Properties

Label 2.1.2.3a1.2-1.8.30h
Base 2.1.2.3a1.2
Degree \(8\)
e \(8\)
f \(1\)
c \(30\)

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Defining polynomial

$x^{8} + \left(b_{31} \pi^{4} + a_{23} \pi^{3}\right) x^{7} + \left(b_{22} \pi^{3} + a_{14} \pi^{2}\right) x^{6} + b_{29} \pi^{4} x^{5} + a_{4} \pi x^{4} + b_{27} \pi^{4} x^{3} + b_{18} \pi^{3} x^{2} + b_{25} \pi^{4} x + c_{32} \pi^{5} + c_{24} \pi^{4} + c_{8} \pi^{2} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $8$
Base field: $\Q_{2}(\sqrt{-2\cdot 5})$
Ramification index $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $30$
Absolute Artin slopes: $[2,3,\frac{7}{2},4]$
Swan slopes: $[1,3,4]$
Means: $\langle\frac{1}{2},\frac{7}{4},\frac{23}{8}\rangle$
Rams: $(1,5,9)$
Field count: $120$ (complete)
Ambiguity: $8$
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: $C_4:C_4$ (show 4), $C_2^2 : C_4$ (show 4), $C_4^2:C_2$ (show 2), $C_4 \times D_4$ (show 8), $C_2 \times (C_2^2:C_4)$ (show 2), $C_4^2:C_2$ (show 2), $C_4^2:C_2$ (show 2), $C_2^2:Q_8$ (show 6), $C_4:D_4$ (show 4), $C_2^2.D_4$ (show 3), $C_2^2\wr C_2$ (show 4), $C_4:D_4$ (show 6), $C_2^2\wr C_2$ (show 1), $C_2^2.D_4$ (show 8), $C_2\wr C_2^2$ (show 4), $C_2\wr C_2^2$ (show 4), $C_2^3.D_4$ (show 4), $C_2^3.D_4$ (show 4), $C_2^4.D_4$ (show 4), $C_2^4.D_4$ (show 4), $C_4^2:D_4$ (show 4), $C_4^2.D_4$ (show 8), $C_4^2:D_4$ (show 8), $C_4^2:D_4$ (show 4), $C_2^4.(C_4\times D_4)$ (show 8), $C_2^4.(C_4\times D_4)$ (show 8)
Hidden Artin slopes: $[\ ]$ (show 8), $[\ ]^{2}$ (show 48), $[2]^{2}$ (show 16), $[2,\frac{7}{2}]^{2}$ (show 24), $[2,2,\frac{7}{2}]^{4}$ (show 16), $[2]^{4}$ (show 8)
Indices of inseparability: $[39,30,20,8,0]$
Associated inertia: $[1,1,1,1]$
Jump Set: $[1,2,4,8,32]$ (show 60), $[1,7,14,32,48]$ (show 60)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 8

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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
2.1.16.54o1.157 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 10 x^{8} + 8 x^{7} + 4 x^{4} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T896) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,\frac{7}{2}]^{4}$ $[1,1,\frac{5}{2}]^{4}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 14, 32, 48]$
2.1.16.54o1.158 $x^{16} + 8 x^{15} + 4 x^{14} + 4 x^{12} + 8 x^{11} + 10 x^{8} + 8 x^{7} + 4 x^{4} + 10$ $C_2^4.(C_4\times D_4)$ (as 16T896) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,\frac{7}{2}]^{4}$ $[1,1,\frac{5}{2}]^{4}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 14, 32, 48]$
2.1.16.54o1.159 $x^{16} + 8 x^{15} + 4 x^{14} + 4 x^{12} + 8 x^{11} + 10 x^{8} + 8 x^{7} + 4 x^{4} + 26$ $C_2^4.(C_4\times D_4)$ (as 16T896) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,\frac{7}{2}]^{4}$ $[1,1,\frac{5}{2}]^{4}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 14, 32, 48]$
2.1.16.54o1.160 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 10 x^{8} + 8 x^{7} + 4 x^{4} + 26$ $C_2^4.(C_4\times D_4)$ (as 16T896) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,\frac{7}{2}]^{4}$ $[1,1,\frac{5}{2}]^{4}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 14, 32, 48]$
2.1.16.54o1.332 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{11} + 8 x^{9} + 10 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 14$ $C_2^4.(C_4\times D_4)$ (as 16T896) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,\frac{7}{2}]^{4}$ $[1,1,\frac{5}{2}]^{4}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.333 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{11} + 8 x^{9} + 10 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 30$ $C_2^4.(C_4\times D_4)$ (as 16T896) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,\frac{7}{2}]^{4}$ $[1,1,\frac{5}{2}]^{4}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.334 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{11} + 8 x^{9} + 10 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 30$ $C_2^4.(C_4\times D_4)$ (as 16T896) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,\frac{7}{2}]^{4}$ $[1,1,\frac{5}{2}]^{4}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
2.1.16.54o1.335 $x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{11} + 8 x^{9} + 10 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 14$ $C_2^4.(C_4\times D_4)$ (as 16T896) $512$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$ $[2,2,\frac{7}{2}]^{4}$ $[1,1,\frac{5}{2}]^{4}$ $[39, 30, 20, 8, 0]$ $[1, 1, 1, 1]$ $z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8, 32]$
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