Defining polynomial
\(x^{4} + 26\)
|
Invariants
Base field: | $\Q_{13}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $3$ |
Discriminant root field: | $\Q_{13}(\sqrt{13\cdot 2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 13 }) }$: | $4$ |
This field is Galois and abelian over $\Q_{13}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{13}(\sqrt{13\cdot 2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{13}$ |
Relative Eisenstein polynomial: |
\( x^{4} + 26 \)
|
Ramification polygon
Not computedInvariants of the Galois closure
Galois group: | $C_4$ (as 4T1) |
Inertia group: | $C_4$ (as 4T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $1$ |
Tame degree: | $4$ |
Wild slopes: | None |
Galois mean slope: | $3/4$ |
Galois splitting model: | $x^{4} - x^{3} - 24 x^{2} + 4 x + 16$ |