Base \(\Q_{17}\)
Degree \(15\)
e \(3\)
f \(5\)
c \(10\)
Galois group $S_3 \times C_5$ (as 15T4)

Related objects

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Defining polynomial

\( x^{15} - 83521 x^{3} + 8519142 \)


Base field: $\Q_{17}$
Degree $d$: $15$
Ramification exponent $e$: $3$
Residue field degree $f$: $5$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{17}(\sqrt{3})$
Root number: $1$
$|\Aut(K/\Q_{ 17 })|$: $5$
This field is not Galois over $\Q_{17}.$

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: $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{5} - x + 6 \)
Relative Eisenstein polynomial:$ x^{3} - 17 t \in\Q_{17}(t)[x]$

Invariants of the Galois closure

Galois group:$C_5\times S_3$ (as 15T4)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$10$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:Not computed