Properties

Label 11.15.14.3
Base \(\Q_{11}\)
Degree \(15\)
e \(15\)
f \(1\)
c \(14\)
Galois group $S_3 \times C_5$ (as 15T4)

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Defining polynomial

\(x^{15} + 22\) Copy content Toggle raw display

Invariants

Base field: $\Q_{11}$
Degree $d$: $15$
Ramification exponent $e$: $15$
Residue field degree $f$: $1$
Discriminant exponent $c$: $14$
Discriminant root field: $\Q_{11}(\sqrt{2})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 11 }) }$: $5$
This field is not Galois over $\Q_{11}.$
Visible slopes:None

Intermediate fields

11.3.2.1, 11.5.4.5

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: \( x^{15} + 22 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{14} + 4z^{13} + 6z^{12} + 4z^{11} + z^{10} + z^{3} + 4z^{2} + 6z + 4$
Associated inertia:$2$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_5\times S_3$ (as 15T4)
Inertia group:$C_{15}$ (as 15T1)
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$15$
Wild slopes:None
Galois mean slope:$14/15$
Galois splitting model:Not computed