Defining polynomial
|
\(x^{13} + 143 x^{3} + 13\)
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Invariants
| Base field: | $\Q_{13}$ |
|
| Degree $d$: | $13$ |
|
| Ramification index $e$: | $13$ |
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| Residue field degree $f$: | $1$ |
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| Discriminant exponent $c$: | $15$ |
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| Discriminant root field: | $\Q_{13}(\sqrt{13\cdot 2})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{13})$: | $C_1$ | |
| This field is not Galois over $\Q_{13}.$ | ||
| Visible Artin slopes: | $[\frac{5}{4}]$ | |
| Visible Swan slopes: | $[\frac{1}{4}]$ | |
| Means: | $\langle\frac{3}{13}\rangle$ | |
| Rams: | $(\frac{1}{4})$ | |
| Jump set: | undefined | |
| Roots of unity: | $12 = (13 - 1)$ |
|
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 13 }$. |
Canonical tower
| Unramified subfield: | $\Q_{13}$ |
|
| Relative Eisenstein polynomial: |
\( x^{13} + 143 x^{3} + 13 \)
|
Ramification polygon
| Residual polynomials: | $z^3 + 6$ |
| Associated inertia: | $3$ |
| Indices of inseparability: | $[3, 0]$ |
Invariants of the Galois closure
| Galois degree: | $156$ |
| Galois group: | $F_{13}$ (as 13T6) |
| Inertia group: | $C_{13}:C_4$ (as 13T4) |
| Wild inertia group: | $C_{13}$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\frac{5}{4}]$ |
| Galois Swan slopes: | $[\frac{1}{4}]$ |
| Galois mean slope: | $1.2115384615384615$ |
| Galois splitting model: | not computed |