Defining polynomial
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\(x^{6} + 2775 x^{5} + 2566998 x^{4} + 791680745 x^{3} + 110476893 x^{2} + 4842384300 x + 27887445532\)
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Invariants
| Base field: | $\Q_{37}$ |
| Degree $d$: | $6$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $3$ |
| Discriminant root field: | $\Q_{37}(\sqrt{37})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 37 }) }$: | $6$ |
| This field is Galois and abelian over $\Q_{37}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{37}(\sqrt{37})$, 37.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 37.3.0.1 $\cong \Q_{37}(t)$ where $t$ is a root of
\( x^{3} + 6 x + 35 \)
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| Relative Eisenstein polynomial: |
\( x^{2} + 925 x + 37 \)
$\ \in\Q_{37}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_6$ (as 6T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $3$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: |
$x^{6} - x^{5} - 47 x^{4} + 10 x^{3} + 495 x^{2} - 162 x - 729$
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