↑

Properties

Label 2.1.2.3a1.2-4.2.8a
Base 2.1.2.3a1.2
Degree \(8\)
e \(2\)
f \(4\)
c \(8\)

Related objects

Downloads

Learn more

Defining polynomial over unramified subextension

$x^{2} + a_{1} \pi x + c_{2} \pi^{2} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $8$
Base field: $\Q_{2}(\sqrt{-2\cdot 5})$
Ramification index $e$: $2$
Residue field degree $f$: $4$
Discriminant exponent $c$: $8$
Absolute Artin slopes: $[2,3]$
Swan slopes: $[1]$
Means: $\langle\frac{1}{2}\rangle$
Rams: $(1)$
Field count: $10$ (complete)
Ambiguity: $8$
Mass: $15$
Absolute Mass: $15/8$

Diagrams

Varying

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

Galois group: $C_4\times C_2^2$ (show 1), $C_8\times C_2$ (show 1), $C_2 \times (C_8:C_2)$ (show 1), $C_2 \times (C_2^2:C_4)$ (show 1), $C_2^4:C_4$ (show 1), $\OD_{16}:C_2^2$ (show 1), $C_2^5:C_4$ (show 4)
Hidden Artin slopes: $[2,2,2]$ (show 4), $[\ ]$ (show 2), $[2]$ (show 2), $[2,2]$ (show 2)
Indices of inseparability: $[5,2,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,2,4]$ (show 1), $[1,2,8]$ (show 1), $[1,3,7]$ (show 8)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
Sort order Select

Showing all 10

  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
2.4.4.32b1.1 $( x^{4} + x + 1 )^{4} + 2 ( x^{4} + x + 1 )^{2} + 4 ( x^{4} + x + 1 ) + 2$ $C_4\times C_2^2$ (as 16T2) $16$ $16$ $[2, 3]^{4}$ $[1,2]^{4}$ $[\ ]$ $[\ ]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4]$
2.4.4.32b1.15 $( x^{4} + x + 1 )^{4} + \left(4 x^{3} + 4 x\right) ( x^{4} + x + 1 )^{3} + 2 ( x^{4} + x + 1 )^{2} + \left(4 x^{2} + 4\right) ( x^{4} + x + 1 ) + 12 x^{3} + 2$ $C_8\times C_2$ (as 16T5) $16$ $16$ $[2, 3]^{4}$ $[1,2]^{4}$ $[\ ]$ $[\ ]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 8]$
2.4.4.32b10.10 $( x^{4} + x + 1 )^{4} + \left(4 x^{2} + 4 x + 2\right) ( x^{4} + x + 1 )^{3} + 2 x ( x^{4} + x + 1 )^{2} + \left(4 x^{2} + 4\right) ( x^{4} + x + 1 ) + 2$ $C_2^5:C_4$ (as 16T261) $128$ $4$ $[2, 2, 2, 2, 3]^{4}$ $[1,1,1,1,2]^{4}$ $[2,2,2]$ $[1,1,1]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + t,t z + (t^2 + 1)$ $[1, 3, 7]$
2.4.4.32b10.46 $( x^{4} + x + 1 )^{4} + \left(4 x^{2} + 4 x + 6\right) ( x^{4} + x + 1 )^{3} + 2 x ( x^{4} + x + 1 )^{2} + \left(4 x^{2} + 4\right) ( x^{4} + x + 1 ) + 6$ $C_2^5:C_4$ (as 16T261) $128$ $4$ $[2, 2, 2, 2, 3]^{4}$ $[1,1,1,1,2]^{4}$ $[2,2,2]$ $[1,1,1]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + t,t z + (t^2 + 1)$ $[1, 3, 7]$
2.4.4.32b25.4 $( x^{4} + x + 1 )^{4} + \left(4 x^{3} + 4 x^{2} + 2\right) ( x^{4} + x + 1 )^{3} + \left(2 x^{2} + 2 x\right) ( x^{4} + x + 1 )^{2} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 ) + 2$ $C_2 \times (C_2^2:C_4)$ (as 16T21) $32$ $8$ $[2, 2, 3]^{4}$ $[1,1,2]^{4}$ $[2]$ $[1]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t),(t^2 + t) z + (t^2 + t + 1)$ $[1, 3, 7]$
2.4.4.32b25.29 $( x^{4} + x + 1 )^{4} + \left(4 x^{3} + 4 x^{2} + 4 x + 6\right) ( x^{4} + x + 1 )^{3} + \left(2 x^{2} + 2 x\right) ( x^{4} + x + 1 )^{2} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 ) + 4 x + 2$ $C_2 \times (C_8:C_2)$ (as 16T15) $32$ $8$ $[2, 2, 3]^{4}$ $[1,1,2]^{4}$ $[2]$ $[1]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^2 + t),(t^2 + t) z + (t^2 + t + 1)$ $[1, 3, 7]$
2.4.4.32b39.8 $( x^{4} + x + 1 )^{4} + \left(4 x^{3} + 2 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{3} + 2 x^{3} ( x^{4} + x + 1 )^{2} + \left(4 x^{3} + 4 x\right) ( x^{4} + x + 1 ) + 2$ $C_2^5:C_4$ (as 16T261) $128$ $4$ $[2, 2, 2, 2, 3]^{4}$ $[1,1,1,1,2]^{4}$ $[2,2,2]$ $[1,1,1]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + t^3,t^3 z + (t^3 + t)$ $[1, 3, 7]$
2.4.4.32b39.44 $( x^{4} + x + 1 )^{4} + \left(4 x^{3} + 2 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 )^{3} + 2 x^{3} ( x^{4} + x + 1 )^{2} + \left(4 x^{3} + 4 x\right) ( x^{4} + x + 1 ) + 6$ $C_2^5:C_4$ (as 16T261) $128$ $4$ $[2, 2, 2, 2, 3]^{4}$ $[1,1,1,1,2]^{4}$ $[2,2,2]$ $[1,1,1]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + t^3,t^3 z + (t^3 + t)$ $[1, 3, 7]$
2.4.4.32b58.16 $( x^{4} + x + 1 )^{4} + \left(6 x^{2} + 6\right) ( x^{4} + x + 1 )^{3} + \left(2 x^{3} + 2 x + 2\right) ( x^{4} + x + 1 )^{2} + \left(4 x^{3} + 4 x^{2} + 4 x\right) ( x^{4} + x + 1 ) + 2$ $C_2^4:C_4$ (as 16T94) $64$ $4$ $[2, 2, 2, 3]^{4}$ $[1,1,1,2]^{4}$ $[2,2]$ $[1,1]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^3 + t + 1),(t^3 + t + 1) z + (t^3 + t^2 + t)$ $[1, 3, 7]$
2.4.4.32b58.37 $( x^{4} + x + 1 )^{4} + \left(2 x^{2} + 6\right) ( x^{4} + x + 1 )^{3} + \left(2 x^{3} + 2 x + 2\right) ( x^{4} + x + 1 )^{2} + \left(4 x^{3} + 4 x^{2} + 4 x\right) ( x^{4} + x + 1 ) + 4 x^{2} + 2$ $\OD_{16}:C_2^2$ (as 16T99) $64$ $4$ $[2, 2, 2, 3]^{4}$ $[1,1,1,2]^{4}$ $[2,2]$ $[1,1]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + (t^3 + t + 1),(t^3 + t + 1) z + (t^3 + t^2 + t)$ $[1, 3, 7]$
  displayed columns for results