Properties

 Label 7.12.9.1 Base $$\Q_{7}$$ Degree $$12$$ e $$4$$ f $$3$$ c $$9$$ Galois group $D_4 \times C_3$ (as 12T14)

Related objects

Defining polynomial

 $$x^{12} - 49 x^{4} + 686$$

Invariants

 Base field: $\Q_{7}$ Degree $d$ : $12$ Ramification exponent $e$ : $4$ Residue field degree $f$ : $3$ Discriminant exponent $c$ : $9$ Discriminant root field: $\Q_{7}(\sqrt{7})$ Root number: $-i$ $|\Aut(K/\Q_{ 7 })|$: $6$ This field is not Galois over $\Q_{7}$.

Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

 Unramified subfield: 7.3.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of $$x^{3} - x + 2$$ Relative Eisenstein polynomial: $x^{4} - 7 t \in\Q_{7}(t)[x]$

Invariants of the Galois closure

 Galois group: $C_3\times D_4$ (as 12T14) Inertia group: Intransitive group isomorphic to $C_4$ Unramified degree: $6$ Tame degree: $4$ Wild slopes: None Galois mean slope: $3/4$ Galois splitting model: $x^{12} + 15 x^{10} - 4 x^{9} + 45 x^{8} - 3 x^{7} - 81 x^{6} + 36 x^{5} + 264 x^{4} + 104 x^{3} + 108 x^{2} + 48 x + 8$