Base \(\Q_{19}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(11\)
Galois group $D_4 \times C_3$ (as 12T14)

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Defining polynomial

\(x^{12} + 95\) Copy content Toggle raw display


Base field: $\Q_{19}$
Degree $d$: $12$
Ramification exponent $e$: $12$
Residue field degree $f$: $1$
Discriminant exponent $c$: $11$
Discriminant root field: $\Q_{19}(\sqrt{19})$
Root number: $i$
$\card{ \Aut(K/\Q_{ 19 }) }$: $6$
This field is not Galois over $\Q_{19}.$
Visible slopes:None

Intermediate fields

$\Q_{19}(\sqrt{19\cdot 2})$,,,

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{19}$
Relative Eisenstein polynomial: \( x^{12} + 95 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{11} + 12z^{10} + 9z^{9} + 11z^{8} + z^{7} + 13z^{6} + 12z^{5} + 13z^{4} + z^{3} + 11z^{2} + 9z + 12$
Associated inertia:$2$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_3\times D_4$ (as 12T14)
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