pp-adic field search results

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Label	$n$	Polynomial	$p$	$e$	$f$	$c$	Galois group	$u$	$t$	Visible slopes	Slope content	Unram. Ext.	Eisen. Poly.	Ind. of Insep.	Assoc. Inertia
7.14.0.1	$14$	$x^{14} + 5 x^{7} + 6 x^{5} + 2 x^{4} + 3 x^{2} + 6 x + 3$	$7$	$1$	$14$	$0$	$C_{14}$ (as 14T1)	$14$	$1$	$[\ ]$	$[\ ]^{14}$	$t^{14} + 5 t^{7} + 6 t^{5} + 2 t^{4} + 3 t^{2} + 6 t + 3$	$x - 7$	$[0]$	$[\ ]$
7.14.7.1	$14$	$x^{14} + 49 x^{12} + 1029 x^{10} + 12017 x^{8} + 8 x^{7} + 82859 x^{6} - 1176 x^{5} + 352947 x^{4} + 13720 x^{3} + 881203 x^{2} - 19160 x + 794999$	$7$	$2$	$7$	$7$	$C_{14}$ (as 14T1)	$7$	$2$	$[\ ]$	$[\ ]_{2}^{7}$	$t^{7} + 6 t + 4$	$x^{2} + 7$	$[0]$	$[1]$
7.14.7.2	$14$	$x^{14} - 756 x^{13} + 407484 x^{12} - 120759324 x^{11} + 18245133144 x^{10} - 1371646043208 x^{9} + 45234910589400 x^{8} - 365498518846944 x^{7} - 160150981540032 x^{6} - 9636001680480 x^{5} - 1806831694584 x^{4} - 187090145760 x^{3} - 8419197738 x^{2} - 138355224 x - 3294172$	$7$	$2$	$7$	$7$	$C_{14}$ (as 14T1)	$7$	$2$	$[\ ]$	$[\ ]_{2}^{7}$	$t^{7} + 6 t + 4$	$x^{2} + \left(21 t^{6} + 42 t^{4} + 42 t^{3} + 42 t^{2} + 42 t\right) x + 7 t$	$[0]$	$[1]$
7.14.14.1	$14$	$x^{14} + 14 x^{9} - 28 x^{8} + 14 x^{7} - 245 x^{4} - 1372 x^{3} - 882 x^{2} - 196 x + 49$	$7$	$7$	$2$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(7 t + 28\right) x^{2} + \left(14 t + 28\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.2	$14$	$x^{14} - 84 x^{9} - 140 x^{8} + 14 x^{7} + 588 x^{4} - 3038 x^{2} - 980 x + 49$	$7$	$7$	$2$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + 14 t x^{2} + \left(35 t + 35\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.3	$14$	$x^{14} + 42 x^{9} + 42 x^{8} + 14 x^{7} + 441 x^{4} + 882 x^{3} + 441 x^{2} + 294 x + 49$	$7$	$7$	$2$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + 21 x^{2} + \left(7 t + 42\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.4	$14$	$x^{14} - 14 x^{9} + 28 x^{8} + 14 x^{7} - 1127 x^{4} - 196 x^{3} + 98 x^{2} + 196 x + 49$	$7$	$7$	$2$	$14$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[7/6]$	$[7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(14 t + 35\right) x^{2} + 14 x + 7$	$[1, 0]$	$[1]$
7.14.14.5	$14$	$x^{14} - 56 x^{9} + 56 x^{8} + 14 x^{7} - 392 x^{4} - 1568 x^{3} + 392 x^{2} + 392 x + 49$	$7$	$7$	$2$	$14$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[7/6]$	$[7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(14 t + 14\right) x^{2} + 28 x + 7$	$[1, 0]$	$[1]$
7.14.14.6	$14$	$x^{14} - 196 x^{9} - 210 x^{8} + 14 x^{7} + 2254 x^{4} + 5880 x^{3} + 2303 x^{2} - 1470 x + 49$	$7$	$7$	$2$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(35 t + 7\right) x^{2} + 35 t x + 7$	$[1, 0]$	$[1]$
7.14.14.7	$14$	$x^{14} - 238 x^{9} - 196 x^{8} + 14 x^{7} + 3577 x^{4} + 2156 x^{3} - 2646 x^{2} - 1372 x + 49$	$7$	$7$	$2$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(42 t + 7\right) x^{2} + \left(42 t + 28\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.8	$14$	$x^{14} - 14 x^{9} - 84 x^{8} + 14 x^{7} - 1127 x^{4} - 4116 x^{3} - 3038 x^{2} - 588 x + 49$	$7$	$7$	$2$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(14 t + 35\right) x^{2} + \left(28 t + 42\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.9	$14$	$x^{14} + 14 x^{9} - 56 x^{8} + 14 x^{7} + 49 x^{4} - 392 x^{3} - 1764 x^{2} - 392 x + 49$	$7$	$7$	$2$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + 7 x^{2} + \left(21 t + 35\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.10	$14$	$x^{14} - 14 x^{9} - 70 x^{8} + 14 x^{7} - 245 x^{4} - 1274 x^{3} - 1519 x^{2} - 490 x + 49$	$7$	$7$	$2$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(7 t + 14\right) x^{2} + \left(21 t + 28\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.11	$14$	$x^{14} - 238 x^{9} - 14 x^{8} + 14 x^{7} + 3577 x^{4} - 1862 x^{3} - 1911 x^{2} - 98 x + 49$	$7$	$7$	$2$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(42 t + 7\right) x^{2} + \left(7 t + 14\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.12	$14$	$x^{14} - 84 x^{8} + 14 x^{7} + 588 x^{2} - 588 x + 49$	$7$	$7$	$2$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + 14 t x + 7$	$[1, 0]$	$[1]$
7.14.14.13	$14$	$x^{14} + 28 x^{9} - 154 x^{8} + 14 x^{7} + 196 x^{4} - 2156 x^{3} - 1225 x^{2} - 1078 x + 49$	$7$	$7$	$2$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + 14 x^{2} + \left(35 t + 28\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.14	$14$	$x^{14} - 14 x^{9} - 140 x^{8} + 14 x^{7} - 245 x^{4} - 1372 x^{3} + 98 x^{2} - 980 x + 49$	$7$	$7$	$2$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(7 t + 14\right) x^{2} + \left(28 t + 14\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.15	$14$	$x^{14} - 168 x^{9} + 14 x^{7} - 3528 x^{4} - 7056 x^{3} - 2352 x^{2} + 49$	$7$	$7$	$2$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(42 t + 42\right) x^{2} + \left(14 t + 42\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.16	$14$	$x^{14} - 42 x^{9} - 126 x^{8} + 14 x^{7} - 735 x^{4} - 882 x^{3} + 1029 x^{2} - 882 x + 49$	$7$	$7$	$2$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(14 t + 21\right) x^{2} + 21 t x + 7$	$[1, 0]$	$[1]$
7.14.14.17	$14$	$x^{14} - 84 x^{9} - 28 x^{8} + 14 x^{7} + 588 x^{4} - 686 x^{2} - 196 x + 49$	$7$	$7$	$2$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + 14 t x^{2} + \left(7 t + 7\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.18	$14$	$x^{14} + 28 x^{8} + 14 x^{7} - 98 x^{2} + 196 x + 49$	$7$	$7$	$2$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(7 t + 35\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.19	$14$	$x^{14} - 42 x^{9} + 42 x^{8} + 14 x^{7} - 735 x^{4} - 882 x^{3} + 147 x^{2} + 294 x + 49$	$7$	$7$	$2$	$14$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[7/6]$	$[7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(14 t + 21\right) x^{2} + 21 x + 7$	$[1, 0]$	$[1]$
7.14.14.20	$14$	$x^{14} - 210 x^{9} - 168 x^{8} + 14 x^{7} + 3675 x^{4} + 2940 x^{3} - 1764 x^{2} - 1176 x + 49$	$7$	$7$	$2$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + 35 t x^{2} + \left(35 t + 21\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.21	$14$	$x^{14} - 14 x^{9} + 14 x^{8} + 14 x^{7} - 1127 x^{4} - 98 x^{3} - 49 x^{2} + 98 x + 49$	$7$	$7$	$2$	$14$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[7/6]$	$[7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(14 t + 35\right) x^{2} + 7 x + 7$	$[1, 0]$	$[1]$
7.14.14.22	$14$	$x^{14} + 14 x^{9} - 182 x^{8} + 14 x^{7} + 49 x^{4} - 1274 x^{3} - 2205 x^{2} - 1274 x + 49$	$7$	$7$	$2$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + 7 x^{2} + \left(42 t + 35\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.23	$14$	$x^{14} - 28 x^{9} - 42 x^{8} + 14 x^{7} - 98 x^{4} - 49 x^{2} - 294 x + 49$	$7$	$7$	$2$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(7 t + 7\right) x^{2} + 7 t x + 7$	$[1, 0]$	$[1]$
7.14.14.24	$14$	$x^{14} - 14 x^{9} + 84 x^{8} + 14 x^{7} - 245 x^{4} - 588 x^{3} + 1666 x^{2} + 588 x + 49$	$7$	$7$	$2$	$14$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[7/6]$	$[7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(7 t + 14\right) x^{2} + 42 x + 7$	$[1, 0]$	$[1]$
7.14.14.25	$14$	$x^{14} - 56 x^{9} + 70 x^{8} + 14 x^{7} - 1862 x^{4} - 1960 x^{3} + 833 x^{2} + 490 x + 49$	$7$	$7$	$2$	$14$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[7/6]$	$[7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(21 t + 35\right) x^{2} + 35 x + 7$	$[1, 0]$	$[1]$
7.14.14.26	$14$	$x^{14} - 154 x^{9} - 112 x^{8} + 14 x^{7} - 1421 x^{4} - 196 x^{3} - 588 x^{2} - 784 x + 49$	$7$	$7$	$2$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(35 t + 28\right) x^{2} + \left(21 t + 7\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.27	$14$	$x^{14} - 70 x^{9} - 98 x^{8} + 14 x^{7} + 49 x^{4} - 1274 x^{3} - 2793 x^{2} - 686 x + 49$	$7$	$7$	$2$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[7/6]$	$[7/6, 7/6]_{6}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(14 t + 7\right) x^{2} + \left(28 t + 35\right) x + 7$	$[1, 0]$	$[1]$
7.14.14.28	$14$	$x^{14} + 28 x^{2} + 14 x + 21$	$7$	$14$	$1$	$14$	$D_7^2:C_6$ (as 14T32)	$2$	$12$	$[13/12]$	$[13/12, 13/12]_{12}^{2}$	$t + 4$	$x^{14} + 28 x^{2} + 14 x + 21$	$[1, 0]$	$[1, 1]$
7.14.14.29	$14$	$x^{14} + 35 x^{2} + 7 x + 21$	$7$	$14$	$1$	$14$	$D_7^2:C_6$ (as 14T32)	$2$	$12$	$[13/12]$	$[13/12, 13/12]_{12}^{2}$	$t + 4$	$x^{14} + 35 x^{2} + 7 x + 21$	$[1, 0]$	$[1, 1]$
7.14.14.30	$14$	$x^{14} + 42 x^{2} + 21 x + 7$	$7$	$14$	$1$	$14$	$D_7^2:C_6$ (as 14T32)	$2$	$12$	$[13/12]$	$[13/12, 13/12]_{12}^{2}$	$t + 4$	$x^{14} + 42 x^{2} + 21 x + 7$	$[1, 0]$	$[1, 1]$
7.14.14.31	$14$	$x^{14} + 35 x^{2} + 14 x + 7$	$7$	$14$	$1$	$14$	$D_7^2:C_6$ (as 14T32)	$2$	$12$	$[13/12]$	$[13/12, 13/12]_{12}^{2}$	$t + 4$	$x^{14} + 35 x^{2} + 14 x + 7$	$[1, 0]$	$[1, 1]$
7.14.14.32	$14$	$x^{14} + 7 x^{2} + 21 x + 21$	$7$	$14$	$1$	$14$	$D_7^2:C_6$ (as 14T32)	$2$	$12$	$[13/12]$	$[13/12, 13/12]_{12}^{2}$	$t + 4$	$x^{14} + 7 x^{2} + 21 x + 21$	$[1, 0]$	$[1, 1]$
7.14.14.33	$14$	$x^{14} + 21 x^{2} + 7 x + 7$	$7$	$14$	$1$	$14$	$D_7^2:C_6$ (as 14T32)	$2$	$12$	$[13/12]$	$[13/12, 13/12]_{12}^{2}$	$t + 4$	$x^{14} + 21 x^{2} + 7 x + 7$	$[1, 0]$	$[1, 1]$
7.14.15.1	$14$	$x^{14} + 14 x^{3} + 7 x^{2} + 21$	$7$	$14$	$1$	$15$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[7/6]$	$[7/6]_{6}^{2}$	$t + 4$	$x^{14} + 14 x^{3} + 7 x^{2} + 21$	$[2, 0]$	$[2, 1]$
7.14.15.2	$14$	$x^{14} + 7 x^{3} + 35 x^{2} + 7$	$7$	$14$	$1$	$15$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[7/6]$	$[7/6]_{6}^{2}$	$t + 4$	$x^{14} + 7 x^{3} + 35 x^{2} + 7$	$[2, 0]$	$[2, 1]$
7.14.15.3	$14$	$x^{14} + 14 x^{3} + 21 x^{2} + 7$	$7$	$14$	$1$	$15$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[7/6]$	$[7/6]_{6}^{2}$	$t + 4$	$x^{14} + 14 x^{3} + 21 x^{2} + 7$	$[2, 0]$	$[2, 1]$
7.14.15.4	$14$	$x^{14} + 14 x^{3} + 14 x^{2} + 21$	$7$	$14$	$1$	$15$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[7/6]$	$[7/6]_{6}^{2}$	$t + 4$	$x^{14} + 14 x^{3} + 14 x^{2} + 21$	$[2, 0]$	$[2, 1]$
7.14.15.5	$14$	$x^{14} + 7 x^{3} + 7 x^{2} + 7$	$7$	$14$	$1$	$15$	$F_7$ (as 14T4)	$1$	$6$	$[7/6]$	$[7/6]_{6}$	$t + 4$	$x^{14} + 7 x^{3} + 7 x^{2} + 7$	$[2, 0]$	$[1, 1]$
7.14.15.6	$14$	$x^{14} + 7 x^{3} + 21 x^{2} + 21$	$7$	$14$	$1$	$15$	$F_7$ (as 14T4)	$1$	$6$	$[7/6]$	$[7/6]_{6}$	$t + 4$	$x^{14} + 7 x^{3} + 21 x^{2} + 21$	$[2, 0]$	$[1, 1]$
7.14.15.7	$14$	$x^{14} + 21 x^{3} + 28 x^{2} + 7$	$7$	$14$	$1$	$15$	$F_7$ (as 14T4)	$1$	$6$	$[7/6]$	$[7/6]_{6}$	$t + 4$	$x^{14} + 21 x^{3} + 28 x^{2} + 7$	$[2, 0]$	$[1, 1]$
7.14.15.8	$14$	$x^{14} + 14 x^{3} + 42 x^{2} + 21$	$7$	$14$	$1$	$15$	$F_7$ (as 14T4)	$1$	$6$	$[7/6]$	$[7/6]_{6}$	$t + 4$	$x^{14} + 14 x^{3} + 42 x^{2} + 21$	$[2, 0]$	$[1, 1]$
7.14.15.9	$14$	$x^{14} + 21 x^{3} + 14 x^{2} + 7$	$7$	$14$	$1$	$15$	$F_7$ (as 14T4)	$1$	$6$	$[7/6]$	$[7/6]_{6}$	$t + 4$	$x^{14} + 21 x^{3} + 14 x^{2} + 7$	$[2, 0]$	$[1, 1]$
7.14.15.10	$14$	$x^{14} + 21 x^{3} + 35 x^{2} + 21$	$7$	$14$	$1$	$15$	$F_7$ (as 14T4)	$1$	$6$	$[7/6]$	$[7/6]_{6}$	$t + 4$	$x^{14} + 21 x^{3} + 35 x^{2} + 21$	$[2, 0]$	$[1, 1]$
7.14.15.11	$14$	$x^{14} + 7 x^{3} + 42 x^{2} + 7$	$7$	$14$	$1$	$15$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[7/6]$	$[7/6]_{6}^{2}$	$t + 4$	$x^{14} + 7 x^{3} + 42 x^{2} + 7$	$[2, 0]$	$[2, 1]$
7.14.15.12	$14$	$x^{14} + 28 x^{2} + 21$	$7$	$14$	$1$	$15$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[7/6]$	$[7/6]_{6}^{2}$	$t + 4$	$x^{14} + 28 x^{2} + 21$	$[2, 0]$	$[2, 1]$
7.14.16.1	$14$	$x^{14} - 182 x^{10} - 210 x^{9} + 14 x^{7} - 2303 x^{6} - 2058 x^{5} + 441 x^{4} - 1274 x^{3} - 1470 x^{2} + 49$	$7$	$7$	$2$	$16$	$C_7:F_7$ (as 14T14)	$2$	$3$	$[4/3]$	$[4/3, 4/3]_{3}^{2}$	$t^{2} + 6 t + 3$	$x^{7} + \left(42 t + 35\right) x^{3} + \left(42 t + 21\right) x^{2} + 7$	$[2, 0]$	$[1]$
7.14.16.2	$14$	$x^{14} + 56 x^{10} + 42 x^{9} + 14 x^{7} + 784 x^{6} + 1176 x^{5} + 147 x^{4} + 392 x^{3} + 294 x^{2} + 49$	$7$	$7$	$2$	$16$	$C_7^2:C_{12}$ (as 14T23)	$4$	$3$	$[4/3]$	$[4/3, 4/3]_{3}^{4}$	$t^{2} + 6 t + 3$	$x^{7} + 28 x^{3} + \left(7 t + 42\right) x^{2} + 7$	$[2, 0]$	$[2]$

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