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Properties

Label 2.1.4.11a1.12-1.4.10b
Base 2.1.4.11a1.12
Degree \(4\)
e \(4\)
f \(1\)
c \(10\)

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Defining polynomial

$x^{4} + \left(b_{11} \pi^{3} + a_{7} \pi^{2}\right) x^{3} + a_{2} \pi x^{2} + b_{9} \pi^{3} x + c_{12} \pi^{4} + c_{4} \pi^{2} + \pi$

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 $c$: $10$
Absolute Artin slopes: $[2,3,\frac{7}{2},4]$
Swan slopes: $[1,3]$
Means: $\langle\frac{1}{2},\frac{7}{4}\rangle$
Rams: $(1,5)$
Field count: $10$ (complete)
Ambiguity: $4$
Mass: $4$
Absolute Mass: $1$

Diagrams

Varying

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

Galois group: $C_4:C_4$ (show 2), $C_2^2 : C_4$ (show 2), $C_4^2:C_2$ (show 1), $C_4 \times D_4$ (show 4), $C_2 \times (C_2^2:C_4)$ (show 1)
Hidden Artin slopes: $[\ ]$ (show 4), $[\ ]^{2}$ (show 6)
Indices of inseparability: $[39,30,20,8,0]$
Associated inertia: $[1,1,1,1]$
Jump Set: $[1,2,4,8,32]$ (show 5), $[1,7,32,48,64]$ (show 5)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 10

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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
2.1.16.54o1.105 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{4} + 18$ $C_4^2:C_2$ (as 16T17) $32$ $8$ $[2, 3, \frac{7}{2}, 4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 32, 48, 64]$
2.1.16.54o1.121 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 18$ $C_4 \times D_4$ (as 16T19) $32$ $8$ $[2, 3, \frac{7}{2}, 4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 32, 48, 64]$
2.1.16.54o1.127 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 2 x^{8} + 8 x^{7} + 4 x^{4} + 8 x^{2} + 18$ $C_4 \times D_4$ (as 16T19) $32$ $8$ $[2, 3, \frac{7}{2}, 4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 32, 48, 64]$
2.1.16.54o1.143 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 18$ $C_2^2 : C_4$ (as 16T10) $16$ $16$ $[2, 3, \frac{7}{2}, 4]$ $[1,2,\frac{5}{2},3]$ $[\ ]$ $[\ ]$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 32, 48, 64]$
2.1.16.54o1.152 $x^{16} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 10 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 18$ $C_2^2 : C_4$ (as 16T10) $16$ $16$ $[2, 3, \frac{7}{2}, 4]$ $[1,2,\frac{5}{2},3]$ $[\ ]$ $[\ ]$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 32, 48, 64]$
2.1.16.54o1.287 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{4} + 14$ $C_4 \times D_4$ (as 16T19) $32$ $8$ $[2, 3, \frac{7}{2}, 4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.297 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 14$ $C_4:C_4$ (as 16T8) $16$ $16$ $[2, 3, \frac{7}{2}, 4]$ $[1,2,\frac{5}{2},3]$ $[\ ]$ $[\ ]$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.306 $x^{16} + 4 x^{14} + 8 x^{13} + 8 x^{9} + 10 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 14$ $C_4:C_4$ (as 16T8) $16$ $16$ $[2, 3, \frac{7}{2}, 4]$ $[1,2,\frac{5}{2},3]$ $[\ ]$ $[\ ]$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.318 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{11} + 2 x^{8} + 8 x^{7} + 4 x^{4} + 8 x^{2} + 14$ $C_2 \times (C_2^2:C_4)$ (as 16T21) $32$ $8$ $[2, 3, \frac{7}{2}, 4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.328 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 14$ $C_4 \times D_4$ (as 16T19) $32$ $8$ $[2, 3, \frac{7}{2}, 4]^{2}$ $[1,2,\frac{5}{2},3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
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