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

Related objects

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Defining polynomial

\( x^{12} - 361 x^{4} + 27436 \)


Base field: $\Q_{19}$
Degree $d$ : $12$
Ramification exponent $e$ : $4$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $9$
Discriminant root field: $\Q_{19}(\sqrt{19})$
Root number: $i$
$|\Aut(K/\Q_{ 19 })|$: $6$
This field is not Galois over $\Q_{19}$.

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 \( x^{3} - x + 4 \)
Relative Eisenstein polynomial:$ x^{4} - 19 t \in\Q_{19}(t)[x]$

Invariants of the Galois closure

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