Defining polynomial
| \( x^{7} + 28 x^{4} + 7 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$ : | $7$ |
| Ramification exponent $e$ : | $7$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $10$ |
| Discriminant root field: | $\Q_{7}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 7 })|$: | $1$ |
| This field is not Galois over $\Q_{7}$. | |
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: | \( x^{7} + 28 x^{4} + 7 \) |
Invariants of the Galois closure
| Galois group: | $C_7:C_3$ (as 7T3) |
| Inertia group: | $C_7:C_3$ |
| Unramified degree: | $1$ |
| Tame degree: | $3$ |
| Wild slopes: | [5/3] |
| Galois mean slope: | $32/21$ |
| Galois splitting model: | $x^{7} - 777 x^{5} - 2331 x^{4} + 117327 x^{3} + 318570 x^{2} - 3217816 x - 8294808$ |