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Properties

Label 7.1.12.11a
Base 7.1.1.0a1.1
Degree \(12\)
e \(12\)
f \(1\)
c \(11\)

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Defining polynomial

$x^{12} + 7d_{0}$

Invariants

Residue field characteristic: $7$
Degree: $12$
Base field: $\Q_{7}$
Ramification index $e$: $12$
Residue field degree $f$: $1$
Discriminant exponent $c$: $11$
Artin slopes: $[\ ]$
Swan slopes: $[\ ]$
Means: $\langle\ \rangle$
Rams: $(\ )$
Field count: $6$ (complete)
Ambiguity: $6$
Mass: $1$
Absolute Mass: $1$

Varying

Indices of inseparability: $[0]$
Associated inertia: $[2]$
Jump Set: undefined (show 5), $[2]$ (show 1)

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 6

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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
7.1.12.11a1.1 6 $x^{12} + 7$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[\ ]_{12}^{2}$ $[\ ]_{12}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{11} + 5 z^{10} + 3 z^9 + 3 z^8 + 5 z^7 + z^6 + z^4 + 5 z^3 + 3 z^2 + 3 z + 5$ $[2]$
7.1.12.11a1.2 6 $x^{12} + 14$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[\ ]_{12}^{2}$ $[\ ]_{12}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{11} + 5 z^{10} + 3 z^9 + 3 z^8 + 5 z^7 + z^6 + z^4 + 5 z^3 + 3 z^2 + 3 z + 5$ undefined
7.1.12.11a1.3 6 $x^{12} + 21$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[\ ]_{12}^{2}$ $[\ ]_{12}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{11} + 5 z^{10} + 3 z^9 + 3 z^8 + 5 z^7 + z^6 + z^4 + 5 z^3 + 3 z^2 + 3 z + 5$ undefined
7.1.12.11a1.4 6 $x^{12} + 28$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[\ ]_{12}^{2}$ $[\ ]_{12}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{11} + 5 z^{10} + 3 z^9 + 3 z^8 + 5 z^7 + z^6 + z^4 + 5 z^3 + 3 z^2 + 3 z + 5$ undefined
7.1.12.11a1.5 6 $x^{12} + 35$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[\ ]_{12}^{2}$ $[\ ]_{12}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{11} + 5 z^{10} + 3 z^9 + 3 z^8 + 5 z^7 + z^6 + z^4 + 5 z^3 + 3 z^2 + 3 z + 5$ undefined
7.1.12.11a1.6 6 $x^{12} + 42$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[\ ]_{12}^{2}$ $[\ ]_{12}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{11} + 5 z^{10} + 3 z^9 + 3 z^8 + 5 z^7 + z^6 + z^4 + 5 z^3 + 3 z^2 + 3 z + 5$ undefined
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