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Properties

Label 2.1.4.11a1.5-1.4.20b
Base 2.1.4.11a1.5
Degree \(4\)
e \(4\)
f \(1\)
c \(20\)

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Defining polynomial

$x^{4} + \left(b_{31} \pi^{8} + b_{27} \pi^{7} + b_{23} \pi^{6} + b_{19} \pi^{5}\right) x^{3} + a_{2} \pi x^{2} + \left(b_{29} \pi^{8} + b_{25} \pi^{7} + b_{21} \pi^{6} + a_{17} \pi^{5}\right) x + c_{32} \pi^{9} + c_{4} \pi^{2} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.11a1.5
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $20$
Absolute Artin slopes: $[2,3,4,5]$
Swan slopes: $[1,8]$
Means: $\langle\frac{1}{2},\frac{17}{4}\rangle$
Rams: $(1,15)$
Field count: $142$ (incomplete)
Ambiguity: $4$
Mass: $128$
Absolute Mass: $64$ ($48$ 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: $QD_{16}$ (show 2), $Q_{16}$ (show 2), $D_8:C_2$ (show 3), $D_8:C_2$ (show 2), $C_4\wr C_2$ (show 4), $D_8:C_2$ (show 2), $D_8:C_2$ (show 2), $D_8:C_2$ (show 2), $C_2\times \SD_{16}$ (show 2), $Q_{16}:C_2$ (show 1), $C_4^2:C_2^2$ (show 2), $C_4^2:C_2^2$ (show 4), $C_4^2:C_2^2$ (show 8), $D_4:D_4$ (show 2), $C_2^3.D_4$ (show 4), $C_2\wr C_4$ (show 4), $C_4^2:C_2^2$ (show 2), $C_4^2:D_4$ (show 2), $C_2^4.D_4$ (show 4), $\OD_{16}:D_4$ (show 4), $C_4^2:D_4$ (show 4), $C_4^2:D_4$ (show 4), $C_4^2.D_4$ (show 8), $C_4^2.D_4$ (show 8), $C_2\wr D_4$ (show 8), $C_4^2:D_4$ (show 2), $C_2\wr D_4$ (show 4), $C_4^2:D_4$ (show 4), $C_4^2:D_4$ (show 2), $C_2^5.(C_2\times D_4)$ (show 16), $C_2^4.(C_4\times D_4)$ (show 16), $C_2^6.D_4$ (show 8) (incomplete)
Hidden Artin slopes: $[\ ]$ (show 4), $[\ ]^{2}$ (show 14), $[\frac{7}{2}]^{2}$ (show 18), $[2,\frac{7}{2}]^{2}$ (show 4), $[\frac{7}{2},\frac{17}{4}]^{2}$ (show 40), $[3,\frac{7}{2}]^{2}$ (show 10), $[\frac{7}{2},\frac{17}{4}]$ (show 8), $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ (show 24), $[2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]^{2}$ (show 16), $[\frac{7}{2}]$ (show 4) (incomplete)
Indices of inseparability: $[49,34,20,8,0]$
Associated inertia: $[1,1,1,1]$
Jump Set: $[1,2,4,8,32]$ (show 88), $[1,2,4,32,48]$ (show 54)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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  displayed columns for results
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
2.1.16.64g1.1200 $x^{16} + 16 x^{15} + 8 x^{10} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 16 x + 26$ $C_2^4.D_4$ (as 16T324) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[\frac{7}{2},\frac{17}{4}]^{2}$ $[\frac{5}{2},\frac{13}{4}]^{2}$ $[49, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 32, 48]$
2.1.16.64g1.1202 $x^{16} + 16 x^{15} + 16 x^{13} + 8 x^{10} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 16 x + 26$ $C_2^4.D_4$ (as 16T324) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[\frac{7}{2},\frac{17}{4}]^{2}$ $[\frac{5}{2},\frac{13}{4}]^{2}$ $[49, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 32, 48]$
2.1.16.64g1.1205 $x^{16} + 8 x^{10} + 16 x^{9} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 16 x + 26$ $C_2^4.D_4$ (as 16T324) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[\frac{7}{2},\frac{17}{4}]^{2}$ $[\frac{5}{2},\frac{13}{4}]^{2}$ $[49, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 32, 48]$
2.1.16.64g1.1208 $x^{16} + 16 x^{13} + 8 x^{10} + 16 x^{9} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 16 x + 26$ $C_2^4.D_4$ (as 16T324) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[\frac{7}{2},\frac{17}{4}]^{2}$ $[\frac{5}{2},\frac{13}{4}]^{2}$ $[49, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 32, 48]$
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