Properties

Label 151.8.4.1
Base \(\Q_{151}\)
Degree \(8\)
e \(2\)
f \(4\)
c \(4\)
Galois group $C_4\times C_2$ (as 8T2)

Related objects

Learn more about

Defining polynomial

\(x^{8} + 273612 x^{4} - 3442951 x^{2} + 18715881636\)  Toggle raw display

Invariants

Base field: $\Q_{151}$
Degree $d$: $8$
Ramification exponent $e$: $2$
Residue field degree $f$: $4$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{151}$
Root number: $-1$
$|\Gal(K/\Q_{ 151 })|$: $8$
This field is Galois and abelian over $\Q_{151}.$

Intermediate fields

$\Q_{151}(\sqrt{3})$, $\Q_{151}(\sqrt{151})$, $\Q_{151}(\sqrt{151\cdot 3})$, 151.4.0.1, 151.4.2.1, 151.4.2.2

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

Unramified/totally ramified tower

Unramified subfield:151.4.0.1 $\cong \Q_{151}(t)$ where $t$ is a root of \( x^{4} - x + 6 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} - 151 t^{2} \)$\ \in\Q_{151}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2\times C_4$ (as 8T2)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$4$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:Not computed