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
8.0.1221964430625.1	x^8 - x^7 + 12x^6 + 30x^5 + 213x^4 - x^3 - 250x^2 - 412x + 664	\( 3^{4}\cdot 5^{4}\cdot 17^{6} \)	$C_4\times C_2$ (as 8T2)	$[120]$
8.0.2291590440000.3	x^8 - 3x^7 - x^6 + 19x^5 - 2x^4 + 10x^3 + 328x^2 - 672x + 384	\( 2^{6}\cdot 3^{4}\cdot 5^{4}\cdot 29^{4} \)	$D_4\times C_2$ (as 8T9)	$[120]$
8.0.2396181823488.1	x^8 - 4x^7 + 54x^6 - 148x^5 + 721x^4 - 1200x^3 + 2880x^2 - 2304x + 2304	\( 2^{16}\cdot 3^{4}\cdot 241\cdot 1873 \)	$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31)	$[2, 52]$
8.0.2647779840000.9	x^8 + 2x^6 - 4x^5 + 87x^4 + 56x^3 + 450x^2 + 248x + 1864	\( 2^{12}\cdot 3^{4}\cdot 5^{4}\cdot 113^{2} \)	$D_4\times C_2$ (as 8T9)	$[120]$
8.0.3555958775625.11	x^8 - 2x^7 + 46x^6 - 66x^5 + 943x^4 - 952x^3 + 9879x^2 - 5481x + 43471	\( 3^{4}\cdot 5^{4}\cdot 17^{4}\cdot 29^{2} \)	$D_4\times C_2$ (as 8T9)	$[120]$
8.0.3619856318464.7	x^8 - 4x^7 + 30x^6 - 76x^5 + 429x^4 - 736x^3 + 2900x^2 - 2544x + 8194	\( 2^{22}\cdot 929^{2} \)	$C_2^2:C_4$ (as 8T10)	$[3, 36]$
8.0.3672007840000.1	x^8 + 63x^6 + 1360x^4 + 11426x^2 + 27289	\( 2^{8}\cdot 5^{4}\cdot 29^{3}\cdot 941 \)	$C_2 \wr C_2\wr C_2$ (as 8T35)	$[2, 52]$
8.0.3694084000000.1	x^8 - 4x^7 + 28x^6 - 70x^5 + 401x^4 - 690x^3 + 2832x^2 - 2498x + 11381	\( 2^{8}\cdot 5^{6}\cdot 31^{4} \)	$C_4\times C_2$ (as 8T2)	$[2, 60]$
8.0.3754541776896.5	x^8 + 18x^6 - 12x^5 + 263x^4 + 240x^3 + 2718x^2 + 2388x + 8542	\( 2^{16}\cdot 3^{4}\cdot 29^{4} \)	$C_2^3$ (as 8T3)	$[3, 3, 12]$
8.0.3843168160000.41	x^8 - 2x^7 + 47x^6 - 70x^5 + 984x^4 - 918x^3 + 10382x^2 - 4392x + 46229	\( 2^{8}\cdot 5^{4}\cdot 13^{4}\cdot 29^{2} \)	$D_4\times C_2$ (as 8T9)	$[8, 16]$
8.0.3960597515625.1	x^8 + 47x^6 + 974x^4 + 9868x^2 + 43681	\( 3^{4}\cdot 5^{6}\cdot 29^{2}\cdot 61^{2} \)	$C_2^2 \wr C_2$ (as 8T18)	$[2, 4, 16]$
8.0.4003584802816.7	x^8 + 44x^6 + 738x^4 + 5728x^2 + 18496	\( 2^{22}\cdot 977^{2} \)	$C_2^2:C_4$ (as 8T10)	$[154]$
8.0.4129938450625.1	x^8 + 34x^6 + 401x^4 + 604x^2 + 3136	\( 5^{4}\cdot 13^{6}\cdot 37^{2} \)	$D_4\times C_2$ (as 8T9)	$[102]$
8.0.4270141210624.7	x^8 + 44x^6 + 770x^4 + 6112x^2 + 18496	\( 2^{22}\cdot 1009^{2} \)	$C_2^2:C_4$ (as 8T10)	$[130]$
8.0.4359744000000.43	x^8 + 54x^6 + 956x^4 + 6264x^2 + 13456	\( 2^{12}\cdot 3^{4}\cdot 5^{6}\cdot 29^{2} \)	$C_2^3: C_4$ (as 8T20)	$[2, 60]$
8.0.4368300642304.1	x^8 - 2x^7 + 23x^6 + 24x^5 + 151x^4 + 430x^3 + 821x^2 + 660x + 900	\( 2^{12}\cdot 17^{4}\cdot 113^{2} \)	$D_4\times C_2$ (as 8T9)	$[112]$
8.0.4406640640000.56	x^8 - 4x^7 + 42x^6 - 112x^5 + 659x^4 - 1136x^3 + 4258x^2 - 3708x + 12679	\( 2^{22}\cdot 5^{4}\cdot 41^{2} \)	$D_4\times C_2$ (as 8T9)	$[6, 18]$
8.0.4477484000000.13	x^8 + 65x^6 + 1430x^4 + 12100x^2 + 33275	\( 2^{8}\cdot 5^{6}\cdot 11^{3}\cdot 29^{2} \)	$C_2 \wr C_2\wr C_2$ (as 8T35)	$[2, 2, 28]$
8.0.4534344360000.16	x^8 + 5x^6 + 83x^4 - 125x^2 + 196	\( 2^{6}\cdot 3^{4}\cdot 5^{4}\cdot 7^{2}\cdot 13^{4} \)	$D_4\times C_2$ (as 8T9)	$[2, 56]$
8.0.4572006150625.3	x^8 - 2x^7 + 22x^6 - 30x^5 + 332x^4 - 346x^3 + 2661x^2 - 1558x + 11444	\( 5^{4}\cdot 31^{4}\cdot 89^{2} \)	$D_4\times C_2$ (as 8T9)	$[2, 66]$
8.0.4700224000000.2	x^8 - 2x^7 + 64x^6 - 8x^5 + 1146x^4 + 1880x^3 + 6720x^2 + 13720x + 15580	\( 2^{12}\cdot 5^{6}\cdot 271^{2} \)	$C_2^2:C_4$ (as 8T10)	$[2, 60]$
8.0.4741941760000.1	x^8 + 34x^6 + 511x^4 + 3574x^2 + 12321	\( 2^{12}\cdot 5^{4}\cdot 1361^{2} \)	$D_4\times C_2$ (as 8T9)	$[120]$
8.0.4902433652736.5	x^8 - 4x^7 + 30x^6 - 76x^5 + 407x^4 - 692x^3 + 2986x^2 - 2652x + 9118	\( 2^{16}\cdot 3^{4}\cdot 31^{4} \)	$C_2^3$ (as 8T3)	$[6, 24]$
8.0.4995336750625.6	x^8 - 4x^7 + 12x^6 - 22x^5 + 207x^4 - 382x^3 + 2220x^2 - 2032x + 8384	\( 5^{4}\cdot 13^{4}\cdot 23^{4} \)	$C_2^3$ (as 8T3)	$[120]$
8.0.5022081000000.2	x^8 - x^7 + 10x^6 - 44x^5 + 164x^4 - 164x^3 + 120x^2 - 216x + 576	\( 2^{6}\cdot 3^{6}\cdot 5^{6}\cdot 83^{2} \)	$C_2^3:S_4$ (as 8T39)	$[2, 60]$
8.0.5108165015625.15	x^8 - 3x^7 + 77x^6 - 240x^5 + 1921x^4 - 5190x^3 + 16968x^2 - 19044x + 24831	\( 3^{4}\cdot 5^{6}\cdot 7^{4}\cdot 41^{2} \)	$C_2^2:C_4$ (as 8T10)	$[2, 84]$
8.0.5263380000000.1	x^8 + 60x^6 + 1230x^4 + 9405x^2 + 16245	\( 2^{8}\cdot 3^{6}\cdot 5^{7}\cdot 19^{2} \)	$C_8:C_2$ (as 8T7)	$[2, 2, 26]$
8.0.5368401903721.1	x^8 - x^7 + 6x^6 - 11x^5 + 33x^4 - 44x^3 + 96x^2 - 64x + 256	\( 53\cdot 173\cdot 24197^{2} \)	$C_2 \wr S_4$ (as 8T44)	$[118]$
8.0.5572394590464.5	x^8 - 4x^7 - 10x^5 + 137x^4 + 154x^3 + 706x^2 + 648x + 1816	\( 2^{8}\cdot 3^{4}\cdot 13^{4}\cdot 97^{2} \)	$D_4\times C_2$ (as 8T9)	$[4, 28]$
8.0.5575945486336.4	x^8 - 4x^7 + 54x^6 - 148x^5 + 845x^4 - 1448x^3 + 4428x^2 - 3728x + 6082	\( 2^{22}\cdot 1153^{2} \)	$C_2^3: C_4$ (as 8T20)	$[134]$
8.0.5790760960000.11	x^8 - 4x^7 + 50x^6 - 136x^5 + 879x^4 - 1536x^3 + 5510x^2 - 4764x + 16911	\( 2^{22}\cdot 5^{4}\cdot 47^{2} \)	$D_4\times C_2$ (as 8T9)	$[112]$
8.0.5861202264064.2	x^8 + 26x^6 + 257x^4 - 424x^2 + 1936	\( 2^{16}\cdot 7^{4}\cdot 193^{2} \)	$D_4\times C_2$ (as 8T9)	$[2, 2, 28]$
8.0.5893241760000.32	x^8 + 24x^6 + 126x^4 + 209x^2 + 81	\( 2^{8}\cdot 3^{2}\cdot 5^{4}\cdot 7^{2}\cdot 17^{4} \)	$D_4\times C_2$ (as 8T9)	$[2, 80]$
8.0.5899691229184.9	x^8 + 40x^6 + 544x^4 + 2952x^2 + 7921	\( 2^{24}\cdot 593^{2} \)	$C_2^2:C_4$ (as 8T10)	$[2, 56]$
8.0.5922408960000.3	x^8 - 2x^7 - 5x^6 + 88x^5 + 321x^4 - 250x^3 + 889x^2 + 164x + 394	\( 2^{12}\cdot 3^{4}\cdot 5^{4}\cdot 13^{4} \)	$C_2^3$ (as 8T3)	$[2, 4, 16]$
8.0.5922408960000.16	x^8 - 4x^7 + 14x^6 - 28x^5 + 329x^4 - 616x^3 + 5356x^2 - 5052x + 21711	\( 2^{12}\cdot 3^{4}\cdot 5^{4}\cdot 13^{4} \)	$C_2^3$ (as 8T3)	$[4, 4, 8]$
8.0.5922408960000.17	x^8 + 26x^6 + 487x^4 + 5694x^2 + 25281	\( 2^{12}\cdot 3^{4}\cdot 5^{4}\cdot 13^{4} \)	$C_2^3$ (as 8T3)	$[2, 4, 16]$
8.0.5922408960000.23	x^8 - 2x^7 + 11x^6 + 64x^5 + 353x^4 + 358x^3 + 649x^2 + 444x + 426	\( 2^{12}\cdot 3^{4}\cdot 5^{4}\cdot 13^{4} \)	$D_4$ (as 8T4)	$[8, 16]$
8.0.5922408960000.25	x^8 - 2x^7 + 5x^6 - 84x^5 + 554x^4 - 1800x^3 + 3320x^2 - 3200x + 1600	\( 2^{12}\cdot 3^{4}\cdot 5^{4}\cdot 13^{4} \)	$D_4$ (as 8T4)	$[8, 16]$
8.0.5922408960000.26	x^8 - 2x^7 + 39x^6 - 86x^5 + 389x^4 - 892x^3 + 1756x^2 - 2640x + 1660	\( 2^{12}\cdot 3^{4}\cdot 5^{4}\cdot 13^{4} \)	$D_4$ (as 8T4)	$[8, 16]$
8.0.5922408960000.27	x^8 - 4x^7 + 2x^6 - 52x^5 + 203x^4 - 124x^3 + 2254x^2 + 600x + 3585	\( 2^{12}\cdot 3^{4}\cdot 5^{4}\cdot 13^{4} \)	$D_4$ (as 8T4)	$[8, 16]$
8.0.5971747888656.6	x^8 + 23x^6 + 252x^4 + 1472x^2 + 4096	\( 2^{4}\cdot 3^{6}\cdot 11^{6}\cdot 17^{2} \)	$C_2^2 \wr C_2$ (as 8T18)	$[2, 56]$
8.0.6020938137600.16	x^8 - 4x^7 + 38x^6 - 100x^5 + 511x^4 - 860x^3 + 2930x^2 - 2516x + 6238	\( 2^{16}\cdot 3^{6}\cdot 5^{2}\cdot 71^{2} \)	$D_4\times C_2$ (as 8T9)	$[2, 2, 40]$
8.0.6108930140625.2	x^8 - x^7 + 16x^6 - 26x^5 + 201x^4 - 325x^3 + 1990x^2 + 1600x + 640	\( 3^{4}\cdot 5^{6}\cdot 13^{6} \)	$C_4\times C_2$ (as 8T2)	$[2, 104]$
8.0.6108930140625.3	x^8 - x^7 + 25x^6 - 114x^5 + 694x^4 - 1754x^3 + 3585x^2 - 3381x + 2401	\( 3^{4}\cdot 5^{6}\cdot 13^{6} \)	$C_4\times C_2$ (as 8T2)	$[2, 52]$
8.0.6108930140625.4	x^8 - x^7 + 44x^6 - 47x^5 + 544x^4 + 501x^3 + 1731x^2 + 1782x + 3636	\( 3^{4}\cdot 5^{6}\cdot 13^{6} \)	$C_4\times C_2$ (as 8T2)	$[2, 2, 26]$
8.0.6122813722624.6	x^8 + 133x^4 + 2116	\( 2^{12}\cdot 23^{2}\cdot 41^{4} \)	$D_4\times C_2$ (as 8T9)	$[2, 56]$
8.0.6134538240000.14	x^8 - 4x^7 - 2x^6 + 129x^4 - 396x^3 + 1548x^2 - 1296x + 1944	\( 2^{16}\cdot 3^{4}\cdot 5^{4}\cdot 43^{2} \)	$D_4\times C_2$ (as 8T9)	$[4, 40]$
8.0.6190144000000.2	x^8 - 2x^7 + 72x^6 + 1466x^4 + 2360x^3 + 9968x^2 + 22144x + 22796	\( 2^{12}\cdot 5^{6}\cdot 311^{2} \)	$C_2^2:C_4$ (as 8T10)	$[2, 56]$
8.0.6358789328896.12	x^8 + 22x^6 - 4x^5 + 313x^4 + 80x^3 + 2612x^2 + 856x + 9247	\( 2^{12}\cdot 31^{4}\cdot 41^{2} \)	$D_4\times C_2$ (as 8T9)	$[120]$

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