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 Num. automorphisms
Ramification index $e$ Residue field degree $f$ Galois tame degree $t$ Galois unram. degree $u$ Intermediate field
Inertia subgroup Indices Assoc. Inertia Jump Set
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Results (8 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
2.1.4.8a1.1 $x^{4} + 4 x + 2$ $2$ $1$ $4$ $8$ $S_4$ (as 4T5) $2$ $3$ $[\frac{8}{3}, \frac{8}{3}]$ $[\frac{5}{3},\frac{5}{3}]$ $[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $t + 1$ $x^{4} + 4 x + 2$ $[5, 4, 0]$ $[1]$ $z + 1$ $[1, 3, 7]$
2.1.4.8a1.2 $x^{4} + 4 x^{2} + 4 x + 2$ $2$ $1$ $4$ $8$ $S_4$ (as 4T5) $2$ $3$ $[\frac{8}{3}, \frac{8}{3}]$ $[\frac{5}{3},\frac{5}{3}]$ $[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $t + 1$ $x^{4} + 4 x^{2} + 4 x + 2$ $[5, 4, 0]$ $[1]$ $z + 1$ $[1, 3, 7]$
2.1.4.8b1.1 $x^{4} + 2 x^{2} + 4 x + 2$ $2$ $1$ $4$ $8$ $C_2^2$ (as 4T2) $1$ $1$ $[2, 3]$ $[1,2]$ $[2, 3]$ $[1,2]$ $[\ ]$ $[\ ]$ $t + 1$ $x^{4} + 2 x^{2} + 4 x + 2$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4]$
2.1.4.8b1.2 $x^{4} + 2 x^{2} + 4 x + 10$ $2$ $1$ $4$ $8$ $C_2^2$ (as 4T2) $1$ $1$ $[2, 3]$ $[1,2]$ $[2, 3]$ $[1,2]$ $[\ ]$ $[\ ]$ $t + 1$ $x^{4} + 2 x^{2} + 4 x + 10$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 8, 12]$
2.1.4.8b1.3 $x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 10$ $2$ $1$ $4$ $8$ $D_{4}$ (as 4T3) $2$ $1$ $[2, 3]$ $[1,2]$ $[2, 3]^{2}$ $[1,2]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $t + 1$ $x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 10$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 7, 11]$
2.1.4.8b1.4 $x^{4} + 2 x^{2} + 4 x + 14$ $2$ $1$ $4$ $8$ $D_{4}$ (as 4T3) $2$ $1$ $[2, 3]$ $[1,2]$ $[2, 3]^{2}$ $[1,2]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $t + 1$ $x^{4} + 2 x^{2} + 4 x + 14$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 8]$
2.1.4.8b1.5 $x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 6$ $2$ $1$ $4$ $8$ $C_2^2$ (as 4T2) $1$ $1$ $[2, 3]$ $[1,2]$ $[2, 3]$ $[1,2]$ $[\ ]$ $[\ ]$ $t + 1$ $x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 6$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 8]$
2.1.4.8b1.6 $x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 14$ $2$ $1$ $4$ $8$ $C_2^2$ (as 4T2) $1$ $1$ $[2, 3]$ $[1,2]$ $[2, 3]$ $[1,2]$ $[\ ]$ $[\ ]$ $t + 1$ $x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 14$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 8]$
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