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Properties

Label 17.1.12.11a
Base 17.1.1.0a1.1
Degree \(12\)
e \(12\)
f \(1\)
c \(11\)

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Defining polynomial

$x^{12} + 17d_{0}$

Invariants

Residue field characteristic: $17$
Degree: $12$
Base field: $\Q_{17}$
Ramification index $e$: $12$
Residue field degree $f$: $1$
Discriminant exponent $c$: $11$
Artin slopes: $[\ ]$
Swan slopes: $[\ ]$
Means: $\langle\ \rangle$
Rams: $(\ )$
Field count: $4$ (complete)
Ambiguity: $4$
Mass: $1$
Absolute Mass: $1$

Varying

Indices of inseparability: $[0]$
Associated inertia: $[2]$
Jump Set: undefined

Galois groups and Hidden Artin slopes

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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Showing all 4

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Label Packet size Polynomial Galois group Galois degree $\#\Aut(K/\Q_p)$ Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
17.1.12.11a1.1 4 $x^{12} + 17$ $S_3 \times C_4$ (as 12T11) $24$ $4$ $[\ ]_{12}^{2}$ $[\ ]_{12}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{11} + 12 z^{10} + 15 z^9 + 16 z^8 + 2 z^7 + 10 z^6 + 6 z^5 + 10 z^4 + 2 z^3 + 16 z^2 + 15 z + 12$ undefined
17.1.12.11a1.2 4 $x^{12} + 51$ $S_3 \times C_4$ (as 12T11) $24$ $4$ $[\ ]_{12}^{2}$ $[\ ]_{12}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{11} + 12 z^{10} + 15 z^9 + 16 z^8 + 2 z^7 + 10 z^6 + 6 z^5 + 10 z^4 + 2 z^3 + 16 z^2 + 15 z + 12$ undefined
17.1.12.11a1.3 4 $x^{12} + 153$ $S_3 \times C_4$ (as 12T11) $24$ $4$ $[\ ]_{12}^{2}$ $[\ ]_{12}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{11} + 12 z^{10} + 15 z^9 + 16 z^8 + 2 z^7 + 10 z^6 + 6 z^5 + 10 z^4 + 2 z^3 + 16 z^2 + 15 z + 12$ undefined
17.1.12.11a1.4 4 $x^{12} + 170$ $S_3 \times C_4$ (as 12T11) $24$ $4$ $[\ ]_{12}^{2}$ $[\ ]_{12}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{11} + 12 z^{10} + 15 z^9 + 16 z^8 + 2 z^7 + 10 z^6 + 6 z^5 + 10 z^4 + 2 z^3 + 16 z^2 + 15 z + 12$ undefined
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