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Properties

Label 2.1.4.9a1.8-1.4.12a
Base 2.1.4.9a1.8
Degree \(4\)
e \(4\)
f \(1\)
c \(12\)

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Defining polynomial

$x^{4} + b_{11} \pi^{3} x^{3} + \left(b_{10} \pi^{3} + b_{6} \pi^{2}\right) x^{2} + a_{9} \pi^{3} x + c_{12} \pi^{4} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.9a1.8
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Absolute Artin slopes: $[2,3,3,\frac{7}{2}]$
Swan slopes: $[3,3]$
Means: $\langle\frac{3}{2},\frac{9}{4}\rangle$
Rams: $(3,3)$
Field count: $12$ (incomplete)
Ambiguity: $2$
Mass: $8$
Absolute Mass: $4$ ($15/4$ currently in the LMFDB)

Diagrams

Varying

The following invariants arise for fields within the LMFDB; since not all fields in this family are stored, it may be incomplete.

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

Galois group: $C_4^2:C_2$ (show 1), $C_4:D_4$ (show 1), $C_4:D_4$ (show 1), $C_2^2.D_4$ (show 1), $C_2^4.D_4$ (show 4), $C_2\wr C_6$ (show 4) (incomplete)
Hidden Artin slopes: $[\ ]^{2}$ (show 4), $[2]^{4}$ (show 4), $[2,2,3]^{3}$ (show 4) (incomplete)
Indices of inseparability: $[33,26,20,8,0]$ (show 4), $[33,26,24,8,0]$ (show 8)
Associated inertia: $[1,2,1]$ (show 8), $[1,3,1]$ (show 4)
Jump Set: $[1,2,4,8,32]$

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
2.1.16.48o1.46 $x^{16} + 4 x^{10} + 10 x^{8} + 8 x + 6$ $C_2^4.D_4$ (as 16T230) $128$ $4$ $[2, 2, 3, 3, \frac{7}{2}]^{4}$ $[1,1,2,2,\frac{5}{2}]^{4}$ $[2]^{4}$ $[1]^{4}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.48o1.48 $x^{16} + 4 x^{10} + 10 x^{8} + 8 x^{7} + 8 x + 6$ $C_2^4.D_4$ (as 16T230) $128$ $4$ $[2, 2, 3, 3, \frac{7}{2}]^{4}$ $[1,1,2,2,\frac{5}{2}]^{4}$ $[2]^{4}$ $[1]^{4}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.48o1.58 $x^{16} + 4 x^{14} + 4 x^{10} + 10 x^{8} + 8 x^{3} + 8 x + 6$ $C_2^2.D_4$ (as 16T54) $32$ $8$ $[2, 3, 3, \frac{7}{2}]^{2}$ $[1,2,2,\frac{5}{2}]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.48o1.65 $x^{16} + 4 x^{14} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{3} + 8 x + 14$ $C_4:D_4$ (as 16T34) $32$ $4$ $[2, 3, 3, \frac{7}{2}]^{2}$ $[1,2,2,\frac{5}{2}]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.48o1.68 $x^{16} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{5} + 8 x + 6$ $C_4:D_4$ (as 16T43) $32$ $8$ $[2, 3, 3, \frac{7}{2}]^{2}$ $[1,2,2,\frac{5}{2}]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.48o1.76 $x^{16} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 8 x + 14$ $C_4^2:C_2$ (as 16T30) $32$ $4$ $[2, 3, 3, \frac{7}{2}]^{2}$ $[1,2,2,\frac{5}{2}]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.48o1.85 $x^{16} + 4 x^{14} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{5} + 8 x^{3} + 8 x + 6$ $C_2^4.D_4$ (as 16T230) $128$ $4$ $[2, 2, 3, 3, \frac{7}{2}]^{4}$ $[1,1,2,2,\frac{5}{2}]^{4}$ $[2]^{4}$ $[1]^{4}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.48o1.87 $x^{16} + 4 x^{14} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 8 x + 6$ $C_2^4.D_4$ (as 16T230) $128$ $4$ $[2, 2, 3, 3, \frac{7}{2}]^{4}$ $[1,1,2,2,\frac{5}{2}]^{4}$ $[2]^{4}$ $[1]^{4}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 2, 4, 8, 32]$
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