Defining polynomial
$( x^{2} + 18 x + 2 )^{3} + 190 ( x^{2} + 18 x + 2 )^{2} + \left(38 x + 39387\right) ( x^{2} + 18 x + 2 ) + 15618 x + 140923$
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Invariants
Base field: | $\Q_{19}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $4$ |
Discriminant root field: | $\Q_{19}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 19 }) }$: | $6$ |
This field is Galois and abelian over $\Q_{19}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{19}(\sqrt{2})$, 19.3.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{19}(\sqrt{2})$ $\cong \Q_{19}(t)$ where $t$ is a root of
\( x^{2} + 18 x + 2 \)
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Relative Eisenstein polynomial: |
\( x^{3} + 95 x + 19 \)
$\ \in\Q_{19}(t)[x]$
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Ramification polygon
Not computedInvariants of the Galois closure
Galois group: | $C_6$ (as 6T1) |
Inertia group: | Intransitive group isomorphic to $C_3$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $3$ |
Wild slopes: | None |
Galois mean slope: | $2/3$ |
Galois splitting model: |
$x^{6} - x^{5} + 20 x^{4} - 23 x^{3} + 186 x^{2} - 168 x + 392$
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