Properties

 Label 89.4.0.1 Base $$\Q_{89}$$ Degree $$4$$ e $$1$$ f $$4$$ c $$0$$ Galois group $C_4$ (as 4T1)

Related objects

Defining polynomial

 $$x^{4} + 4 x^{2} + 72 x + 3$$ x^4 + 4*x^2 + 72*x + 3

Invariants

 Base field: $\Q_{89}$ Degree $d$: $4$ Ramification exponent $e$: $1$ Residue field degree $f$: $4$ Discriminant exponent $c$: $0$ Discriminant root field: $\Q_{89}(\sqrt{3})$ Root number: $1$ $\card{ \Gal(K/\Q_{ 89 }) }$: $4$ This field is Galois and abelian over $\Q_{89}.$ Visible slopes: None

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: 89.4.0.1 $\cong \Q_{89}(t)$ where $t$ is a root of $$x^{4} + 4 x^{2} + 72 x + 3$$ x^4 + 4*x^2 + 72*x + 3 Relative Eisenstein polynomial: $$x - 89$$ x - 89 $\ \in\Q_{89}(t)[x]$

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

 Galois group: $C_4$ (as 4T1) Inertia group: trivial Wild inertia group: $C_1$ Unramified degree: $4$ Tame degree: $1$ Wild slopes: None Galois mean slope: $0$ Galois splitting model: $x^{4} - x^{3} + 2 x^{2} + 4 x + 3$