Number field search results

Degree	Signature	Class number	Class group structure	CM field
Ramified prime count	Ramified	Unramified primes	Galois group	Is Galois
Discriminant	Root discriminant	Regulator	Intermediate field	$p$-adic completions
Monogenic	Index	Ines. \(p\)

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Label	Polynomial	Degree	Signature	Discriminant	Ram. prime count	Root discriminant	CM field	Galois	Monogenic	Galois group	Class group	Unit group torsion	Unit group rank	Regulator
8.0.1257728.1	x^8 - 2x^7 + 4x^5 - 4x^4 + 3x^2 - 2x + 1	$8$	[0,4]	$2^{8}\cdot 17^{3}$	$$	$5.78693149384$			✓	$C_4\wr C_2$ (as 8T17)	trivial	$4$	$3$	$0.618886617067$
8.0.1265625.1	x^8 - x^7 + x^5 - x^4 + x^3 - x + 1	$8$	[0,4]	$3^{4}\cdot 5^{6}$	$$	$5.79146092644$	✓	✓	✓	$C_4\times C_2$ (as 8T2)	trivial	$30$	$3$	$4.6618207773$
8.0.1282789.1	x^8 - x^7 + 2x^6 - 3x^5 + 3x^4 - 3x^3 + 3x^2 - 2x + 1	$8$	[0,4]	$1103\cdot 1163$	$$	$5.80122090621$			✓	$S_8$ (as 8T50)	trivial	$2$	$3$	$0.313539838403$
8.0.1327833.1	x^8 - x^7 + x^6 - 2x^5 + 3x^4 - 4x^3 + 4x^2 - 2x + 1	$8$	[0,4]	$3^{4}\cdot 13^{2}\cdot 97$	$$	$5.82630119728$			✓	$C_2 \wr C_2\wr C_2$ (as 8T35)	trivial	$6$	$3$	$0.963083858701$
8.0.1342413.1	x^8 - x^7 + x^6 + x^4 - 2x^3 + 3x^2 - 3x + 1	$8$	[0,4]	$3^{4}\cdot 16573$	$$	$5.83425985742$			✓	$S_4\wr C_2$ (as 8T47)	trivial	$6$	$3$	$0.970727427757$
8.0.1361513.1	x^8 - x^7 + 2x^6 - 3x^5 + 3x^4 - 3x^3 + 2x^2 - x + 1	$8$	[0,4]	$17\cdot 283^{2}$	$$	$5.84457214701$			✓	$C_2 \wr S_4$ (as 8T44)	trivial	$2$	$3$	$0.32641205382$
8.0.1385533.1	x^8 - 2x^7 + 3x^6 - 3x^5 + x^4 + 1	$8$	[0,4]	$29\cdot 47777$	$$	$5.85736259192$			✓	$S_8$ (as 8T50)	trivial	$2$	$3$	$0.331112529152$
8.0.1424293.1	x^8 - 2x^7 + 2x^6 - x^5 + x^4 - x^3 + 2x^2 - 2x + 1	$8$	[0,4]	$13\cdot 331^{2}$	$$	$5.87759852004$			✓	$C_2 \wr S_4$ (as 8T44)	trivial	$2$	$3$	$0.336709575883$
8.0.1474013.1	x^8 - x^7 + x^6 - x^4 + x^3 - x^2 + 1	$8$	[0,4]	$617\cdot 2389$	$$	$5.90286246256$			✓	$S_8$ (as 8T50)	trivial	$2$	$3$	$0.345105135911$
8.0.1492101.1	x^8 - x^7 + x^6 - 3x^5 + x^4 - 2x^3 + 3x^2 + 1	$8$	[0,4]	$3^{4}\cdot 13^{2}\cdot 109$	$$	$5.91186866834$			✓	$C_2 \wr C_2\wr C_2$ (as 8T35)	trivial	$6$	$3$	$1.04325557006$
8.0.1513728.1	x^8 - 2x^7 + x^6 + 2x^5 - 3x^4 - 2x^3 + x^2 + 2x + 1	$8$	[0,4]	$2^{8}\cdot 3^{4}\cdot 73$	$$	$5.92251242534$			✓	$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31)	trivial	$12$	$3$	$2.10694405423$
8.0.1520789.1	x^8 - x^7 - x^6 + x^4 - x^2 + x + 1	$8$	[0,4]	$29\cdot 229^{2}$	$$	$5.9259586988$			✓	$C_2 \wr S_4$ (as 8T44)	trivial	$2$	$3$	$0.353845256559$
8.0.1578125.1	x^8 - 2x^7 + 3x^5 - x^4 - 3x^3 + 2x + 1	$8$	[0,4]	$5^{6}\cdot 101$	$$	$5.95343580388$			✓	$((C_8 : C_2):C_2):C_2$ (as 8T27)	trivial	$10$	$3$	$1.81195906066$
8.0.1590773.1	x^8 - x^7 + 2x^5 - 2x^4 + x^3 + x^2 - 2x + 1	$8$	[0,4]	$179\cdot 8887$	$$	$5.95937927669$			✓	$S_8$ (as 8T50)	trivial	$2$	$3$	$0.363609073488$
8.0.1601613.1	x^8 - 2x^6 - 3x^5 + 3x^4 + 3x^3 - 2x^2 + 1	$8$	[0,4]	$3^{6}\cdot 13^{3}$	$$	$5.9644403303$			✓	$C_4\wr C_2$ (as 8T17)	trivial	$6$	$3$	$1.10095849199$
8.0.1740113.1	x^8 - 2x^7 + 3x^6 - 4x^5 + 4x^4 - 4x^3 + 3x^2 - x + 1	$8$	[0,4]	$1740113$	$$	$6.0265974216$			✓	$S_8$ (as 8T50)	trivial	$2$	$3$	$0.38876666774366264$
8.0.1750329.1	x^8 - 3x^7 + 4x^6 - 3x^5 + 3x^4 - 3x^3 + 4x^2 - 3x + 1	$8$	[0,4]	$3^{6}\cdot 7^{4}$	$$	$6.03100878452$		✓	not computed	$D_4$ (as 8T4)	trivial	$6$	$3$	$1.1656993933$
8.0.1763584.1	x^8 - 2x^7 + x^6 + 2x^5 - 3x^4 + 2x^3 + x^2 - 2x + 1	$8$	[0,4]	$2^{8}\cdot 83^{2}$	$$	$6.03669895858$			✓	$S_4\times C_2$ (as 8T24)	trivial	$4$	$3$	$0.781195313952$
8.0.1797309.1	x^8 - 3x^7 + 6x^6 - 8x^5 + 10x^4 - 9x^3 + 7x^2 - 4x + 1	$8$	[0,4]	$3^{4}\cdot 22189$	$$	$6.05100962346$			✓	$S_4\wr C_2$ (as 8T47)	trivial	$6$	$3$	$1.20014144807$
8.0.1820637.1	x^8 - x^7 + x^5 - x^4 - 2x^3 + 2x + 1	$8$	[0,4]	$3^{4}\cdot 7\cdot 13^{2}\cdot 19$	$$	$6.06077163648$			✓	$C_2 \wr C_2\wr C_2$ (as 8T35)	trivial	$6$	$3$	$1.21081359453$
8.0.1867553.1	x^8 - x^7 + x^5 - 2x^4 + x^3 + x^2 - x + 1	$8$	[0,4]	$1867553$	$$	$6.08007752654$			✓	$S_8$ (as 8T50)	trivial	$2$	$3$	$0.40885972396034426$
8.0.1890625.1	x^8 - 3x^7 + 6x^6 - 9x^5 + 9x^4 - 7x^3 + 4x^2 - x + 1	$8$	[0,4]	$5^{6}\cdot 11^{2}$	$$	$6.08941642815$			✓	$C_2^2:C_4$ (as 8T10)	trivial	$10$	$3$	$2.04169489066$
8.0.2067245.1	x^8 - x^7 + x^6 - x^4 + x^2 - x + 1	$8$	[0,4]	$5\cdot 643^{2}$	$$	$6.15777751924$			✓	$C_2 \wr S_4$ (as 8T44)	trivial	$2$	$3$	$0.453534943932$
8.0.2105893.1	x^8 - x^7 + 2x^6 - x^5 + x^4 - x^3 + x^2 - 2x + 1	$8$	[0,4]	$29\cdot 72617$	$$	$6.17205144936$			✓	$S_8$ (as 8T50)	trivial	$2$	$3$	$0.44069755405256955$
8.0.2118069.1	x^8 - x^7 - x^3 + 2x^2 - x + 1	$8$	[0,4]	$3^{4}\cdot 79\cdot 331$	$$	$6.17650095624$			✓	$S_4\wr C_2$ (as 8T47)	trivial	$6$	$3$	$1.33864883478$
8.0.2149173.1	x^8 - x^7 - 2x^6 + 2x^5 + x^3 + x^2 - 2x + 1	$8$	[0,4]	$3^{4}\cdot 13^{2}\cdot 157$	$$	$6.18776657236$			✓	$C_2 \wr C_2\wr C_2$ (as 8T35)	trivial	$6$	$3$	$1.34468819349$
8.0.2178981.1	x^8 - x^7 - x^6 + 2x^5 + x^4 - 2x^3 - x^2 + x + 1	$8$	[0,4]	$3^{6}\cdot 7^{2}\cdot 61$	$$	$6.19842970625$			✓	$C_2 \wr C_2\wr C_2$ (as 8T35)	trivial	$6$	$3$	$1.34805397566$
8.0.2193125.1	x^8 - 2x^7 + 4x^6 - 5x^5 + 5x^4 - 5x^3 + 4x^2 - 2x + 1	$8$	[0,4]	$5^{4}\cdot 11^{2}\cdot 29$	$$	$6.20344481555$			✓	$C_2 \wr C_2\wr C_2$ (as 8T35)	trivial	$2$	$3$	$0.454073872008$
8.0.2217213.1	x^8 - x^7 - x^6 - x^5 + 3x^4 - x + 1	$8$	[0,4]	$3^{4}\cdot 31\cdot 883$	$$	$6.21192104402$			✓	$S_4\wr C_2$ (as 8T47)	trivial	$6$	$3$	$1.37910404568$
8.0.2313441.1	x^8 - x^7 + 2x^6 + 3x^5 - x^4 + 3x^3 + 2x^2 - x + 1	$8$	[0,4]	$3^{4}\cdot 13^{4}$	$$	$6.2449979984$		✓	not computed	$D_4$ (as 8T4)	trivial	$6$	$3$	$1.41187738145$
8.0.2314413.1	x^8 - x^6 - x^5 - x^3 + 2x + 1	$8$	[0,4]	$3^{4}\cdot 28573$	$$	$6.24532592023$			✓	$S_4\wr C_2$ (as 8T47)	trivial	$6$	$3$	$1.42029894182$
8.0.2477709.1	x^8 - x^7 + 3x^5 - x^3 + 2x^2 + 2x + 1	$8$	[0,4]	$3^{4}\cdot 13^{2}\cdot 181$	$$	$6.2987777876$			✓	$C_2 \wr C_2\wr C_2$ (as 8T35)	trivial	$6$	$3$	$1.51551812359$
8.0.2560000.1	x^8 - 2x^7 + 2x^6 + 2x^5 - 2x^4 + 2x^3 + 2x^2 - 2x + 1	$8$	[0,4]	$2^{12}\cdot 5^{4}$	$$	$6.32455532034$		✓	not computed	$D_4$ (as 8T4)	trivial	$4$	$3$	$1.0839823354$
8.0.2671805.1	x^8 - x^5 + x^4 - x^3 + 1	$8$	[0,4]	$5\cdot 17^{2}\cdot 43^{2}$	$$	$6.35844025272$			✓	$C_2 \wr S_4$ (as 8T44)	trivial	$2$	$3$	$0.542256876594$
8.0.2708009.1	x^8 - x^7 - x^6 + x^5 + x^4 - x^3 - x^2 + x + 1	$8$	[0,4]	$41\cdot 257^{2}$	$$	$6.36914685919$			✓	$C_2 \wr S_4$ (as 8T44)	trivial	$2$	$3$	$0.563263410972$
8.0.2712933.1	x^8 - 3x^7 + 3x^6 - x^5 + x^4 - 2x^2 + x + 1	$8$	[0,4]	$3^{4}\cdot 33493$	$$	$6.37059334437$			✓	$S_4\wr C_2$ (as 8T47)	trivial	$6$	$3$	$1.57666375612$
8.0.2824713.1	x^8 - 3x^7 + 2x^6 + 3x^5 - 4x^4 + x^2 + 1	$8$	[0,4]	$3^{4}\cdot 43\cdot 811$	$$	$6.40282734558$			✓	$S_4\wr C_2$ (as 8T47)	trivial	$6$	$3$	$1.7400253268$
8.0.2828125.1	x^8 - x^7 - x^6 - 2x^5 + 4x^4 + x^3 + x^2 - 3x + 1	$8$	[0,4]	$5^{6}\cdot 181$	$$	$6.40379359023$			✓	$((C_8 : C_2):C_2):C_2$ (as 8T27)	trivial	$10$	$3$	$2.66425859071$
8.0.2844928.1	x^8 - 2x^7 + 2x^6 - 2x^4 + 2x^3 + x^2 - 2x + 1	$8$	[0,4]	$2^{8}\cdot 11113$	$$	$6.40853720479$			✓	$S_4\wr C_2$ (as 8T47)	trivial	$4$	$3$	$1.11974748455$
8.0.2963293.1	x^8 - 3x^7 + 5x^6 - 6x^5 + 7x^4 - 6x^3 + 5x^2 - 3x + 1	$8$	[0,4]	$37\cdot 283^{2}$	$$	$6.44127475268$			✓	$C_2 \wr S_4$ (as 8T44)	trivial	$2$	$3$	$0.545793348928$
8.0.2970513.1	x^8 - 2x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^2 + x + 1	$8$	[0,4]	$3^{4}\cdot 7\cdot 13^{2}\cdot 31$	$$	$6.44323441823$			✓	$C_2 \wr C_2\wr C_2$ (as 8T35)	trivial	$6$	$3$	$1.66725651269$
8.0.2985984.1	x^8 - 2x^7 + 2x^6 - 2x^5 + 7x^4 - 10x^3 + 8x^2 - 4x + 1	$8$	[0,4]	$2^{12}\cdot 3^{6}$	$$	$6.44741959094$		✓	not computed	$D_4$ (as 8T4)	trivial	$12$	$3$	$3.64163813098$
8.0.3057993.1	x^8 - 2x^7 + 3x^6 - 2x^5 + x^3 + 2x^2 - 3x + 1	$8$	[0,4]	$3^{4}\cdot 19\cdot 1987$	$$	$6.46665305207$			not computed	$S_4\wr C_2$ (as 8T47)	trivial	$6$	$3$	$1.83028307754$
8.0.3134053.1	x^8 - x^7 + x^6 - 2x^5 + 3x^4 - 2x^3 + x^2 - x + 1	$8$	[0,4]	$13\cdot 491^{2}$	$$	$6.48654286035$			✓	$C_2 \wr S_4$ (as 8T44)	trivial	$2$	$3$	$0.570147165225$
8.0.3167829.1	x^8 - 3x^7 + 4x^6 - 4x^4 + 3x^3 + 2x^2 - 3x + 1	$8$	[0,4]	$3^{4}\cdot 7\cdot 37\cdot 151$	$$	$6.49524019945$			✓	$S_4\wr C_2$ (as 8T47)	trivial	$6$	$3$	$1.77871765724$
8.0.3198933.1	x^8 - x^7 - x^6 + 4x^4 - x^3 - 3x^2 + x + 1	$8$	[0,4]	$3^{4}\cdot 73\cdot 541$	$$	$6.50317802496$			✓	$S_4\wr C_2$ (as 8T47)	trivial	$6$	$3$	$2.04684730841$
8.0.3203125.1	x^8 - x^7 + 2x^6 + 2x^5 + 3x^3 + 2x^2 + x + 1	$8$	[0,4]	$5^{7}\cdot 41$	$$	$6.50424266534$			✓	$((C_8 : C_2):C_2):C_2$ (as 8T27)	trivial	$10$	$3$	$3.04388544877$
8.0.3207681.1	x^8 - 3x^7 + 4x^6 - 3x^5 + 2x^4 - x^2 + 1	$8$	[0,4]	$3^{4}\cdot 199^{2}$	$$	$6.50539836897$			✓	$S_4\times C_2$ (as 8T24)	trivial	$6$	$3$	$1.89392987116$
8.0.3211264.1	x^8 - 4x^7 + 8x^6 - 8x^5 + 7x^4 - 8x^3 + 8x^2 - 4x + 1	$8$	[0,4]	$2^{16}\cdot 7^{2}$	$$	$6.50630624679$			not computed	$D_4\times C_2$ (as 8T9)	trivial	$8$	$3$	$2.5180422861$
8.0.3240000.1	x^8 - 3x^7 + 3x^6 + x^5 - 2x^4 - 3x^3 + 7x^2 - 4x + 1	$8$	[0,4]	$2^{6}\cdot 3^{4}\cdot 5^{4}$	$$	$6.51355562433$			✓	$Q_8:C_2$ (as 8T11)	trivial	$6$	$3$	$1.89812936734$

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