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Properties

Label 2.2.4.16b1.4-1.2.4a
Base 2.2.4.16b1.4
Degree \(2\)
e \(2\)
f \(1\)
c \(4\)

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Defining polynomial

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

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.2.4.16b1.4
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $4$
Absolute Artin slopes: $[2,3,3]$
Swan slopes: $[3]$
Means: $\langle\frac{3}{2}\rangle$
Rams: $(3)$
Field count: $6$ (complete)
Ambiguity: $2$
Mass: $12$
Absolute Mass: $3/2$

Diagrams

Varying

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

Galois group: $C_2^3:C_4$ (show 2), $C_2^2.D_4$ (show 2), $C_2^4.D_4$ (show 2)
Hidden Artin slopes: $[\ ]^{2}$ (show 2), $[2]$ (show 2), $[2,3]^{2}$ (show 2)
Indices of inseparability: $[13,10,4,0]$ (show 2), $[13,12,4,0]$ (show 4)
Associated inertia: $[1,1]$ (show 4), $[1,2]$ (show 2)
Jump Set: $[1,5,15,23]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 6

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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
2.2.8.40d1.6 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 2$ $C_2^3:C_4$ (as 16T33) $32$ $8$ $[2, 3, 3]^{4}$ $[1,2,2]^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $[13, 12, 4, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 5, 15, 23]$
2.2.8.40d1.14 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 2$ $C_2^2.D_4$ (as 16T54) $32$ $8$ $[2, 2, 3, 3]^{2}$ $[1,1,2,2]^{2}$ $[2]$ $[1]$ $[13, 12, 4, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 5, 15, 23]$
2.2.8.40d1.16 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 10$ $C_2^2.D_4$ (as 16T54) $32$ $8$ $[2, 2, 3, 3]^{2}$ $[1,1,2,2]^{2}$ $[2]$ $[1]$ $[13, 12, 4, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 5, 15, 23]$
2.2.8.40d1.17 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 2$ $C_2^3:C_4$ (as 16T33) $32$ $8$ $[2, 3, 3]^{4}$ $[1,2,2]^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $[13, 12, 4, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 5, 15, 23]$
2.2.8.40d6.4 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_4$ (as 16T324) $128$ $2$ $[2, 2, 3, 3, 3]^{4}$ $[1,1,2,2,2]^{4}$ $[2,3]^{2}$ $[1,2]^{2}$ $[13, 10, 4, 0]$ $[1, 2]$ $z^4 + 1,z^3 + t z + t$ $[1, 5, 15, 23]$
2.2.8.40d6.9 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_4$ (as 16T324) $128$ $2$ $[2, 2, 3, 3, 3]^{4}$ $[1,1,2,2,2]^{4}$ $[2,3]^{2}$ $[1,2]^{2}$ $[13, 10, 4, 0]$ $[1, 2]$ $z^4 + 1,z^3 + t z + t$ $[1, 5, 15, 23]$
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