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Properties

Label 2.1.18.18a
Base 2.1.1.0a1.1
Degree \(18\)
e \(18\)
f \(1\)
c \(18\)

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Defining polynomial

$x^{18} + 2 c_{2} x^{2} + 2 a_{1} x + 2$

Invariants

Residue field characteristic: $2$
Degree: $18$
Base field: $\Q_{2}$
Ramification index $e$: $18$
Residue field degree $f$: $1$
Discriminant exponent $c$: $18$
Artin slopes: $[\frac{10}{9}]$
Swan slopes: $[\frac{1}{9}]$
Means: $\langle\frac{1}{18}\rangle$
Rams: $(1)$
Field count: $2$ (complete)
Ambiguity: $2$
Mass: $1$
Absolute Mass: $1$

Diagrams

Varying

Indices of inseparability: $[1,0]$
Associated inertia: $[6,1]$
Jump Set: $[9,19]$

Galois groups and Hidden Artin slopes

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
2.1.18.18a1.1 1 $x^{18} + 2 x + 2$ $C_2^6:C_9:C_6$ (as 18T433) $3456$ $2$ $[\frac{10}{9}, \frac{10}{9}, \frac{10}{9}, \frac{10}{9}, \frac{10}{9}, \frac{10}{9}]_{9}^{6}$ $[\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9}]_{9}^{6}$ $[\frac{10}{9},\frac{10}{9},\frac{10}{9},\frac{10}{9},\frac{10}{9}]^{6}$ $[\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9}]^{6}$ $[1, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 19]$
2.1.18.18a1.2 1 $x^{18} + 2 x^{2} + 2 x + 2$ $C_2^6:C_9:C_6$ (as 18T434) $3456$ $2$ $[\frac{10}{9}, \frac{10}{9}, \frac{10}{9}, \frac{10}{9}, \frac{10}{9}, \frac{10}{9}]_{9}^{6}$ $[\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9}]_{9}^{6}$ $[\frac{10}{9},\frac{10}{9},\frac{10}{9},\frac{10}{9},\frac{10}{9}]^{6}$ $[\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9}]^{6}$ $[1, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 19]$
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