Global number field search results

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Degree	Signature	Galois group	Class number	Class group
Regulator	Num. ramified primes	Ramified primes		Unramified primes
CM field	Discriminant		Root discriminant

Results (displaying matches 1-50 of )

Label	Polynomial	Discriminant	Galois group	Class group
8.0.206356816265625.6	x^8 - x^7 + 196x^6 - 111x^5 + 10891x^4 + 2605x^3 + 169260x^2 + 177855x + 801445	\( 3^{4}\cdot 5^{6}\cdot 113^{4} \)	$C_4\times C_2$ (as 8T2)	$[1020]$
8.0.206763233181696.27	x^8 - 4x^7 + 78x^6 - 220x^5 + 2471x^4 - 4580x^3 + 37690x^2 - 35436x + 229678	\( 2^{16}\cdot 3^{4}\cdot 79^{4} \)	$C_2^3$ (as 8T3)	$[10, 120]$
8.0.219087362560000.26	x^8 - 2x^7 + 139x^6 - 188x^5 + 7627x^4 - 5976x^3 + 194639x^2 - 67790x + 1953899	\( 2^{12}\cdot 5^{4}\cdot 11^{2}\cdot 29^{4} \)	$C_2^2 \wr C_2$ (as 8T18)	$[4, 280]$ (GRH)
8.0.220873842330625.5	x^8 - 2x^7 + 41x^6 + 174x^5 + 1035x^4 + 626x^3 + 7859x^2 + 2018x + 10264	\( 5^{4}\cdot 11^{4}\cdot 17^{6} \)	$C_4\times C_2$ (as 8T2)	$[1904]$
8.0.253800743765625.26	x^8 - 2x^7 + 112x^6 - 170x^5 + 5181x^4 - 4870x^3 + 115303x^2 - 46621x + 1038811	\( 3^{4}\cdot 5^{6}\cdot 7^{4}\cdot 17^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 520]$
8.0.274299844000000.84	x^8 - 4x^7 + 88x^6 - 250x^5 + 3221x^4 - 6030x^3 + 56442x^2 - 53468x + 418076	\( 2^{8}\cdot 5^{6}\cdot 7^{4}\cdot 13^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 520]$
8.0.277102632960000.225	x^8 + 74x^6 - 12x^5 + 2503x^4 + 576x^3 + 45054x^2 + 17676x + 322254	\( 2^{16}\cdot 3^{4}\cdot 5^{4}\cdot 17^{4} \)	$C_2^3$ (as 8T3)	$[2, 12, 48]$
8.0.315635197515625.11	x^8 + 157x^6 + 9489x^4 + 256673x^2 + 2686321	\( 5^{6}\cdot 13^{4}\cdot 29^{4} \)	$C_2^2:C_4$ (as 8T10)	$[2, 6, 84]$ (GRH)
8.0.323194101760000.29	x^8 - 4x^7 + 106x^6 - 304x^5 + 4417x^4 - 8332x^3 + 85976x^2 - 81860x + 653129	\( 2^{16}\cdot 5^{4}\cdot 53^{4} \)	$C_2^3$ (as 8T3)	$[6, 168]$
8.0.335563778560000.11	x^8 + 94x^6 + 3989x^4 + 87940x^2 + 792100	\( 2^{12}\cdot 5^{4}\cdot 107^{4} \)	$C_2^3$ (as 8T3)	$[3, 3, 126]$
8.0.361879703274496.13	x^8 + 91x^6 + 2538x^4 + 21680x^2 + 92416	\( 2^{10}\cdot 11^{4}\cdot 17^{6} \)	$C_2^2:C_4$ (as 8T10)	$[4, 272]$
8.0.368041203360000.44	x^8 - 4x^7 + 72x^6 - 202x^5 + 2117x^4 - 3902x^3 + 30370x^2 - 28452x + 172084	\( 2^{8}\cdot 3^{4}\cdot 5^{4}\cdot 73^{4} \)	$C_2^3$ (as 8T3)	$[4, 280]$
8.0.368041203360000.64	x^8 - 4x^7 - 40x^6 + 62x^5 + 737x^4 + 170x^3 - 3158x^2 + 9432x + 43224	\( 2^{8}\cdot 3^{4}\cdot 5^{4}\cdot 73^{4} \)	$D_4\times C_2$ (as 8T9)	$[4, 280]$
8.0.372726462515625.9	x^8 - 2x^7 + 124x^6 - 188x^5 + 6291x^4 - 5950x^3 + 152575x^2 - 62965x + 1488895	\( 3^{4}\cdot 5^{6}\cdot 131^{4} \)	$C_4\times C_2$ (as 8T2)	$[3, 6, 60]$ (GRH)
8.0.390971529000000.7	x^8 - 31x^6 + 206x^4 + 1384x^2 + 3136	\( 2^{6}\cdot 3^{4}\cdot 5^{6}\cdot 13^{6} \)	$C_2^2:C_4$ (as 8T10)	$[2, 520]$
8.0.396014975015625.5	x^8 - x^7 + 21x^6 - 101x^5 - 129x^4 + 245x^3 + 6730x^2 - 6120x + 1920	\( 3^{4}\cdot 5^{6}\cdot 7^{4}\cdot 19^{4} \)	$D_4\times C_2$ (as 8T9)	$[2, 16, 32]$
8.0.396014975015625.24	x^8 - x^7 + 231x^6 - 131x^5 + 15146x^4 + 3565x^3 + 278685x^2 + 290630x + 1488895	\( 3^{4}\cdot 5^{6}\cdot 7^{4}\cdot 19^{4} \)	$C_4\times C_2$ (as 8T2)	$[4, 8, 32]$ (GRH)
8.0.408243035250625.8	x^8 - 4x^7 + 4x^6 + 2x^5 + 451x^4 - 910x^3 + 4496x^2 - 4040x + 52880	\( 5^{4}\cdot 29^{4}\cdot 31^{4} \)	$C_2^3$ (as 8T3)	$[2, 2, 336]$
8.0.408243035250625.15	x^8 - 4x^7 + 172x^6 - 502x^5 + 11122x^4 - 21412x^3 + 320545x^2 - 309922x + 3557609	\( 5^{4}\cdot 29^{4}\cdot 31^{4} \)	$D_4\times C_2$ (as 8T9)	$[2, 2, 336]$ (GRH)
8.0.418382806630464.7	x^8 - 4x^7 + 32x^6 - 82x^5 + 585x^4 - 1038x^3 + 6486x^2 - 5980x + 28792	\( 2^{6}\cdot 3^{4}\cdot 13^{4}\cdot 41^{4} \)	$D_4\times C_2$ (as 8T9)	$[1008]$
8.0.432373800960000.30	x^8 + 28x^6 + 553x^4 + 1008x^2 + 1296	\( 2^{16}\cdot 3^{4}\cdot 5^{4}\cdot 19^{4} \)	$C_2^3$ (as 8T3)	$[2, 4, 4, 48]$
8.0.432373800960000.208	x^8 - 4x^7 + 114x^6 - 328x^5 + 5093x^4 - 9644x^3 + 105916x^2 - 101148x + 857761	\( 2^{16}\cdot 3^{4}\cdot 5^{4}\cdot 19^{4} \)	$C_2^3$ (as 8T3)	$[2, 8, 8, 8]$
8.0.432373800960000.273	x^8 - 4x^7 + 94x^6 - 268x^5 + 3543x^4 - 6644x^3 + 63466x^2 - 60188x + 450526	\( 2^{16}\cdot 3^{4}\cdot 5^{4}\cdot 19^{4} \)	$C_2^3$ (as 8T3)	$[4, 8, 32]$
8.0.432972864000000.134	x^8 - 2x^7 + 129x^6 - 188x^5 + 6776x^4 - 6970x^3 + 170620x^2 - 97980x + 1720945	\( 2^{12}\cdot 3^{4}\cdot 5^{6}\cdot 17^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 816]$
8.0.444638964023296.21	x^8 + 42x^6 + 669x^4 + 2492x^2 + 12996	\( 2^{16}\cdot 7^{4}\cdot 41^{4} \)	$C_2^3$ (as 8T3)	$[2, 4, 168]$
8.0.445848503765625.3	x^8 - x^7 + 238x^6 - 135x^5 + 16081x^4 + 3775x^3 + 305082x^2 + 317763x + 1667191	\( 3^{4}\cdot 5^{6}\cdot 137^{4} \)	$C_4\times C_2$ (as 8T2)	$[1092]$
8.0.449151726240000.68	x^8 + 222x^6 + 14670x^4 + 378189x^2 + 3337929	\( 2^{8}\cdot 3^{4}\cdot 5^{4}\cdot 7^{2}\cdot 29^{4} \)	$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29)	$[4, 8, 32]$ (GRH)
8.0.450203524000000.2	x^8 - 4x^7 + 100x^6 - 286x^5 + 4109x^4 - 7746x^3 + 80340x^2 - 76514x + 653801	\( 2^{8}\cdot 5^{6}\cdot 103^{4} \)	$C_4\times C_2$ (as 8T2)	$[1110]$
8.0.453614388150625.5	x^8 - 4x^7 + 60x^6 - 166x^5 + 1863x^4 - 3454x^3 + 32532x^2 - 30832x + 236864	\( 5^{4}\cdot 13^{4}\cdot 71^{4} \)	$C_2^3$ (as 8T3)	$[1260]$
8.0.472072610381824.13	x^8 - 4x^7 + 102x^6 - 292x^5 + 4153x^4 - 7824x^3 + 79568x^2 - 75704x + 608606	\( 2^{22}\cdot 103^{4} \)	$C_4\times C_2$ (as 8T2)	$[5, 260]$
8.0.472459130015625.4	x^8 - 2x^7 + 132x^6 - 200x^5 + 7091x^4 - 6730x^3 + 181423x^2 - 75741x + 1861021	\( 3^{4}\cdot 5^{6}\cdot 139^{4} \)	$C_4\times C_2$ (as 8T2)	$[3, 3, 6, 24]$
8.0.482823627669504.85	x^8 + 100x^6 + 3705x^4 + 61168x^2 + 394384	\( 2^{16}\cdot 3^{6}\cdot 11^{2}\cdot 17^{4} \)	$D_4\times C_2$ (as 8T9)	$[2, 16, 32]$ (GRH)
8.0.484654236938496.8	x^8 - 6x^6 + 245x^4 + 3984x^2 + 13924	\( 2^{8}\cdot 3^{4}\cdot 17^{4}\cdot 23^{4} \)	$C_2^3$ (as 8T3)	$[2, 6, 84]$
8.0.513366837760000.49	x^8 - 4x^7 - 32x^6 + 110x^5 + 308x^4 - 804x^3 - 719x^2 + 1140x + 11610	\( 2^{12}\cdot 5^{4}\cdot 7^{4}\cdot 17^{4} \)	$D_4\times C_2$ (as 8T9)	$[20, 60]$
8.0.513366837760000.82	x^8 - 2x^7 - 9x^6 + 164x^5 + 413x^4 - 122x^3 + 11549x^2 + 35784x + 63326	\( 2^{12}\cdot 5^{4}\cdot 7^{4}\cdot 17^{4} \)	$D_4$ (as 8T4)	$[2, 12, 60]$
8.0.513366837760000.109	x^8 + 106x^6 + 4967x^4 + 119134x^2 + 1164241	\( 2^{12}\cdot 5^{4}\cdot 7^{4}\cdot 17^{4} \)	$C_2^3$ (as 8T3)	$[4, 20, 20]$ (GRH)
8.0.524318404000000.2	x^8 - 4x^7 + 104x^6 - 298x^5 + 4429x^4 - 8366x^3 + 89586x^2 - 85452x + 750956	\( 2^{8}\cdot 5^{6}\cdot 107^{4} \)	$C_4\times C_2$ (as 8T2)	$[1050]$
8.0.537859958308864.20	x^8 + 66x^6 - 4x^5 + 1883x^4 + 160x^3 + 27082x^2 + 4252x + 157682	\( 2^{16}\cdot 17^{2}\cdot 73^{4} \)	$D_4\times C_2$ (as 8T9)	$[8, 160]$
8.0.552395897634816.13	x^8 + 90x^6 - 12x^5 + 3575x^4 + 672x^3 + 73422x^2 + 24636x + 601534	\( 2^{16}\cdot 3^{4}\cdot 101^{4} \)	$C_2^3$ (as 8T3)	$[3, 420]$
8.0.566660933138881.4	x^8 - 4x^7 - 44x^6 + 146x^5 + 643x^4 - 1534x^3 + 1496x^2 - 704x + 8192	\( 7^{4}\cdot 17^{4}\cdot 41^{4} \)	$C_2^3$ (as 8T3)	$[2730]$
8.0.567125647360000.45	x^8 - 4x^7 + 122x^6 - 352x^5 + 5817x^4 - 11052x^3 + 128720x^2 - 123252x + 1107009	\( 2^{16}\cdot 5^{4}\cdot 61^{4} \)	$C_2^3$ (as 8T3)	$[6, 240]$
8.0.568987363143936.85	x^8 + 14x^6 + 433x^4 + 7572x^2 + 36864	\( 2^{8}\cdot 3^{4}\cdot 11^{4}\cdot 37^{4} \)	$C_2^3$ (as 8T3)	$[4, 8, 32]$
8.0.574130719050625.22	x^8 - 4x^7 + 102x^6 - 292x^5 + 4212x^4 - 7942x^3 + 81635x^2 - 77712x + 666304	\( 5^{4}\cdot 11^{4}\cdot 89^{4} \)	$D_4\times C_2$ (as 8T9)	$[16, 96]$ (GRH)
8.0.580851645546496.2	x^8 + 160x^6 - 216x^5 + 7560x^4 - 11392x^3 + 124720x^2 - 158864x + 691934	\( 2^{28}\cdot 1471^{2} \)	$C_2^3: C_4$ (as 8T20)	$[2, 532]$ (GRH)
8.0.585949800960000.136	x^8 + 110x^6 + 5317x^4 + 130980x^2 + 1313316	\( 2^{12}\cdot 3^{4}\cdot 5^{4}\cdot 41^{4} \)	$C_2^3$ (as 8T3)	$[2, 2, 2, 6, 30]$
8.0.585949800960000.164	x^8 - 14x^6 + 317x^4 - 1876x^2 + 37636	\( 2^{12}\cdot 3^{4}\cdot 5^{4}\cdot 41^{4} \)	$D_4\times C_2$ (as 8T9)	$[2, 12, 60]$
8.0.589770161913856.75	x^8 + 152x^6 + 7220x^4 + 109744x^2 + 638401	\( 2^{24}\cdot 7^{4}\cdot 11^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 24, 24]$
8.0.597466897514496.16	x^8 - 4x^7 + 102x^6 - 292x^5 + 4151x^4 - 7820x^3 + 79882x^2 - 76020x + 607150	\( 2^{16}\cdot 3^{4}\cdot 103^{4} \)	$C_2^3$ (as 8T3)	$[30, 60]$
8.0.598885164000000.4	x^8 + 201x^6 + 12921x^4 + 281961x^2 + 1848411	\( 2^{8}\cdot 3^{6}\cdot 5^{6}\cdot 59^{3} \)	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$[2, 2, 2, 160]$ (GRH)
8.0.600953971539968.31	x^8 + 184x^6 + 10580x^4 + 194672x^2 + 559682	\( 2^{31}\cdot 23^{4} \)	$C_8$ (as 8T1)	$[2, 514]$

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