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Properties

Label 7.2.8.14a
Base 7.1.1.0a1.1
Degree \(16\)
e \(8\)
f \(2\)
c \(14\)

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Defining polynomial over unramified subextension

$x^{8} + 7d_{0}$

Invariants

Residue field characteristic: $7$
Degree: $16$
Base field: $\Q_{7}$
Ramification index $e$: $8$
Residue field degree $f$: $2$
Discriminant exponent $c$: $14$
Artin slopes: $[\ ]$
Swan slopes: $[\ ]$
Means: $\langle\ \rangle$
Rams: $(\ )$
Field count: $5$ (complete)
Ambiguity: $16$
Mass: $1$
Absolute Mass: $1/2$

Varying

Indices of inseparability: $[0]$
Associated inertia: $[1]$
Jump Set: undefined

Galois groups and Hidden Artin slopes

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Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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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
7.2.8.14a1.1 2 $( x^{2} + 6 x + 3 )^{8} + 7 x$ $C_8.C_8$ (as 16T124) $64$ $8$ $[\ ]_{8}^{8}$ $[\ ]_{8}^{8}$ $[\ ]^{4}$ $[\ ]^{4}$ $[0]$ $[1]$ $z^7 + z^6 + 1$ undefined
7.2.8.14a1.2 1 $( x^{2} + 6 x + 3 )^{8} + 7$ $D_{8}$ (as 16T13) $16$ $16$ $[\ ]_{8}^{2}$ $[\ ]_{8}^{2}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^7 + z^6 + 1$ undefined
7.2.8.14a1.3 2 $( x^{2} + 6 x + 3 )^{8} + 7 x + 7$ $C_8.C_8$ (as 16T124) $64$ $8$ $[\ ]_{8}^{8}$ $[\ ]_{8}^{8}$ $[\ ]^{4}$ $[\ ]^{4}$ $[0]$ $[1]$ $z^7 + z^6 + 1$ undefined
7.2.8.14a1.4 1 $( x^{2} + 6 x + 3 )^{8} + 7 x + 28$ $C_8.C_4$ (as 16T49) $32$ $8$ $[\ ]_{8}^{4}$ $[\ ]_{8}^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[1]$ $z^7 + z^6 + 1$ undefined
7.2.8.14a1.5 1 $( x^{2} + 6 x + 3 )^{8} + 14 x + 42$ $Q_{16}$ (as 16T14) $16$ $16$ $[\ ]_{8}^{2}$ $[\ ]_{8}^{2}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^7 + z^6 + 1$ undefined
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