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Properties

Label 2.1.4.10a1.2-2.2.8a
Base 2.1.4.10a1.2
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: 2.1.4.10a1.2
Ramification index $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $8$
Absolute Artin slopes: $[3,\frac{7}{2},\frac{7}{2}]$
Swan slopes: $[3]$
Means: $\langle\frac{3}{2}\rangle$
Rams: $(3)$
Field count: $10$ (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: $C_2\wr C_2^2$ (show 2), $C_2\wr C_2^2$ (show 4), $C_2^6:(C_2\times C_4)$ (show 4)
Hidden Artin slopes: $[2,2]$ (show 6), $[2,2,2,\frac{7}{2}]^{2}$ (show 4)
Indices of inseparability: $[17,14,8,0]$ (show 4), $[17,16,8,0]$ (show 6)
Associated inertia: $[1,1]$ (show 6), $[1,2]$ (show 4)
Jump Set: $[1,3,7,15]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 10

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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
2.2.8.48b1.77 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 ) + 2$ $C_2\wr C_2^2$ (as 16T149) $64$ $8$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{2}$ $[2,2]$ $[1,1]$ $[17, 16, 8, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 3, 7, 15]$
2.2.8.48b1.79 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 ) + 2$ $C_2\wr C_2^2$ (as 16T149) $64$ $8$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{2}$ $[2,2]$ $[1,1]$ $[17, 16, 8, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 3, 7, 15]$
2.2.8.48b1.88 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 2$ $C_2\wr C_2^2$ (as 16T128) $64$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{2}$ $[2,2]$ $[1,1]$ $[17, 16, 8, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 3, 7, 15]$
2.2.8.48b1.90 $( x^{2} + x + 1 )^{8} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 2$ $C_2\wr C_2^2$ (as 16T128) $64$ $4$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{2}$ $[2,2]$ $[1,1]$ $[17, 16, 8, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 3, 7, 15]$
2.2.8.48b1.91 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 2$ $C_2\wr C_2^2$ (as 16T149) $64$ $8$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{2}$ $[2,2]$ $[1,1]$ $[17, 16, 8, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 3, 7, 15]$
2.2.8.48b1.93 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 2$ $C_2\wr C_2^2$ (as 16T149) $64$ $8$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{2}$ $[2,2]$ $[1,1]$ $[17, 16, 8, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 3, 7, 15]$
2.2.8.48b6.36 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 12 x ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 ) + 2$ $C_2^6:(C_2\times C_4)$ (as 16T821) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2}]^{4}$ $[2,2,2,\frac{7}{2}]^{2}$ $[1,1,1,\frac{5}{2}]^{2}$ $[17, 14, 8, 0]$ $[1, 2]$ $z^4 + 1,z^3 + t z + t$ $[1, 3, 7, 15]$
2.2.8.48b6.40 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 12 x ( x^{2} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 ) + 2$ $C_2^6:(C_2\times C_4)$ (as 16T821) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2}]^{4}$ $[2,2,2,\frac{7}{2}]^{2}$ $[1,1,1,\frac{5}{2}]^{2}$ $[17, 14, 8, 0]$ $[1, 2]$ $z^4 + 1,z^3 + t z + t$ $[1, 3, 7, 15]$
2.2.8.48b6.44 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 12 x ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 2$ $C_2^6:(C_2\times C_4)$ (as 16T821) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2}]^{4}$ $[2,2,2,\frac{7}{2}]^{2}$ $[1,1,1,\frac{5}{2}]^{2}$ $[17, 14, 8, 0]$ $[1, 2]$ $z^4 + 1,z^3 + t z + t$ $[1, 3, 7, 15]$
2.2.8.48b6.48 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 12 x ( x^{2} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 2$ $C_2^6:(C_2\times C_4)$ (as 16T821) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2}]^{4}$ $[2,2,2,\frac{7}{2}]^{2}$ $[1,1,1,\frac{5}{2}]^{2}$ $[17, 14, 8, 0]$ $[1, 2]$ $z^4 + 1,z^3 + t z + t$ $[1, 3, 7, 15]$
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