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Properties

Label 2.1.18.20a
Base 2.1.1.0a1.1
Degree \(18\)
e \(18\)
f \(1\)
c \(20\)

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Defining polynomial

$x^{18} + 2 c_{6} x^{6} + 2 b_{5} x^{5} + 2 a_{3} x^{3} + 2$

Invariants

Residue field characteristic: $2$
Degree: $18$
Base field: $\Q_{2}$
Ramification index $e$: $18$
Residue field degree $f$: $1$
Discriminant exponent $c$: $20$
Artin slopes: $[\frac{4}{3}]$
Swan slopes: $[\frac{1}{3}]$
Means: $\langle\frac{1}{6}\rangle$
Rams: $(3)$
Field count: $4$ (complete)
Ambiguity: $2$
Mass: $2$
Absolute Mass: $2$

Diagrams

Varying

Indices of inseparability: $[3,0]$
Associated inertia: $[6,1]$
Jump Set: $[9,21]$

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
2.1.18.20a1.1 1 $x^{18} + 2 x^{3} + 2$ $C_3^2.S_4$ (as 18T98) $216$ $2$ $[\frac{4}{3}, \frac{4}{3}]_{9}^{6}$ $[\frac{1}{3},\frac{1}{3}]_{9}^{6}$ $[\frac{4}{3}]^{6}$ $[\frac{1}{3}]^{6}$ $[3, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 21]$
2.1.18.20a1.2 1 $x^{18} + 2 x^{6} + 2 x^{3} + 2$ $C_3^2.S_4$ (as 18T101) $216$ $2$ $[\frac{4}{3}, \frac{4}{3}]_{9}^{6}$ $[\frac{1}{3},\frac{1}{3}]_{9}^{6}$ $[\frac{4}{3}]^{6}$ $[\frac{1}{3}]^{6}$ $[3, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 21]$
2.1.18.20a1.3 1 $x^{18} + 2 x^{5} + 2 x^{3} + 2$ $C_2^2:A_4^2.S_4$ (as 18T592) $13824$ $2$ $[\frac{10}{9}, \frac{10}{9}, \frac{10}{9}, \frac{10}{9}, \frac{10}{9}, \frac{10}{9}, \frac{4}{3}, \frac{4}{3}]_{9}^{6}$ $[\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{3},\frac{1}{3}]_{9}^{6}$ $[\frac{10}{9},\frac{10}{9},\frac{10}{9},\frac{10}{9},\frac{10}{9},\frac{10}{9},\frac{4}{3}]^{6}$ $[\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{3}]^{6}$ $[3, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 21]$
2.1.18.20a1.4 1 $x^{18} + 2 x^{6} + 2 x^{5} + 2 x^{3} + 2$ $C_2^2:A_4^2.S_4$ (as 18T588) $13824$ $2$ $[\frac{10}{9}, \frac{10}{9}, \frac{10}{9}, \frac{10}{9}, \frac{10}{9}, \frac{10}{9}, \frac{4}{3}, \frac{4}{3}]_{9}^{6}$ $[\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{3},\frac{1}{3}]_{9}^{6}$ $[\frac{10}{9},\frac{10}{9},\frac{10}{9},\frac{10}{9},\frac{10}{9},\frac{10}{9},\frac{4}{3}]^{6}$ $[\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{3}]^{6}$ $[3, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 21]$
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