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Properties

Label 2.1.8.24c1.46-1.2.14a
Base 2.1.8.24c1.46
Degree \(2\)
e \(2\)
f \(1\)
c \(14\)

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Defining polynomial

$x^{2} + \left(b_{25} \pi^{13} + b_{23} \pi^{12} + b_{21} \pi^{11} + b_{19} \pi^{10} + b_{17} \pi^{9} + b_{15} \pi^{8} + a_{13} \pi^{7}\right) x + c_{26} \pi^{14} + \pi$

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 $c$: $14$
Absolute Artin slopes: $[2,3,4,\frac{19}{4}]$
Swan slopes: $[13]$
Means: $\langle\frac{13}{2}\rangle$
Rams: $(13)$
Field count: $16$ (complete)
Ambiguity: $2$
Mass: $64$
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: $C_2^6:Q_8$ (show 8), $C_2^6.Q_8$ (show 8)
Hidden Artin slopes: $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$
Indices of inseparability: $[47,34,20,8,0]$
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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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
2.1.16.62h1.1649 $x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.62h1.1650 $x^{16} + 8 x^{15} + 8 x^{14} + 20 x^{12} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.62h1.1651 $x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 16 x^{11} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.62h1.1652 $x^{16} + 8 x^{15} + 8 x^{14} + 20 x^{12} + 16 x^{11} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.62h1.1653 $x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 16 x^{9} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.62h1.1654 $x^{16} + 8 x^{15} + 8 x^{14} + 20 x^{12} + 16 x^{9} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.62h1.1655 $x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 16 x^{11} + 16 x^{9} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.62h1.1656 $x^{16} + 8 x^{15} + 8 x^{14} + 20 x^{12} + 16 x^{11} + 16 x^{9} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 22$ $C_2^6.Q_8$ (as 16T968) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.62h1.1657 $x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 16 x + 22$ $C_2^6:Q_8$ (as 16T958) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.62h1.1658 $x^{16} + 8 x^{15} + 8 x^{14} + 20 x^{12} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 16 x + 22$ $C_2^6:Q_8$ (as 16T958) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.62h1.1659 $x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 16 x^{11} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 16 x + 22$ $C_2^6:Q_8$ (as 16T958) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.62h1.1660 $x^{16} + 8 x^{15} + 8 x^{14} + 20 x^{12} + 16 x^{11} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 16 x + 22$ $C_2^6:Q_8$ (as 16T958) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.62h1.1661 $x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 16 x^{9} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 16 x + 22$ $C_2^6:Q_8$ (as 16T958) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.62h1.1662 $x^{16} + 8 x^{15} + 8 x^{14} + 20 x^{12} + 16 x^{9} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 16 x + 22$ $C_2^6:Q_8$ (as 16T958) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.62h1.1663 $x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 16 x^{11} + 16 x^{9} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 16 x + 22$ $C_2^6:Q_8$ (as 16T958) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.62h1.1664 $x^{16} + 8 x^{15} + 8 x^{14} + 20 x^{12} + 16 x^{11} + 16 x^{9} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 16 x + 22$ $C_2^6:Q_8$ (as 16T958) $512$ $2$ $[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4}]^{2}$ $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ $[2,\frac{5}{2},3,\frac{13}{4}]^{2}$ $[47, 34, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
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