# Properties

 Label 2.8.12.14 Base $$\Q_{2}$$ Degree $$8$$ e $$4$$ f $$2$$ c $$12$$ Galois group $D_4$ (as 8T4)

# Related objects

## Defining polynomial

 $$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 95 x^{4} + 130 x^{3} + 58 x^{2} - 58 x + 13$$ x^8 + 8*x^7 + 28*x^6 + 58*x^5 + 95*x^4 + 130*x^3 + 58*x^2 - 58*x + 13

## Invariants

 Base field: $\Q_{2}$ Degree $d$: $8$ Ramification exponent $e$: $4$ Residue field degree $f$: $2$ Discriminant exponent $c$: $12$ Discriminant root field: $\Q_{2}$ Root number: $1$ $\card{ \Gal(K/\Q_{ 2 }) }$: $8$ This field is Galois over $\Q_{2}.$ Visible slopes: $[2, 2]$

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Unramified/totally ramified tower

 Unramified subfield: $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of $$x^{2} + x + 1$$ x^2 + x + 1 Relative Eisenstein polynomial: $$x^{4} + 2 x^{3} + 6$$ x^4 + 2*x^3 + 6 $\ \in\Q_{2}(t)[x]$

## Ramification polygon

 Residual polynomials: $z^{3} + 1$ Associated inertia: $1$ Indices of inseparability: $[3, 3, 0]$

## Invariants of the Galois closure

 Galois group: $D_4$ (as 8T4) Inertia group: Intransitive group isomorphic to $C_2^2$ Wild inertia group: $C_2^2$ Unramified degree: $2$ Tame degree: $1$ Wild slopes: $[2, 2]$ Galois mean slope: $3/2$ Galois splitting model: $x^{8} + 12 x^{4} + 144$ x^8 + 12*x^4 + 144