Properties

Label 17.12.11.1
Base \(\Q_{17}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(11\)
Galois group $S_3 \times C_4$ (as 12T11)

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Defining polynomial

\(x^{12} + 17\) Copy content Toggle raw display

Invariants

Base field: $\Q_{17}$
Degree $d$: $12$
Ramification exponent $e$: $12$
Residue field degree $f$: $1$
Discriminant exponent $c$: $11$
Discriminant root field: $\Q_{17}(\sqrt{17})$
Root number: $-1$
$\card{ \Aut(K/\Q_{ 17 }) }$: $4$
This field is not Galois over $\Q_{17}.$
Visible slopes:None

Intermediate fields

$\Q_{17}(\sqrt{17})$, 17.3.2.1, 17.4.3.1, 17.6.5.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{17}$
Relative Eisenstein polynomial: \( x^{12} + 17 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{11} + 12z^{10} + 15z^{9} + 16z^{8} + 2z^{7} + 10z^{6} + 6z^{5} + 10z^{4} + 2z^{3} + 16z^{2} + 15z + 12$
Associated inertia:$2$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_4\times S_3$ (as 12T11)
Inertia group:$C_{12}$ (as 12T1)
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$12$
Wild slopes:None
Galois mean slope:$11/12$
Galois splitting model:Not computed