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Properties

Label 2.2.2.6a1.4-1.2.5a
Base 2.2.2.6a1.4
Degree \(2\)
e \(2\)
f \(1\)
c \(5\)

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Defining polynomial

$x^{2} + \left(b_{7} \pi^{4} + b_{5} \pi^{3}\right) x + c_{8} \pi^{5} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.2.2.6a1.4
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $5$
Absolute Artin slopes: $[3,4]$
Swan slopes: $[4]$
Means: $\langle2\rangle$
Rams: $(4)$
Field count: $16$ (complete)
Ambiguity: $2$
Mass: $16$
Absolute Mass: $8$

Diagrams

Varying

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

Galois group: $Z_8 : Z_8^\times$ (show 4), $(((C_4 \times C_2): C_2):C_2):C_2$ (show 4), $(((C_4 \times C_2): C_2):C_2):C_2$ (show 4), $(((C_4 \times C_2): C_2):C_2):C_2$ (show 4)
Hidden Artin slopes: $[2,2,\frac{7}{2}]$ (show 8), $[2,3]^{2}$ (show 4), $[2,2]$ (show 4)
Indices of inseparability: $[8,4,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,3,7]$

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
2.2.4.22a1.39 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 2, 3, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.40 $( x^{2} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 2, 3, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.41 $( x^{2} + x + 1 )^{4} + 12 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 2, 3, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.42 $( x^{2} + x + 1 )^{4} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 2, 3, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.43 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 8 x + 2$ $Z_8 : Z_8^\times$ (as 8T15) $32$ $2$ $[2, 2, 3, 4]^{2}$ $[1,1,2,3]^{2}$ $[2,2]$ $[1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.44 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 24 x + 2$ $Z_8 : Z_8^\times$ (as 8T15) $32$ $2$ $[2, 2, 3, 4]^{2}$ $[1,1,2,3]^{2}$ $[2,2]$ $[1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.45 $( x^{2} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 8 x + 2$ $Z_8 : Z_8^\times$ (as 8T15) $32$ $2$ $[2, 2, 3, 4]^{2}$ $[1,1,2,3]^{2}$ $[2,2]$ $[1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.46 $( x^{2} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 24 x + 2$ $Z_8 : Z_8^\times$ (as 8T15) $32$ $2$ $[2, 2, 3, 4]^{2}$ $[1,1,2,3]^{2}$ $[2,2]$ $[1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.47 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) $64$ $2$ $[2, 2, 3, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.48 $( x^{2} + x + 1 )^{4} + 12 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) $64$ $2$ $[2, 2, 3, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.77 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T30) $64$ $2$ $[2, 3, 3, 4]^{4}$ $[1,2,2,3]^{4}$ $[2,3]^{2}$ $[1,2]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.78 $( x^{2} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T30) $64$ $2$ $[2, 3, 3, 4]^{4}$ $[1,2,2,3]^{4}$ $[2,3]^{2}$ $[1,2]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.79 $( x^{2} + x + 1 )^{4} + 12 ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T30) $64$ $2$ $[2, 3, 3, 4]^{4}$ $[1,2,2,3]^{4}$ $[2,3]^{2}$ $[1,2]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.80 $( x^{2} + x + 1 )^{4} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T30) $64$ $2$ $[2, 3, 3, 4]^{4}$ $[1,2,2,3]^{4}$ $[2,3]^{2}$ $[1,2]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.81 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) $64$ $2$ $[2, 2, 3, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.82 $( x^{2} + x + 1 )^{4} + 12 ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T28) $64$ $2$ $[2, 2, 3, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
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