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Properties

Label 2.2.2.4a2.1-1.4.8a
Base 2.2.2.4a2.1
Degree \(4\)
e \(4\)
f \(1\)
c \(8\)

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Defining polynomial

$x^{4} + b_{6} \pi^{2} x^{2} + a_{5} \pi^{2} x + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.2.2.4a2.1
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Absolute Artin slopes: $[2,\frac{7}{3},\frac{7}{3}]$
Swan slopes: $[\frac{5}{3},\frac{5}{3}]$
Means: $\langle\frac{5}{6},\frac{5}{4}\rangle$
Rams: $(\frac{5}{3},\frac{5}{3})$
Field count: $6$ (complete)
Ambiguity: $1$
Mass: $12$
Absolute Mass: $6$

Diagrams

Varying

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

Galois group: $C_2^6:S_4$ (show 2), $C_2^6.A_4\wr C_2$ (show 4) (incomplete)
Hidden Artin slopes: not computed (show 2), $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,\frac{7}{3},\frac{7}{3}]^{3}_{3}$ (show 2), $[\frac{4}{3},\frac{4}{3},2,\frac{13}{6},\frac{13}{6}]_{3}$ (show 2) (incomplete)
Indices of inseparability: $[9,9,4,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,3,7,15]$

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
2.2.8.32d3.1 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 ) + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[9, 9, 4, 0]$ $[1, 1]$ $z^4 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.32d3.2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 4 ( x^{2} + x + 1 ) + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ not computed not computed not computed not computed $[9, 9, 4, 0]$ $[1, 1]$ $z^4 + t,t z + 1$ $[1, 3, 7, 15]$
2.2.8.32d4.1 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 ) + 2$ $C_2^6:S_4$ (as 16T1315) $1536$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, 2, \frac{13}{6}, \frac{13}{6}, \frac{7}{3}, \frac{7}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,1,\frac{7}{6},\frac{7}{6},\frac{4}{3},\frac{4}{3}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{13}{6},\frac{13}{6}]_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{7}{6},\frac{7}{6}]_{3}$ $[9, 9, 4, 0]$ $[1, 1]$ $z^4 + t,t z + t$ $[1, 3, 7, 15]$
2.2.8.32d4.2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 4 x ( x^{2} + x + 1 ) + 2$ $C_2^6:S_4$ (as 16T1315) $1536$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, 2, \frac{13}{6}, \frac{13}{6}, \frac{7}{3}, \frac{7}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,1,\frac{7}{6},\frac{7}{6},\frac{4}{3},\frac{4}{3}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{13}{6},\frac{13}{6}]_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{7}{6},\frac{7}{6}]_{3}$ $[9, 9, 4, 0]$ $[1, 1]$ $z^4 + t,t z + t$ $[1, 3, 7, 15]$
2.2.8.32d5.1 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, \frac{7}{3}, \frac{7}{3}, \frac{7}{3}, \frac{7}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,\frac{7}{3},\frac{7}{3}]^{3}_{3}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,\frac{4}{3},\frac{4}{3}]^{3}_{3}$ $[9, 9, 4, 0]$ $[1, 1]$ $z^4 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.32d5.2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{2} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 2$ $C_2^6.A_4\wr C_2$ (as 16T1783) $18432$ $1$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, \frac{7}{3}, \frac{7}{3}, \frac{7}{3}, \frac{7}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,\frac{7}{3},\frac{7}{3}]^{3}_{3}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,\frac{4}{3},\frac{4}{3}]^{3}_{3}$ $[9, 9, 4, 0]$ $[1, 1]$ $z^4 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
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