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Properties

Label 2.1.4.10a1.2-1.4.8b
Base 2.1.4.10a1.2
Degree \(4\)
e \(4\)
f \(1\)
c \(8\)

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Defining polynomial

$x^{4} + b_{7} \pi^{2} x^{3} + a_{2} \pi x^{2} + a_{5} \pi^{2} x + c_{8} \pi^{3} + c_{4} \pi^{2} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.10a1.2
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Absolute Artin slopes: $[2,3,3,\frac{7}{2}]$
Swan slopes: $[1,2]$
Means: $\langle\frac{1}{2},\frac{5}{4}\rangle$
Rams: $(1,3)$
Field count: $5$ (incomplete)
Ambiguity: $4$
Mass: $2$
Absolute Mass: $1$ ($3/4$ 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: $C_4:D_4$ (show 1), $C_4:D_4$ (show 3), $C_2^2.D_4$ (show 1) (incomplete)
Hidden Artin slopes: $[\ ]^{2}$ (incomplete)
Indices of inseparability: $[33,26,24,8,0]$
Associated inertia: $[1,2,1]$
Jump Set: $[1,2,4,8,32]$ (show 3), $[1,5,17,33,49]$ (show 2)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 5

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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
2.1.16.48o1.3 $x^{16} + 4 x^{10} + 10 x^{8} + 8 x^{7} + 8 x + 2$ $C_2^2.D_4$ (as 16T54) $32$ $8$ $[2, 3, 3, \frac{7}{2}]^{2}$ $[1,2,2,\frac{5}{2}]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 17, 33, 49]$
2.1.16.48o1.37 $x^{16} + 4 x^{14} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 8 x + 2$ $C_4:D_4$ (as 16T43) $32$ $8$ $[2, 3, 3, \frac{7}{2}]^{2}$ $[1,2,2,\frac{5}{2}]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 17, 33, 49]$
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) $32$ $4$ $[2, 3, 3, \frac{7}{2}]^{2}$ $[1,2,2,\frac{5}{2}]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.48o1.74 $x^{16} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{5} + 8 x + 14$ $C_4:D_4$ (as 16T43) $32$ $8$ $[2, 3, 3, \frac{7}{2}]^{2}$ $[1,2,2,\frac{5}{2}]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.48o1.75 $x^{16} + 4 x^{12} + 4 x^{10} + 10 x^{8} + 8 x^{5} + 8 x + 14$ $C_4:D_4$ (as 16T43) $32$ $8$ $[2, 3, 3, \frac{7}{2}]^{2}$ $[1,2,2,\frac{5}{2}]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[33, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 2, 4, 8, 32]$
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