Note: Search results may be incomplete due to uncomputed quantities: Class number (201181 objects)
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Label | Polynomial | Discriminant | Galois group | Class group |
---|---|---|---|---|
8.0.64000000.3 | $x^{8} - 4 x^{7} + 8 x^{6} - 10 x^{5} + 11 x^{4} - 10 x^{3} + 2 x^{2} + 2 x + 1$ | $2^{12}\cdot 5^{6}$ | $D_4$ (as 8T4) | $[2]$ |
8.0.76488065.1 | $x^{8} - 4 x^{7} + 8 x^{6} - 10 x^{5} + 7 x^{4} - 2 x^{3} + x^{2} - x + 1$ | $5\cdot 37\cdot 643^{2}$ | $C_2 \wr S_4$ (as 8T44) | $[2]$ |
8.0.81000000.1 | $x^{8} - 3 x^{7} + 6 x^{6} - 9 x^{5} + 14 x^{4} - 27 x^{3} + 39 x^{2} - 36 x + 16$ | $2^{6}\cdot 3^{4}\cdot 5^{6}$ | $D_4\times C_2$ (as 8T9) | $[2]$ |
8.0.81450625.1 | $x^{8} - x^{7} + 8 x^{6} + 7 x^{5} + 19 x^{4} + 7 x^{3} + 8 x^{2} - x + 1$ | $5^{4}\cdot 19^{4}$ | $D_4$ (as 8T4) | $[2]$ |
8.0.93000677.1 | $x^{8} - 3 x^{7} + 4 x^{6} - 3 x^{5} + 5 x^{4} - 7 x^{3} + 7 x^{2} - 4 x + 1$ | $7^{3}\cdot 11\cdot 157^{2}$ | $C_2 \wr S_4$ (as 8T44) | $[2]$ |
8.0.96059601.1 | $x^{8} - x^{7} - 2 x^{6} - 5 x^{5} + 11 x^{4} + x^{3} + 4 x^{2} - x + 1$ | $3^{8}\cdot 11^{4}$ | $S_4$ (as 8T14) | $[2]$ |
8.0.97831881.1 | $x^{8} - 2 x^{7} + x^{6} - 2 x^{5} + 6 x^{4} - 13 x^{3} + 16 x^{2} - 7 x + 1$ | $3^{4}\cdot 7^{2}\cdot 157^{2}$ | $A_4\wr C_2$ (as 8T42) | $[2]$ |
8.0.100618240.2 | $x^{8} - 2 x^{7} + 8 x^{6} - 4 x^{5} + 13 x^{4} + 8 x^{3} + 10 x^{2} + 6 x + 1$ | $2^{12}\cdot 5\cdot 17^{3}$ | $C_2 \wr C_2\wr C_2$ (as 8T35) | $[2]$ |
8.0.101875968.1 | $x^{8} - 2 x^{7} + 4 x^{6} - 8 x^{4} + 8 x^{3} + 15 x^{2} + 6 x + 1$ | $2^{8}\cdot 3^{4}\cdot 17^{3}$ | $C_4\wr C_2$ (as 8T17) | $[2]$ |
8.0.102468645.1 | $x^{8} - 2 x^{7} + 3 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{2} + 2 x + 1$ | $3^{4}\cdot 5\cdot 503^{2}$ | $C_2 \wr S_4$ (as 8T44) | $[2]$ |
8.0.104934441.1 | $x^{8} - 3 x^{7} + 4 x^{6} - 10 x^{5} + 21 x^{4} - 20 x^{3} + 20 x^{2} - 27 x + 15$ | $3\cdot 23\cdot 29\cdot 229^{2}$ | $C_2 \wr S_4$ (as 8T44) | $[2]$ |
8.0.106172416.1 | $x^{8} - 2 x^{7} + 3 x^{6} - 2 x^{5} + 8 x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1$ | $2^{12}\cdot 7^{2}\cdot 23^{2}$ | $C_2^2 \wr C_2$ (as 8T18) | $[2]$ |
8.0.106770069.1 | $x^{8} - 2 x^{7} + 7 x^{5} - 11 x^{4} - 3 x^{3} + 30 x^{2} - 42 x + 21$ | $3^{6}\cdot 7^{4}\cdot 61$ | $C_2 \wr C_2\wr C_2$ (as 8T35) | $[2]$ |
8.0.110502144.1 | $x^{8} - 4 x^{7} + 11 x^{6} - 14 x^{5} + 7 x^{4} + 18 x^{3} - 18 x^{2} - 36 x + 36$ | $2^{8}\cdot 3^{4}\cdot 73^{2}$ | $D_4\times C_2$ (as 8T9) | $[2]$ |
8.0.110502144.2 | $x^{8} - 9 x^{6} + 29 x^{4} - 36 x^{2} + 16$ | $2^{8}\cdot 3^{4}\cdot 73^{2}$ | $D_4\times C_2$ (as 8T9) | $[2]$ |
8.0.110502144.3 | $x^{8} + x^{6} - 8 x^{5} - 4 x^{4} - 4 x^{3} + 21 x^{2} + 20 x + 25$ | $2^{8}\cdot 3^{4}\cdot 73^{2}$ | $D_4\times C_2$ (as 8T9) | $[2]$ |
8.0.111017597.1 | $x^{8} - 5 x^{6} - 2 x^{5} + 12 x^{4} + 5 x^{3} - 9 x^{2} - 10 x + 9$ | $29\cdot 73\cdot 229^{2}$ | $C_2 \wr S_4$ (as 8T44) | $[2]$ |
8.0.112021056.1 | $x^{8} - x^{7} - x^{6} - 7 x^{5} + 22 x^{4} - 23 x^{3} + 11 x^{2} - 2 x + 1$ | $2^{6}\cdot 3^{6}\cdot 7^{4}$ | $Q_8:C_2$ (as 8T11) | $[2]$ |
8.0.113396024.1 | $x^{8} - 3 x^{7} + 8 x^{6} - 10 x^{5} + 12 x^{4} - 10 x^{3} + 8 x^{2} - 3 x + 1$ | $2^{3}\cdot 7\cdot 1423^{2}$ | $C_2 \wr S_4$ (as 8T44) | $[2]$ |
8.0.119771136.5 | $x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{4} - 6 x^{3} + 8 x^{2} + 4 x + 1$ | $2^{12}\cdot 3^{4}\cdot 19^{2}$ | $S_4\times C_2$ (as 8T24) | $[2]$ |
8.0.121999616.1 | $x^{8} - 4 x^{7} + 7 x^{6} - 6 x^{5} - 4 x^{4} + 4 x^{3} + 11 x^{2} + 6 x + 1$ | $2^{8}\cdot 17^{3}\cdot 97$ | $C_2 \wr C_2\wr C_2$ (as 8T35) | $[2]$ |
8.0.123183909.1 | $x^{8} - 3 x^{7} + 6 x^{6} - 4 x^{5} + 5 x^{4} - 15 x^{3} + 25 x^{2} - x + 1$ | $3^{4}\cdot 29\cdot 229^{2}$ | $C_2 \wr S_4$ (as 8T44) | $[2]$ |
8.0.125333229.1 | $x^{8} + 4 x^{6} - 3 x^{5} + 5 x^{4} - 3 x^{3} + 4 x^{2} + 1$ | $3\cdot 7^{3}\cdot 349^{2}$ | $C_2 \wr S_4$ (as 8T44) | $[2]$ |
8.0.126376848.1 | $x^{8} - 2 x^{7} - 4 x^{6} + 8 x^{5} + 5 x^{4} - 10 x^{3} + 2 x^{2} + 1$ | $2^{4}\cdot 3^{4}\cdot 13^{2}\cdot 577$ | $S_4\wr C_2$ (as 8T47) | $[2]$ |
8.0.126773829.1 | $x^{8} - x^{7} - 2 x^{6} - 4 x^{5} + 7 x^{4} + 5 x^{3} + 7 x^{2} + 2 x + 1$ | $3^{7}\cdot 7^{3}\cdot 13^{2}$ | $C_2 \wr C_2\wr C_2$ (as 8T35) | $[2]$ |
8.0.127828125.1 | $x^{8} - 2 x^{7} + 12 x^{6} - 20 x^{5} + 51 x^{4} - 65 x^{3} + 83 x^{2} - 66 x + 31$ | $3^{4}\cdot 5^{6}\cdot 101$ | $((C_8 : C_2):C_2):C_2$ (as 8T27) | $[2]$ |
8.0.128307240.1 | $x^{8} - 3 x^{7} + 6 x^{6} - 6 x^{5} + 8 x^{4} - 6 x^{3} + 6 x^{2} - 3 x + 1$ | $2^{3}\cdot 3^{4}\cdot 5\cdot 199^{2}$ | $C_2 \wr S_4$ (as 8T44) | $[2]$ |
8.0.128777104.2 | $x^{8} - x^{7} + 2 x^{6} - x^{5} + 6 x^{4} + x^{3} + 2 x^{2} + x + 1$ | $2^{4}\cdot 2837^{2}$ | $C_2^3:S_4$ (as 8T39) | $[2]$ |
8.0.128799801.1 | $x^{8} - 2 x^{7} - 4 x^{6} + 29 x^{4} + 6 x^{3} - 43 x^{2} - 35 x + 49$ | $3^{4}\cdot 13^{2}\cdot 97^{2}$ | $C_2^2 \wr C_2$ (as 8T18) | $[2]$ |
8.0.129267065.1 | $x^{8} - x^{7} + x^{6} + 3 x^{5} + 2 x^{4} + 3 x^{3} + 13 x^{2} + 20 x + 9$ | $5\cdot 17\cdot 29\cdot 229^{2}$ | $C_2 \wr S_4$ (as 8T44) | $[2]$ |
8.0.131974144.1 | $x^{8} + x^{6} + 4 x^{4} - 3 x^{2} + 1$ | $2^{10}\cdot 359^{2}$ | $C_2^3:S_4$ (as 8T39) | $[2]$ |
8.0.132710400.3 | $x^{8} - 4 x^{5} + 13 x^{4} - 24 x^{3} + 22 x^{2} - 8 x + 1$ | $2^{16}\cdot 3^{4}\cdot 5^{2}$ | $Q_8:C_2$ (as 8T11) | $[2]$ |
8.0.132917841.2 | $x^{8} - x^{7} - 6 x^{6} + 11 x^{5} + 37 x^{4} + 9 x^{3} - 30 x^{2} - 9 x + 9$ | $3^{6}\cdot 7^{2}\cdot 61^{2}$ | $C_2^2 \wr C_2$ (as 8T18) | $[2]$ |
8.0.135644301.1 | $x^{8} - x^{7} + x^{6} - 8 x^{5} + 5 x^{4} - 2 x^{3} + 7 x^{2} + 5 x + 1$ | $3^{7}\cdot 13^{2}\cdot 367$ | $S_4\wr C_2$ (as 8T47) | $[2]$ |
8.0.138515453.1 | $x^{8} - 3 x^{7} + 5 x^{6} - 4 x^{5} + 4 x^{4} - 4 x^{3} + 4 x^{2} + 5 x + 1$ | $19\cdot 23\cdot 563^{2}$ | $C_2 \wr S_4$ (as 8T44) | $[2]$ |
8.0.140955633.1 | $x^{8} - x^{7} - 5 x^{6} + 10 x^{5} - 19 x^{3} + 22 x^{2} - 8 x + 1$ | $3^{4}\cdot 7\cdot 13^{2}\cdot 1471$ | $S_4\wr C_2$ (as 8T47) | $[2]$ |
8.0.142850304.2 | $x^{8} + 3 x^{6} - 4 x^{5} + 8 x^{4} - 6 x^{3} + 7 x^{2} + 2 x + 1$ | $2^{8}\cdot 3^{4}\cdot 83^{2}$ | $S_4\times C_2$ (as 8T24) | $[2]$ |
8.0.144733797.1 | $x^{8} - 4 x^{7} + 7 x^{6} - 7 x^{5} - 3 x^{4} + 13 x^{3} - 2 x^{2} - 5 x + 43$ | $3^{4}\cdot 13^{2}\cdot 97\cdot 109$ | $C_2 \wr C_2\wr C_2$ (as 8T35) | $[2]$ |
8.0.144733797.2 | $x^{8} - 2 x^{6} - 5 x^{5} - 7 x^{4} + 41 x^{3} - 76 x + 49$ | $3^{4}\cdot 13^{2}\cdot 97\cdot 109$ | $C_2 \wr C_2\wr C_2$ (as 8T35) | $[2]$ |
8.0.146313216.1 | $x^{8} - 2 x^{7} + 4 x^{5} + 10 x^{4} - 48 x^{3} + 72 x^{2} - 48 x + 12$ | $2^{12}\cdot 3^{6}\cdot 7^{2}$ | $C_2^2 \wr C_2$ (as 8T18) | $[2]$ |
8.0.146313216.2 | $x^{8} - 2 x^{7} - 4 x^{6} + 16 x^{5} - 14 x^{4} - 16 x^{3} + 56 x^{2} - 64 x + 28$ | $2^{12}\cdot 3^{6}\cdot 7^{2}$ | $C_2^2 \wr C_2$ (as 8T18) | $[2]$ |
8.0.148777965.1 | $x^{8} + 2 x^{6} - 3 x^{5} + 6 x^{4} - 3 x^{3} + 2 x^{2} + 3 x + 1$ | $3^{6}\cdot 5\cdot 7^{4}\cdot 17$ | $C_2 \wr C_2\wr C_2$ (as 8T35) | $[2]$ |
8.0.151253973.1 | $x^{8} - 3 x^{7} + 3 x^{6} - x^{4} + 3 x^{2} - 3 x + 1$ | $3^{4}\cdot 13\cdot 379^{2}$ | $C_2 \wr S_4$ (as 8T44) | $[2]$ |
8.0.151725312.1 | $x^{8} - 4 x^{7} + 13 x^{6} - 22 x^{5} + 28 x^{4} - 16 x^{3} - 2 x^{2} + 2 x + 1$ | $2^{8}\cdot 3^{7}\cdot 271$ | $S_4\wr C_2$ (as 8T47) | $[2]$ |
8.0.151807041.1 | $x^{8} - 5 x^{6} + 28 x^{4} + 15 x^{2} + 9$ | $3^{4}\cdot 37^{4}$ | $D_4$ (as 8T4) | $[2]$ |
8.0.153140625.1 | $x^{8} - x^{7} - 9 x^{5} + 9 x^{4} - 4 x^{3} + 20 x^{2} - 16 x + 16$ | $3^{4}\cdot 5^{6}\cdot 11^{2}$ | $D_4\times C_2$ (as 8T9) | $[2]$ |
8.0.153140625.2 | $x^{8} - 3 x^{7} - x^{6} + 13 x^{5} + 6 x^{4} - 35 x^{3} - 15 x^{2} + 25 x + 25$ | $3^{4}\cdot 5^{6}\cdot 11^{2}$ | $C_2^2:C_4$ (as 8T10) | $[2]$ |
8.0.153364736.1 | $x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} + 16 x^{4} - 10 x^{3} + 5 x^{2} - 2 x + 1$ | $2^{8}\cdot 7\cdot 23\cdot 61^{2}$ | $C_2 \wr S_4$ (as 8T44) | $[2]$ |
8.0.153760000.1 | $x^{8} + 7 x^{6} + 18 x^{4} + 17 x^{2} + 1$ | $2^{8}\cdot 5^{4}\cdot 31^{2}$ | $C_2^2 \wr C_2$ (as 8T18) | $[2]$ |
8.0.155356461.1 | $x^{8} - 2 x^{7} - 7 x^{6} + 16 x^{5} + 22 x^{4} - 43 x^{3} - 37 x^{2} + 44 x + 37$ | $3^{6}\cdot 13^{3}\cdot 97$ | $C_2 \wr C_2\wr C_2$ (as 8T35) | $[2]$ |