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Properties

Label 2.2.4.16b2.1-1.2.2a
Base 2.2.4.16b2.1
Degree \(2\)
e \(2\)
f \(1\)
c \(2\)

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Defining polynomial

$x^{2} + a_{1} \pi x + c_{2} \pi^{2} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.2.4.16b2.1
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $2$
Absolute Artin slopes: $[2,2,3]$
Swan slopes: $[1]$
Means: $\langle\frac{1}{2}\rangle$
Rams: $(1)$
Field count: $4$ (complete)
Ambiguity: $2$
Mass: $3$
Absolute Mass: $3/4$

Diagrams

Varying

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

Galois group: $C_2^2.D_4$ (show 2), $C_2^4.D_4$ (show 2)
Hidden Artin slopes: $[3]$ (show 2), $[2,3]^{2}$ (show 2)
Indices of inseparability: $[11,6,4,0]$ (show 2), $[11,6,6,0]$ (show 2)
Associated inertia: $[1,1]$ (show 2), $[2,1]$ (show 2)
Jump Set: $[1,3,11,19]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 4

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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
2.2.8.36b2.12 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{3} + 10$ $C_2^2.D_4$ (as 16T54) $32$ $8$ $[2, 2, 3, 3]^{2}$ $[1,1,2,2]^{2}$ $[3]$ $[2]$ $[11, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + t$ $[1, 3, 11, 19]$
2.2.8.36b2.39 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{3} + 14$ $C_2^2.D_4$ (as 16T54) $32$ $8$ $[2, 2, 3, 3]^{2}$ $[1,1,2,2]^{2}$ $[3]$ $[2]$ $[11, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + t$ $[1, 3, 11, 19]$
2.2.8.36b14.5 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + \left(2 x + 2\right) ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 10$ $C_2^4.D_4$ (as 16T274) $128$ $4$ $[2, 2, 2, 3, 3]^{4}$ $[1,1,1,2,2]^{4}$ $[2,3]^{2}$ $[1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + t z^2 + t,t z + 1$ $[1, 3, 11, 19]$
2.2.8.36b15.8 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + \left(2 x + 2\right) ( x^{2} + x + 1 )^{6} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^4.D_4$ (as 16T274) $128$ $4$ $[2, 2, 2, 3, 3]^{4}$ $[1,1,1,2,2]^{4}$ $[2,3]^{2}$ $[1,2]^{2}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + t z^2 + t,t z + t$ $[1, 3, 11, 19]$
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