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Properties

Label 2.1.2.2a1.2-2.2.10a
Base 2.1.2.2a1.2
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: $\Q_{2}(\sqrt{-5})$
Ramification index $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $10$
Absolute Artin slopes: $[2,\frac{7}{2}]$
Swan slopes: $[4]$
Means: $\langle2\rangle$
Rams: $(4)$
Field count: $12$ (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: $D_4\times C_2$ (show 4), $Q_8:C_2$ (show 4), $C_2^3: C_4$ (show 4)
Hidden Artin slopes: $[3]$ (show 8), $[2,3]$ (show 4)
Indices of inseparability: $[6,2,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,5,9]$

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
2.2.4.18a1.1 $( x^{2} + x + 1 )^{4} + 2 ( x^{2} + x + 1 )^{2} + 2$ $D_4\times C_2$ (as 8T9) $16$ $4$ $[2, 3, \frac{7}{2}]^{2}$ $[1,2,\frac{5}{2}]^{2}$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
2.2.4.18a1.2 $( x^{2} + x + 1 )^{4} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{2} + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, \frac{7}{2}]^{2}$ $[1,2,\frac{5}{2}]^{2}$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
2.2.4.18a1.3 $( x^{2} + x + 1 )^{4} + 2 ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 2$ $C_2^3: C_4$ (as 8T21) $32$ $2$ $[2, 2, 3, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[2,3]$ $[1,2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
2.2.4.18a1.4 $( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2 ( x^{2} + x + 1 )^{2} + 2$ $D_4\times C_2$ (as 8T9) $16$ $4$ $[2, 3, \frac{7}{2}]^{2}$ $[1,2,\frac{5}{2}]^{2}$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
2.2.4.18a1.5 $( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{2} + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, \frac{7}{2}]^{2}$ $[1,2,\frac{5}{2}]^{2}$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
2.2.4.18a1.6 $( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2 ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 2$ $C_2^3: C_4$ (as 8T21) $32$ $2$ $[2, 2, 3, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[2,3]$ $[1,2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
2.2.4.18a1.7 $( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 2 ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, \frac{7}{2}]^{2}$ $[1,2,\frac{5}{2}]^{2}$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
2.2.4.18a1.8 $( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 2$ $D_4\times C_2$ (as 8T9) $16$ $4$ $[2, 3, \frac{7}{2}]^{2}$ $[1,2,\frac{5}{2}]^{2}$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
2.2.4.18a1.9 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2 ( x^{2} + x + 1 )^{2} + 2$ $D_4\times C_2$ (as 8T9) $16$ $4$ $[2, 3, \frac{7}{2}]^{2}$ $[1,2,\frac{5}{2}]^{2}$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
2.2.4.18a1.10 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{2} + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, \frac{7}{2}]^{2}$ $[1,2,\frac{5}{2}]^{2}$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
2.2.4.18a1.11 $( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2 ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 2$ $C_2^3: C_4$ (as 8T21) $32$ $2$ $[2, 2, 3, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[2,3]$ $[1,2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
2.2.4.18a1.12 $( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 2 ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 2$ $C_2^3: C_4$ (as 8T21) $32$ $2$ $[2, 2, 3, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2}]^{2}$ $[2,3]$ $[1,2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
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