Defining polynomial
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\(x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875\)
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Invariants
| Base field: | $\Q_{197}$ |
| Degree $d$: | $4$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $2$ |
| Discriminant root field: | $\Q_{197}$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 197 }) }$: | $4$ |
| This field is Galois and abelian over $\Q_{197}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{197}(\sqrt{2})$, $\Q_{197}(\sqrt{197})$, $\Q_{197}(\sqrt{197\cdot 2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{197}(\sqrt{2})$ $\cong \Q_{197}(t)$ where $t$ is a root of
\( x^{2} + 192 x + 2 \)
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| Relative Eisenstein polynomial: |
\( x^{2} + 33293 x + 197 \)
$\ \in\Q_{197}(t)[x]$
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Ramification polygon
Data 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: |
$x^{4} + 985 x^{2} + 349281$
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