Defining polynomial
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\(x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232\)
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Invariants
| Base field: | $\Q_{89}$ |
| Degree $d$: | $4$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $2$ |
| Discriminant root field: | $\Q_{89}$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 89 }) }$: | $4$ |
| This field is Galois and abelian over $\Q_{89}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{89}(\sqrt{3})$, $\Q_{89}(\sqrt{89})$, $\Q_{89}(\sqrt{89\cdot 3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{89}(\sqrt{3})$ $\cong \Q_{89}(t)$ where $t$ is a root of
\( x^{2} + 82 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 6052 x + 89 \)
$\ \in\Q_{89}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants 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: |
$x^{4} + 979 x^{2} + 285156$
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