$p$-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 795 matches)

 

Label	$n$	Polynomial	$p$	$e$	$f$	$c$	Galois group	$u$	$t$	Slope content
3.9.0.1	$9$	x^9 - x^3 + x^2 + 1	$3$	$1$	$9$	$0$	$C_9$ (as 9T1)	$9$	$1$	$[\ ]^{9}$
3.9.9.1	$9$	x^9 + 54x^5 + 27x^3 + 189	$3$	$3$	$3$	$9$	$S_3\times C_3$ (as 9T4)	$3$	$2$	$[3/2]_{2}^{3}$
3.9.9.10	$9$	x^9 + 6x^6 + 9x^4 + 27	$3$	$3$	$3$	$9$	$(C_3^3:C_3):C_2$ (as 9T22)	$3$	$2$	$[3/2, 3/2, 3/2]_{2}^{3}$
3.9.9.11	$9$	x^9 + 3x + 3	$3$	$9$	$1$	$9$	$(C_3^2:C_8):C_2$ (as 9T19)	$2$	$8$	$[9/8, 9/8]_{8}^{2}$
3.9.9.12	$9$	x^9 + 6x + 6	$3$	$9$	$1$	$9$	$(C_3^2:C_8):C_2$ (as 9T19)	$2$	$8$	$[9/8, 9/8]_{8}^{2}$
3.9.9.2	$9$	x^9 + 18x^3 + 27x + 27	$3$	$3$	$3$	$9$	$C_3^2 : S_3 $ (as 9T13)	$3$	$2$	$[3/2, 3/2]_{2}^{3}$
3.9.9.3	$9$	x^9 + 9x^4 + 18x^3 + 54	$3$	$3$	$3$	$9$	$(C_3^3:C_3):C_2$ (as 9T22)	$3$	$2$	$[3/2, 3/2, 3/2]_{2}^{3}$
3.9.9.4	$9$	x^9 + 3x^6 + 9x^4 + 54	$3$	$3$	$3$	$9$	$(C_3^3:C_3):C_2$ (as 9T22)	$3$	$2$	$[3/2, 3/2, 3/2]_{2}^{3}$
3.9.9.5	$9$	x^9 + 3x^7 + 3x^6 + 54	$3$	$3$	$3$	$9$	$(C_3^3:C_3):C_2$ (as 9T22)	$3$	$2$	$[3/2, 3/2, 3/2]_{2}^{3}$
3.9.9.6	$9$	x^9 + 3x^7 + 3x^6 + 18x^4 + 54	$3$	$3$	$3$	$9$	$S_3\times C_3$ (as 9T4)	$3$	$2$	$[3/2]_{2}^{3}$
3.9.9.7	$9$	x^9 + 18x^3 + 54x + 27	$3$	$3$	$3$	$9$	$C_3^2 : S_3 $ (as 9T13)	$3$	$2$	$[3/2, 3/2]_{2}^{3}$
3.9.9.8	$9$	x^9 + 6x^7 + 18x^3 + 27	$3$	$3$	$3$	$9$	$(C_3^3:C_3):C_2$ (as 9T22)	$3$	$2$	$[3/2, 3/2, 3/2]_{2}^{3}$
3.9.9.9	$9$	x^9 + 18x^5 + 27x^2 + 54	$3$	$3$	$3$	$9$	$(C_3^3:C_3):C_2$ (as 9T22)	$3$	$2$	$[3/2, 3/2, 3/2]_{2}^{3}$
3.9.10.1	$9$	x^9 + 3x^2 + 3	$3$	$9$	$1$	$10$	$C_3^2:Q_8$ (as 9T14)	$2$	$4$	$[5/4, 5/4]_{4}^{2}$
3.9.10.2	$9$	x^9 + 3x^2 + 6	$3$	$9$	$1$	$10$	$S_3^2:C_2$ (as 9T16)	$2$	$4$	$[5/4, 5/4]_{4}^{2}$
3.9.12.1	$9$	x^9 + 18x^5 + 18x^3 + 27x^2 + 216	$3$	$3$	$3$	$12$	$C_3^2$ (as 9T2)	$3$	$1$	$[2]^{3}$
3.9.12.10	$9$	x^9 + 24x^6 + 18x^5 + 27	$3$	$3$	$3$	$12$	$C_3 \wr C_3 $ (as 9T17)	$3$	$1$	$[2, 2, 2]^{3}$
3.9.12.11	$9$	x^9 + 21x^6 + 54x^2 + 54	$3$	$3$	$3$	$12$	$C_3 \wr C_3 $ (as 9T17)	$3$	$1$	$[2, 2, 2]^{3}$
3.9.12.12	$9$	x^9 + 3x^6 + 18x^5 + 54	$3$	$3$	$3$	$12$	$C_3 \wr C_3 $ (as 9T17)	$3$	$1$	$[2, 2, 2]^{3}$
3.9.12.13	$9$	x^9 + 3x^8 + 72x^3 + 27	$3$	$3$	$3$	$12$	$C_3 \wr C_3 $ (as 9T17)	$3$	$1$	$[2, 2, 2]^{3}$
3.9.12.14	$9$	x^9 + 3x^8 + 45x^3 + 27	$3$	$3$	$3$	$12$	$C_3 \wr C_3 $ (as 9T17)	$3$	$1$	$[2, 2, 2]^{3}$
3.9.12.15	$9$	x^9 + 72x^3 + 54x^2 + 54	$3$	$3$	$3$	$12$	$C_3 \wr C_3 $ (as 9T17)	$3$	$1$	$[2, 2, 2]^{3}$
3.9.12.16	$9$	x^9 + 9x^5 + 18x^3 + 27x^2 + 27	$3$	$3$	$3$	$12$	$S_3\times C_3$ (as 9T4)	$6$	$1$	$[2]^{6}$
3.9.12.17	$9$	x^9 + 6x^8 + 6x^6 + 27	$3$	$3$	$3$	$12$	$C_3^2 : S_3 $ (as 9T13)	$6$	$1$	$[2, 2]^{6}$
3.9.12.18	$9$	x^9 + 6x^8 + 18x^3 + 27	$3$	$3$	$3$	$12$	$(C_3^3:C_3):C_2$ (as 9T22)	$6$	$1$	$[2, 2, 2]^{6}$
3.9.12.19	$9$	x^9 + 6x^8 + 18x^5 + 18x^3 + 27	$3$	$3$	$3$	$12$	$(C_3^3:C_3):C_2$ (as 9T22)	$6$	$1$	$[2, 2, 2]^{6}$
3.9.12.2	$9$	x^9 + 18x^5 + 18x^3 + 27x^2 + 54	$3$	$3$	$3$	$12$	$C_9$ (as 9T1)	$3$	$1$	$[2]^{3}$
3.9.12.20	$9$	x^9 + 6x^6 + 54x^2 + 27	$3$	$3$	$3$	$12$	$(C_3^3:C_3):C_2$ (as 9T22)	$6$	$1$	$[2, 2, 2]^{6}$
3.9.12.21	$9$	x^9 + 3x^4 + 6	$3$	$9$	$1$	$12$	$C_3^2:C_4$ (as 9T9)	$2$	$2$	$[3/2, 3/2]_{2}^{2}$
3.9.12.22	$9$	x^9 + 6x^4 + 6x^3 + 3	$3$	$9$	$1$	$12$	$C_3^2 : C_6$ (as 9T11)	$3$	$2$	$[3/2, 3/2]_{2}^{3}$
3.9.12.23	$9$	x^9 + 6x^4 + 3x^3 + 3	$3$	$9$	$1$	$12$	$C_3^2 : C_6$ (as 9T11)	$3$	$2$	$[3/2, 3/2]_{2}^{3}$
3.9.12.24	$9$	x^9 + 3x^4 + 3	$3$	$9$	$1$	$12$	$S_3^2$ (as 9T8)	$2$	$2$	$[3/2, 3/2]_{2}^{2}$
3.9.12.25	$9$	x^9 + 3x^4 + 6x^3 + 3	$3$	$9$	$1$	$12$	$C_3^2:C_8$ (as 9T15)	$4$	$2$	$[3/2, 3/2]_{2}^{4}$
3.9.12.26	$9$	x^9 + 3x^4 + 3x^3 + 3	$3$	$9$	$1$	$12$	$C_3^2:C_8$ (as 9T15)	$4$	$2$	$[3/2, 3/2]_{2}^{4}$
3.9.12.3	$9$	x^9 + 6x^8 + 3x^6 + 9x^3 + 135	$3$	$3$	$3$	$12$	$C_9$ (as 9T1)	$3$	$1$	$[2]^{3}$
3.9.12.4	$9$	x^9 + 6x^6 + 18x^5 + 36x^3 + 27	$3$	$3$	$3$	$12$	$C_3^2:C_3$ (as 9T7)	$3$	$1$	$[2, 2]^{3}$
3.9.12.5	$9$	x^9 + 18x^5 + 18x^3 + 27x^2 + 27	$3$	$3$	$3$	$12$	$C_9:C_3$ (as 9T6)	$3$	$1$	$[2, 2]^{3}$
3.9.12.6	$9$	x^9 + 3x^8 + 6x^6 + 27	$3$	$3$	$3$	$12$	$C_9:C_3$ (as 9T6)	$3$	$1$	$[2, 2]^{3}$
3.9.12.7	$9$	x^9 + 18x^5 + 72x^3 + 54	$3$	$3$	$3$	$12$	$C_3 \wr C_3 $ (as 9T17)	$3$	$1$	$[2, 2, 2]^{3}$
3.9.12.8	$9$	x^9 + 12x^6 + 54x^2 + 54	$3$	$3$	$3$	$12$	$C_3 \wr C_3 $ (as 9T17)	$3$	$1$	$[2, 2, 2]^{3}$
3.9.12.9	$9$	x^9 + 15x^6 + 18x^5 + 27	$3$	$3$	$3$	$12$	$C_3 \wr C_3 $ (as 9T17)	$3$	$1$	$[2, 2, 2]^{3}$
3.9.13.1	$9$	x^9 + 6x^5 + 6	$3$	$9$	$1$	$13$	$(C_3^2:C_8):C_2$ (as 9T19)	$2$	$8$	$[13/8, 13/8]_{8}^{2}$
3.9.13.10	$9$	x^9 + 3x^6 + 3x^5 + 3x^3 + 6	$3$	$9$	$1$	$13$	$C_3 \wr S_3 $ (as 9T20)	$3$	$2$	$[3/2, 3/2, 5/3]_{2}^{3}$
3.9.13.2	$9$	x^9 + 6x^5 + 3x^3 + 3	$3$	$9$	$1$	$13$	$C_3^2 : D_{6} $ (as 9T18)	$2$	$2$	$[3/2, 3/2, 5/3]_{2}^{2}$
3.9.13.3	$9$	x^9 + 3x^6 + 6x^5 + 6x^3 + 3	$3$	$9$	$1$	$13$	$C_3 \wr S_3 $ (as 9T20)	$3$	$2$	$[3/2, 3/2, 5/3]_{2}^{3}$
3.9.13.4	$9$	x^9 + 6x^5 + 6x^3 + 3	$3$	$9$	$1$	$13$	$C_3 \wr S_3 $ (as 9T20)	$3$	$2$	$[3/2, 3/2, 5/3]_{2}^{3}$
3.9.13.5	$9$	x^9 + 6x^6 + 6x^5 + 6x^3 + 3	$3$	$9$	$1$	$13$	$C_3 \wr S_3 $ (as 9T20)	$3$	$2$	$[3/2, 3/2, 5/3]_{2}^{3}$
3.9.13.6	$9$	x^9 + 3x^5 + 3	$3$	$9$	$1$	$13$	$(C_3^2:C_8):C_2$ (as 9T19)	$2$	$8$	$[13/8, 13/8]_{8}^{2}$
3.9.13.7	$9$	x^9 + 3x^5 + 6x^3 + 3	$3$	$9$	$1$	$13$	$C_3^2 : D_{6} $ (as 9T18)	$2$	$2$	$[3/2, 3/2, 5/3]_{2}^{2}$
3.9.13.8	$9$	x^9 + 3x^5 + 3x^3 + 6	$3$	$9$	$1$	$13$	$C_3 \wr S_3 $ (as 9T20)	$3$	$2$	$[3/2, 3/2, 5/3]_{2}^{3}$