Defining polynomial
\(x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445\)
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Invariants
Base field: | $\Q_{19}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $2$ |
Discriminant root field: | $\Q_{19}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 19 }) }$: | $4$ |
This field is Galois and abelian over $\Q_{19}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{19}(\sqrt{19})$, $\Q_{19}(\sqrt{19\cdot 2})$, $\Q_{19}(\sqrt{2})$ |
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: |
\( x^{2} + 19 \)
$\ \in\Q_{19}(t)[x]$
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Ramification polygon
Not computedInvariants of the Galois closure
Galois group: | $C_2^2$ (as 4T2) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: |
$x^{4} + 57 x^{2} + 1444$
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