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Properties

Label 2.1.2.2a1.2-1.4.16a
Base 2.1.2.2a1.2
Degree \(4\)
e \(4\)
f \(1\)
c \(16\)

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Defining polynomial

$x^{4} + \left(b_{19} \pi^{5} + b_{15} \pi^{4}\right) x^{3} + \left(b_{10} \pi^{3} + a_{6} \pi^{2}\right) x^{2} + \left(b_{17} \pi^{5} + a_{13} \pi^{4}\right) x + c_{20} \pi^{6} + c_{12} \pi^{4} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: $\Q_{2}(\sqrt{-5})$
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $16$
Absolute Artin slopes: $[2,3,4]$
Swan slopes: $[3,5]$
Means: $\langle\frac{3}{2},\frac{13}{4}\rangle$
Rams: $(3,7)$
Field count: $32$ (complete)
Ambiguity: $4$
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: $C_4\times C_2$ (show 4), $Q_8$ (show 4), $QD_{16}$ (show 2), $D_4\times C_2$ (show 6), $Q_8:C_2$ (show 6), $Z_8 : Z_8^\times$ (show 2), $C_4\wr C_2$ (show 8)
Hidden Artin slopes: $[\ ]$ (show 8), $[\ ]^{2}$ (show 14), $[\ ]^{4}$ (show 4), $[2]^{2}$ (show 6)
Indices of inseparability: $[17,10,4,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,2,4,16]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 32

  displayed columns for results
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
2.1.8.24c1.33 $x^{8} + 2 x^{4} + 4 x^{2} + 8 x + 6$ $Z_8 : Z_8^\times$ (as 8T15) $32$ $2$ $[2, 2, 3, 4]^{2}$ $[1,1,2,3]^{2}$ $[2]^{2}$ $[1]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.34 $x^{8} + 8 x^{7} + 2 x^{4} + 4 x^{2} + 8 x + 6$ $Z_8 : Z_8^\times$ (as 8T15) $32$ $2$ $[2, 2, 3, 4]^{2}$ $[1,1,2,3]^{2}$ $[2]^{2}$ $[1]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.35 $x^{8} + 8 x^{5} + 2 x^{4} + 4 x^{2} + 8 x + 6$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 2, 3, 4]^{2}$ $[1,1,2,3]^{2}$ $[2]^{2}$ $[1]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.36 $x^{8} + 8 x^{5} + 2 x^{4} + 4 x^{2} + 8 x + 22$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 2, 3, 4]^{2}$ $[1,1,2,3]^{2}$ $[2]^{2}$ $[1]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.37 $x^{8} + 8 x^{7} + 8 x^{5} + 2 x^{4} + 4 x^{2} + 8 x + 6$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 2, 3, 4]^{2}$ $[1,1,2,3]^{2}$ $[2]^{2}$ $[1]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.38 $x^{8} + 8 x^{7} + 8 x^{5} + 2 x^{4} + 4 x^{2} + 8 x + 22$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 2, 3, 4]^{2}$ $[1,1,2,3]^{2}$ $[2]^{2}$ $[1]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.39 $x^{8} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 6$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[\ ]^{4}$ $[\ ]^{4}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.40 $x^{8} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 22$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[\ ]^{4}$ $[\ ]^{4}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.41 $x^{8} + 8 x^{7} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 6$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[\ ]^{4}$ $[\ ]^{4}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.42 $x^{8} + 8 x^{7} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 22$ $C_4\wr C_2$ (as 8T17) $32$ $4$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[\ ]^{4}$ $[\ ]^{4}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.43 $x^{8} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 6$ $QD_{16}$ (as 8T8) $16$ $2$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.44 $x^{8} + 8 x^{7} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 6$ $QD_{16}$ (as 8T8) $16$ $2$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.45 $x^{8} + 4 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 6$ $Q_8$ (as 8T5) $8$ $8$ $[2, 3, 4]$ $[1,2,3]$ $[\ ]$ $[\ ]$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.46 $x^{8} + 4 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 22$ $Q_8$ (as 8T5) $8$ $8$ $[2, 3, 4]$ $[1,2,3]$ $[\ ]$ $[\ ]$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.47 $x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 6$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.48 $x^{8} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 4 x^{2} + 8 x + 6$ $D_4\times C_2$ (as 8T9) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.49 $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 4 x^{2} + 8 x + 6$ $D_4\times C_2$ (as 8T9) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.50 $x^{8} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 6$ $C_4\times C_2$ (as 8T2) $8$ $8$ $[2, 3, 4]$ $[1,2,3]$ $[\ ]$ $[\ ]$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.51 $x^{8} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 22$ $C_4\times C_2$ (as 8T2) $8$ $8$ $[2, 3, 4]$ $[1,2,3]$ $[\ ]$ $[\ ]$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.52 $x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 6$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.53 $x^{8} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 6$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.54 $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 6$ $D_4\times C_2$ (as 8T9) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.55 $x^{8} + 4 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 14$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.56 $x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 14$ $Q_8$ (as 8T5) $8$ $8$ $[2, 3, 4]$ $[1,2,3]$ $[\ ]$ $[\ ]$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.57 $x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 30$ $Q_8$ (as 8T5) $8$ $8$ $[2, 3, 4]$ $[1,2,3]$ $[\ ]$ $[\ ]$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.58 $x^{8} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 4 x^{2} + 8 x + 14$ $D_4\times C_2$ (as 8T9) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.59 $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 4 x^{2} + 8 x + 14$ $D_4\times C_2$ (as 8T9) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.60 $x^{8} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.61 $x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$ $C_4\times C_2$ (as 8T2) $8$ $8$ $[2, 3, 4]$ $[1,2,3]$ $[\ ]$ $[\ ]$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.62 $x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 30$ $C_4\times C_2$ (as 8T2) $8$ $8$ $[2, 3, 4]$ $[1,2,3]$ $[\ ]$ $[\ ]$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.63 $x^{8} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$ $D_4\times C_2$ (as 8T9) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
2.1.8.24c1.64 $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
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