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Properties

Label 7.2.6.10a
Base 7.1.1.0a1.1
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)

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Defining polynomial over unramified subextension

$x^{6} + 7d_{0}$

Invariants

Residue field characteristic: $7$
Degree: $12$
Base field: $\Q_{7}$
Ramification index $e$: $6$
Residue field degree $f$: $2$
Discriminant exponent $c$: $10$
Artin slopes: $[\ ]$
Swan slopes: $[\ ]$
Means: $\langle\ \rangle$
Rams: $(\ )$
Field count: $6$ (complete)
Ambiguity: $12$
Mass: $1$
Absolute Mass: $1/2$

Varying

Indices of inseparability: $[0]$
Associated inertia: $[1]$
Jump Set: undefined (show 5), $[1]$ (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
7.2.6.10a1.1 3 $( x^{2} + 6 x + 3 )^{6} + 7 x$ $C_{12}$ (as 12T1) $12$ $12$ $[\ ]_{6}^{2}$ $[\ ]_{6}^{2}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$ undefined
7.2.6.10a1.2 3 $( x^{2} + 6 x + 3 )^{6} + 7$ $C_6\times C_2$ (as 12T2) $12$ $12$ $[\ ]_{6}^{2}$ $[\ ]_{6}^{2}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$ $[1]$
7.2.6.10a1.3 3 $( x^{2} + 6 x + 3 )^{6} + 7 x + 7$ $C_{12}$ (as 12T1) $12$ $12$ $[\ ]_{6}^{2}$ $[\ ]_{6}^{2}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$ undefined
7.2.6.10a1.4 3 $( x^{2} + 6 x + 3 )^{6} + 7 x + 28$ $C_6\times C_2$ (as 12T2) $12$ $12$ $[\ ]_{6}^{2}$ $[\ ]_{6}^{2}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$ undefined
7.2.6.10a1.5 3 $( x^{2} + 6 x + 3 )^{6} + 35 x + 28$ $C_{12}$ (as 12T1) $12$ $12$ $[\ ]_{6}^{2}$ $[\ ]_{6}^{2}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$ undefined
7.2.6.10a1.6 3 $( x^{2} + 6 x + 3 )^{6} + 14 x + 42$ $C_6\times C_2$ (as 12T2) $12$ $12$ $[\ ]_{6}^{2}$ $[\ ]_{6}^{2}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$ undefined
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