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Properties

Label 2.1.4.11a1.12-1.4.26c
Base 2.1.4.11a1.12
Degree \(4\)
e \(4\)
f \(1\)
c \(26\)

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Defining polynomial

$x^{4} + \left(b_{31} \pi^{8} + b_{27} \pi^{7} + a_{23} \pi^{6}\right) x^{3} + \left(b_{26} \pi^{7} + b_{22} \pi^{6} + b_{18} \pi^{5} + a_{14} \pi^{4}\right) x^{2} + \left(b_{29} \pi^{8} + b_{25} \pi^{7}\right) x + c_{32} \pi^{9} + c_{28} \pi^{8} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.11a1.12
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $26$
Absolute Artin slopes: $[3,4,\frac{19}{4},5]$
Swan slopes: $[7,8]$
Means: $\langle\frac{7}{2},\frac{23}{4}\rangle$
Rams: $(7,9)$
Field count: $96$ (complete)
Ambiguity: $4$
Mass: $128$
Absolute Mass: $32$

Diagrams

Varying

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

Galois group: $C_2^4:C_8$ (show 32), $C_2^4:C_8$ (show 16), $C_2^4:C_8$ (show 16), $C_2\wr (C_2\times C_4)$ (show 32)
Hidden Artin slopes: $[2,\frac{7}{2},\frac{17}{4}]$ (show 64), $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4}]^{2}$ (show 32)
Indices of inseparability: $[55,46,32,16,0]$
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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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
2.1.16.70f1.1645 $x^{16} + 8 x^{14} + 16 x^{13} + 16 x^{11} + 4 x^{8} + 16 x^{7} + 18$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70f1.1646 $x^{16} + 8 x^{14} + 16 x^{13} + 16 x^{11} + 4 x^{8} + 16 x^{7} + 50$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70f1.1647 $x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 16 x^{11} + 4 x^{8} + 16 x^{7} + 18$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70f1.1648 $x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 16 x^{11} + 4 x^{8} + 16 x^{7} + 50$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70f1.1653 $x^{16} + 8 x^{14} + 16 x^{13} + 16 x^{12} + 16 x^{11} + 4 x^{8} + 16 x^{7} + 18$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70f1.1654 $x^{16} + 8 x^{14} + 16 x^{13} + 16 x^{12} + 16 x^{11} + 4 x^{8} + 16 x^{7} + 50$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70f1.1655 $x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 16 x^{12} + 16 x^{11} + 4 x^{8} + 16 x^{7} + 18$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70f1.1656 $x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 16 x^{12} + 16 x^{11} + 4 x^{8} + 16 x^{7} + 50$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70f1.1665 $x^{16} + 8 x^{14} + 16 x^{11} + 16 x^{10} + 4 x^{8} + 16 x^{7} + 18$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70f1.1666 $x^{16} + 8 x^{14} + 16 x^{11} + 16 x^{10} + 4 x^{8} + 16 x^{7} + 50$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70f1.1667 $x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{11} + 16 x^{10} + 4 x^{8} + 16 x^{7} + 18$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70f1.1668 $x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{11} + 16 x^{10} + 4 x^{8} + 16 x^{7} + 50$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70f1.1673 $x^{16} + 8 x^{14} + 16 x^{12} + 16 x^{11} + 16 x^{10} + 4 x^{8} + 16 x^{7} + 18$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70f1.1674 $x^{16} + 8 x^{14} + 16 x^{12} + 16 x^{11} + 16 x^{10} + 4 x^{8} + 16 x^{7} + 50$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70f1.1675 $x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{12} + 16 x^{11} + 16 x^{10} + 4 x^{8} + 16 x^{7} + 18$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.70f1.1676 $x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{12} + 16 x^{11} + 16 x^{10} + 4 x^{8} + 16 x^{7} + 50$ $C_2^4:C_8$ (as 16T258) $128$ $4$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, 5]$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4},4]$ $[2,\frac{7}{2},\frac{17}{4}]$ $[1,\frac{5}{2},\frac{13}{4}]$ $[55, 46, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
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