Defining polynomial
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\(x^{8} + 4 x^{7} + 8 x^{4} + 4 x^{2} + 8 x + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $8$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]$ |
| Visible Swan slopes: | $[\frac{5}{3},\frac{5}{3},\frac{5}{2}]$ |
| Means: | $\langle\frac{5}{6}, \frac{5}{4}, \frac{15}{8}\rangle$ |
| Rams: | $(\frac{5}{3}, \frac{5}{3}, 5)$ |
| Jump set: | $[1, 3, 7, 15]$ |
| Roots of unity: | $2$ |
Intermediate fields
| 2.1.4.8a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{8} + 4 x^{7} + 8 x^{4} + 4 x^{2} + 8 x + 2 \)
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Ramification polygon
| Residual polynomials: | $z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[15, 10, 8, 0]$ |
Invariants of the Galois closure
| Galois degree: | $48$ |
| Galois group: | $\GL(2,3)$ (as 8T23) |
| Inertia group: | $\SL(2,3)$ (as 8T12) |
| Wild inertia group: | $Q_8$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{8}{3}, \frac{8}{3}, \frac{7}{2}]$ |
| Galois Swan slopes: | $[\frac{5}{3},\frac{5}{3},\frac{5}{2}]$ |
| Galois mean slope: | $2.8333333333333335$ |
| Galois splitting model: | $x^{8} - 8 x^{6} + 18 x^{4} - 3$ |