Defining polynomial
\(x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243\)
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Invariants
Base field: | $\Q_{37}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $2$ |
Discriminant root field: | $\Q_{37}$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 37 }) }$: | $4$ |
This field is Galois and abelian over $\Q_{37}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{37}(\sqrt{2})$, $\Q_{37}(\sqrt{37})$, $\Q_{37}(\sqrt{37\cdot 2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{37}(\sqrt{2})$ $\cong \Q_{37}(t)$ where $t$ is a root of
\( x^{2} + 33 x + 2 \)
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Relative Eisenstein polynomial: |
\( x^{2} + 925 x + 37 \)
$\ \in\Q_{37}(t)[x]$
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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: |
$x^{4} + 333 x^{2} + 34225$
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