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Properties

Label 2.1.2.3a1.3-1.4.19a
Base 2.1.2.3a1.3
Degree \(4\)
e \(4\)
f \(1\)
c \(19\)

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Defining polynomial

$x^{4} + \left(b_{23} \pi^{6} + b_{19} \pi^{5}\right) x^{3} + \left(b_{14} \pi^{4} + b_{10} \pi^{3}\right) x^{2} + \left(b_{21} \pi^{6} + b_{17} \pi^{5}\right) x + c_{24} \pi^{7} + c_{16} \pi^{5} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: $\Q_{2}(\sqrt{2})$
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $19$
Absolute Artin slopes: $[3,4,5]$
Swan slopes: $[4,6]$
Means: $\langle2,4\rangle$
Rams: $(4,8)$
Field count: $76$ (complete)
Ambiguity: $4$
Mass: $64$
Absolute Mass: $32$

Diagrams

Varying

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

Galois group: $C_8$ (show 8), $D_{8}$ (show 4), $C_8:C_2$ (show 12), $QD_{16}$ (show 4), $(C_8:C_2):C_2$ (show 8), $(C_4^2 : C_2):C_2$ (show 8), $((C_8 : C_2):C_2):C_2$ (show 16), $(((C_4 \times C_2): C_2):C_2):C_2$ (show 16)
Hidden Artin slopes: $[\ ]$ (show 8), $[\ ]^{2}$ (show 4), $[2]$ (show 16), $[2,\frac{7}{2}]$ (show 8), $[2,\frac{7}{2},\frac{17}{4}]$ (show 32), $[2,\frac{7}{2}]^{2}$ (show 8)
Indices of inseparability: $[24,16,8,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,3,7,15]$

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
2.1.8.31a1.207 $x^{8} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 16 x + 2$ $(C_4^2 : C_2):C_2$ (as 8T26) $64$ $2$ $[2, 3, \frac{7}{2}, 4, 5]^{2}$ $[1,2,\frac{5}{2},3,4]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.208 $x^{8} + 16 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 16 x + 2$ $(C_4^2 : C_2):C_2$ (as 8T26) $64$ $2$ $[2, 3, \frac{7}{2}, 4, 5]^{2}$ $[1,2,\frac{5}{2},3,4]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.209 $x^{8} + 8 x^{6} + 16 x^{5} + 4 x^{4} + 8 x^{2} + 16 x + 2$ $(C_4^2 : C_2):C_2$ (as 8T26) $64$ $2$ $[2, 3, \frac{7}{2}, 4, 5]^{2}$ $[1,2,\frac{5}{2},3,4]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.210 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 4 x^{4} + 8 x^{2} + 16 x + 2$ $(C_4^2 : C_2):C_2$ (as 8T26) $64$ $2$ $[2, 3, \frac{7}{2}, 4, 5]^{2}$ $[1,2,\frac{5}{2},3,4]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.215 $x^{8} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 16 x + 18$ $(C_4^2 : C_2):C_2$ (as 8T26) $64$ $2$ $[2, 3, \frac{7}{2}, 4, 5]^{2}$ $[1,2,\frac{5}{2},3,4]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.216 $x^{8} + 16 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 16 x + 18$ $(C_4^2 : C_2):C_2$ (as 8T26) $64$ $2$ $[2, 3, \frac{7}{2}, 4, 5]^{2}$ $[1,2,\frac{5}{2},3,4]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.217 $x^{8} + 8 x^{6} + 16 x^{5} + 4 x^{4} + 8 x^{2} + 16 x + 18$ $(C_4^2 : C_2):C_2$ (as 8T26) $64$ $2$ $[2, 3, \frac{7}{2}, 4, 5]^{2}$ $[1,2,\frac{5}{2},3,4]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.218 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 4 x^{4} + 8 x^{2} + 16 x + 18$ $(C_4^2 : C_2):C_2$ (as 8T26) $64$ $2$ $[2, 3, \frac{7}{2}, 4, 5]^{2}$ $[1,2,\frac{5}{2},3,4]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
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