Base \(\Q_{19}\)
Degree \(12\)
e \(4\)
f \(3\)
c \(9\)
Galois group $D_4 \times C_3$ (as 12T14)

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Defining polynomial

\(x^{12} + 1444 x^{4} - 116603\) Copy content Toggle raw display


Base field: $\Q_{19}$
Degree $d$: $12$
Ramification exponent $e$: $4$
Residue field degree $f$: $3$
Discriminant exponent $c$: $9$
Discriminant root field: $\Q_{19}(\sqrt{19\cdot 2})$
Root number: $-i$
$\card{ \Aut(K/\Q_{ 19 }) }$: $6$
This field is not Galois over $\Q_{19}.$
Visible slopes:None

Intermediate fields


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

Unramified/totally ramified tower

Unramified subfield: $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{3} + 4 x + 17 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{4} + 19 t \) $\ \in\Q_{19}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_3\times D_4$ (as 12T14)
Inertia group:Intransitive group isomorphic to $C_4$
Wild inertia group:$C_1$
Unramified degree:$6$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:Not computed