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Properties

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

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Defining polynomial

$x^{4} + b_{11} \pi^{3} x^{3} + \left(b_{10} \pi^{3} + b_{6} \pi^{2}\right) x^{2} + a_{9} \pi^{3} x + c_{12} \pi^{4} + \pi$

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: $[3,\frac{7}{2},\frac{7}{2}]$
Swan slopes: $[3,3]$
Means: $\langle\frac{3}{2},\frac{9}{4}\rangle$
Rams: $(3,3)$
Field count: $6$ (complete)
Ambiguity: $2$
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: $(((C_4 \times C_2): C_2):C_2):C_2$ (show 4), $C_2^4:C_6$ (show 2)
Hidden Artin slopes: $[2,2]^{2}$ (show 4), $[2,2]^{3}$ (show 2)
Indices of inseparability: $[17,14,8,0]$ (show 2), $[17,16,8,0]$ (show 4)
Associated inertia: $[1,2]$ (show 4), $[1,3]$ (show 2)
Jump Set: $[1,3,7,15]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 6

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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
2.1.8.24b1.9 $x^{8} + 4 x^{4} + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{2}$ $[2,2]^{2}$ $[1,1]^{2}$ $[17, 16, 8, 0]$ $[1, 2]$ $z^4 + 1,z^3 + 1$ $[1, 3, 7, 15]$
2.1.8.24b1.10 $x^{8} + 12 x^{4} + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{2}$ $[2,2]^{2}$ $[1,1]^{2}$ $[17, 16, 8, 0]$ $[1, 2]$ $z^4 + 1,z^3 + 1$ $[1, 3, 7, 15]$
2.1.8.24b1.11 $x^{8} + 4 x^{4} + 8 x^{2} + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{2}$ $[2,2]^{2}$ $[1,1]^{2}$ $[17, 16, 8, 0]$ $[1, 2]$ $z^4 + 1,z^3 + 1$ $[1, 3, 7, 15]$
2.1.8.24b1.12 $x^{8} + 12 x^{4} + 8 x^{2} + 8 x + 2$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $64$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{2}$ $[2,2]^{2}$ $[1,1]^{2}$ $[17, 16, 8, 0]$ $[1, 2]$ $z^4 + 1,z^3 + 1$ $[1, 3, 7, 15]$
2.1.8.24b2.1 $x^{8} + 4 x^{6} + 8 x + 2$ $C_2^4:C_6$ (as 8T33) $96$ $1$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2]^{3}$ $[1,1]^{3}$ $[17, 14, 8, 0]$ $[1, 3]$ $z^4 + 1,z^3 + z + 1$ $[1, 3, 7, 15]$
2.1.8.24b2.2 $x^{8} + 4 x^{6} + 8 x^{3} + 8 x + 2$ $C_2^4:C_6$ (as 8T33) $96$ $1$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2]^{3}$ $[1,1]^{3}$ $[17, 14, 8, 0]$ $[1, 3]$ $z^4 + 1,z^3 + z + 1$ $[1, 3, 7, 15]$
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