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Properties

Label 2.1.2.3a1.2-1.4.19a
Base 2.1.2.3a1.2
Degree \(4\)
e \(4\)
f \(1\)
c \(19\)

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Defining polynomial

$x^{4} + \left(b_{23} \pi^{6} + b_{19} \pi^{5}\right) x^{3} + \left(b_{14} \pi^{4} + b_{10} \pi^{3}\right) x^{2} + \left(b_{21} \pi^{6} + b_{17} \pi^{5}\right) x + c_{24} \pi^{7} + c_{16} \pi^{5} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: $\Q_{2}(\sqrt{-2\cdot 5})$
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $19$
Absolute Artin slopes: $[3,4,5]$
Swan slopes: $[4,6]$
Means: $\langle2,4\rangle$
Rams: $(4,8)$
Field count: $72$ (complete)
Ambiguity: $4$
Mass: $64$
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: $D_{8}$ (show 2), $QD_{16}$ (show 6), $Z_8 : Z_8^\times$ (show 8), $C_4\wr C_2$ (show 16), $(C_4^2 : C_2):C_2$ (show 8), $C_2 \wr C_2\wr C_2$ (show 32)
Hidden Artin slopes: $[2]$ (show 4), $[2]^{2}$ (show 8), $[2,\frac{7}{2}]$ (show 16), $[2,\frac{7}{2},\frac{17}{4}]^{2}$ (show 32), $[\ ]^{2}$ (show 4), $[2,\frac{7}{2}]^{2}$ (show 8)
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 32

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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
2.1.8.31a1.78 $x^{8} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.79 $x^{8} + 16 x^{7} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.80 $x^{8} + 16 x^{5} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.81 $x^{8} + 16 x^{7} + 16 x^{5} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.82 $x^{8} + 16 x^{3} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.83 $x^{8} + 16 x^{7} + 16 x^{3} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.89 $x^{8} + 16 x + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.90 $x^{8} + 16 x^{7} + 16 x + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.91 $x^{8} + 16 x^{5} + 16 x + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.92 $x^{8} + 16 x^{7} + 16 x^{5} + 16 x + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.93 $x^{8} + 16 x^{3} + 16 x + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.94 $x^{8} + 16 x^{7} + 16 x^{3} + 16 x + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.102 $x^{8} + 8 x^{6} + 16 x^{3} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.103 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{3} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.111 $x^{8} + 8 x^{6} + 16 x^{3} + 16 x + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.112 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{3} + 16 x + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.113 $x^{8} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.114 $x^{8} + 8 x^{2} + 42$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.115 $x^{8} + 16 x^{7} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.116 $x^{8} + 16 x^{7} + 8 x^{2} + 42$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.117 $x^{8} + 16 x^{3} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.118 $x^{8} + 16 x^{7} + 16 x^{3} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.119 $x^{8} + 16 x^{5} + 16 x^{3} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.120 $x^{8} + 16 x^{7} + 16 x^{5} + 16 x^{3} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.129 $x^{8} + 8 x^{6} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.130 $x^{8} + 8 x^{6} + 8 x^{2} + 42$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.131 $x^{8} + 16 x^{7} + 8 x^{6} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.132 $x^{8} + 16 x^{7} + 8 x^{6} + 8 x^{2} + 42$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.133 $x^{8} + 8 x^{6} + 16 x^{3} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.134 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{3} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.135 $x^{8} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.31a1.136 $x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
  displayed columns for results