Defining polynomial
|
\(x^{13} + 7\)
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $13$ |
| Ramification exponent $e$: | $13$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | None |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: |
\( x^{13} + 7 \)
|
Ramification polygon
| Residual polynomials: | $z^{12} + 6z^{11} + z^{10} + 6z^{9} + z^{8} + 6z^{7} + z^{6} + z^{5} + 6z^{4} + z^{3} + 6z^{2} + z + 6$ |
| Associated inertia: | $12$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $F_{13}$ (as 13T6) |
| Inertia group: | $C_{13}$ (as 13T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $12$ |
| Tame degree: | $13$ |
| Wild slopes: | None |
| Galois mean slope: | $12/13$ |
| Galois splitting model: |
$x^{13} - 26 x^{12} + 312 x^{11} - 2288 x^{10} + 11440 x^{9} - 41184 x^{8} + 109824 x^{7} - 219648 x^{6} + 329472 x^{5} - 366080 x^{4} + 292864 x^{3} - 159744 x^{2} + 53248 x - 8199$
|