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Properties

Label 2.1.2.3a1.3-2.2.10a
Base 2.1.2.3a1.3
Degree \(4\)
e \(2\)
f \(2\)
c \(10\)

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Defining polynomial over unramified subextension

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

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: $\Q_{2}(\sqrt{2})$
Ramification index $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $10$
Absolute Artin slopes: $[3,4]$
Swan slopes: $[4]$
Means: $\langle2\rangle$
Rams: $(4)$
Field count: $14$ (complete)
Ambiguity: $4$
Mass: $16$
Absolute Mass: $4$

Diagrams

Varying

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

Galois group: $C_4\times C_2$ (show 4), $D_4\times C_2$ (show 2), $Q_8:C_2$ (show 4), $(((C_4 \times C_2): C_2):C_2):C_2$ (show 4)
Hidden Artin slopes: $[\ ]$ (show 4), $[2]$ (show 6), $[2,2,\frac{7}{2}]$ (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
2.2.4.22a1.49 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_4\times C_2$ (as 8T2) $8$ $8$ $[3, 4]^{2}$ $[2,3]^{2}$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.50 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 16 x + 2$ $C_4\times C_2$ (as 8T2) $8$ $8$ $[3, 4]^{2}$ $[2,3]^{2}$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.51 $( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.52 $( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 16 x + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.53 $( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_4\times C_2$ (as 8T2) $8$ $8$ $[3, 4]^{2}$ $[2,3]^{2}$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.54 $( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 16 x + 2$ $C_4\times C_2$ (as 8T2) $8$ $8$ $[3, 4]^{2}$ $[2,3]^{2}$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.55 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31) $64$ $2$ $[2, 2, 3, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.56 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 16 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31) $64$ $2$ $[2, 2, 3, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.57 $( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31) $64$ $2$ $[2, 2, 3, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.58 $( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 16 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T31) $64$ $2$ $[2, 2, 3, \frac{7}{2}, 4]^{2}$ $[1,1,2,\frac{5}{2},3]^{2}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.59 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 2$ $D_4\times C_2$ (as 8T9) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.60 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 16 x + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.61 $( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.2.4.22a1.62 $( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 16 x + 2$ $D_4\times C_2$ (as 8T9) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
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