Properties

Label 11.14.13.2
Base \(\Q_{11}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(13\)
Galois group $(C_7:C_3) \times C_2$ (as 14T5)

Related objects

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Defining polynomial

\( x^{14} + 33 \)

Invariants

Base field: $\Q_{11}$
Degree $d$: $14$
Ramification exponent $e$: $14$
Residue field degree $f$: $1$
Discriminant exponent $c$: $13$
Discriminant root field: $\Q_{11}(\sqrt{11\cdot 2})$
Root number: $-i$
$|\Aut(K/\Q_{ 11 })|$: $2$
This field is not Galois over $\Q_{11}.$

Intermediate fields

$\Q_{11}(\sqrt{11\cdot 2})$, 11.7.6.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{11}$
Relative Eisenstein polynomial:\( x^{14} + 33 \)

Invariants of the Galois closure

Galois group:$C_2\times C_7:C_3$ (as 14T5)
Inertia group:$C_{14}$
Unramified degree:$3$
Tame degree:$14$
Wild slopes:None
Galois mean slope:$13/14$
Galois splitting model:Not computed