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Properties

Label 2.1.4.8b1.2-1.4.22c
Base 2.1.4.8b1.2
Degree \(4\)
e \(4\)
f \(1\)
c \(22\)

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Defining polynomial

$x^{4} + \left(b_{27} \pi^{7} + b_{23} \pi^{6} + a_{19} \pi^{5}\right) x^{3} + \left(b_{18} \pi^{5} + b_{14} \pi^{4} + a_{10} \pi^{3}\right) x^{2} + \left(b_{25} \pi^{7} + b_{21} \pi^{6}\right) x + c_{28} \pi^{8} + c_{20} \pi^{6} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.8b1.2
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $22$
Absolute Artin slopes: $[2,3,\frac{7}{2},4]$
Swan slopes: $[5,7]$
Means: $\langle\frac{5}{2},\frac{19}{4}\rangle$
Rams: $(5,9)$
Field count: $60$ (complete)
Ambiguity: $4$
Mass: $64$
Absolute Mass: $16$

Diagrams

Varying

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

Galois group: $C_2^2 : C_4$ (show 4), $C_4^2:C_2$ (show 2), $C_4 \times D_4$ (show 4), $C_4^2:C_2$ (show 1), $C_4^2:C_2$ (show 1), $C_4:D_4$ (show 1), $C_2^2.D_4$ (show 3), $C_2^2\wr C_2$ (show 2), $C_4:D_4$ (show 4), $C_2^2.D_4$ (show 6), $C_2\wr C_2^2$ (show 2), $C_2\wr C_2^2$ (show 2), $C_2^3.D_4$ (show 2), $C_2^3.D_4$ (show 2), $C_2^4.D_4$ (show 2), $C_2^4.D_4$ (show 2), $C_4^2:D_4$ (show 2), $C_4^2.D_4$ (show 4), $C_4^2:D_4$ (show 4), $C_4^2:D_4$ (show 2), $C_2^4.(C_4\times D_4)$ (show 4), $C_2^4.(C_4\times D_4)$ (show 4)
Hidden Artin slopes: $[\ ]^{2}$ (show 24), $[2]^{2}$ (show 8), $[2,2,\frac{7}{2}]^{4}$ (show 8), $[2]^{4}$ (show 4), $[2,\frac{7}{2}]^{2}$ (show 12), $[\ ]$ (show 4)
Indices of inseparability: $[39,30,20,8,0]$
Associated inertia: $[1,1,1,1]$
Jump Set: $[1,7,14,32,48]$

Fields

Galois group Galois Artin Indices Hidden content
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
2.1.16.54o1.201 $x^{16} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 10$ $C_2^2 : C_4$ (as 16T10) $16$ $16$ $[2, 3, \frac{7}{2}, 4]$ $[1,2,\frac{5}{2},3]$ $[\ ]$ $[\ ]$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 14, 32, 48]$
2.1.16.54o1.202 $x^{16} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 26$ $C_2^2 : C_4$ (as 16T10) $16$ $16$ $[2, 3, \frac{7}{2}, 4]$ $[1,2,\frac{5}{2},3]$ $[\ ]$ $[\ ]$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 14, 32, 48]$
2.1.16.54o1.212 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 10 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 10$ $C_2^2 : C_4$ (as 16T10) $16$ $16$ $[2, 3, \frac{7}{2}, 4]$ $[1,2,\frac{5}{2},3]$ $[\ ]$ $[\ ]$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 14, 32, 48]$
2.1.16.54o1.213 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 10 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 26$ $C_2^2 : C_4$ (as 16T10) $16$ $16$ $[2, 3, \frac{7}{2}, 4]$ $[1,2,\frac{5}{2},3]$ $[\ ]$ $[\ ]$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 14, 32, 48]$
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