$p$-adic field search results

Degree	Residue characteristic	Galois group	Galois unramified degree
Ramification index	Discriminant exponent	Inertia subgroup	Galois tame degree
Residue field degree	Top slope	Wild inertia subgroup

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Results (1-50 of 5493 matches)

 

Label	$n$	Polynomial	$p$	$e$	$f$	$c$	Galois group	$u$	$t$	Slope content
2.12.0.1	$12$	x^12 - 26x^10 + 275x^8 - 1500x^6 + 4375x^4 - 6250x^2 + 7221	$2$	$1$	$12$	$0$	$C_{12}$ (as 12T1)	$12$	$1$	$[\ ]^{12}$
2.12.8.1	$12$	x^12 - 6x^9 + 12x^6 - 8x^3 + 16	$2$	$3$	$4$	$8$	$C_3 : C_4$ (as 12T5)	$4$	$3$	$[\ ]_{3}^{4}$
2.12.8.2	$12$	x^12 - 8x^3 + 16	$2$	$3$	$4$	$8$	$C_3\times (C_3 : C_4)$ (as 12T19)	$12$	$3$	$[\ ]_{3}^{12}$
2.12.12.1	$12$	x^12 - 48x^10 + 49x^8 + 8x^6 + 19x^4 - 24x^2 + 59	$2$	$2$	$6$	$12$	$C_2^4:C_3.D_4$ (as 12T134)	$6$	$1$	$[2, 2, 2, 2, 2, 2]^{6}$
2.12.12.10	$12$	x^12 - 6x^10 + 23x^8 - 28x^6 - 9x^4 - 30x^2 - 15	$2$	$2$	$6$	$12$	$C_2^2\wr C_2:C_3$ (as 12T58)	$6$	$1$	$[2, 2, 2, 2]^{6}$
2.12.12.11	$12$	x^12 - 6x^10 - 73x^8 + 140x^6 + 79x^4 - 6x^2 + 57	$2$	$2$	$6$	$12$	$A_4 \times C_2$ (as 12T7)	$6$	$1$	$[2, 2]^{6}$
2.12.12.12	$12$	x^12 + 66x^10 - 93x^8 - 68x^6 - 41x^4 + 66x^2 - 123	$2$	$2$	$6$	$12$	$C_4\times A_4$ (as 12T29)	$6$	$1$	$[2, 2, 2]^{6}$
2.12.12.13	$12$	x^12 - 18x^10 - 13x^8 - 44x^6 + 55x^4 + 62x^2 + 21	$2$	$2$	$6$	$12$	$C_2^5.C_6$ (as 12T105)	$6$	$1$	$[2, 2, 2, 2, 2]^{6}$
2.12.12.14	$12$	x^12 + 4x^10 + 21x^8 - 16x^6 + 43x^4 + 12x^2 - 1	$2$	$2$	$6$	$12$	$C_2^4:C_3.D_4$ (as 12T134)	$6$	$1$	$[2, 2, 2, 2, 2, 2]^{6}$
2.12.12.15	$12$	x^12 - 28x^10 - 63x^8 - 32x^6 + 19x^4 + 60x^2 - 21	$2$	$2$	$6$	$12$	$C_2^4:C_3.D_4$ (as 12T134)	$6$	$1$	$[2, 2, 2, 2, 2, 2]^{6}$
2.12.12.16	$12$	x^12 - 16x^10 - 23x^8 + 24x^6 - 29x^4 - 8x^2 - 13	$2$	$2$	$6$	$12$	$C_2^4:C_3.D_4$ (as 12T134)	$6$	$1$	$[2, 2, 2, 2, 2, 2]^{6}$
2.12.12.17	$12$	x^12 + 22x^10 + 75x^8 - 12x^6 - 89x^4 + 54x^2 - 115	$2$	$2$	$6$	$12$	$C_4\times A_4$ (as 12T29)	$12$	$1$	$[2, 2]^{12}$
2.12.12.18	$12$	x^12 + 80x^10 + 81x^8 - 160x^6 - 117x^4 + 80x^2 + 227	$2$	$2$	$6$	$12$	$D_4 \times C_3$ (as 12T14)	$6$	$1$	$[2, 2]^{6}$
2.12.12.19	$12$	x^12 - 6x^10 + 27x^8 - 4x^6 + 7x^4 + 10x^2 + 29	$2$	$2$	$6$	$12$	$C_2^5.C_6$ (as 12T105)	$12$	$1$	$[2, 2, 2, 2]^{12}$
2.12.12.2	$12$	x^12 - 6x^10 - 13x^8 - 28x^6 + 15x^4 - 30x^2 - 3	$2$	$2$	$6$	$12$	$C_2^5.C_6$ (as 12T105)	$6$	$1$	$[2, 2, 2, 2, 2]^{6}$
2.12.12.20	$12$	x^12 - 18x^10 - 49x^8 - 52x^6 + 39x^4 + 6x^2 + 9	$2$	$2$	$6$	$12$	$C_2\times C_2^2\wr C_2:C_3$ (as 12T87)	$6$	$1$	$[2, 2, 2, 2, 2]^{6}$
2.12.12.21	$12$	x^12 + 44x^10 + 45x^8 - 48x^6 + 59x^4 - 60x^2 + 23	$2$	$2$	$6$	$12$	$C_2^4:C_3.D_4$ (as 12T134)	$6$	$1$	$[2, 2, 2, 2, 2, 2]^{6}$
2.12.12.22	$12$	x^12 - 52x^10 - 7x^8 + 32x^6 + 35x^4 - 44x^2 - 29	$2$	$2$	$6$	$12$	$C_2^4:C_3.D_4$ (as 12T134)	$6$	$1$	$[2, 2, 2, 2, 2, 2]^{6}$
2.12.12.23	$12$	x^12 - 2x^10 - 65x^8 + 100x^6 - 97x^4 - 98x^2 + 97	$2$	$2$	$6$	$12$	$C_2^2 \times A_4$ (as 12T25)	$6$	$1$	$[2, 2, 2]^{6}$
2.12.12.24	$12$	x^12 - 100x^10 - 59x^8 + 104x^6 + 387x^4 + 444x^2 + 439	$2$	$2$	$6$	$12$	$D_4 \times C_3$ (as 12T14)	$6$	$1$	$[2, 2]^{6}$
2.12.12.25	$12$	x^12 - 78x^10 - 1621x^8 + 460x^6 - 1977x^4 + 866x^2 + 749	$2$	$2$	$6$	$12$	$C_{12}$ (as 12T1)	$6$	$1$	$[2]^{6}$
2.12.12.26	$12$	x^12 - 162x^10 + 26423x^8 + 125508x^6 - 64481x^4 - 122498x^2 - 86071	$2$	$2$	$6$	$12$	$C_6\times C_2$ (as 12T2)	$6$	$1$	$[2]^{6}$
2.12.12.27	$12$	x^12 - 18x^10 + 171x^8 + 116x^6 - 313x^4 + 190x^2 + 877	$2$	$6$	$2$	$12$	$A_4:C_4$ (as 12T30)	$4$	$3$	$[4/3, 4/3]_{3}^{4}$
2.12.12.28	$12$	x^12 - x^10 + 2x^8 - x^6 - 2x^4 + 3x^2 + 1	$2$	$6$	$2$	$12$	$S_4$ (as 12T9)	$2$	$3$	$[4/3, 4/3]_{3}^{2}$
2.12.12.29	$12$	x^12 + 6x^10 + 51x^8 - 252x^6 - 393x^4 - 234x^2 - 203	$2$	$6$	$2$	$12$	$A_4^2.C_4$ (as 12T159)	$12$	$3$	$[4/3, 4/3, 4/3, 4/3]_{3}^{12}$
2.12.12.3	$12$	x^12 - 16x^10 - 51x^8 - 8x^6 + 43x^4 + 24x^2 - 57	$2$	$2$	$6$	$12$	$C_2^4:C_3.D_4$ (as 12T134)	$6$	$1$	$[2, 2, 2, 2, 2, 2]^{6}$
2.12.12.30	$12$	x^12 - 6x^10 + 15x^8 - 52x^6 + 111x^4 - 102x^2 - 991	$2$	$6$	$2$	$12$	$A_4\wr C_2$ (as 12T126)	$6$	$3$	$[4/3, 4/3, 4/3, 4/3]_{3}^{6}$
2.12.12.31	$12$	x^12 + 4x^11 - 6x^10 + 8x^9 - 4x^8 + 8x^7 - 4x^6 + 4x^5 - 4x^4 + 8x + 8	$2$	$4$	$3$	$12$	12T205	$6$	$3$	$[4/3, 4/3, 4/3, 4/3, 4/3, 4/3]_{3}^{6}$
2.12.12.32	$12$	x^12 + 6x^11 - 4x^10 - 2x^9 + 4x^8 - 8x^7 - 8x^6 - 8x^5 - 4x^4 - 8	$2$	$4$	$3$	$12$	$A_4\wr C_2$ (as 12T129)	$6$	$3$	$[4/3, 4/3, 4/3, 4/3]_{3}^{6}$
2.12.12.33	$12$	x^12 + 6x^11 - 4x^9 - 2x^8 + 8x^7 + 8x^6 - 4x^5 + 8x^3 + 8x^2 + 8	$2$	$4$	$3$	$12$	$C_3\times S_4$ (as 12T45)	$6$	$3$	$[4/3, 4/3]_{3}^{6}$
2.12.12.34	$12$	x^12 + 2x^11 + 2x^9 + 2x^7 + 2x^5 + 2x^3 + 2x + 2	$2$	$12$	$1$	$12$	12T254	$6$	$9$	$[10/9, 10/9, 10/9, 10/9, 10/9, 10/9]_{9}^{6}$
2.12.12.4	$12$	x^12 - 6x^10 + 15x^8 - 20x^6 + 15x^4 - 38x^2 - 31	$2$	$2$	$6$	$12$	$C_2\times C_2^2\wr C_2:C_3$ (as 12T87)	$6$	$1$	$[2, 2, 2, 2, 2]^{6}$
2.12.12.5	$12$	x^12 + 52x^10 - 11x^8 - 8x^6 - 45x^4 - 44x^2 - 9	$2$	$2$	$6$	$12$	$D_4\times A_4$ (as 12T51)	$6$	$1$	$[2, 2, 2, 2]^{6}$
2.12.12.6	$12$	x^12 - 18x^10 + 11x^8 - 52x^6 - x^4 + 6x^2 - 11	$2$	$2$	$6$	$12$	$C_2^5.C_6$ (as 12T105)	$12$	$1$	$[2, 2, 2, 2]^{12}$
2.12.12.7	$12$	x^12 - 48x^10 + 53x^8 + 40x^6 + 27x^4 - 56x^2 + 47	$2$	$2$	$6$	$12$	$C_2^4:C_3.D_4$ (as 12T134)	$6$	$1$	$[2, 2, 2, 2, 2, 2]^{6}$
2.12.12.8	$12$	x^12 + 8x^10 - 31x^8 + 64x^6 - 53x^4 - 8x^2 - 45	$2$	$2$	$6$	$12$	$D_4\times A_4$ (as 12T51)	$6$	$1$	$[2, 2, 2, 2]^{6}$
2.12.12.9	$12$	x^12 - 18x^10 + 7x^8 - 28x^6 - x^4 - 18x^2 - 7	$2$	$2$	$6$	$12$	$C_2^2\wr C_2:C_3$ (as 12T58)	$6$	$1$	$[2, 2, 2, 2]^{6}$
2.12.14.1	$12$	x^12 + 2x^3 + 2	$2$	$12$	$1$	$14$	$S_4$ (as 12T8)	$2$	$3$	$[4/3, 4/3]_{3}^{2}$
2.12.14.2	$12$	x^12 + 2x^4 + 2x^3 + 2	$2$	$12$	$1$	$14$	$A_4:C_4$ (as 12T27)	$4$	$3$	$[4/3, 4/3]_{3}^{4}$
2.12.14.3	$12$	x^12 + 2x^3 + 2x^2 + 2	$2$	$12$	$1$	$14$	$A_4\wr C_2$ (as 12T128)	$6$	$3$	$[4/3, 4/3, 4/3, 4/3]_{3}^{6}$
2.12.16.1	$12$	x^12 + 48x^10 + 17x^8 - 128x^6 + 171x^4 - 176x^2 + 3	$2$	$6$	$2$	$16$	12T208	$6$	$3$	$[4/3, 4/3, 4/3, 4/3, 2, 2]_{3}^{6}$
2.12.16.10	$12$	x^12 + 54x^10 - 257x^8 - 492x^6 - 945x^4 + 342x^2 + 81	$2$	$6$	$2$	$16$	$(C_2\times A_4^2).C_2$ (as 12T158)	$6$	$3$	$[4/3, 4/3, 4/3, 4/3, 2]_{3}^{6}$
2.12.16.11	$12$	x^12 + 20x^10 - 44x^8 - 4x^6 - 16x^4 - 48	$2$	$6$	$2$	$16$	$C_3\times (C_3 : C_4)$ (as 12T19)	$6$	$3$	$[2]_{3}^{6}$
2.12.16.12	$12$	x^12 + 18x^10 + 171x^8 - 404x^6 - 281x^4 - 286x^2 + 461	$2$	$6$	$2$	$16$	$A_4^2.C_4$ (as 12T159)	$6$	$3$	$[4/3, 4/3, 4/3, 4/3, 2]_{3}^{6}$
2.12.16.13	$12$	x^12 + 12x^10 + 12x^8 + 8x^6 + 32x^4 - 16x^2 + 16	$2$	$6$	$2$	$16$	$D_6$ (as 12T3)	$2$	$3$	$[2]_{3}^{2}$
2.12.16.14	$12$	x^12 + 36x^10 + 37x^8 + 88x^6 - 61x^4 + 68x^2 - 105	$2$	$6$	$2$	$16$	12T208	$6$	$3$	$[4/3, 4/3, 4/3, 4/3, 2, 2]_{3}^{6}$
2.12.16.15	$12$	x^12 - 71x^8 + 123x^4 - 245	$2$	$6$	$2$	$16$	$\GL(2,\Z/4)$ (as 12T50)	$2$	$3$	$[4/3, 4/3, 2, 2]_{3}^{2}$
2.12.16.16	$12$	x^12 - 54x^10 - 509x^8 - 964x^6 - 777x^4 - 934x^2 + 357	$2$	$6$	$2$	$16$	$A_4:C_4$ (as 12T30)	$2$	$3$	$[4/3, 4/3, 2]_{3}^{2}$
2.12.16.17	$12$	x^12 + 9x^8 - 224x^6 + 187x^4 - 32x^2 - 133	$2$	$6$	$2$	$16$	12T208	$6$	$3$	$[4/3, 4/3, 4/3, 4/3, 2, 2]_{3}^{6}$
2.12.16.18	$12$	x^12 + x^10 + 6x^8 - 3x^6 + 6x^4 + x^2 - 3	$2$	$6$	$2$	$16$	$C_3 : C_4$ (as 12T5)	$2$	$3$	$[2]_{3}^{2}$