6x+9=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13888.1.
13968
1
-1:
20
comment: $x^4-2x^3-7x^2+8x+4=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13968.1.
13968
1
-3:
6440874/5
comment: $x^4-2x^3-7x^2+8x+4=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13968.1.
13968
1
-5:
315257566198060/21
comment: $x^4-2x^3-7x^2+8x+4=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.13968.1.
14013
1
-1:
44/3
comment: $x^4-x^3-6x^2+6x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.14013.1.
14013
1
-3:
17618654/15
comment: $x^4-x^3-6x^2+6x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.14013.1.
14013
1
-5:
935383371334964/63
comment: $x^4-x^3-6x^2+6x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.14013.1.
14197
1
-1:
12
comment: $x^4-6x^2-3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.14197.1.
14197
1
-3:
5995886/5
comment: $x^4-6x^2-3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.14197.1.
14197
1
-5:
334067400980692/21
comment: $x^4-6x^2-3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.14197.1.
14272
1
-1:
14
comment: $x^4-2x^3-5x^2+2x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.14272.1.
14272
1
-3:
6204179/5
comment: $x^4-2x^3-5x^2+2x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.14272.1.
14272
1
-5:
344442459910634/21
comment: $x^4-2x^3-5x^2+2x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.14272.1.
14336
1
-1:
49/3
comment: $x^4-8x^2+14=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.14336.1.
14336
1
-3:
39724091/30
comment: $x^4-8x^2+14=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.14336.1.
14336
1
-5:
1074170393202649/63
comment: $x^4-8x^2+14=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.14336.1.
14400
1
-1:
68/5
comment: $x^4-2x^3-13x^2+14x+19=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.14400.1.
14400
1
-3:
6342474/5
comment: $x^4-2x^3-13x^2+14x+19=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.14400.1.
14400
1
-5:
258092617601908/15
comment: $x^4-2x^3-13x^2+14x+19=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.14400.1.
14656
1
-1:
56/3
comment: $x^4-2x^3-4x^2+4x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.14656.1.
14656
1
-3:
21734588/15
comment: $x^4-2x^3-4x^2+4x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.14656.1.
14656
1
-5:
1214555398237016/63
comment: $x^4-2x^3-4x^2+4x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.14656.1.
14725
1
-1:
188/15
comment: $x^4-x^3-13x^2+11x+31=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.14725.1.
14725
1
-3:
20433334/15
comment: $x^4-x^3-13x^2+11x+31=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.14725.1.
14725
1
-5:
6125548958895236/315
comment: $x^4-x^3-13x^2+11x+31=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.14725.1.
15125
1
-1:
40/3
comment: $x^4-x^3-14x^2+14x+31=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.15125.1.
15125
1
-3:
4494500/3
comment: $x^4-x^3-14x^2+14x+31=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.15125.1.
15125
1
-5:
1419781907968600/63
comment: $x^4-x^3-14x^2+14x+31=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.15125.1.
15188
1
-1:
68/3
comment: $x^4-x^3-7x^2+x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.15188.1.
15188
1
-3:
25901714/15
comment: $x^4-x^3-7x^2+x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.15188.1.
15188
1
-5:
214142720065124/9
comment: $x^4-x^3-7x^2+x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.15188.1.
15317
1
-1:
24
comment: $x^4-2x^3-4x^2+5x+2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.15317.1.
15317
1
-3:
8926684/5
comment: $x^4-2x^3-4x^2+5x+2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.15317.1.
15317
1
-5:
523585666420424/21
comment: $x^4-2x^3-4x^2+5x+2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.15317.1.
15529
1
-1:
56/3
comment: $x^4-x^3-6x^2-x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.15529.1.
15529
1
-3:
26297228/15
comment: $x^4-x^3-6x^2-x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.15529.1.
15529
1
-5:
1667356227050456/63
comment: $x^4-x^3-6x^2-x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.15529.1.
15952
1
-1:
62/3
comment: $x^4-6x^2-2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.15952.1.
15952
1
-3:
5838925/3
comment: $x^4-6x^2-2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.15952.1.
15952
1
-5:
1935477580690622/63
comment: $x^4-6x^2-2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.15952.1.
16225
1
-1:
244/15
comment: $x^4-x^3-13x^2+6x+36=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.16225.1.
16225
1
-3:
28917242/15
comment: $x^4-x^3-13x^2+6x+36=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.16225.1.
16225
1
-5:
10448851243665868/315
comment: $x^4-x^3-13x^2+6x+36=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.16225.1.
16317
1
-1:
16
comment: $x^4-2x^3-4x^2+5x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.16317.1.
16317
1
-3:
9792552/5
comment: $x^4-2x^3-4x^2+5x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.16317.1.
16317
1
-5:
718344880547696/21
comment: $x^4-2x^3-4x^2+5x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.16317.1.
16357
1
-1:
52/3
comment: $x^4-6x^2-x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.16357.1.
16357
1
-3:
29974018/15
comment: $x^4-6x^2-x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.16357.1.
16357
1
-5:
312458861087476/9
comment: $x^4-6x^2-x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.16357.1.
16400
1
-1:
248/15
comment: $x^4-13x^2+41=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.16400.1.
16400
1
-3:
29995204/15
comment: $x^4-13x^2+41=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.16400.1.
16400
1
-5:
11082608452686536/315
comment: $x^4-13x^2+41=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.16400.1.
16448
1
-1:
26
comment: $x^4-2x^3-7x^2+8x+14=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.16448.1.
16448
1
-3:
11419177/5
comment: $x^4-2x^3-7x^2+8x+14=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.16448.1.
16448
1
-5:
258187038330482/7
comment: $x^4-2x^3-7x^2+8x+14=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.16448.1.
16448
2
-1:
62/3
comment: $x^4-2x^3-6x^2+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.16448.2.
16448
2
-3:
32196449/15
comment: $x^4-2x^3-6x^2+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.16448.2.
16448
2
-5:
326804513642066/9
comment: $x^4-2x^3-6x^2+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.16448.2.
16609
1
-1:
68/3
comment: $x^4-7x^2-x+9=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.16609.1.
16609
1
-3:
33680126/15
comment: $x^4-7x^2-x+9=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.16609.1.
16609
1
-5:
2416700826756068/63
comment: $x^4-7x^2-x+9=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.16609.1.
16997
1
-1:
50/3
comment: $x^4-6x^2-x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.16997.1.
16997
1
-3:
33873377/15
comment: $x^4-6x^2-x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.16997.1.
16997
1
-5:
2697588428577710/63
comment: $x^4-6x^2-x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.16997.1.
17069
1
-1:
58/3
comment: $x^4-x^3-8x^2-4x+3=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.17069.1.
17069
1
-3:
35119957/15
comment: $x^4-x^3-8x^2-4x+3=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.17069.1.
17069
1
-5:
35987334226278574/819
comment: $x^4-x^3-8x^2-4x+3=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.17069.1.
17417
1
-1:
68/3
comment: $x^4-2x^3-5x^2+3x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.17417.1.
17417
1
-3:
39345446/15
comment: $x^4-2x^3-5x^2+3x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.17417.1.
17417
1
-5:
3134143639701188/63
comment: $x^4-2x^3-5x^2+3x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.17417.1.
17424
1
-1:
28
comment: $x^4-7x^2+4=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.17424.1.
17424
1
-3:
13964358/5
comment: $x^4-7x^2+4=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.17424.1.
17424
1
-5:
1063508156680388/21
comment: $x^4-7x^2+4=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.17424.1.
17428
1
-1:
92/3
comment: $x^4-x^3-6x^2+4x+6=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.17428.1.
17428
1
-3:
42438566/15
comment: $x^4-x^3-6x^2+4x+6=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.17428.1.
17428
1
-5:
3198933478784132/63
comment: $x^4-x^3-6x^2+4x+6=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.17428.1.
17600
1
-1:
86/5
comment: $x^4-14x^2+44=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.17600.1.
17600
1
-3:
12761963/5
comment: $x^4-14x^2+44=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.17600.1.
17600
1
-5:
1815610402384854/35
comment: $x^4-14x^2+44=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.17600.1.
17609
1
-1:
68/3
comment: $x^4-x^3-7x^2+10x-1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.17609.1.
17609
1
-3:
8167558/3
comment: $x^4-x^3-7x^2+10x-1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.17609.1.
17609
1
-5:
3328758896639588/63
comment: $x^4-x^3-7x^2+10x-1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.17609.1.
17725
1
-1:
82/5
comment: $x^4-2x^3-12x^2+13x+41=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.17725.1.
17725
1
-3:
13032461/5
comment: $x^4-2x^3-12x^2+13x+41=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.17725.1.
17725
1
-5:
5661655888243534/105
comment: $x^4-2x^3-12x^2+13x+41=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.17725.1.
17989
1
-1:
58/3
comment: $x^4-x^3-7x^2-3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.17989.1.
17989
1
-3:
41746549/15
comment: $x^4-x^3-7x^2-3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.17989.1.
17989
1
-5:
3690044846081638/63
comment: $x^4-x^3-7x^2-3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.17989.1.
18097
1
-1:
64/3
comment: $x^4-x^3-7x^2+6x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.18097.1.
18097
1
-3:
8582024/3
comment: $x^4-x^3-7x^2+6x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.18097.1.
18097
1
-5:
3815196714904864/63
comment: $x^4-x^3-7x^2+6x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.18097.1.
18432
1
-1:
73/3
comment: $x^4-12x^2+18=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.18432.1.
18432
1
-3:
95783699/30
comment: $x^4-12x^2+18=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.18432.1.
18432
1
-5:
4279294971376273/63
comment: $x^4-12x^2+18=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.18432.1.
18496
1
-1:
92/3
comment: $x^4-2x^3-11x^2+12x+2=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.18496.1.
18496
1
-3:
51631382/15
comment: $x^4-2x^3-11x^2+12x+2=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.18496.1.
18496
1
-5:
4430749437475652/63
comment: $x^4-2x^3-11x^2+12x+2=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.18496.1.
18625
1
-1:
104/5
comment: $x^4-x^3-14x^2+9x+41=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.18625.1.
18625
1
-3:
15647372/5
comment: $x^4-x^3-14x^2+9x+41=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.18625.1.
18625
1
-5:
354217230250968/5
comment: $x^4-x^3-14x^2+9x+41=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.18625.1.
18688
1
-1:
73/3
comment: $x^4-10x^2-4x+14=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.18688.1.
18688
1
-3:
100455443/30
comment: $x^4-10x^2-4x+14=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.18688.1.
18688
1
-5:
4616498458549393/63
comment: $x^4-10x^2-4x+14=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.18688.1.
18736
1
-1:
22
comment: $x^4-2x^3-5x^2+4x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.18736.1.
18736
1
-3:
16113599/5
comment: $x^4-2x^3-5x^2+4x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.18736.1.
18736
1
-5:
73279902928922
comment: $x^4-2x^3-5x^2+4x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.18736.1.
19025
1
-1:
328/15
comment: $x^4-2x^3-13x^2+14x+44=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.19025.1.
19025
1
-3:
50632124/15
comment: $x^4-2x^3-13x^2+14x+44=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.19025.1.
19025
1
-5:
25083752494021336/315
comment: $x^4-2x^3-13x^2+14x+44=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.19025.1.
19225
1
-1:
316/15
comment: $x^4-x^3-15x^2+2x+44=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.19225.1.
19225
1
-3:
52368038/15
comment: $x^4-x^3-15x^2+2x+44=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.19225.1.
19225
1
-5:
26565488743662052/315
comment: $x^4-x^3-15x^2+2x+44=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.19225.1.
19429
1
-1:
22
comment: $x^4-x^3-7x^2-x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.19429.1.
19429
1
-3:
18226907/5
comment: $x^4-x^3-7x^2-x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.19429.1.
19429
1
-5:
626226588992014/7
comment: $x^4-x^3-7x^2-x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.19429.1.
19525
1
-1:
308/15
comment: $x^4-2x^3-14x^2+15x+45=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.19525.1.
19525
1
-3:
55026994/15
comment: $x^4-2x^3-14x^2+15x+45=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.19525.1.
19525
1
-5:
4130940046532948/45
comment: $x^4-2x^3-14x^2+15x+45=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.19525.1.
19600
1
-1:
304/15
comment: $x^4-2x^3-15x^2+16x+29=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.19600.1.
19600
1
-3:
55806632/15
comment: $x^4-2x^3-15x^2+16x+29=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.19600.1.
19600
1
-5:
29536211719301968/315
comment: $x^4-2x^3-15x^2+16x+29=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.19600.1.
19664
1
-1:
104/3
comment: $x^4-2x^3-5x^2+2x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.19664.1.
19664
1
-3:
64080524/15
comment: $x^4-2x^3-5x^2+2x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.19664.1.
19664
1
-5:
6205464202852664/63
comment: $x^4-2x^3-5x^2+2x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.19664.1.
19773
1
-1:
24
comment: $x^4-x^3-11x^2-9x+3=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.19773.1.
19773
1
-3:
19584396/5
comment: $x^4-x^3-11x^2-9x+3=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.19773.1.
19773
1
-5:
8977618187208344/91
comment: $x^4-x^3-11x^2-9x+3=0$; Galois group $C_4$ (4T1); $h_K=1$; LMFDB 4.4.19773.1.
19796
1
-1:
104/3
comment: $x^4-x^3-7x^2+x+8=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.19796.1.
19796
1
-3:
65575028/15
comment: $x^4-x^3-7x^2+x+8=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.19796.1.
19796
1
-5:
6438006184332824/63
comment: $x^4-x^3-7x^2+x+8=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.19796.1.
19821
1
-1:
22
comment: $x^4-x^3-8x^2+6x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.19821.1.
19821
1
-3:
19517979/5
comment: $x^4-x^3-8x^2+6x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.19821.1.
19821
1
-5:
2096712182117162/21
comment: $x^4-x^3-8x^2+6x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.19821.1.
20032
1
-1:
65/3
comment: $x^4-2x^3-9x^2+10x+17=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.20032.1.
20032
1
-3:
24112517/6
comment: $x^4-2x^3-9x^2+10x+17=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.20032.1.
20032
1
-5:
6659896676425115/63
comment: $x^4-2x^3-9x^2+10x+17=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.20032.1.
20225
1
-1:
112/5
comment: $x^4-x^3-15x^2+12x+44=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.20225.1.
20225
1
-3:
20840256/5
comment: $x^4-x^3-15x^2+12x+44=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.20225.1.
20225
1
-5:
11703532280649584/105
comment: $x^4-x^3-15x^2+12x+44=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.20225.1.
20308
1
-1:
116/3
comment: $x^4-x^3-8x^2+4x+12=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.20308.1.
20308
1
-3:
72482618/15
comment: $x^4-x^3-8x^2+4x+12=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.20308.1.
20308
1
-5:
7418447735672396/63
comment: $x^4-x^3-8x^2+4x+12=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.20308.1.
20808
1
-1:
88/3
comment: $x^4-x^3-7x^2+10x-2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.20808.1.
20808
1
-3:
73381636/15
comment: $x^4-x^3-7x^2+10x-2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.20808.1.
20808
1
-5:
1191198175343944/9
comment: $x^4-x^3-7x^2+10x-2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.20808.1.
21025
1
-1:
128/5
comment: $x^4-17x^2+36=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=2$; LMFDB 4.4.21025.1.
21025
1
-3:
23950664/5
comment: $x^4-17x^2+36=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=2$; LMFDB 4.4.21025.1.
21025
1
-5:
689946580224736/5
comment: $x^4-17x^2+36=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=2$; LMFDB 4.4.21025.1.
21056
1
-1:
70/3
comment: $x^4-2x^3-11x^2+2x+17=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21056.1.
21056
1
-3:
71781787/15
comment: $x^4-2x^3-11x^2+2x+17=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21056.1.
21056
1
-5:
8760988999286050/63
comment: $x^4-2x^3-11x^2+2x+17=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21056.1.
21200
1
-1:
92/3
comment: $x^4-6x^2-2x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.21200.1.
21200
1
-3:
78298418/15
comment: $x^4-6x^2-2x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.21200.1.
21200
1
-5:
9238785575253212/63
comment: $x^4-6x^2-2x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.21200.1.
21208
1
-1:
100/3
comment: $x^4-x^3-8x^2+13x-3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.21208.1.
21208
1
-3:
15881174/3
comment: $x^4-x^3-8x^2+13x-3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.21208.1.
21208
1
-5:
9271692071145460/63
comment: $x^4-x^3-8x^2+13x-3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.21208.1.
21308
1
-1:
92/3
comment: $x^4-x^3-8x^2+3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.21308.1.
21308
1
-3:
79764026/15
comment: $x^4-x^3-8x^2+3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.21308.1.
21308
1
-5:
9501736873529132/63
comment: $x^4-x^3-8x^2+3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.21308.1.
21312
1
-1:
106/3
comment: $x^4-2x^3-8x^2+6=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21312.1.
21312
1
-3:
81465931/15
comment: $x^4-2x^3-8x^2+6=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21312.1.
21312
1
-5:
9535286039623786/63
comment: $x^4-2x^3-8x^2+6=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21312.1.
21469
1
-1:
74/3
comment: $x^4-x^3-7x^2+9x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.21469.1.
21469
1
-3:
77429933/15
comment: $x^4-x^3-7x^2+9x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.21469.1.
21469
1
-5:
1394234137548602/9
comment: $x^4-x^3-7x^2+9x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.21469.1.
21568
1
-1:
116/3
comment: $x^4-2x^3-11x^2+12x+18=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21568.1.
21568
1
-3:
17681026/3
comment: $x^4-2x^3-11x^2+12x+18=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21568.1.
21568
1
-5:
10315814257138316/63
comment: $x^4-2x^3-11x^2+12x+18=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21568.1.
21725
1
-1:
332/15
comment: $x^4-x^3-15x^2+7x+49=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21725.1.
21725
1
-3:
79690126/15
comment: $x^4-x^3-15x^2+7x+49=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21725.1.
21725
1
-5:
7430537720483372/45
comment: $x^4-x^3-15x^2+7x+49=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21725.1.
21737
1
-1:
76/3
comment: $x^4-x^3-9x^2+4x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.21737.1.
21737
1
-3:
80496838/15
comment: $x^4-x^3-9x^2+4x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.21737.1.
21737
1
-5:
10439571291976276/63
comment: $x^4-x^3-9x^2+4x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.21737.1.
21801
1
-1:
28
comment: $x^4-x^3-7x^2+2x+4=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21801.1.
21801
1
-3:
27438582/5
comment: $x^4-x^3-7x^2+2x+4=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21801.1.
21801
1
-5:
46038392474932244/273
comment: $x^4-x^3-7x^2+2x+4=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21801.1.
21964
1
-1:
32
comment: $x^4-x^3-11x^2+12x+8=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21964.1.
21964
1
-3:
29560016/5
comment: $x^4-x^3-11x^2+12x+8=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21964.1.
21964
1
-5:
3742032951832352/21
comment: $x^4-x^3-11x^2+12x+8=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.21964.1.
22000
1
-1:
76/3
comment: $x^4-15x^2+55=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.22000.1.
22000
1
-3:
16752014/3
comment: $x^4-15x^2+55=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.22000.1.
22000
1
-5:
11151406063515460/63
comment: $x^4-15x^2+55=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.22000.1.
22221
1
-1:
86/3
comment: $x^4-x^3-6x^2+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.22221.1.
22221
1
-3:
88403411/15
comment: $x^4-x^3-6x^2+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.22221.1.
22221
1
-5:
11810306672356106/63
comment: $x^4-x^3-6x^2+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.22221.1.
22545
1
-1:
36
comment: $x^4-x^3-6x^2+5x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.22545.1.
22545
1
-3:
32714406/5
comment: $x^4-x^3-6x^2+5x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.22545.1.
22545
1
-5:
1441675812319052/7
comment: $x^4-x^3-6x^2+5x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.22545.1.
22592
1
-1:
42
comment: $x^4-2x^3-9x^2+8x+16=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.22592.1.
22592
1
-3:
34680481/5
comment: $x^4-2x^3-9x^2+8x+16=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.22592.1.
22592
1
-5:
633995772865226/3
comment: $x^4-2x^3-9x^2+8x+16=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.22592.1.
22676
1
-1:
128/3
comment: $x^4-x^3-6x^2+6x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.22676.1.
22676
1
-3:
105492872/15
comment: $x^4-x^3-6x^2+6x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.22676.1.
22676
1
-5:
13589113663997408/63
comment: $x^4-x^3-6x^2+6x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.22676.1.
22784
1
-1:
97/3
comment: $x^4-12x^2-8x+17=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.22784.1.
22784
1
-3:
200951531/30
comment: $x^4-12x^2-8x+17=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.22784.1.
22784
1
-5:
13730312173619017/63
comment: $x^4-12x^2-8x+17=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.22784.1.
22896
1
-1:
98/3
comment: $x^4-9x^2-4x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.22896.1.
22896
1
-3:
98699789/15
comment: $x^4-9x^2-4x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.22896.1.
22896
1
-5:
13927493244389318/63
comment: $x^4-9x^2-4x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.22896.1.
23252
1
-1:
44
comment: $x^4-x^3-6x^2+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23252.1.
23252
1
-3:
38360038/5
comment: $x^4-x^3-6x^2+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23252.1.
23252
1
-5:
1733182812205468/7
comment: $x^4-x^3-6x^2+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23252.1.
23297
1
-1:
100/3
comment: $x^4-2x^3-5x^2+3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23297.1.
23297
1
-3:
108602734/15
comment: $x^4-2x^3-5x^2+3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23297.1.
23297
1
-5:
15518995169028100/63
comment: $x^4-2x^3-5x^2+3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23297.1.
23301
1
-1:
88/3
comment: $x^4-x^3-6x^2+2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23301.1.
23301
1
-3:
103334428/15
comment: $x^4-x^3-6x^2+2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23301.1.
23301
1
-5:
15313003017797608/63
comment: $x^4-x^3-6x^2+2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23301.1.
23377
1
-1:
112/3
comment: $x^4-x^3-7x^2+6x+7=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23377.1.
23377
1
-3:
22256864/3
comment: $x^4-x^3-7x^2+6x+7=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23377.1.
23377
1
-5:
15836090801849872/63
comment: $x^4-x^3-7x^2+6x+7=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23377.1.
23525
1
-1:
134/5
comment: $x^4-2x^3-14x^2+15x+55=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.23525.1.
23525
1
-3:
35207767/5
comment: $x^4-2x^3-14x^2+15x+55=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.23525.1.
23525
1
-5:
26864980662146618/105
comment: $x^4-2x^3-14x^2+15x+55=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.23525.1.
23552
1
-1:
35
comment: $x^4-10x^2+23=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.23552.1.
23552
1
-3:
15057053/2
comment: $x^4-10x^2+23=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.23552.1.
23552
1
-5:
5492303284945595/21
comment: $x^4-10x^2+23=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.23552.1.
23600
1
-1:
134/5
comment: $x^4-16x^2+59=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.23600.1.
23600
1
-3:
35631827/5
comment: $x^4-16x^2+59=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.23600.1.
23600
1
-5:
9114214220584006/35
comment: $x^4-16x^2+59=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.23600.1.
23665
1
-1:
116/3
comment: $x^4-x^3-7x^2+8x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23665.1.
23665
1
-3:
116291414/15
comment: $x^4-x^3-7x^2+8x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23665.1.
23665
1
-5:
16940238787659476/63
comment: $x^4-x^3-7x^2+8x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23665.1.
23724
1
-1:
112/3
comment: $x^4-x^3-7x^2+4x+10=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23724.1.
23724
1
-3:
116354584/15
comment: $x^4-x^3-7x^2+4x+10=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23724.1.
23724
1
-5:
17154579297740272/63
comment: $x^4-x^3-7x^2+4x+10=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.23724.1.
24197
1
-1:
48
comment: $x^4-6x^2-x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.24197.1.
24197
1
-3:
8847032
comment: $x^4-6x^2-x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.24197.1.
24197
1
-5:
6474717522733328/21
comment: $x^4-6x^2-x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.24197.1.
24336
1
-1:
104/3
comment: $x^4-2x^3-11x^2+12x-3=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.24336.1.
24336
1
-3:
24398860/3
comment: $x^4-2x^3-11x^2+12x-3=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.24336.1.
24336
1
-5:
36172881455293256/117
comment: $x^4-2x^3-11x^2+12x-3=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.24336.1.
24400
1
-1:
458/15
comment: $x^4-17x^2+61=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.24400.1.
24400
1
-3:
120527869/15
comment: $x^4-17x^2+61=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.24400.1.
24400
1
-5:
98547698337877886/315
comment: $x^4-17x^2+61=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.24400.1.
24417
1
-1:
36
comment: $x^4-x^3-7x^2-2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.24417.1.
24417
1
-3:
42672582/5
comment: $x^4-x^3-7x^2-2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.24417.1.
24417
1
-5:
2232438685667532/7
comment: $x^4-x^3-7x^2-2x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.24417.1.
24437
1
-1:
86/3
comment: $x^4-x^3-7x^2+7x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.24437.1.
24437
1
-3:
120701219/15
comment: $x^4-x^3-7x^2+7x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.24437.1.
24437
1
-5:
2838529755616358/9
comment: $x^4-x^3-7x^2+7x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.24437.1.
24525
1
-1:
144/5
comment: $x^4-x^3-16x^2+10x+55=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.24525.1.
24525
1
-3:
40736232/5
comment: $x^4-x^3-16x^2+10x+55=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.24525.1.
24525
1
-5:
11259074801544656/35
comment: $x^4-x^3-16x^2+10x+55=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.24525.1.
24749
1
-1:
82/3
comment: $x^4-x^3-7x^2+7x-1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.24749.1.
24749
1
-3:
125794249/15
comment: $x^4-x^3-7x^2+7x-1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.24749.1.
24749
1
-5:
21302967065195182/63
comment: $x^4-x^3-7x^2+7x-1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.24749.1.
24832
1
-1:
112/3
comment: $x^4-14x^2-12x+18=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.24832.1.
24832
1
-3:
135836536/15
comment: $x^4-14x^2-12x+18=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.24832.1.
24832
1
-5:
22043548516922032/63
comment: $x^4-14x^2-12x+18=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.24832.1.
24917
1
-1:
104/3
comment: $x^4-2x^3-6x^2+x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.24917.1.
24917
1
-3:
132050708/15
comment: $x^4-2x^3-6x^2+x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.24917.1.
24917
1
-5:
22171355983174424/63
comment: $x^4-2x^3-6x^2+x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.24917.1.
25088
1
-1:
136/3
comment: $x^4-6x^2+2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.25088.1.
25088
1
-3:
28839860/3
comment: $x^4-6x^2+2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.25088.1.
25088
1
-5:
23386458575388136/63
comment: $x^4-6x^2+2=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.25088.1.
25225
1
-1:
476/15
comment: $x^4-2x^3-15x^2+16x+59=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.25225.1.
25225
1
-3:
135502678/15
comment: $x^4-2x^3-15x^2+16x+59=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.25225.1.
25225
1
-5:
118338039978739652/315
comment: $x^4-2x^3-15x^2+16x+59=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.25225.1.
25488
1
-1:
164/3
comment: $x^4-9x^2-6x+6=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.25488.1.
25488
1
-3:
32109058/3
comment: $x^4-9x^2-6x+6=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.25488.1.
25488
1
-5:
25881564537540284/63
comment: $x^4-9x^2-6x+6=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.25488.1.
25492
1
-1:
164/3
comment: $x^4-8x^2-2x+8=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.25492.1.
25492
1
-3:
160634306/15
comment: $x^4-8x^2-2x+8=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.25492.1.
25492
1
-5:
25903916320457564/63
comment: $x^4-8x^2-2x+8=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.25492.1.
25525
1
-1:
458/15
comment: $x^4-x^3-17x^2+3x+59=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.25525.1.
25525
1
-3:
140559169/15
comment: $x^4-x^3-17x^2+3x+59=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.25525.1.
25525
1
-5:
126243080609176886/315
comment: $x^4-x^3-17x^2+3x+59=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.25525.1.
25717
1
-1:
32
comment: $x^4-x^3-6x^2+6x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.25717.1.
25717
1
-3:
48537704/5
comment: $x^4-x^3-6x^2+6x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.25717.1.
25717
1
-5:
8781240915587072/21
comment: $x^4-x^3-6x^2+6x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.25717.1.
25808
1
-1:
142/3
comment: $x^4-10x^2-6x+9=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.25808.1.
25808
1
-3:
159211561/15
comment: $x^4-10x^2-6x+9=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.25808.1.
25808
1
-5:
27324370882705102/63
comment: $x^4-10x^2-6x+9=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.25808.1.
25857
1
-1:
40
comment: $x^4-x^3-8x^2+9x+3=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.25857.1.
25857
1
-3:
50470788/5
comment: $x^4-x^3-8x^2+9x+3=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.25857.1.
25857
1
-5:
117835168002005240/273
comment: $x^4-x^3-8x^2+9x+3=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.25857.1.
25893
1
-1:
110/3
comment: $x^4-6x^2-x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.25893.1.
25893
1
-3:
151050503/15
comment: $x^4-6x^2-x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.25893.1.
25893
1
-5:
27388499906777810/63
comment: $x^4-6x^2-x+3=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.25893.1.
25961
1
-1:
124/3
comment: $x^4-9x^2-7x-1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.25961.1.
25961
1
-3:
158964826/15
comment: $x^4-9x^2-7x-1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.25961.1.
25961
1
-5:
4021749585832612/9
comment: $x^4-9x^2-7x-1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.25961.1.
26032
1
-1:
106/3
comment: $x^4-10x^2-4x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.26032.1.
26032
1
-3:
152647993/15
comment: $x^4-10x^2-4x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.26032.1.
26032
1
-5:
28175378087996926/63
comment: $x^4-10x^2-4x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.26032.1.
26125
1
-1:
464/15
comment: $x^4-2x^3-16x^2+17x+61=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.26125.1.
26125
1
-3:
152252392/15
comment: $x^4-2x^3-16x^2+17x+61=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.26125.1.
26125
1
-5:
143442725125320368/315
comment: $x^4-2x^3-16x^2+17x+61=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.26125.1.
26176
1
-1:
97/3
comment: $x^4-2x^3-9x^2+10x+23=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.26176.1.
26176
1
-3:
307491833/30
comment: $x^4-2x^3-9x^2+10x+23=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.26176.1.
26176
1
-5:
29003593006173307/63
comment: $x^4-2x^3-9x^2+10x+23=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.26176.1.
26224
1
-1:
106/3
comment: $x^4-7x^2-2x+7=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.26224.1.
26224
1
-3:
156572281/15
comment: $x^4-7x^2-2x+7=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.26224.1.
26224
1
-5:
29337226959529726/63
comment: $x^4-7x^2-2x+7=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.26224.1.
26225
1
-1:
100/3
comment: $x^4-x^3-18x^2+x+61=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.26225.1.
26225
1
-3:
31045298/3
comment: $x^4-x^3-18x^2+x+61=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.26225.1.
26225
1
-5:
29310078305199964/63
comment: $x^4-x^3-18x^2+x+61=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.26225.1.
26525
1
-1:
162/5
comment: $x^4-x^3-17x^2+13x+59=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.26525.1.
26525
1
-3:
53595781/5
comment: $x^4-x^3-17x^2+13x+59=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.26525.1.
26525
1
-5:
51985076326752574/105
comment: $x^4-x^3-17x^2+13x+59=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.26525.1.
26541
1
-1:
116/3
comment: $x^4-x^3-6x^2+4x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.26541.1.
26541
1
-3:
164885186/15
comment: $x^4-x^3-6x^2+4x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.26541.1.
26541
1
-5:
31378635546501356/63
comment: $x^4-x^3-6x^2+4x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.26541.1.
26569
1
-1:
136/3
comment: $x^4-x^3-7x^2+2x+9=0$; Galois group $A_4$ (4T4); $h_K=1$; LMFDB 4.4.26569.1.
26569
1
-3:
34836092/3
comment: $x^4-x^3-7x^2+2x+9=0$; Galois group $A_4$ (4T4); $h_K=1$; LMFDB 4.4.26569.1.
26569
1
-5:
32016474356549896/63
comment: $x^4-x^3-7x^2+2x+9=0$; Galois group $A_4$ (4T4); $h_K=1$; LMFDB 4.4.26569.1.
26825
1
-1:
44
comment: $x^4-2x^3-5x^2+5x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.26825.1.
26825
1
-3:
59487106/5
comment: $x^4-2x^3-5x^2+5x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.26825.1.
26825
1
-5:
3745373019639268/7
comment: $x^4-2x^3-5x^2+5x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.26825.1.
26873
1
-1:
128/3
comment: $x^4-x^3-8x^2+9x-2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.26873.1.
26873
1
-3:
35833672/3
comment: $x^4-x^3-8x^2+9x-2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.26873.1.
26873
1
-5:
4862525599446944/9
comment: $x^4-x^3-8x^2+9x-2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.26873.1.
27004
1
-1:
148/3
comment: $x^4-x^3-6x^2+3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.27004.1.
27004
1
-3:
185251846/15
comment: $x^4-x^3-6x^2+3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.27004.1.
27004
1
-5:
35018105542450468/63
comment: $x^4-x^3-6x^2+3x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.27004.1.
27225
1
-1:
176/5
comment: $x^4-19x^2+49=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.27225.1.
27225
1
-3:
58984248/5
comment: $x^4-19x^2+49=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.27225.1.
27225
1
-5:
60014226083349712/105
comment: $x^4-19x^2+49=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.27225.1.
27329
1
-1:
44
comment: $x^4-x^3-6x^2+x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.27329.1.
27329
1
-3:
63396202/5
comment: $x^4-x^3-6x^2+x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.27329.1.
27329
1
-5:
592695484722524
comment: $x^4-x^3-6x^2+x+4=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.27329.1.
27472
1
-1:
142/3
comment: $x^4-8x^2-2x+11=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.27472.1.
27472
1
-3:
39151349/3
comment: $x^4-8x^2-2x+11=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.27472.1.
27472
1
-5:
38477322035955022/63
comment: $x^4-8x^2-2x+11=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.27472.1.
27648
1
-1:
50
comment: $x^4-6x^2+3=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.27648.1.
27648
1
-3:
66905871/5
comment: $x^4-6x^2+3=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.27648.1.
27648
1
-5:
1897977380224990/3
comment: $x^4-6x^2+3=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.27648.1.
27725
1
-1:
94/3
comment: $x^4-x^3-18x^2+16x+61=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.27725.1.
27725
1
-3:
37414859/3
comment: $x^4-x^3-18x^2+16x+61=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.27725.1.
27725
1
-5:
39779772596980354/63
comment: $x^4-x^3-18x^2+16x+61=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.27725.1.
27792
1
-1:
128/3
comment: $x^4-2x^3-11x^2+12x+9=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.27792.1.
27792
1
-3:
194195072/15
comment: $x^4-2x^3-11x^2+12x+9=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.27792.1.
27792
1
-5:
40432127343444608/63
comment: $x^4-2x^3-11x^2+12x+9=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.27792.1.
28025
1
-1:
196/5
comment: $x^4-x^3-17x^2+8x+64=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.28025.1.
28025
1
-3:
65475938/5
comment: $x^4-x^3-17x^2+8x+64=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.28025.1.
28025
1
-5:
70386037860386812/105
comment: $x^4-x^3-17x^2+8x+64=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.28025.1.
28224
1
-1:
36
comment: $x^4-2x^3-13x^2+14x+7=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.28224.1.
28224
1
-3:
13340250
comment: $x^4-2x^3-13x^2+14x+7=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.28224.1.
28224
1
-5:
4876885769547332/7
comment: $x^4-2x^3-13x^2+14x+7=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.28224.1.
28224
2
-1:
40
comment: $x^4-10x^2+4=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.28224.2.
28224
2
-3:
67070388/5
comment: $x^4-10x^2+4=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.28224.2.
28224
2
-5:
2090593147126280/3
comment: $x^4-10x^2+4=0$; Galois group $C_2 \times C_2$ (4T2); $h_K=1$; LMFDB 4.4.28224.2.
28400
1
-1:
572/15
comment: $x^4-17x^2+71=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.28400.1.
28400
1
-3:
205007206/15
comment: $x^4-17x^2+71=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.28400.1.
28400
1
-5:
227118328332134804/315
comment: $x^4-17x^2+71=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.28400.1.
28473
1
-1:
152/3
comment: $x^4-2x^3-5x^2+3x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.28473.1.
28473
1
-3:
221954132/15
comment: $x^4-2x^3-5x^2+3x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.28473.1.
28473
1
-5:
46848254710981592/63
comment: $x^4-2x^3-5x^2+3x+2=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.28473.1.
28669
1
-1:
36
comment: $x^4-x^3-6x^2+2x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.28669.1.
28669
1
-3:
70280186/5
comment: $x^4-x^3-6x^2+2x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.28669.1.
28669
1
-5:
15942261769864636/21
comment: $x^4-x^3-6x^2+2x+5=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.28669.1.
28677
1
-1:
38
comment: $x^4-x^3-8x^2+12x-1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.28677.1.
28677
1
-3:
71078451/5
comment: $x^4-x^3-8x^2+12x-1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.28677.1.
28677
1
-5:
15987496471877338/21
comment: $x^4-x^3-8x^2+12x-1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.28677.1.
28749
1
-1:
40
comment: $x^4-x^3-7x^2-x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.28749.1.
28749
1
-3:
71790852/5
comment: $x^4-x^3-7x^2-x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.28749.1.
28749
1
-5:
2315703900302120/3
comment: $x^4-x^3-7x^2-x+1=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.28749.1.
29237
1
-1:
130/3
comment: $x^4-2x^3-6x^2+7x+9=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.29237.1.
29237
1
-3:
230985841/15
comment: $x^4-2x^3-6x^2+7x+9=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.29237.1.
29237
1
-5:
99205927635835690/117
comment: $x^4-2x^3-6x^2+7x+9=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.29237.1.
29248
1
-1:
37
comment: $x^4-2x^3-11x^2+6x+23=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.29248.1.
29248
1
-3:
151022509/10
comment: $x^4-2x^3-11x^2+6x+23=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.29248.1.
29248
1
-5:
5932843317862189/7
comment: $x^4-2x^3-11x^2+6x+23=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.29248.1.
29268
1
-1:
68
comment: $x^4-x^3-9x^2+5x+16=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.29268.1.
29268
1
-3:
17372370
comment: $x^4-x^3-9x^2+5x+16=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.29268.1.
29268
1
-5:
2636876740166404/3
comment: $x^4-x^3-9x^2+5x+16=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.29268.1.
29813
1
-1:
112/3
comment: $x^4-x^3-6x^2+4x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.29813.1.
29813
1
-3:
241525648/15
comment: $x^4-x^3-6x^2+4x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.29813.1.
29813
1
-5:
59307740195142832/63
comment: $x^4-x^3-6x^2+4x+1=0$; Galois group $S_4$ (4T5); $h_K=1$; LMFDB 4.4.29813.1.
29952
1
-1:
178/3
comment: $x^4-8x^2+13=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.29952.1.
29952
1
-3:
53623235/3
comment: $x^4-8x^2+13=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.29952.1.
29952
1
-5:
61978780650712978/63
comment: $x^4-8x^2+13=0$; Galois group $D_4$ (4T3); $h_K=1$; LMFDB 4.4.29952.1.
Definition
Let $K$ be a totally real quartic field [5] with discriminant $D$, and let $\zeta_K(s)$ be its Dedekind zeta function [4], taken by analytic continuation at negative integers. Listed is $\zeta_K(s)$ at negative odd integers $s$.
Parameters
$D$
—   discriminant of $K$ ($D$ the positive discriminant of a totally real quartic field)
$k$
—   index of $K$ among the totally real quartic fields of discriminant $D$ ($k\geq 1$)
$s$
—   argument ($s$ a negative odd integer)
Formulas
(1)
$\zeta_K(2m)=\frac{2^{8m}m^4\pi^{8m}}{((2m)!)^4D^{2m-1/2}}\, \zeta_K(1-2m)$ for $m\geq 1$, from $\Lambda_K(s)=D^{s/2}(\pi^{-s/2}\Gamma(s/2))^4\zeta_K(s)=\Lambda_K(1-s)$. In particular $\zeta_K(2)=16\pi^8\zeta_K(-1)/D^{3/2}$.
(2)
If $K/\mathbb{Q}$ is abelian and $X$ is the group of Dirichlet characters belonging to $K$, then $\zeta_K(1-2m)=\prod_{\chi\in X}L(1-2m,\chi)$, with $L(1-2m,\chi)=-B_{2m,\chi}/(2m)$ and $B_{2m,\chi}$ a generalized Bernoulli number.
Comments
(3)
Non-isomorphic quartic fields can share a discriminant, so $D$ is not always enough to identify $K$. The fields of discriminant $D$ are numbered $k=1,2,\ldots$ by lexicographic order of the coefficient vector $(a_3,a_2,a_1,a_0)$ of PARI's polredabs defining polynomial $x^4+a_3x^3+a_2x^2+a_1x+a_0$. For every field listed here, the LMFDB label [6] is $4.4.D.k$, so $k$ is the last component of that label. Here $D=16448$ and $D=28224$ are the only discriminants carrying two fields; every other $D$ has $k=1$.
(4)
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 quartic field, the values at negative even integers vanish, while $\zeta_K(1-2m)$ has positive sign by the functional equation (1).
(5)
The comment on each entry gives the reduced polynomial $f$, so that $K=\mathbb{Q}(a)$ with $f(a)=0$; the Galois group of the Galois closure as a 4T label; the class number $h_K$; and the LMFDB label.
(6)
Only totally real fields are listed, so every field here has signature $(4,0)$ and $w_K=2$, where $w_K$ is the number of roots of unity in $K$. If $K$ has a complex place, then $\zeta_K(1-2m)=0$ for every $m\geq 1$, because the gamma factor in the functional equation has a pole.
(7)
For a totally real quartic field, Serre's denominator estimates [3] imply that the denominator divides $30$ at $s=-1$, $1020$ at $s=-3$, and $8190$ at $s=-5$.
(8)
Formula (2) gives the abelian factorisation. The factor $\zeta(1-2m)$ is in the table of Riemann zeta values, and the values $B_{2m,\chi}$ are indexed in the generalized-Bernoulli table.
Programs
(P1)
PARI/GP
default(realprecision, 80);
bestappr(lfun(lfuncreate(x^4 - x^3 - 3*x^2 + x + 1), -1), 30)    \\ 2/15, for D = 725
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]
Jean-Pierre Serre, "Congruences et formes modulaires", Seminaire Bourbaki 416 (1971/72), 319-338.
Links
Similar tables
Values of Dedekind zeta functions of totally real cubic fields at negative odd integers —   the degree-three table at the same negative odd arguments
Values of Dedekind zeta functions of real quadratic fields at negative odd integers —   the degree-two table at the same negative odd arguments
Values of the Riemann zeta function at rational numbers —   the factor $\zeta(1-2m)=-B_{2m}/(2m)$ in the abelian factorisation
Bernoulli numbers —   the Bernoulli numbers $B_{2m}$ in $\zeta(1-2m)=-B_{2m}/(2m)$
Generalized Bernoulli numbers —   the generalized Bernoulli numbers $B_{2m,\chi}$ in the abelian factorisation
Data properties
Entries are of type: rational number
How well the digits are known: exact
Table is complete: no (it holds every totally real quartic field with $D\leq 30000$, at $s=-1,-3,-5$)
How they were obtained:

The generator enumerates the totally real quartic fields with $D\leq 30000$ by PARI's nflist [7] over the five transitive groups of degree $4$, and refuses the run unless it finds $204$ fields.

more

Each value is recognised from PARI lfun computations at several working precisions, with the denominator bounded by Serre's theorem [3]: in degree four the possible denominators divide $30$, $1020$ and $8190$ at $s=-1,-3,-5$. The generator refuses a value unless the same rational is obtained at all working precisions and the functional equation predicts PARI's computations of $\zeta_K(2)$, $\zeta_K(4)$ and $\zeta_K(6)$ to relative error below $10^{-60}$. For abelian quartic fields the value is also checked in exact rational arithmetic from the Dirichlet-character factorisation (2).



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