Properties

Label 7.14.7.2
Base \(\Q_{7}\)
Degree \(14\)
e \(2\)
f \(7\)
c \(7\)
Galois group $C_{14}$ (as 14T1)

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Defining polynomial

\( x^{14} - 686 x^{8} + 117649 x^{2} - 3294172 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $14$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $7$
Discriminant exponent $c$ : $7$
Discriminant root field: $\Q_{7}(\sqrt{7})$
Root number: $i$
$|\Gal(K/\Q_{ 7 })|$: $14$
This field is Galois and abelian over $\Q_{7}$.

Intermediate fields

$\Q_{7}(\sqrt{7})$, 7.7.0.1

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

Unramified/totally ramified tower

Unramified subfield:7.7.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{7} - x + 2 \)
Relative Eisenstein polynomial:$ x^{2} + \left(42 t^{6} + 35 t^{5} + 28 t^{4} + 42 t^{3} + 21 t^{2} + 28\right) x + 35 t^{4} + 35 t^{3} + 14 t^{2} + 21 t + 42 \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{14}$ (as 14T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$7$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{14} - 5 x^{13} + 12 x^{12} + 31 x^{11} + 722 x^{10} - 4645 x^{9} + 23535 x^{8} - 41683 x^{7} + 272781 x^{6} - 1040258 x^{5} + 7172534 x^{4} - 22736110 x^{3} + 85299995 x^{2} - 143854350 x + 314845525$