Properties

Label 11.10.9.8
Base \(\Q_{11}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(9\)
Galois group $C_{10}$ (as 10T1)

Related objects

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Defining polynomial

\( x^{10} + 33 \)

Invariants

Base field: $\Q_{11}$
Degree $d$ : $10$
Ramification exponent $e$ : $10$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $9$
Discriminant root field: $\Q_{11}(\sqrt{11*})$
Root number: $i$
$|\Gal(K/\Q_{ 11 })|$: $10$
This field is Galois and abelian over $\Q_{11}$.

Intermediate fields

$\Q_{11}(\sqrt{11*})$, 11.5.4.3

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{11}$
Relative Eisenstein polynomial:\( x^{10} + 33 \)

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:$C_{10}$
Unramified degree:$1$
Tame degree:$10$
Wild slopes:None
Galois mean slope:$9/10$
Galois splitting model:$x^{10} - 385 x^{7} - 330 x^{6} + 3344 x^{5} + 47300 x^{4} + 107580 x^{3} + 138050 x^{2} + 69740 x + 13343$