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Properties

Label 2.1.4.8b1.4-1.4.20a
Base 2.1.4.8b1.4
Degree \(4\)
e \(4\)
f \(1\)
c \(20\)

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Defining polynomial

$x^{4} + b_{19} \pi^{5} x^{3} + \left(b_{22} \pi^{6} + b_{18} \pi^{5} + b_{14} \pi^{4}\right) x^{2} + \left(b_{21} \pi^{6} + a_{17} \pi^{5}\right) x + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.8b1.4
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $20$
Absolute Artin slopes: $[2,3,\frac{11}{3},\frac{11}{3}]$
Swan slopes: $[\frac{17}{3},\frac{17}{3}]$
Means: $\langle\frac{17}{6},\frac{17}{4}\rangle$
Rams: $(\frac{17}{3},\frac{17}{3})$
Field count: $16$ (complete)
Ambiguity: $1$
Mass: $32$
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: $C_2^6.\GL(2,\mathbb{Z}/4)$ (incomplete)
Hidden Artin slopes: $[\frac{4}{3},\frac{4}{3},\frac{7}{3},\frac{7}{3},\frac{19}{6},\frac{19}{6}]^{2}_{3}$ (show 4), not computed (show 12) (incomplete)
Indices of inseparability: $[37,34,20,8,0]$ (show 8), $[37,36,20,8,0]$ (show 8)
Associated inertia: $[1,1,1]$
Jump Set: $[1,2,4,8,32]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 16

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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
2.1.16.52n1.33 $x^{16} + 2 x^{8} + 8 x^{5} + 4 x^{4} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{7}{3}, \frac{7}{3}, 3, \frac{19}{6}, \frac{19}{6}, \frac{11}{3}, \frac{11}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{4}{3},\frac{4}{3},2,\frac{13}{6},\frac{13}{6},\frac{8}{3},\frac{8}{3}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},\frac{7}{3},\frac{7}{3},\frac{19}{6},\frac{19}{6}]^{2}_{3}$ $[\frac{1}{3},\frac{1}{3},\frac{4}{3},\frac{4}{3},\frac{13}{6},\frac{13}{6}]^{2}_{3}$ $[37, 36, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.52n1.34 $x^{16} + 8 x^{10} + 2 x^{8} + 8 x^{5} + 4 x^{4} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{7}{3}, \frac{7}{3}, 3, \frac{19}{6}, \frac{19}{6}, \frac{11}{3}, \frac{11}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{4}{3},\frac{4}{3},2,\frac{13}{6},\frac{13}{6},\frac{8}{3},\frac{8}{3}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},\frac{7}{3},\frac{7}{3},\frac{19}{6},\frac{19}{6}]^{2}_{3}$ $[\frac{1}{3},\frac{1}{3},\frac{4}{3},\frac{4}{3},\frac{13}{6},\frac{13}{6}]^{2}_{3}$ $[37, 36, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.52n1.35 $x^{16} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{7}{3}, \frac{7}{3}, 3, \frac{19}{6}, \frac{19}{6}, \frac{11}{3}, \frac{11}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{4}{3},\frac{4}{3},2,\frac{13}{6},\frac{13}{6},\frac{8}{3},\frac{8}{3}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},\frac{7}{3},\frac{7}{3},\frac{19}{6},\frac{19}{6}]^{2}_{3}$ $[\frac{1}{3},\frac{1}{3},\frac{4}{3},\frac{4}{3},\frac{13}{6},\frac{13}{6}]^{2}_{3}$ $[37, 36, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.52n1.36 $x^{16} + 8 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, \frac{7}{3}, \frac{7}{3}, 3, \frac{19}{6}, \frac{19}{6}, \frac{11}{3}, \frac{11}{3}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{4}{3},\frac{4}{3},2,\frac{13}{6},\frac{13}{6},\frac{8}{3},\frac{8}{3}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},\frac{7}{3},\frac{7}{3},\frac{19}{6},\frac{19}{6}]^{2}_{3}$ $[\frac{1}{3},\frac{1}{3},\frac{4}{3},\frac{4}{3},\frac{13}{6},\frac{13}{6}]^{2}_{3}$ $[37, 36, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.52n1.37 $x^{16} + 2 x^{8} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ not computed not computed not computed not computed $[37, 36, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.52n1.38 $x^{16} + 8 x^{10} + 2 x^{8} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ not computed not computed not computed not computed $[37, 36, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.52n1.39 $x^{16} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ not computed not computed not computed not computed $[37, 36, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.52n1.40 $x^{16} + 8 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ not computed not computed not computed not computed $[37, 36, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.52n1.41 $x^{16} + 2 x^{8} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ not computed not computed not computed not computed $[37, 34, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.52n1.42 $x^{16} + 8 x^{10} + 2 x^{8} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ not computed not computed not computed not computed $[37, 34, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.52n1.43 $x^{16} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ not computed not computed not computed not computed $[37, 34, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.52n1.44 $x^{16} + 8 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ not computed not computed not computed not computed $[37, 34, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.52n1.45 $x^{16} + 2 x^{8} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ not computed not computed not computed not computed $[37, 34, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.52n1.46 $x^{16} + 8 x^{10} + 2 x^{8} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ not computed not computed not computed not computed $[37, 34, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.52n1.47 $x^{16} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ not computed not computed not computed not computed $[37, 34, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.52n1.48 $x^{16} + 8 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 6$ $C_2^6.\GL(2,\mathbb{Z}/4)$ (as 16T1684) $6144$ $1$ not computed not computed not computed not computed $[37, 34, 20, 8, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 2, 4, 8, 32]$
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