Properties

Label 151.4.2.1
Base \(\Q_{151}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(2\)
Galois group $C_2^2$ (as 4T2)

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Defining polynomial

\(x^{4} + 298 x^{3} + 22515 x^{2} + 46786 x + 3373376\) Copy content Toggle raw display

Invariants

Base field: $\Q_{151}$
Degree $d$: $4$
Ramification exponent $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $2$
Discriminant root field: $\Q_{151}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 151 }) }$: $4$
This field is Galois and abelian over $\Q_{151}.$
Visible slopes:None

Intermediate fields

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

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{151}(\sqrt{3})$ $\cong \Q_{151}(t)$ where $t$ is a root of \( x^{2} + 149 x + 6 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + 151 \) $\ \in\Q_{151}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Data not computed

Invariants of the Galois closure

Galois group: $C_2^2$ (as 4T2)
Inertia group: Intransitive group isomorphic to $C_2$
Wild inertia group: $C_1$
Unramified degree: $2$
Tame degree: $2$
Wild slopes: None
Galois mean slope: $1/2$
Galois splitting model: $x^{4} + 3473 x^{2} + 3283344$ Copy content Toggle raw display