Properties

Label 41.2.1.2
Base \(\Q_{41}\)
Degree \(2\)
e \(2\)
f \(1\)
c \(1\)
Galois group $C_2$ (as 2T1)

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Defining polynomial

\(x^{2} + 123\) Copy content Toggle raw display

Invariants

Base field: $\Q_{41}$
Degree $d$: $2$
Ramification exponent $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $1$
Discriminant root field: $\Q_{41}(\sqrt{41\cdot 3})$
Root number: $-1$
$\card{ \Gal(K/\Q_{ 41 }) }$: $2$
This field is Galois and abelian over $\Q_{41}.$
Visible slopes:None

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 41 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{41}$
Relative Eisenstein polynomial: \( x^{2} + 123 \) Copy content Toggle raw display

Ramification polygon

Data not computed

Invariants of the Galois closure

Galois group: $C_2$ (as 2T1)
Inertia group: $C_2$ (as 2T1)
Wild inertia group: $C_1$
Unramified degree: $1$
Tame degree: $2$
Wild slopes: None
Galois mean slope: $1/2$
Galois splitting model:$x^{2} + 246$