Properties

Label 2.1.4.11a1.4-1.2.9a
Base 2.1.4.11a1.4
Degree \(2\)
e \(2\)
f \(1\)
c \(9\)

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Defining polynomial

$x^{2} + \left(b_{15} \pi^{8} + b_{13} \pi^{7} + b_{11} \pi^{6} + b_{9} \pi^{5}\right) x + c_{16} \pi^{9} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.1.4.11a1.4
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $9$
Absolute Artin slopes: $[3,4,5]$
Swan slopes: $[8]$
Means: $\langle4\rangle$
Rams: $(8)$
Field count: $16$ (complete)
Ambiguity: $2$
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: $QD_{16}$ (show 2), $Z_8 : Z_8^\times$ (show 2), $(C_4^2 : C_2):C_2$ (show 4), $C_2 \wr C_2\wr C_2$ (show 8)
Hidden Artin slopes: $[2,\frac{7}{2},\frac{17}{4}]^{2}$ (show 8), $[\ ]^{2}$ (show 2), $[2]^{2}$ (show 2), $[2,\frac{7}{2}]^{2}$ (show 4)
Indices of inseparability: $[24,16,8,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,3,7,15]$

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
2.1.8.31a1.57 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 8 x^{2} + 50$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.58 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 8 x^{2} + 18$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.59 $x^{8} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 8 x^{2} + 50$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.60 $x^{8} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 8 x^{2} + 18$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.61 $x^{8} + 8 x^{6} + 16 x^{5} + 8 x^{2} + 50$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.62 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 8 x^{2} + 50$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.63 $x^{8} + 8 x^{6} + 16 x^{5} + 8 x^{2} + 18$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.64 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 8 x^{2} + 18$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.65 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 8 x^{2} + 16 x + 18$ $(C_4^2 : C_2):C_2$ (as 8T26) $64$ $2$ $[2, 3, \frac{7}{2}, 4, 5]^{2}$ $[1,2,\frac{5}{2},3,4]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.66 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 8 x^{2} + 16 x + 50$ $(C_4^2 : C_2):C_2$ (as 8T26) $64$ $2$ $[2, 3, \frac{7}{2}, 4, 5]^{2}$ $[1,2,\frac{5}{2},3,4]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.67 $x^{8} + 16 x^{7} + 8 x^{6} + 8 x^{2} + 16 x + 50$ $(C_4^2 : C_2):C_2$ (as 8T26) $64$ $2$ $[2, 3, \frac{7}{2}, 4, 5]^{2}$ $[1,2,\frac{5}{2},3,4]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.68 $x^{8} + 16 x^{7} + 8 x^{6} + 8 x^{2} + 16 x + 18$ $(C_4^2 : C_2):C_2$ (as 8T26) $64$ $2$ $[2, 3, \frac{7}{2}, 4, 5]^{2}$ $[1,2,\frac{5}{2},3,4]^{2}$ $[2,\frac{7}{2}]^{2}$ $[1,\frac{5}{2}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.69 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{3} + 8 x^{2} + 16 x + 50$ $QD_{16}$ (as 8T8) $16$ $2$ $[3, 4, 5]^{2}$ $[2,3,4]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.70 $x^{8} + 8 x^{6} + 16 x^{3} + 8 x^{2} + 16 x + 50$ $QD_{16}$ (as 8T8) $16$ $2$ $[3, 4, 5]^{2}$ $[2,3,4]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.71 $x^{8} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 8 x^{2} + 16 x + 50$ $Z_8 : Z_8^\times$ (as 8T15) $32$ $2$ $[2, 3, 4, 5]^{2}$ $[1,2,3,4]^{2}$ $[2]^{2}$ $[1]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.72 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 8 x^{2} + 16 x + 50$ $Z_8 : Z_8^\times$ (as 8T15) $32$ $2$ $[2, 3, 4, 5]^{2}$ $[1,2,3,4]^{2}$ $[2]^{2}$ $[1]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
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