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Properties

Label 2.1.14.14a
Base 2.1.1.0a1.1
Degree \(14\)
e \(14\)
f \(1\)
c \(14\)

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Defining polynomial

$x^{14} + 2 c_{2} x^{2} + 2 a_{1} x + 2$

Invariants

Residue field characteristic: $2$
Degree: $14$
Base field: $\Q_{2}$
Ramification index $e$: $14$
Residue field degree $f$: $1$
Discriminant exponent $c$: $14$
Artin slopes: $[\frac{8}{7}]$
Swan slopes: $[\frac{1}{7}]$
Means: $\langle\frac{1}{14}\rangle$
Rams: $(1)$
Field count: $2$ (complete)
Ambiguity: $2$
Mass: $1$
Absolute Mass: $1$

Diagrams

Varying

Indices of inseparability: $[1,0]$
Associated inertia: $[3,1]$
Jump Set: $[7,15]$

Galois groups and Hidden Artin slopes

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Fields

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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
2.1.14.14a1.1 1 $x^{14} + 2 x + 2$ $F_8:C_6$ (as 14T18) $336$ $2$ $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}]_{7}^{6}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7}]_{7}^{6}$ $[\frac{8}{7},\frac{8}{7}]^{6}$ $[\frac{1}{7},\frac{1}{7}]^{6}$ $[1, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 15]$
2.1.14.14a1.2 1 $x^{14} + 2 x^{2} + 2 x + 2$ $F_8:C_3$ (as 14T11) $168$ $2$ $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}]_{7}^{3}$ $[\frac{1}{7},\frac{1}{7},\frac{1}{7}]_{7}^{3}$ $[\frac{8}{7},\frac{8}{7}]^{3}$ $[\frac{1}{7},\frac{1}{7}]^{3}$ $[1, 0]$ $[3, 1]$ $z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$ $[7, 15]$
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