LMFDB - p-adic field search results

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Label	Polynomial	$p$	$e$	$f$	$c$	Galois group	Slope content
2.8.0.1	x⁸ + x⁴ + x³ + x + 1	$2$	$1$	$8$	$0$	$C_8$ (as 8T1)	$ [\ ]^{8}$
2.8.8.1	x⁸ + 28x⁴ + 144	$2$	$2$	$4$	$8$	$C_4\times C_2$ (as 8T2)	$ [2]^{4}$
2.8.8.10	x⁸ + 2x⁶ + 8x³ + 16	$2$	$2$	$4$	$8$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$ [2, 2, 2, 2]^{4}$
2.8.8.11	x⁸ + 20x² + 4	$2$	$4$	$2$	$8$	$S_4$ (as 8T14)	$ [4/3, 4/3]_{3}^{2}$
2.8.8.12	x⁸ + 2x⁵ + 2x⁴ + 4	$2$	$4$	$2$	$8$	$A_4\wr C_2$ (as 8T42)	$ [4/3, 4/3, 4/3, 4/3]_{3}^{6}$
2.8.8.13	x⁸ + 2x + 2	$2$	$8$	$1$	$8$	$C_2^3:(C_7: C_3)$ (as 8T36)	$ [8/7, 8/7, 8/7]_{7}^{3}$
2.8.8.2	x⁸ + 2x⁷ + 8x² + 48	$2$	$2$	$4$	$8$	$C_2^2:C_4$ (as 8T10)	$ [2, 2]^{4}$
2.8.8.3	x⁸ + 2x⁷ + 2x⁶ + 16	$2$	$2$	$4$	$8$	$C_2^3: C_4$ (as 8T20)	$ [2, 2, 2]^{4}$
2.8.8.4	x⁸ + 2x⁷ + 2x⁶ + 8x³ + 48	$2$	$2$	$4$	$8$	$C_8$ (as 8T1)	$ [2]^{4}$
2.8.8.5	x⁸ + 2x⁷ + 8x² + 16	$2$	$2$	$4$	$8$	$C_8:C_2$ (as 8T7)	$ [2, 2]^{4}$
2.8.8.6	x⁸ + 2x⁷ + 2x⁶ + 16x² + 16	$2$	$2$	$4$	$8$	$(C_8:C_2):C_2$ (as 8T16)	$ [2, 2, 2]^{4}$
2.8.8.7	x⁸ + 2x⁶ + 4x⁵ + 16	$2$	$2$	$4$	$8$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$ [2, 2, 2, 2]^{4}$
2.8.8.8	x⁸ + 4x⁵ + 8x² + 48	$2$	$2$	$4$	$8$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$ [2, 2, 2, 2]^{4}$
2.8.8.9	x⁸ + 6x⁶ + 4x⁵ + 16	$2$	$2$	$4$	$8$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$ [2, 2, 2, 2]^{4}$
2.8.10.1	x⁸ + 2x³ + 2	$2$	$8$	$1$	$10$	$C_2^3:(C_7: C_3)$ (as 8T36)	$ [10/7, 10/7, 10/7]_{7}^{3}$
2.8.10.2	x⁸ + 20x² + 20	$2$	$8$	$1$	$10$	$\textrm{GL(2,3)}$ (as 8T23)	$ [4/3, 4/3, 3/2]_{3}^{2}$
2.8.10.3	x⁸ + 4x² + 20	$2$	$8$	$1$	$10$	$\textrm{GL(2,3)}$ (as 8T23)	$ [4/3, 4/3, 3/2]_{3}^{2}$
2.8.12.1	x⁸ + 6x⁶ + 8x⁵ + 16	$2$	$2$	$4$	$12$	$C_4\times C_2$ (as 8T2)	$ [3]^{4}$
2.8.12.10	x⁸ + 2x⁶ + 112	$2$	$2$	$4$	$12$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$ [2, 2, 2, 3]^{4}$
2.8.12.11	x⁸ + 6x⁶ + 48	$2$	$2$	$4$	$12$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$ [2, 2, 2, 3]^{4}$
2.8.12.12	x⁸ + 2x⁶ + 48	$2$	$2$	$4$	$12$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$ [2, 2, 2, 3]^{4}$
2.8.12.13	x⁸ + 12x⁴ + 16	$2$	$4$	$2$	$12$	$D_4$ (as 8T4)	$ [2, 2]^{2}$
2.8.12.14	x⁸ + 12x⁴ + 144	$2$	$4$	$2$	$12$	$D_4$ (as 8T4)	$ [2, 2]^{2}$
2.8.12.15	x⁸ + 2x⁷ + 2x⁴ + 12	$2$	$4$	$2$	$12$	$C_2^2:C_4$ (as 8T10)	$ [2, 2]^{4}$
2.8.12.16	x⁸ + 24x² + 4	$2$	$4$	$2$	$12$	$A_4\times C_2$ (as 8T13)	$ [2, 2]^{6}$
2.8.12.17	x⁸ + 2x⁷ + 2x⁶ + 2x⁴ + 4	$2$	$4$	$2$	$12$	$C_2^4:C_6$ (as 8T33)	$ [2, 2, 2, 2]^{6}$
2.8.12.18	x⁸ + 8x⁷ + 48	$2$	$4$	$2$	$12$	$C_2^4:C_6$ (as 8T33)	$ [2, 2, 2, 2]^{6}$
2.8.12.19	x⁸ + 12x⁴ + 80	$2$	$4$	$2$	$12$	$(C_8:C_2):C_2$ (as 8T16)	$ [2, 2, 2]^{4}$
2.8.12.2	x⁸ + 2x⁶ + 8x⁴ + 16	$2$	$2$	$4$	$12$	$C_4\times C_2$ (as 8T2)	$ [3]^{4}$
2.8.12.20	x⁸ + 8x⁶ + 12x⁴ + 80	$2$	$4$	$2$	$12$	$C_2^3: C_4$ (as 8T21)	$ [2, 2, 2]^{4}$
2.8.12.21	x⁸ + 12x⁶ + 12x⁴ + 80	$2$	$4$	$2$	$12$	$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28)	$ [2, 2, 2, 2]^{4}$
2.8.12.22	x⁸ + 4x⁷ + 16x³ + 48	$2$	$4$	$2$	$12$	$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28)	$ [2, 2, 2, 2]^{4}$
2.8.12.23	x⁸ + 8x⁵ + 4x⁴ + 48	$2$	$4$	$2$	$12$	$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28)	$ [2, 2, 2, 2]^{4}$
2.8.12.24	x⁸ + 4x⁶ + 28x⁴ + 80	$2$	$4$	$2$	$12$	$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28)	$ [2, 2, 2, 2]^{4}$
2.8.12.25	x⁸ + 28x² + 4	$2$	$8$	$1$	$12$	$S_4\times C_2$ (as 8T24)	$ [4/3, 4/3, 2]_{3}^{2}$
2.8.12.26	x⁸ + 12x² + 4	$2$	$8$	$1$	$12$	$S_4\times C_2$ (as 8T24)	$ [4/3, 4/3, 2]_{3}^{2}$
2.8.12.27	x⁸ + 4x⁵ + 4	$2$	$8$	$1$	$12$	$C_2^3:(C_7: C_3)$ (as 8T36)	$ [12/7, 12/7, 12/7]_{7}^{3}$
2.8.12.28	x⁸ + 2x⁵ + 2	$2$	$8$	$1$	$12$	$C_2^3:(C_7: C_3)$ (as 8T36)	$ [12/7, 12/7, 12/7]_{7}^{3}$
2.8.12.29	x⁸ + 28x² + 20	$2$	$8$	$1$	$12$	$\textrm{GL(2,3)}$ (as 8T23)	$ [4/3, 4/3, 2]_{3}^{2}$
2.8.12.3	x⁸ + 2x⁶ + 16x² + 16	$2$	$2$	$4$	$12$	$C_2^2:C_4$ (as 8T10)	$ [2, 3]^{4}$
2.8.12.30	x⁸ + 12x² + 20	$2$	$8$	$1$	$12$	$\textrm{GL(2,3)}$ (as 8T23)	$ [4/3, 4/3, 2]_{3}^{2}$
2.8.12.4	x⁸ + 2x⁶ + 16	$2$	$2$	$4$	$12$	$C_2^3: C_4$ (as 8T20)	$ [2, 2, 3]^{4}$
2.8.12.5	x⁸ + 6x⁶ + 8x⁵ + 80	$2$	$2$	$4$	$12$	$C_8$ (as 8T1)	$ [3]^{4}$
2.8.12.6	x⁸ + 2x⁶ + 8x⁴ + 80	$2$	$2$	$4$	$12$	$C_8$ (as 8T1)	$ [3]^{4}$
2.8.12.7	x⁸ + 4x⁶ + 8x² + 80	$2$	$2$	$4$	$12$	$C_8:C_2$ (as 8T7)	$ [2, 3]^{4}$
2.8.12.8	x⁸ + 2x⁶ + 80	$2$	$2$	$4$	$12$	$(C_8:C_2):C_2$ (as 8T16)	$ [2, 2, 3]^{4}$
2.8.12.9	x⁸ + 6x⁶ + 112	$2$	$2$	$4$	$12$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$ [2, 2, 2, 3]^{4}$
2.8.14.1	x⁸ + 2x⁷ + 6	$2$	$8$	$1$	$14$	$A_4\times C_2$ (as 8T13)	$ [2, 2, 2]^{3}$
2.8.14.10	x⁸ + 4x² + 28	$2$	$8$	$1$	$14$	$C_2 \wr S_4$ (as 8T44)	$ [4/3, 4/3, 2, 7/3, 7/3, 5/2]_{3}^{2}$
2.8.14.11	x⁸ + 20x² + 12	$2$	$8$	$1$	$14$	$C_2 \wr S_4$ (as 8T44)	$ [4/3, 4/3, 2, 7/3, 7/3, 5/2]_{3}^{2}$

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