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Properties

Label 2.2.4.12a4.2-1.2.6a
Base 2.2.4.12a4.2
Degree \(2\)
e \(2\)
f \(1\)
c \(6\)

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Defining polynomial

$x^{2} + \left(b_{9} \pi^{5} + b_{7} \pi^{4} + a_{5} \pi^{3}\right) x + c_{10} \pi^{6} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.2.4.12a4.2
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $6$
Absolute Artin slopes: $[2,2,3]$
Swan slopes: $[5]$
Means: $\langle\frac{5}{2}\rangle$
Rams: $(5)$
Field count: $60$ (complete)
Ambiguity: $2$
Mass: $48$
Absolute Mass: $24$

Diagrams

Varying

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

Galois group: $C_2^5:C_4$ (show 2), $C_2^5:C_4$ (show 2), $C_2^5.D_4$ (show 8), $(C_4\times C_8).D_4$ (show 2), $C_2^2.C_2\wr C_4$ (show 6), $D_4^2:C_4$ (show 4), $C_2^4.Q_{16}$ (show 6), $C_2^5.\OD_{16}$ (show 4), $C_2^3.C_2\wr C_4$ (show 6), $C_2^6:C_8$ (show 2), $Q_8^2:C_8$ (show 4), $C_4^3.D_4$ (show 6), $D_4^2:C_8$ (show 8)
Hidden Artin slopes: $[2,2,\frac{9}{4}]^{4}$ (show 2), $[2,2,3]^{4}$ (show 20), $[2,2,3]^{2}$ (show 20), $[2,2,\frac{5}{2}]^{4}$ (show 6), $[2,2,\frac{9}{4}]^{2}$ (show 2), $[2,2,\frac{5}{2}]^{2}$ (show 4), $[2,2,2]^{4}$ (show 2), $[2,2]^{2}$ (show 4)
Indices of inseparability: $[11,6,4,0]$
Associated inertia: $[2,1]$
Jump Set: $[1,2,7,15]$

Fields

Galois group Galois Artin Indices Hidden content
Assoc. Inertia Jump Set
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Showing all 6

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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
2.2.8.36b10.23 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 6$ $C_2^3.C_2\wr C_4$ (as 16T914) $512$ $2$ $[2, 2, 2, 2, \frac{5}{2}, 3]^{8}$ $[1,1,1,1,\frac{3}{2},2]^{8}$ $[2,2,\frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
2.2.8.36b10.24 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 6$ $C_2^3.C_2\wr C_4$ (as 16T914) $512$ $2$ $[2, 2, 2, 2, \frac{5}{2}, 3]^{8}$ $[1,1,1,1,\frac{3}{2},2]^{8}$ $[2,2,\frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
2.2.8.36b10.36 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 6$ $Q_8^2:C_8$ (as 16T929) $512$ $4$ $[2, 2, 2, 2, \frac{5}{2}, 3]^{8}$ $[1,1,1,1,\frac{3}{2},2]^{8}$ $[2,2,\frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
2.2.8.36b10.37 $( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 8 x + 6$ $Q_8^2:C_8$ (as 16T929) $512$ $4$ $[2, 2, 2, 2, \frac{5}{2}, 3]^{8}$ $[1,1,1,1,\frac{3}{2},2]^{8}$ $[2,2,\frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
2.2.8.36b10.38 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 6$ $Q_8^2:C_8$ (as 16T929) $512$ $4$ $[2, 2, 2, 2, \frac{5}{2}, 3]^{8}$ $[1,1,1,1,\frac{3}{2},2]^{8}$ $[2,2,\frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
2.2.8.36b10.39 $( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 8 x + 6$ $Q_8^2:C_8$ (as 16T929) $512$ $4$ $[2, 2, 2, 2, \frac{5}{2}, 3]^{8}$ $[1,1,1,1,\frac{3}{2},2]^{8}$ $[2,2,\frac{5}{2}]^{4}$ $[1,1,\frac{3}{2}]^{4}$ $[11, 6, 4, 0]$ $[2, 1]$ $z^6 + z^2 + t,t z + (t + 1)$ $[1, 2, 7, 15]$
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