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Properties

Label 7.1.7.12a
Base 7.1.1.0a1.1
Degree \(7\)
e \(7\)
f \(1\)
c \(12\)

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Defining polynomial

$x^{7} + 7 a_{6} x^{6} + 49 c_{7} + 7$

Invariants

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

Diagrams

Varying

Indices of inseparability: $[6,0]$
Associated inertia: $[1]$ (show 7), $[2]$ (show 1), $[3]$ (show 2), $[6]$ (show 2)
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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Showing all 12

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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
7.1.7.12a1.1 1 $x^{7} + 7 x^{6} + 7$ $D_{7}$ (as 7T2) $14$ $1$ $[2]^{2}$ $[1]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[6, 0]$ $[2]$ $z^6 + 1$ undefined
7.1.7.12a2.1 1 $x^{7} + 14 x^{6} + 7$ $F_7$ (as 7T4) $42$ $1$ $[2]^{6}$ $[1]^{6}$ $[\ ]^{6}$ $[\ ]^{6}$ $[6, 0]$ $[6]$ $z^6 + 2$ undefined
7.1.7.12a3.1 1 $x^{7} + 21 x^{6} + 7$ $C_7:C_3$ (as 7T3) $21$ $1$ $[2]^{3}$ $[1]^{3}$ $[\ ]^{3}$ $[\ ]^{3}$ $[6, 0]$ $[3]$ $z^6 + 3$ undefined
7.1.7.12a4.1 1 $x^{7} + 28 x^{6} + 7$ $F_7$ (as 7T4) $42$ $1$ $[2]^{6}$ $[1]^{6}$ $[\ ]^{6}$ $[\ ]^{6}$ $[6, 0]$ $[6]$ $z^6 + 4$ undefined
7.1.7.12a5.1 1 $x^{7} + 35 x^{6} + 7$ $C_7:C_3$ (as 7T3) $21$ $1$ $[2]^{3}$ $[1]^{3}$ $[\ ]^{3}$ $[\ ]^{3}$ $[6, 0]$ $[3]$ $z^6 + 5$ undefined
7.1.7.12a6.1 7 $x^{7} + 42 x^{6} + 7$ $C_7$ (as 7T1) $7$ $7$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[6, 0]$ $[1]$ $z^6 + 6$ undefined
7.1.7.12a6.2 7 $x^{7} + 42 x^{6} + 56$ $C_7$ (as 7T1) $7$ $7$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[6, 0]$ $[1]$ $z^6 + 6$ undefined
7.1.7.12a6.3 7 $x^{7} + 42 x^{6} + 105$ $C_7$ (as 7T1) $7$ $7$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[6, 0]$ $[1]$ $z^6 + 6$ undefined
7.1.7.12a6.4 7 $x^{7} + 42 x^{6} + 154$ $C_7$ (as 7T1) $7$ $7$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[6, 0]$ $[1]$ $z^6 + 6$ undefined
7.1.7.12a6.5 7 $x^{7} + 42 x^{6} + 203$ $C_7$ (as 7T1) $7$ $7$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[6, 0]$ $[1]$ $z^6 + 6$ undefined
7.1.7.12a6.6 7 $x^{7} + 42 x^{6} + 252$ $C_7$ (as 7T1) $7$ $7$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[6, 0]$ $[1]$ $z^6 + 6$ undefined
7.1.7.12a6.7 7 $x^{7} + 42 x^{6} + 301$ $C_7$ (as 7T1) $7$ $7$ $[2]$ $[1]$ $[\ ]$ $[\ ]$ $[6, 0]$ $[1]$ $z^6 + 6$ undefined
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