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Properties

Label 2.1.4.11a1.14-1.4.16a
Base 2.1.4.11a1.14
Degree \(4\)
e \(4\)
f \(1\)
c \(16\)

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Defining polynomial

$x^{4} + b_{15} \pi^{4} x^{3} + \left(b_{14} \pi^{4} + b_{10} \pi^{3}\right) x^{2} + \left(b_{17} \pi^{5} + a_{13} \pi^{4}\right) x + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.11a1.14
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $16$
Absolute Artin slopes: $[3,4,\frac{49}{12},\frac{49}{12}]$
Swan slopes: $[\frac{13}{3},\frac{13}{3}]$
Means: $\langle\frac{13}{6},\frac{13}{4}\rangle$
Rams: $(\frac{13}{3},\frac{13}{3})$
Field count: $8$ (complete)
Ambiguity: $1$
Mass: $16$
Absolute Mass: $8$

Diagrams

Varying

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

Galois group: $C_2^6:(C_2\times S_4)$
Hidden Artin slopes: $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}_{3}$
Indices of inseparability: $[45,42,32,16,0]$ (show 4), $[45,45,32,16,0]$ (show 4)
Associated inertia: $[1,1,1]$
Jump Set: $[1,3,7,15,31]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 8

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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
2.1.16.60j1.65 $x^{16} + 8 x^{13} + 4 x^{8} + 2$ $C_2^6:(C_2\times S_4)$ (as 16T1519) $3072$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4, \frac{49}{12}, \frac{49}{12}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3,\frac{37}{12},\frac{37}{12}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}_{3}$ $[45, 45, 32, 16, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.60j1.67 $x^{16} + 8 x^{14} + 8 x^{13} + 4 x^{8} + 2$ $C_2^6:(C_2\times S_4)$ (as 16T1519) $3072$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4, \frac{49}{12}, \frac{49}{12}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3,\frac{37}{12},\frac{37}{12}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}_{3}$ $[45, 45, 32, 16, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.60j1.69 $x^{16} + 8 x^{13} + 8 x^{12} + 4 x^{8} + 2$ $C_2^6:(C_2\times S_4)$ (as 16T1519) $3072$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4, \frac{49}{12}, \frac{49}{12}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3,\frac{37}{12},\frac{37}{12}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}_{3}$ $[45, 45, 32, 16, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.60j1.71 $x^{16} + 8 x^{14} + 8 x^{13} + 8 x^{12} + 4 x^{8} + 2$ $C_2^6:(C_2\times S_4)$ (as 16T1519) $3072$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4, \frac{49}{12}, \frac{49}{12}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3,\frac{37}{12},\frac{37}{12}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}_{3}$ $[45, 45, 32, 16, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.60j1.90 $x^{16} + 8 x^{13} + 8 x^{10} + 4 x^{8} + 8 x^{4} + 18$ $C_2^6:(C_2\times S_4)$ (as 16T1519) $3072$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4, \frac{49}{12}, \frac{49}{12}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3,\frac{37}{12},\frac{37}{12}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}_{3}$ $[45, 42, 32, 16, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.60j1.92 $x^{16} + 8 x^{15} + 8 x^{13} + 8 x^{10} + 4 x^{8} + 8 x^{4} + 18$ $C_2^6:(C_2\times S_4)$ (as 16T1519) $3072$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4, \frac{49}{12}, \frac{49}{12}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3,\frac{37}{12},\frac{37}{12}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}_{3}$ $[45, 42, 32, 16, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.60j1.94 $x^{16} + 8 x^{14} + 8 x^{13} + 8 x^{10} + 4 x^{8} + 8 x^{4} + 18$ $C_2^6:(C_2\times S_4)$ (as 16T1519) $3072$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4, \frac{49}{12}, \frac{49}{12}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3,\frac{37}{12},\frac{37}{12}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}_{3}$ $[45, 42, 32, 16, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.60j1.96 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 8 x^{10} + 4 x^{8} + 8 x^{4} + 18$ $C_2^6:(C_2\times S_4)$ (as 16T1519) $3072$ $1$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4, \frac{49}{12}, \frac{49}{12}]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3,\frac{37}{12},\frac{37}{12}]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}_{3}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}_{3}$ $[45, 42, 32, 16, 0]$ $[1, 1, 1]$ $z^8 + 1,z^4 + 1,z + 1$ $[1, 3, 7, 15, 31]$
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