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Properties

Label 2.1.8.24c1.46-1.2.12a
Base 2.1.8.24c1.46
Degree \(2\)
e \(2\)
f \(1\)
c \(12\)

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Defining polynomial

$x^{2} + \left(b_{21} \pi^{11} + b_{19} \pi^{10} + b_{17} \pi^{9} + b_{15} \pi^{8} + b_{13} \pi^{7} + a_{11} \pi^{6}\right) x + c_{22} \pi^{12} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.1.8.24c1.46
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Absolute Artin slopes: $[2,3,4,\frac{9}{2}]$
Swan slopes: $[11]$
Means: $\langle\frac{11}{2}\rangle$
Rams: $(11)$
Field count: $8$ (complete)
Ambiguity: $2$
Mass: $32$
Absolute Mass: $4$

Diagrams

Varying

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

Galois group: $C_2^6.Q_8$
Hidden Artin slopes: $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$
Indices of inseparability: $[45,34,20,8,0]$
Associated inertia: $[1,1,1,1]$
Jump Set: $[1,2,4,8,32]$

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
2.1.16.60l1.805 $x^{16} + 8 x^{13} + 4 x^{12} + 8 x^{10} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[45, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.60l1.806 $x^{16} + 8 x^{13} + 4 x^{12} + 8 x^{10} + 18 x^{8} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[45, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.60l1.807 $x^{16} + 8 x^{13} + 4 x^{12} + 8 x^{10} + 2 x^{8} + 16 x^{7} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[45, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.60l1.808 $x^{16} + 8 x^{13} + 4 x^{12} + 8 x^{10} + 18 x^{8} + 16 x^{7} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[45, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.60l1.829 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{10} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[45, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.60l1.830 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{10} + 18 x^{8} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[45, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.60l1.831 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{10} + 2 x^{8} + 16 x^{7} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[45, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.60l1.832 $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{10} + 18 x^{8} + 16 x^{7} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{9}{2}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{7}{2}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[45, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
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