$p$-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

Sort order

Select

Results (17 matches)

 

Label	$n$	Polynomial	$p$	$e$	$f$	$c$	Galois group	$u$	$t$	Slope content
17.12.0.1	$12$	x^12 + 3x^2 - 2x + 5	$17$	$1$	$12$	$0$	$C_{12}$ (as 12T1)	$12$	$1$	$[\ ]^{12}$
17.12.6.1	$12$	x^12 + 117912x^6 - 1419857x^2 + 3475809936	$17$	$2$	$6$	$6$	$C_6\times C_2$ (as 12T2)	$6$	$2$	$[\ ]_{2}^{6}$
17.12.6.2	$12$	x^12 - 1419857x^2 + 289650828	$17$	$2$	$6$	$6$	$C_{12}$ (as 12T1)	$6$	$2$	$[\ ]_{2}^{6}$
17.12.8.1	$12$	x^12 - 51x^9 + 867x^6 - 4913x^3 + 111166451	$17$	$3$	$4$	$8$	$C_3 : C_4$ (as 12T5)	$4$	$3$	$[\ ]_{3}^{4}$
17.12.8.2	$12$	x^12 - 4913x^3 + 918731	$17$	$3$	$4$	$8$	$C_3\times (C_3 : C_4)$ (as 12T19)	$12$	$3$	$[\ ]_{3}^{12}$
17.12.9.1	$12$	x^12 - 34x^8 - 10115x^4 - 397953	$17$	$4$	$3$	$9$	$C_{12}$ (as 12T1)	$3$	$4$	$[\ ]_{4}^{3}$
17.12.9.2	$12$	x^12 - 34x^8 + 289x^4 - 44217	$17$	$4$	$3$	$9$	$C_{12}$ (as 12T1)	$3$	$4$	$[\ ]_{4}^{3}$
17.12.9.3	$12$	x^12 - 289x^4 + 14739	$17$	$4$	$3$	$9$	$C_{12}$ (as 12T1)	$3$	$4$	$[\ ]_{4}^{3}$
17.12.9.4	$12$	x^12 + 153x^8 + 7514x^4 + 132651	$17$	$4$	$3$	$9$	$C_{12}$ (as 12T1)	$3$	$4$	$[\ ]_{4}^{3}$
17.12.10.1	$12$	x^12 - 170x^6 + 210681	$17$	$6$	$2$	$10$	$D_6$ (as 12T3)	$2$	$6$	$[\ ]_{6}^{2}$
17.12.10.2	$12$	x^12 + 85x^6 + 2601	$17$	$6$	$2$	$10$	$C_6\times S_3$ (as 12T18)	$6$	$6$	$[\ ]_{6}^{6}$
17.12.10.3	$12$	x^12 + 136x^6 + 7803	$17$	$6$	$2$	$10$	$C_3 : C_4$ (as 12T5)	$2$	$6$	$[\ ]_{6}^{2}$
17.12.10.4	$12$	x^12 - 17x^6 + 867	$17$	$6$	$2$	$10$	$C_3\times (C_3 : C_4)$ (as 12T19)	$6$	$6$	$[\ ]_{6}^{6}$
17.12.11.1	$12$	x^12 - 17	$17$	$12$	$1$	$11$	$S_3 \times C_4$ (as 12T11)	$2$	$12$	$[\ ]_{12}^{2}$
17.12.11.2	$12$	x^12 - 153	$17$	$12$	$1$	$11$	$S_3 \times C_4$ (as 12T11)	$2$	$12$	$[\ ]_{12}^{2}$
17.12.11.3	$12$	x^12 + 51	$17$	$12$	$1$	$11$	$S_3 \times C_4$ (as 12T11)	$2$	$12$	$[\ ]_{12}^{2}$
17.12.11.4	$12$	x^12 + 459	$17$	$12$	$1$	$11$	$S_3 \times C_4$ (as 12T11)	$2$	$12$	$[\ ]_{12}^{2}$