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Properties

Label 2.1.8.28b1.18-1.2.6a
Base 2.1.8.28b1.18
Degree \(2\)
e \(2\)
f \(1\)
c \(6\)

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Defining polynomial

$x^{2} + \left(b_{9} \pi^{5} + b_{7} \pi^{4} + a_{5} \pi^{3}\right) x + c_{10} \pi^{6} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.1.8.28b1.18
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $6$
Absolute Artin slopes: $[3,4,\frac{17}{4},\frac{17}{4}]$
Swan slopes: $[5]$
Means: $\langle\frac{5}{2}\rangle$
Rams: $(5)$
Field count: $4$ (complete)
Ambiguity: $2$
Mass: $4$
Absolute Mass: $2$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.

Galois group: $C_2^6:D_4$
Hidden Artin slopes: $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$
Indices of inseparability: $[47,46,32,16,0]$ (show 2), $[47,47,32,16,0]$ (show 2)
Associated inertia: $[1,1,2]$
Jump Set: $[1,3,7,15,31]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Label Polynomial $/ \Q_p$ Galois group $/ \Q_p$ Galois degree $/ \Q_p$ $\#\Aut(K/\Q_p)$ Artin slope content $/ \Q_p$ Swan slope content $/ \Q_p$ Hidden Artin slopes $/ \Q_p$ Hidden Swan slopes $/ \Q_p$ Ind. of Insep. $/ \Q_p$ Assoc. Inertia $/ \Q_p$ Resid. Poly Jump Set
2.1.16.62g1.109 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{4} + 16 x + 10$ $C_2^6:D_4$ (as 16T960) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 46, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.111 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{4} + 16 x^{2} + 16 x + 10$ $C_2^6:D_4$ (as 16T960) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 46, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.113 $x^{16} + 8 x^{15} + 8 x^{12} + 8 x^{4} + 10$ $C_2^6:D_4$ (as 16T960) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 47, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
2.1.16.62g1.115 $x^{16} + 8 x^{15} + 8 x^{12} + 8 x^{4} + 16 x^{2} + 10$ $C_2^6:D_4$ (as 16T960) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[47, 47, 32, 16, 0]$ $[1, 1, 2]$ $z^8 + 1,z^4 + 1,z^3 + 1$ $[1, 3, 7, 15, 31]$
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