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Properties

Label 2.1.2.3a1.3-1.8.42g
Base 2.1.2.3a1.3
Degree \(8\)
e \(8\)
f \(1\)
c \(42\)

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Defining polynomial

$x^{8} + b_{39} \pi^{5} x^{7} + b_{30} \pi^{4} x^{6} + b_{37} \pi^{5} x^{5} + \left(c_{44} \pi^{6} + c_{36} \pi^{5} + b_{28} \pi^{4} + b_{20} \pi^{3}\right) x^{4} + \left(b_{43} \pi^{6} + a_{35} \pi^{5}\right) x^{3} + \left(b_{34} \pi^{5} + a_{26} \pi^{4}\right) x^{2} + b_{41} \pi^{6} x + c_{32} \pi^{5} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $8$
Base field: $\Q_{2}(\sqrt{2})$
Ramification index $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $42$
Absolute Artin slopes: $[3,4,\frac{17}{4},\frac{19}{4}]$
Swan slopes: $[4,\frac{9}{2},\frac{11}{2}]$
Means: $\langle2,\frac{13}{4},\frac{35}{8}\rangle$
Rams: $(4,5,9)$
Field count: $320$ (complete)
Ambiguity: $8$
Mass: $256$
Absolute Mass: $128$

Diagrams

Varying

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

Galois group: $C_2\wr C_4$ (show 16), $C_2\wr C_4$ (show 16), $C_2^5:C_4$ (show 32), $C_4^2:D_4$ (show 8), $C_4^2:D_4$ (show 8), $C_2\wr D_4$ (show 8), $C_4^2:D_4$ (show 8), $C_2^6:(C_2\times C_4)$ (show 64), $C_2^6.(C_2\times C_4)$ (show 32), $C_2^6.D_4$ (show 64), $C_2^6:D_4$ (show 32), $C_2^5.(C_2\times D_4)$ (show 32)
Hidden Artin slopes: $[2,\frac{7}{2}]$ (show 32), $[2,\frac{7}{2}]^{2}$ (show 64), $[2,3,\frac{7}{2},4]^{2}$ (show 160), $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ (show 64)
Indices of inseparability: $[51,42,32,16,0]$
Associated inertia: $[1,1,1,1]$
Jump Set: $[1,3,7,15,31]$

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
2.1.16.66j1.647 $x^{16} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 16 x^{7} + 16 x^{3} + 2$ $C_4^2:D_4$ (as 16T399) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.648 $x^{16} + 16 x^{12} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 16 x^{7} + 16 x^{3} + 2$ $C_4^2:D_4$ (as 16T399) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.667 $x^{16} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 16 x^{7} + 16 x^{3} + 18$ $C_4^2:D_4$ (as 16T399) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.668 $x^{16} + 16 x^{12} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 16 x^{7} + 16 x^{3} + 18$ $C_4^2:D_4$ (as 16T399) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.727 $x^{16} + 8 x^{12} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 16 x^{7} + 16 x^{3} + 2$ $C_4^2:D_4$ (as 16T399) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.728 $x^{16} + 24 x^{12} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 16 x^{7} + 16 x^{3} + 2$ $C_4^2:D_4$ (as 16T399) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.743 $x^{16} + 8 x^{12} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 16 x^{7} + 16 x^{3} + 18$ $C_4^2:D_4$ (as 16T399) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.66j1.744 $x^{16} + 24 x^{12} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 16 x^{7} + 16 x^{3} + 18$ $C_4^2:D_4$ (as 16T399) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[51, 42, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
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