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Properties

Label 2.1.2.3a1.3-1.4.12b
Base 2.1.2.3a1.3
Degree \(4\)
e \(4\)
f \(1\)
c \(12\)

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Defining polynomial

$x^{4} + \left(b_{15} \pi^{4} + b_{11} \pi^{3}\right) x^{3} + a_{2} \pi x^{2} + \left(b_{13} \pi^{4} + a_{9} \pi^{3}\right) x + c_{16} \pi^{5} + c_{4} \pi^{2} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: $\Q_{2}(\sqrt{2})$
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Absolute Artin slopes: $[2,3,4]$
Swan slopes: $[1,4]$
Means: $\langle\frac{1}{2},\frac{9}{4}\rangle$
Rams: $(1,7)$
Field count: $20$ (complete)
Ambiguity: $4$
Mass: $8$
Absolute Mass: $4$

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), $D_4$ (show 2), $Q_8$ (show 2), $D_4\times C_2$ (show 4), $Q_8:C_2$ (show 8)
Hidden Artin slopes: $[\ ]$ (show 8), $[\ ]^{2}$ (show 12)
Indices of inseparability: $[17,10,4,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,2,4,8]$ (show 1), $[1,2,4,16]$ (show 10), $[1,11,19,27]$ (show 5), $[1,13,21,29]$ (show 2), $[1,15,23,31]$ (show 1), $[1,16,24,32]$ (show 1)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 20

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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
2.1.8.24c1.1 $x^{8} + 2 x^{4} + 4 x^{2} + 8 x + 2$ $C_4\times C_2$ (as 8T2) $8$ $8$ $[2, 3, 4]$ $[1,2,3]$ $[\ ]$ $[\ ]$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 8]$
2.1.8.24c1.2 $x^{8} + 2 x^{4} + 4 x^{2} + 8 x + 18$ $C_4\times C_2$ (as 8T2) $8$ $8$ $[2, 3, 4]$ $[1,2,3]$ $[\ ]$ $[\ ]$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 16, 24, 32]$
2.1.8.24c1.3 $x^{8} + 8 x^{7} + 2 x^{4} + 4 x^{2} + 8 x + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 15, 23, 31]$
2.1.8.24c1.4 $x^{8} + 8 x^{5} + 2 x^{4} + 4 x^{2} + 8 x + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 13, 21, 29]$
2.1.8.24c1.5 $x^{8} + 8 x^{7} + 8 x^{5} + 2 x^{4} + 4 x^{2} + 8 x + 2$ $D_4\times C_2$ (as 8T9) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 13, 21, 29]$
2.1.8.24c1.6 $x^{8} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 11, 19, 27]$
2.1.8.24c1.7 $x^{8} + 8 x^{7} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 11, 19, 27]$
2.1.8.24c1.8 $x^{8} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 2$ $Q_8:C_2$ (as 8T11) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 11, 19, 27]$
2.1.8.24c1.9 $x^{8} + 8 x^{7} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 2$ $D_4$ (as 8T4) $8$ $8$ $[2, 3, 4]$ $[1,2,3]$ $[\ ]$ $[\ ]$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 11, 19, 27]$
2.1.8.24c1.10 $x^{8} + 8 x^{7} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 18$ $D_4$ (as 8T4) $8$ $8$ $[2, 3, 4]$ $[1,2,3]$ $[\ ]$ $[\ ]$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 11, 19, 27]$
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) $16$ $4$ $[2, 3, 4]^{2}$ $[1,2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[17, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 2, 4, 16]$
  displayed columns for results