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
6.0.9747.1	x^6 - x^5 + x^4 - 2x^3 + 4x^2 - 3x + 1	$6$	[0,3]	$-\,3^{3}\cdot 19^{2}$	$$	$4.62180723059$			✓	$S_3\times C_3$ (as 6T5)	trivial	$6$	$2$	$0.601543105945$
6.0.10051.1	x^6 - x^5 + 2x^4 - 2x^3 + 2x^2 - 2x + 1	$6$	[0,3]	$-\,19\cdot 23^{2}$	$$	$4.64552582629$			✓	$S_4\times C_2$ (as 6T11)	trivial	$2$	$2$	$0.205216461048$
6.0.10571.1	x^6 - x^5 - x^4 + 2x^3 - x + 1	$6$	[0,3]	$-\,11\cdot 31^{2}$	$$	$4.68474560175$			✓	$S_4\times C_2$ (as 6T11)	trivial	$2$	$2$	$0.21320935989$
6.0.10816.1	x^6 - x^4 - 2x^3 + 2x + 1	$6$	[0,3]	$-\,2^{6}\cdot 13^{2}$	$$	$4.70266937544$			✓	$S_3\times C_3$ (as 6T5)	trivial	$4$	$2$	$0.434147804699$
6.0.11691.1	x^6 - x^5 - x^2 + x + 1	$6$	[0,3]	$-\,3^{3}\cdot 433$	$$	$4.76403865609$			✓	$C_3^2:D_4$ (as 6T13)	trivial	$6$	$2$	$0.691359666578$
6.0.12167.1	x^6 - 3x^5 + 5x^4 - 5x^3 + 5x^2 - 3x + 1	$6$	[0,3]	$-\,23^{3}$	$$	$4.79583152331$		✓	✓	$S_3$ (as 6T2)	trivial	$2$	$2$	$0.237219601798$
6.0.14283.1	x^6 - x^5 + x^4 - 2x^3 + x^2 + 1	$6$	[0,3]	$-\,3^{3}\cdot 23^{2}$	$$	$4.92572209907$			✓	$D_{6}$ (as 6T3)	trivial	$6$	$2$	$0.803563966653$
6.0.14731.1	x^6 - x^5 + x^3 - x^2 + 1	$6$	[0,3]	$-\,14731$	$$	$4.95114187916$			✓	$S_6$ (as 6T16)	trivial	$2$	$2$	$0.277754459978$
6.0.16551.1	x^6 - x^5 - x^4 + 3x^3 - 2x + 1	$6$	[0,3]	$-\,3^{3}\cdot 613$	$$	$5.04820953884$			✓	$C_3^2:D_4$ (as 6T13)	trivial	$6$	$2$	$0.932873082902$
6.0.16807.1	x^6 - x^5 + x^4 - x^3 + x^2 - x + 1	$6$	[0,3]	$-\,7^{5}$	$$	$5.0611401848$	✓	✓	✓	$C_6$ (as 6T1)	trivial	$14$	$2$	$2.10181872849$
6.0.18515.1	x^6 + 2x^4 - x^3 + 2x^2 + 1	$6$	[0,3]	$-\,5\cdot 7\cdot 23^{2}$	$$	$5.14344325363$			✓	$S_4\times C_2$ (as 6T11)	trivial	$2$	$2$	$0.328780614749$
6.0.19683.1	x^6 - x^3 + 1	$6$	[0,3]	$-\,3^{9}$	$$	$5.19615242271$	✓	✓	✓	$C_6$ (as 6T1)	trivial	$18$	$2$	$3.39714980258$
6.0.20627.1	x^6 - x^5 + x^4 - 2x^3 + 2x^2 - x + 1	$6$	[0,3]	$-\,20627$	$$	$5.23688067144$			✓	$S_6$ (as 6T16)	trivial	$2$	$2$	$0.390200557232$
6.0.21168.1	x^6 - 3x^5 + 4x^4 - x^3 - 2x^2 + x + 1	$6$	[0,3]	$-\,2^{4}\cdot 3^{3}\cdot 7^{2}$	$$	$5.25952638099$			✓	$S_3\times C_3$ (as 6T5)	trivial	$6$	$2$	$1.11901604193$
6.0.21296.1	x^6 - x^5 + 2x^4 - 3x^3 + 2x^2 - x + 1	$6$	[0,3]	$-\,2^{4}\cdot 11^{3}$	$$	$5.26481368119$		✓	✓	$S_3$ (as 6T2)	trivial	$2$	$2$	$0.371341380446$
6.0.22291.1	x^6 + x^4 - x^3 - x + 1	$6$	[0,3]	$-\,22291$	$$	$5.30503506276$			✓	$S_6$ (as 6T16)	trivial	$2$	$2$	$0.373661196485$
6.0.22592.1	x^6 - 2x^5 + 2x^4 - x^2 + 1	$6$	[0,3]	$-\,2^{6}\cdot 353$	$$	$5.316907603$			✓	$C_3^2:D_4$ (as 6T13)	trivial	$4$	$2$	$0.752120362427$
6.0.22707.1	x^6 - x^5 + 4x^4 - 4x^3 + 5x^2 - 3x + 1	$6$	[0,3]	$-\,3^{3}\cdot 29^{2}$	$$	$5.32140883904$			✓	$D_{6}$ (as 6T3)	trivial	$6$	$2$	$1.25603528677$
6.0.22747.1	x^6 - x^5 - 2x^3 + x^2 + x + 1	$6$	[0,3]	$-\,23^{2}\cdot 43$	$$	$5.32297003373$			✓	$S_4\times C_2$ (as 6T11)	trivial	$2$	$2$	$0.388951542699$
6.0.23031.1	x^6 + x^4 - x^3 + x^2 - 2x + 1	$6$	[0,3]	$-\,3^{3}\cdot 853$	$$	$5.33398920661$			✓	$C_3^2:D_4$ (as 6T13)	trivial	$6$	$2$	$1.18590232532$
6.0.24003.1	x^6 - x^5 + 3x^3 - x^2 - 2x + 1	$6$	[0,3]	$-\,3^{3}\cdot 7\cdot 127$	$$	$5.3708652218$			✓	$C_3^2:D_4$ (as 6T13)	trivial	$6$	$2$	$1.15781781619$
6.0.25747.1	x^6 - x^5 + 2x^4 - 2x^3 + 3x^2 - x + 1	$6$	[0,3]	$-\,25747$	$$	$5.43401841081$			✓	$S_6$ (as 6T16)	trivial	$2$	$2$	$0.457055178835$
6.0.25947.1	x^6 - x^4 - 2x^3 + x^2 + x + 1	$6$	[0,3]	$-\,3^{3}\cdot 31^{2}$	$$	$5.44103089586$			✓	$D_{6}$ (as 6T3)	trivial	$6$	$2$	$1.37763890943$
6.0.27556.1	x^6 - x^5 + 2x^3 - x + 1	$6$	[0,3]	$-\,2^{2}\cdot 83^{2}$	$$	$5.4958646601$			✓	$S_4\times C_2$ (as 6T11)	trivial	$2$	$2$	$0.594769419961$
6.0.27848.1	x^6 - x^5 + 2x^4 + 2x^2 - x + 1	$6$	[0,3]	$-\,2^{3}\cdot 59^{2}$	$$	$5.5055283296$			not computed	$S_4\times C_2$ (as 6T11)	trivial	$2$	$2$	$0.599228830431$
6.0.27971.1	x^6 - x^5 + x^4 - 2x^3 + 3x^2 - 2x + 1	$6$	[0,3]	$-\,83\cdot 337$	$$	$5.50957372539$			✓	$S_6$ (as 6T16)	trivial	$2$	$2$	$0.436494860807$
6.2.28037.1	x^6 - 2x^5 + 3x^3 - 2x - 1	$6$	[2,2]	$23^{2}\cdot 53$	$$	$5.51173831771$			✓	$S_4\times C_2$ (as 6T11)	trivial	$2$	$3$	$0.478924705547$
6.2.29077.1	x^6 - x^5 + x^4 - x^2 + 2x - 1	$6$	[2,2]	$29077$	$$	$5.54529854883$			✓	$S_6$ (as 6T16)	trivial	$2$	$3$	$0.491602105294$
6.0.29095.1	x^6 - x^5 + 3x^4 - x^3 + 3x^2 - x + 1	$6$	[0,3]	$-\,5\cdot 11\cdot 23^{2}$	$$	$5.54587053377$			✓	$S_4\times C_2$ (as 6T11)	trivial	$2$	$2$	$0.446620658377$
6.2.29189.1	x^6 - 2x^5 + x^4 + x^3 - 2x^2 + x + 1	$6$	[2,2]	$17^{2}\cdot 101$	$$	$5.54885278411$			✓	$S_6$ (as 6T16)	trivial	$2$	$3$	$0.49291662671$
6.0.29791.1	x^6 - 3x^5 + 7x^4 - 9x^3 + 7x^2 - 3x + 1	$6$	[0,3]	$-\,31^{3}$	$$	$5.56776436283$		✓	✓	$S_3$ (as 6T2)	trivial	$2$	$2$	$0.438333916947$
6.2.30125.1	x^6 - x^5 + x^4 - 2x^2 + x - 1	$6$	[2,2]	$5^{3}\cdot 241$	$$	$5.57811986838$			✓	$C_3^2:D_4$ (as 6T13)	trivial	$2$	$3$	$0.50437245294$
6.0.30808.1	x^6 - x^4 - x^3 + x^2 + x + 1	$6$	[0,3]	$-\,2^{3}\cdot 3851$	$$	$5.59900150547$			✓	$S_6$ (as 6T16)	trivial	$2$	$2$	$0.643679055537$
6.0.30976.1	x^6 - 2x^5 + 2x^4 - 2x^3 + 4x^2 - 4x + 2	$6$	[0,3]	$-\,2^{8}\cdot 11^{2}$	$$	$5.60407866131$			✓	$D_{6}$ (as 6T3)	trivial	$4$	$2$	$1.29138778296$
6.2.31133.1	x^6 - 2x^4 - x^3 + 2x + 1	$6$	[2,2]	$163\cdot 191$	$$	$5.60880268418$			✓	$S_6$ (as 6T16)	trivial	$2$	$3$	$0.516354684031$
6.0.31211.1	x^6 - x^4 - x^3 + 2x^2 + 3x + 1	$6$	[0,3]	$-\,23^{2}\cdot 59$	$$	$5.61114227353$			✓	$S_4\times C_2$ (as 6T11)	trivial	$2$	$2$	$0.482550424804$
6.2.31213.1	x^6 - 2x^5 + 2x^4 - 3x^3 + 2x^2 - 2x + 1	$6$	[2,2]	$7^{4}\cdot 13$	$$	$5.6112021989$			✓	$A_4\times C_2$ (as 6T6)	trivial	$2$	$3$	$0.517040346258$
6.0.31223.1	x^6 - x^5 + x^4 - x^3 + 2x^2 - x + 1	$6$	[0,3]	$-\,31223$	$$	$5.61150177778$			✓	$S_6$ (as 6T16)	trivial	$2$	$2$	$0.649982527739$
6.0.31347.1	x^6 - 3x^4 - 2x^3 + 3x^2 + 3x + 1	$6$	[0,3]	$-\,3^{6}\cdot 43$	$$	$5.61520992871$			✓	$C_3^2:D_4$ (as 6T13)	trivial	$6$	$2$	$1.57963516749$
6.2.31709.1	x^6 - x^5 + 2x^3 - 2x^2 + 1	$6$	[2,2]	$37\cdot 857$	$$	$5.62596584169$			✓	$S_6$ (as 6T16)	trivial	$2$	$3$	$0.522986177098$
6.0.32171.1	x^6 - x^5 + 2x^4 - x^3 + x^2 + 1	$6$	[0,3]	$-\,53\cdot 607$	$$	$5.63954535118$			✓	$S_6$ (as 6T16)	trivial	$2$	$2$	$0.536093401482$
6.2.32269.1	x^6 - 3x^5 + 4x^4 - 3x^3 + x^2 - 1	$6$	[2,2]	$23^{2}\cdot 61$	$$	$5.64240494139$			✓	$S_4\times C_2$ (as 6T11)	trivial	$2$	$3$	$0.529910039105$
6.0.32911.1	x^6 - x^3 - x^2 + x + 1	$6$	[0,3]	$-\,32911$	$$	$5.6609612195$			✓	$S_6$ (as 6T16)	trivial	$2$	$2$	$0.518066133218$
6.0.33791.1	x^6 - x^5 + x^4 + x^3 + x + 1	$6$	[0,3]	$-\,33791$	$$	$5.68591252578$			✓	$S_6$ (as 6T16)	trivial	$2$	$2$	$0.688322549696$
6.0.33856.1	x^6 - x^4 + 1	$6$	[0,3]	$-\,2^{6}\cdot 23^{2}$	$$	$5.6877339597$			✓	$S_4\times C_2$ (as 6T11)	trivial	$2$	$2$	$0.480272990261$
6.0.33856.2	x^6 + x^4 + 2x^2 + 1	$6$	[0,3]	$-\,2^{6}\cdot 23^{2}$	$$	$5.6877339597$			✓	$D_{6}$ (as 6T3)	trivial	$4$	$2$	$1.02428142886$
6.2.33856.1	x^6 + x^4 - 1	$6$	[2,2]	$2^{6}\cdot 23^{2}$	$$	$5.6877339597$			✓	$S_4$ (as 6T7)	trivial	$2$	$3$	$0.550118007418$
6.0.34371.1	x^6 - x^5 - 2x^3 + 2x + 1	$6$	[0,3]	$-\,3^{3}\cdot 19\cdot 67$	$$	$5.70206321806$			✓	$C_3^2:D_4$ (as 6T13)	trivial	$6$	$2$	$1.68719062958$
6.0.34992.1	x^6 - 3x^5 + 5x^3 - 3x + 1	$6$	[0,3]	$-\,2^{4}\cdot 3^{7}$	$$	$5.71910575798$		✓	✓	$S_3$ (as 6T2)	trivial	$6$	$2$	$1.81542571879$
6.0.35099.1	x^6 + x^4 + x^2 - x + 1	$6$	[0,3]	$-\,35099$	$$	$5.72201673807$			✓	$S_6$ (as 6T16)	trivial	$2$	$2$	$0.585021824316$

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