Defining polynomial
\(x^{6} + 38\)
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Invariants
Base field: | $\Q_{19}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $5$ |
Discriminant root field: | $\Q_{19}(\sqrt{19})$ |
Root number: | $i$ |
$\card{ \Gal(K/\Q_{ 19 }) }$: | $6$ |
This field is Galois and abelian over $\Q_{19}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{19}(\sqrt{19})$, 19.3.2.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{19}$ |
Relative Eisenstein polynomial: |
\( x^{6} + 38 \)
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