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
8.0.13932219912780625.3	x^8 - 2x^7 - 14x^6 + 44x^5 + 673x^4 - 1522x^3 + 2284x^2 - 6776x + 108736	\( 5^{4}\cdot 17^{6}\cdot 31^{4} \)	$C_4\times C_2$ (as 8T2)	$[6, 1746]$
8.0.19698518008285281.1	x^8 - 4x^7 - 52x^6 + 170x^5 + 839x^4 - 1966x^3 + 7900x^2 - 6888x + 10272	\( 3^{4}\cdot 11^{4}\cdot 359^{4} \)	$D_4$ (as 8T4)	$[9, 3420]$
8.0.21105297771390625.4	x^8 - 2x^7 - 24x^6 + 54x^5 + 499x^4 - 1108x^3 + 8084x^2 - 7504x + 37376	\( 5^{6}\cdot 13^{6}\cdot 23^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 5340]$ (GRH)
8.0.22622316185780224.21	x^8 - 4x^7 + 270x^6 - 796x^5 + 28009x^4 - 54696x^3 + 1321928x^2 - 1294712x + 23971358	\( 2^{22}\cdot 271^{4} \)	$C_4\times C_2$ (as 8T2)	$[4, 4, 660]$ (GRH)
8.0.23977322427515625.10	x^8 - 2x^7 + 364x^6 - 548x^5 + 51171x^4 - 50230x^3 + 3286975x^2 - 1538965x + 81361795	\( 3^{4}\cdot 5^{6}\cdot 7^{4}\cdot 53^{4} \)	$C_4\times C_2$ (as 8T2)	$[10, 1120]$ (GRH)
8.0.28456893697490944.138	x^8 - 4x^7 + 286x^6 - 844x^5 + 31385x^4 - 61368x^3 + 1565000x^2 - 1534456x + 29943358	\( 2^{22}\cdot 7^{4}\cdot 41^{4} \)	$C_4\times C_2$ (as 8T2)	$[4, 4, 728]$ (GRH)
8.0.32135409961317441.4	x^8 - 2x^7 + 36x^6 - 206x^5 + 1127x^4 - 6256x^3 + 21556x^2 - 11904x + 69376	\( 3^{4}\cdot 17^{4}\cdot 41^{6} \)	$C_4\times C_2$ (as 8T2)	$[2, 6630]$ (GRH)
8.0.34900526103080625.25	x^8 - 2x^7 - 6x^6 + 32x^5 + 797x^4 - 1726x^3 + 6024x^2 - 11840x + 169984	\( 3^{4}\cdot 5^{4}\cdot 13^{4}\cdot 17^{6} \)	$C_4\times C_2$ (as 8T2)	$[2, 2, 12, 480]$ (GRH)
8.0.35353337448628224.54	x^8 - 4x^7 + 302x^6 - 892x^5 + 34953x^4 - 68424x^3 + 1836168x^2 - 1802104x + 36967006	\( 2^{22}\cdot 3^{4}\cdot 101^{4} \)	$C_4\times C_2$ (as 8T2)	$[10680]$ (GRH)
8.0.36393705691566336.1	x^8 - 26x^6 + 745x^4 + 6324x^2 + 82944	\( 2^{8}\cdot 3^{4}\cdot 1151^{4} \)	$D_4$ (as 8T4)	$[14, 1148]$
8.0.36466485444000000.13	x^8 + 721x^6 + 148526x^4 + 8741816x^2 + 112550881	\( 2^{8}\cdot 3^{4}\cdot 5^{6}\cdot 103^{4} \)	$C_4\times C_2$ (as 8T2)	$[12, 888]$ (GRH)
8.0.39008669412515625.7	x^8 - 2x^7 + 412x^6 - 620x^5 + 65331x^4 - 64270x^3 + 4719103x^2 - 2226421x + 130966711	\( 3^{4}\cdot 5^{6}\cdot 419^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 6, 6, 144]$ (GRH)
8.0.39569302579314688.45	x^8 - 4x^7 + 590x^6 - 684x^5 + 72253x^4 - 9616x^3 + 3058644x^2 + 1406944x + 39051074	\( 2^{22}\cdot 2113^{3} \)	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$[2, 6338]$ (GRH)
8.0.40904253504000000.104	x^8 - 2x^7 + 417x^6 - 620x^5 + 66896x^4 - 67450x^3 + 4889788x^2 - 2564556x + 137261521	\( 2^{12}\cdot 3^{4}\cdot 5^{6}\cdot 53^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 6888]$ (GRH)
8.0.41421204484000000.110	x^8 - 4x^7 + 316x^6 - 934x^5 + 38561x^4 - 75570x^3 + 2148000x^2 - 2110370x + 46274405	\( 2^{8}\cdot 5^{6}\cdot 11^{4}\cdot 29^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 4, 4, 520]$ (GRH)
8.0.41421204484000000.120	x^8 - 2x^7 + 545x^6 - 652x^5 + 110229x^4 - 63918x^3 + 9782735x^2 - 1970538x + 323501201	\( 2^{8}\cdot 5^{6}\cdot 11^{4}\cdot 29^{4} \)	$C_2^2:C_4$ (as 8T10)	$[2, 2, 4, 4, 208]$ (GRH)
8.0.42285999987360000.25	x^8 - 2x^7 + 127x^6 + 110x^5 + 4322x^4 + 17864x^3 - 2744x^2 - 38048x + 49984	\( 2^{8}\cdot 3^{4}\cdot 5^{4}\cdot 239^{4} \)	$D_4$ (as 8T4)	$[4, 16, 240]$ (GRH)
8.0.42522399132160000.2	x^8 - 36x^6 + 1284x^4 + 40160x^2 + 230400	\( 2^{12}\cdot 5^{4}\cdot 359^{4} \)	$D_4$ (as 8T4)	$[8, 2128]$
8.0.43433280913014784.35	x^8 - 4x^7 + 318x^6 - 940x^5 + 38713x^4 - 75864x^3 + 2136968x^2 - 2099192x + 45159038	\( 2^{22}\cdot 11^{4}\cdot 29^{4} \)	$C_4\times C_2$ (as 8T2)	$[5, 5, 520]$ (GRH)
8.0.43673110606265625.2	x^8 - 2x^7 + 424x^6 - 638x^5 + 69141x^4 - 68050x^3 + 5132575x^2 - 2425465x + 146290645	\( 3^{4}\cdot 5^{6}\cdot 431^{4} \)	$C_4\times C_2$ (as 8T2)	$[6, 2100]$ (GRH)
8.0.45735244164000000.42	x^8 - 2x^7 + 429x^6 - 638x^5 + 70751x^4 - 71320x^3 + 5312995x^2 - 2782980x + 153124945	\( 2^{8}\cdot 3^{4}\cdot 5^{6}\cdot 109^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 4, 1632]$ (GRH)
8.0.45799819979655625.7	x^8 - 7x^6 + 911x^4 + 35635x^2 + 562500	\( 5^{4}\cdot 13^{4}\cdot 37^{6} \)	$D_4$ (as 8T4)	$[43, 344]$ (GRH)
8.0.46138325148368896.5	x^8 + 456x^6 + 64980x^4 + 2963088x^2 + 45252529	\( 2^{24}\cdot 229^{4} \)	$C_4\times C_2$ (as 8T2)	$[11570]$ (GRH)
8.0.46156279310015625.25	x^8 - x^7 + 763x^6 - 435x^5 + 166006x^4 + 36625x^3 + 10264857x^2 + 10395438x + 151685491	\( 3^{4}\cdot 5^{6}\cdot 19^{4}\cdot 23^{4} \)	$C_4\times C_2$ (as 8T2)	$[4, 8, 336]$ (GRH)
8.0.46390687886520576.1	x^8 - 98x^6 + 3697x^4 - 48828x^2 + 419904	\( 2^{8}\cdot 3^{4}\cdot 1223^{4} \)	$D_4$ (as 8T4)	$[7, 14, 210]$
8.0.46714881157697536.8	x^8 - 68x^6 + 1652x^4 + 1504x^2 + 61504	\( 2^{12}\cdot 7^{4}\cdot 41^{6} \)	$D_4\times C_2$ (as 8T9)	$[8, 1400]$ (GRH)
8.0.47007063923765625.4	x^8 - 2x^7 + 432x^6 - 650x^5 + 71741x^4 - 70630x^3 + 5421223x^2 - 2564541x + 157228171	\( 3^{4}\cdot 5^{6}\cdot 439^{4} \)	$C_4\times C_2$ (as 8T2)	$[10, 1200]$ (GRH)
8.0.47956879384510464.30	x^8 - 4x^7 + 326x^6 - 964x^5 + 40665x^4 - 79728x^3 + 2298960x^2 - 2259256x + 49731358	\( 2^{22}\cdot 3^{4}\cdot 109^{4} \)	$C_4\times C_2$ (as 8T2)	$[17808]$ (GRH)
8.0.48014450884000000.9	x^8 - 4x^7 + 328x^6 - 970x^5 + 41501x^4 - 81390x^3 + 2394882x^2 - 2354348x + 53385356	\( 2^{8}\cdot 5^{6}\cdot 331^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 5412]$ (GRH)
8.0.48597320704000000.14	x^8 + 664x^6 + 130891x^4 + 6861444x^2 + 47458321	\( 2^{16}\cdot 5^{6}\cdot 83^{4} \)	$C_4\times C_2$ (as 8T2)	$[10, 1060]$ (GRH)
8.0.49753187290350625.8	x^8 - 4x^7 + 568x^6 - 1690x^5 + 119932x^4 - 237052x^3 + 11157049x^2 - 11038804x + 388332179	\( 5^{4}\cdot 29^{4}\cdot 103^{4} \)	$D_4\times C_2$ (as 8T9)	$[30, 480]$ (GRH)
8.0.51575965646730625.6	x^8 - 2x^7 + 217x^6 + 590x^5 + 12795x^4 + 15666x^3 + 261955x^2 + 111746x + 1180136	\( 5^{4}\cdot 17^{6}\cdot 43^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 2, 5740]$ (GRH)
8.0.51713670760960000.177	x^8 - 2x^7 + 567x^6 - 980x^5 + 120649x^4 - 157308x^3 + 11407937x^2 - 8114182x + 405473849	\( 2^{12}\cdot 5^{4}\cdot 13^{4}\cdot 29^{4} \)	$D_4\times C_2$ (as 8T9)	$[4, 4, 1260]$ (GRH)
8.0.52161356264177664.105	x^8 + 476x^6 + 42483x^4 + 962948x^2 + 4092529	\( 2^{16}\cdot 3^{4}\cdot 7^{6}\cdot 17^{4} \)	$D_4$ (as 8T4)	$[2, 4, 36, 36]$ (GRH)
8.0.52361395482515625.47	x^8 - 2x^7 + 444x^6 - 668x^5 + 75731x^4 - 74590x^3 + 5874175x^2 - 2782965x + 174770095	\( 3^{4}\cdot 5^{6}\cdot 11^{4}\cdot 41^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 12, 12, 60]$ (GRH)
8.0.52430660701675776.32	x^8 - 4x^7 + 208x^6 - 274x^5 + 11301x^4 + 11338x^3 + 127102x^2 - 137576x + 3412048	\( 2^{8}\cdot 3^{4}\cdot 13^{4}\cdot 97^{4} \)	$D_4\times C_2$ (as 8T9)	$[2, 4, 1680]$ (GRH)
8.0.55060156955384064.46	x^8 + 431x^6 - 204x^5 + 40569x^4 + 31212x^3 + 1181948x^2 + 655248x + 10316944	\( 2^{8}\cdot 3^{6}\cdot 11^{4}\cdot 67^{4} \)	$D_4\times C_2$ (as 8T9)	$[2, 2, 2720]$ (GRH)
8.0.57779667567968256.198	x^8 - 4x^7 + 282x^6 - 832x^5 + 26883x^4 - 52384x^3 + 993434x^2 - 967380x + 15658114	\( 2^{16}\cdot 3^{4}\cdot 17^{4}\cdot 19^{4} \)	$D_4\times C_2$ (as 8T9)	$[4, 12, 288]$ (GRH)
8.0.58160617516265625.1	x^8 - 2x^7 + 456x^6 - 686x^5 + 79829x^4 - 78658x^3 + 6351679x^2 - 3013449x + 193740301	\( 3^{4}\cdot 5^{6}\cdot 463^{4} \)	$C_4\times C_2$ (as 8T2)	$[5, 3150]$ (GRH)
8.0.58664715264000000.212	x^8 + 696x^6 + 143811x^4 + 7902036x^2 + 57289761	\( 2^{16}\cdot 3^{4}\cdot 5^{6}\cdot 29^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 2, 4, 816]$ (GRH)
8.0.69668698738130944.20	x^8 - 4x^7 + 358x^6 - 1060x^5 + 48953x^4 - 96144x^3 + 3028688x^2 - 2980792x + 71572318	\( 2^{22}\cdot 359^{4} \)	$C_4\times C_2$ (as 8T2)	$[19380]$ (GRH)
8.0.73190108211183616.41	x^8 + 512x^6 + 81920x^4 + 4194304x^2 + 71385601	\( 2^{24}\cdot 257^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 4, 16, 80]$ (GRH)
8.0.76089400370397184.11	x^8 - 4x^7 + 366x^6 - 1084x^5 + 51145x^4 - 100488x^3 + 3232520x^2 - 3182456x + 78004958	\( 2^{22}\cdot 367^{4} \)	$C_4\times C_2$ (as 8T2)	$[4, 20, 180]$ (GRH)
8.0.76600525824000000.16	x^8 + 744x^6 + 164331x^4 + 9652284x^2 + 74805201	\( 2^{16}\cdot 3^{4}\cdot 5^{6}\cdot 31^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 4, 12, 120]$ (GRH)
8.0.81824822383869952.9	x^8 + 556x^6 - 472x^5 + 93514x^4 - 43152x^3 + 5460652x^2 + 1871256x + 79418063	\( 2^{28}\cdot 673^{3} \)	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$[2, 2, 2568]$ (GRH)
8.0.83644662035775488.51	x^8 + 632x^6 + 124820x^4 + 7888624x^2 + 77900162	\( 2^{31}\cdot 79^{4} \)	$C_8$ (as 8T1)	$[2, 2, 54, 54]$ (GRH)
8.0.90251618821341184.12	x^8 - 4x^7 + 382x^6 - 1132x^5 + 55673x^4 - 109464x^3 + 3667208x^2 - 3612664x + 92160766	\( 2^{22}\cdot 383^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 2, 5304]$ (GRH)
8.0.95805052718831569.4	x^8 - 2x^7 + 240x^6 - 354x^5 + 23086x^4 - 23500x^3 + 1051985x^2 - 574924x + 18900752	\( 17^{6}\cdot 251^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 8, 952]$ (GRH)
8.0.102865042331926528.42	x^8 + 460x^6 - 496x^5 + 73084x^4 - 123512x^3 + 4713712x^2 - 6499440x + 106212478	\( 2^{26}\cdot 1153^{3} \)	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$[2, 2, 2508]$ (GRH)
8.0.104603249369153536.23	x^8 + 560x^6 + 98000x^4 + 5488000x^2 + 101626561	\( 2^{24}\cdot 281^{4} \)	$C_4\times C_2$ (as 8T2)	$[10, 1040]$ (GRH)

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