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Properties

Label 2.1.8.31a1.8-1.2.17a
Base 2.1.8.31a1.8
Degree \(2\)
e \(2\)
f \(1\)
c \(17\)

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Defining polynomial

$x^{2} + \left(b_{31} \pi^{16} + b_{29} \pi^{15} + b_{27} \pi^{14} + b_{25} \pi^{13} + b_{23} \pi^{12} + b_{21} \pi^{11} + b_{19} \pi^{10} + b_{17} \pi^{9}\right) x + c_{32} \pi^{17} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.1.8.31a1.8
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $17$
Absolute Artin slopes: $[3,4,5,6]$
Swan slopes: $[16]$
Means: $\langle8\rangle$
Rams: $(16)$
Field count: $256$ (complete)
Ambiguity: $2$
Mass: $256$
Absolute Mass: $128$

Diagrams

Varying

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

Galois group: $Q_8^2:D_4$ (show 4), $C_8^2:D_4$ (show 4), $D_4^2:D_4$ (show 4), $C_8^2:D_4$ (show 4), $Q_8^2:D_8$ (show 8), $D_4^2:D_8$ (show 8), $\OD_{16}^2:D_4$ (show 16), $C_4^4:D_4$ (show 16), $C_2^5.C_2\wr D_4$ (show 32), $C_2\wr D_8$ (show 32), $C_2\wr D_4.C_4$ (show 64), $C_2^6.C_2\wr D_4$ (show 64) (incomplete)
Hidden Artin slopes: $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ (show 16), $[2,2,\frac{7}{2},\frac{17}{4},\frac{19}{4},\frac{21}{4}]^{2}$ (show 16), $[2,\frac{7}{2},\frac{17}{4},\frac{19}{4},\frac{41}{8},\frac{43}{8},\frac{45}{8}]^{2}$ (show 32), not computed (show 112), $[2,3,\frac{7}{2},4,\frac{17}{4},\frac{19}{4},\frac{41}{8},\frac{43}{8}]^{2}$ (show 32), $[2,3,\frac{7}{2},\frac{17}{4},\frac{19}{4},\frac{41}{8},\frac{43}{8},\frac{45}{8}]^{2}$ (show 32), $[2,\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ (show 16) (incomplete)
Indices of inseparability: $[64,48,32,16,0]$
Associated inertia: $[1,1,1,1]$
Jump Set: $[1,3,7,15,31]$

Fields

Galois group Galois Artin Indices Hidden content
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
2.1.16.79a1.1853 $x^{16} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.79a1.1854 $x^{16} + 32 x^{15} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.79a1.1855 $x^{16} + 32 x^{11} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.79a1.1856 $x^{16} + 32 x^{15} + 32 x^{11} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.79a1.1857 $x^{16} + 32 x^{9} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.79a1.1858 $x^{16} + 32 x^{15} + 32 x^{9} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.79a1.1859 $x^{16} + 32 x^{11} + 32 x^{9} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.79a1.1860 $x^{16} + 32 x^{15} + 32 x^{11} + 32 x^{9} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.79a1.6574 $x^{16} + 16 x^{14} + 8 x^{12} + 32 x^{7} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.79a1.6575 $x^{16} + 32 x^{15} + 16 x^{14} + 8 x^{12} + 32 x^{7} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.79a1.6576 $x^{16} + 16 x^{14} + 32 x^{13} + 8 x^{12} + 32 x^{11} + 32 x^{7} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.79a1.6577 $x^{16} + 32 x^{15} + 16 x^{14} + 32 x^{13} + 8 x^{12} + 32 x^{11} + 32 x^{7} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.79a1.6578 $x^{16} + 16 x^{14} + 8 x^{12} + 32 x^{9} + 32 x^{7} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.79a1.6579 $x^{16} + 32 x^{15} + 16 x^{14} + 8 x^{12} + 32 x^{9} + 32 x^{7} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.79a1.6580 $x^{16} + 16 x^{14} + 32 x^{13} + 8 x^{12} + 32 x^{11} + 32 x^{9} + 32 x^{7} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
2.1.16.79a1.6581 $x^{16} + 32 x^{15} + 16 x^{14} + 32 x^{13} + 8 x^{12} + 32 x^{11} + 32 x^{9} + 32 x^{7} + 16 x^{6} + 16 x^{2} + 32 x + 2$ $\OD_{16}^2:D_4$ (as 16T1466) $2048$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5, \frac{21}{4}, 6]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4,\frac{17}{4},5]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{21}{4}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4},\frac{17}{4}]^{2}$ $[64, 48, 32, 16, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15, 31]$
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