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The results below are complete, since the LMFDB contains all p-adic fields of degree at most 23 and residue characteristic at most 199

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Degree $n$ Residue characteristic $p$ Discriminant exponent $c$ Galois group
Ramification index $e$ Residue field degree $f$ Galois tame degree $t$ Galois unram. degree $u$
Num. automorphisms Inertia subgroup Indices Assoc. Inertia Jump Set
Top Artin slope Galois Artin Visible Artin Wild inertia subgroup
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Results (13 matches)

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Label Polynomial $p$ $f$ $e$ $c$ Galois group $u$ $t$ Visible Artin slopes Visible Swan slopes Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Unram. Ext. Eisen. Poly. Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
7.16.1.0a1.1 $x^{16} + 4 x^{8} + 5 x^{7} + 3 x^{6} + 4 x^{5} + x^{4} + 6 x^{3} + 2 x^{2} + 4 x + 3$ $7$ $16$ $1$ $0$ $C_{16}$ (as 16T1) $16$ $1$ $[\ ]$ $[\ ]$ $[\ ]^{16}$ $[\ ]^{16}$ $[\ ]$ $[\ ]$ $t^{16} + 4 t^{8} + 5 t^{7} + 3 t^{6} + 4 t^{5} + t^{4} + 6 t^{3} + 2 t^{2} + 4 t + 3$ $x - 7$ $[0]$ $[\ ]$ undefined
7.8.2.8a1.1 $( x^{8} + 4 x^{3} + 6 x^{2} + 2 x + 3 )^{2} + 7 x$ $7$ $8$ $2$ $8$ $C_{16}$ (as 16T1) $8$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{8}$ $[\ ]_{2}^{8}$ $[\ ]$ $[\ ]$ $t^{8} + 4 t^{3} + 6 t^{2} + 2 t + 3$ $x^{2} + 7 t$ $[0]$ $[1]$ $z + 2$ undefined
7.8.2.8a1.2 $( x^{8} + 4 x^{3} + 6 x^{2} + 2 x + 3 )^{2} + 7$ $7$ $8$ $2$ $8$ $C_8\times C_2$ (as 16T5) $8$ $2$ $[\ ]$ $[\ ]$ $[\ ]_{2}^{8}$ $[\ ]_{2}^{8}$ $[\ ]$ $[\ ]$ $t^{8} + 4 t^{3} + 6 t^{2} + 2 t + 3$ $x^{2} + 7$ $[0]$ $[1]$ $z + 2$ undefined
7.4.4.12a1.1 $( x^{4} + 5 x^{2} + 4 x + 3 )^{4} + 7 x^{2}$ $7$ $4$ $4$ $12$ $C_8: C_2$ (as 16T6) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 5 t^{2} + 4 t + 3$ $x^{4} + 7 t^{2}$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
7.4.4.12a1.2 $( x^{4} + 5 x^{2} + 4 x + 3 )^{4} + 7 x$ $7$ $4$ $4$ $12$ $C_{16} : C_2$ (as 16T22) $8$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{8}$ $[\ ]_{4}^{8}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{4} + 5 t^{2} + 4 t + 3$ $x^{4} + 7 t^{3}$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
7.4.4.12a1.3 $( x^{4} + 5 x^{2} + 4 x + 3 )^{4} + 7$ $7$ $4$ $4$ $12$ $C_4:C_4$ (as 16T8) $4$ $4$ $[\ ]$ $[\ ]$ $[\ ]_{4}^{4}$ $[\ ]_{4}^{4}$ $[\ ]$ $[\ ]$ $t^{4} + 5 t^{2} + 4 t + 3$ $x^{4} + 7$ $[0]$ $[1]$ $z^3 + 4 z^2 + 6 z + 4$ undefined
7.2.8.14a1.1 $( x^{2} + 6 x + 3 )^{8} + 7 x$ $7$ $2$ $8$ $14$ $C_8.C_8$ (as 16T124) $8$ $8$ $[\ ]$ $[\ ]$ $[\ ]_{8}^{8}$ $[\ ]_{8}^{8}$ $[\ ]^{4}$ $[\ ]^{4}$ $t^{2} + 6 t + 3$ $x^{8} + 42 t + 7$ $[0]$ $[1]$ $z^7 + z^6 + 1$ undefined
7.2.8.14a1.2 $( x^{2} + 6 x + 3 )^{8} + 7$ $7$ $2$ $8$ $14$ $D_{8}$ (as 16T13) $2$ $8$ $[\ ]$ $[\ ]$ $[\ ]_{8}^{2}$ $[\ ]_{8}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 6 t + 3$ $x^{8} + 7$ $[0]$ $[1]$ $z^7 + z^6 + 1$ undefined
7.2.8.14a1.3 $( x^{2} + 6 x + 3 )^{8} + 7 x + 7$ $7$ $2$ $8$ $14$ $C_8.C_8$ (as 16T124) $8$ $8$ $[\ ]$ $[\ ]$ $[\ ]_{8}^{8}$ $[\ ]_{8}^{8}$ $[\ ]^{4}$ $[\ ]^{4}$ $t^{2} + 6 t + 3$ $x^{8} + 35 t + 28$ $[0]$ $[1]$ $z^7 + z^6 + 1$ undefined
7.2.8.14a1.4 $( x^{2} + 6 x + 3 )^{8} + 7 x + 28$ $7$ $2$ $8$ $14$ $C_8.C_4$ (as 16T49) $4$ $8$ $[\ ]$ $[\ ]$ $[\ ]_{8}^{4}$ $[\ ]_{8}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $t^{2} + 6 t + 3$ $x^{8} + 14 t + 28$ $[0]$ $[1]$ $z^7 + z^6 + 1$ undefined
7.2.8.14a1.5 $( x^{2} + 6 x + 3 )^{8} + 14 x + 42$ $7$ $2$ $8$ $14$ $Q_{16}$ (as 16T14) $2$ $8$ $[\ ]$ $[\ ]$ $[\ ]_{8}^{2}$ $[\ ]_{8}^{2}$ $[\ ]$ $[\ ]$ $t^{2} + 6 t + 3$ $x^{8} + 14 t + 42$ $[0]$ $[1]$ $z^7 + z^6 + 1$ undefined
7.1.16.15a1.1 $x^{16} + 7$ $7$ $1$ $16$ $15$ $\SD_{32}$ (as 16T55) $2$ $16$ $[\ ]$ $[\ ]$ $[\ ]_{16}^{2}$ $[\ ]_{16}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $t + 4$ $x^{16} + 7$ $[0]$ $[2]$ $z^{15} + 2 z^{14} + z^{13} + 2 z^8 + 4 z^7 + 2 z^6 + z + 2$ undefined
7.1.16.15a1.2 $x^{16} + 21$ $7$ $1$ $16$ $15$ $\SD_{32}$ (as 16T55) $2$ $16$ $[\ ]$ $[\ ]$ $[\ ]_{16}^{2}$ $[\ ]_{16}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $t + 4$ $x^{16} + 21$ $[0]$ $[2]$ $z^{15} + 2 z^{14} + z^{13} + 2 z^8 + 4 z^7 + 2 z^6 + z + 2$ undefined
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