Defining polynomial
|
\(x^{12} + 3605\)
|
Invariants
| Base field: | $\Q_{103}$ |
|
| Degree $d$: | $12$ |
|
| Ramification index $e$: | $12$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $11$ |
|
| Discriminant root field: | $\Q_{103}(\sqrt{103\cdot 3})$ | |
| Root number: | $i$ | |
| $\Aut(K/\Q_{103})$: | $C_6$ | |
| This field is not Galois over $\Q_{103}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $102 = (103 - 1)$ |
|
Intermediate fields
| $\Q_{103}(\sqrt{103})$, $\Q_{103}(\sqrt[3]{103 \cdot 25})$, $\Q_{103}(\sqrt[4]{103})$, $\Q_{103}(\sqrt[6]{103 \cdot 25})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{103}$ |
|
| Relative Eisenstein polynomial: |
\( x^{12} + 3605 \)
|
Ramification polygon
| Residual polynomials: | $z^{11} + 12 z^{10} + 66 z^9 + 14 z^8 + 83 z^7 + 71 z^6 + 100 z^5 + 71 z^4 + 83 z^3 + 14 z^2 + 66 z + 12$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $24$ |
| Galois group: | $C_3\times D_4$ (as 12T14) |
| Inertia group: | $C_{12}$ (as 12T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $12$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9166666666666666$ |
| Galois splitting model: | not computed |