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Properties

Label 2.1.18.24a
Base 2.1.1.0a1.1
Degree \(18\)
e \(18\)
f \(1\)
c \(24\)

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Defining polynomial

$x^{18} + 2 c_{14} x^{14} + 2 b_{13} x^{13} + 2 b_{11} x^{11} + 2 b_{9} x^{9} + 2 a_{7} x^{7} + 2$

Invariants

Residue field characteristic: $2$
Degree: $18$
Base field: $\Q_{2}$
Ramification index $e$: $18$
Residue field degree $f$: $1$
Discriminant exponent $c$: $24$
Artin slopes: $[\frac{16}{9}]$
Swan slopes: $[\frac{7}{9}]$
Means: $\langle\frac{7}{18}\rangle$
Rams: $(7)$
Field count: $16$ (complete)
Ambiguity: $2$
Mass: $8$
Absolute Mass: $8$

Diagrams

Varying

Indices of inseparability: $[7,0]$
Associated inertia: $[6,1]$
Jump Set: $[9,25]$

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 16

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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
2.1.18.24a1.1 4 $x^{18} + 2 x^{7} + 2$ $C_2^6:C_9:C_6$ (as 18T434) $3456$ $2$ $[\frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
2.1.18.24a1.2 4 $x^{18} + 2 x^{14} + 2 x^{7} + 2$ $C_2^6:C_9:C_6$ (as 18T433) $3456$ $2$ $[\frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
2.1.18.24a1.3 4 $x^{18} + 2 x^{13} + 2 x^{7} + 2$ $C_2^6:C_9:C_6$ (as 18T434) $3456$ $2$ $[\frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
2.1.18.24a1.4 4 $x^{18} + 2 x^{14} + 2 x^{13} + 2 x^{7} + 2$ $C_2^6:C_9:C_6$ (as 18T433) $3456$ $2$ $[\frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
2.1.18.24a1.5 4 $x^{18} + 2 x^{11} + 2 x^{7} + 2$ $C_2^2:A_4^2.S_4$ (as 18T592) $13824$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
2.1.18.24a1.6 4 $x^{18} + 2 x^{14} + 2 x^{11} + 2 x^{7} + 2$ $C_2^2:A_4^2.S_4$ (as 18T588) $13824$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
2.1.18.24a1.7 4 $x^{18} + 2 x^{13} + 2 x^{11} + 2 x^{7} + 2$ $C_2^2:A_4^2.S_4$ (as 18T592) $13824$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
2.1.18.24a1.8 4 $x^{18} + 2 x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{7} + 2$ $C_2^2:A_4^2.S_4$ (as 18T588) $13824$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
2.1.18.24a1.9 4 $x^{18} + 2 x^{9} + 2 x^{7} + 2$ $C_2^2:A_4^2.S_4$ (as 18T592) $13824$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
2.1.18.24a1.10 4 $x^{18} + 2 x^{14} + 2 x^{9} + 2 x^{7} + 2$ $C_2^2:A_4^2.S_4$ (as 18T588) $13824$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
2.1.18.24a1.11 4 $x^{18} + 2 x^{13} + 2 x^{9} + 2 x^{7} + 2$ $C_2^2:A_4^2.S_4$ (as 18T592) $13824$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
2.1.18.24a1.12 4 $x^{18} + 2 x^{14} + 2 x^{13} + 2 x^{9} + 2 x^{7} + 2$ $C_2^2:A_4^2.S_4$ (as 18T588) $13824$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
2.1.18.24a1.13 4 $x^{18} + 2 x^{11} + 2 x^{9} + 2 x^{7} + 2$ $C_2^6:C_9:C_6$ (as 18T434) $3456$ $2$ $[\frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
2.1.18.24a1.14 4 $x^{18} + 2 x^{14} + 2 x^{11} + 2 x^{9} + 2 x^{7} + 2$ $C_2^6:C_9:C_6$ (as 18T433) $3456$ $2$ $[\frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
2.1.18.24a1.15 4 $x^{18} + 2 x^{13} + 2 x^{11} + 2 x^{9} + 2 x^{7} + 2$ $C_2^6:C_9:C_6$ (as 18T434) $3456$ $2$ $[\frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
2.1.18.24a1.16 4 $x^{18} + 2 x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{9} + 2 x^{7} + 2$ $C_2^6:C_9:C_6$ (as 18T433) $3456$ $2$ $[\frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}, \frac{16}{9}]_{9}^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]_{9}^{6}$ $[\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9},\frac{16}{9}]^{6}$ $[\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9},\frac{7}{9}]^{6}$ $[7, 0]$ $[6, 1]$ $z^{16} + z^{14} + 1,z + 1$ $[9, 25]$
  displayed columns for results