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Properties

Label 2.1.2.3a1.2-2.2.8a
Base 2.1.2.3a1.2
Degree \(4\)
e \(2\)
f \(2\)
c \(8\)

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Defining polynomial over unramified subextension

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

Invariants

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

Diagrams

Varying

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

Galois group: $D_4\times C_2$ (show 2), $Q_8:C_2$ (show 2), $(((C_4 \times C_2): C_2):C_2):C_2$ (show 4)
Hidden Artin slopes: $[2]$ (show 4), $[2,2,\frac{7}{2}]$ (show 4)
Indices of inseparability: $[7,4,0]$
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
2.2.4.20a1.13 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 2$ $D_4\times C_2$ (as 8T9) $16$ $4$ $[2, 3, \frac{7}{2}]^{2}$ $[1,2,\frac{5}{2}]^{2}$ $[2]$ $[1]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.20a1.14 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{2} + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, \frac{7}{2}]^{2}$ $[1,2,\frac{5}{2}]^{2}$ $[2]$ $[1]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.20a1.15 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 2$ $D_4\times C_2$ (as 8T9) $16$ $4$ $[2, 3, \frac{7}{2}]^{2}$ $[1,2,\frac{5}{2}]^{2}$ $[2]$ $[1]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.20a1.16 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, \frac{7}{2}]^{2}$ $[1,2,\frac{5}{2}]^{2}$ $[2]$ $[1]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.20a2.25 $( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{2} + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31) $64$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7]$
2.2.4.20a2.26 $( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + \left(4 x + 12\right) ( x^{2} + x + 1 )^{2} + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31) $64$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7]$
2.2.4.20a2.27 $( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31) $64$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7]$
2.2.4.20a2.28 $( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + \left(4 x + 12\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31) $64$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + 1,z + t$ $[1, 3, 7]$
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