Properties

Label 29.8.4.1
Base \(\Q_{29}\)
Degree \(8\)
e \(2\)
f \(4\)
c \(4\)
Galois group $C_4\times C_2$ (as 8T2)

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Defining polynomial

\(x^{8} + 2784 x^{7} + 2906616 x^{6} + 1348864734 x^{5} + 234834277018 x^{4} + 41857830864 x^{3} + 492109772617 x^{2} + 3561769809750 x + 616658760166\) Copy content Toggle raw display

Invariants

Base field: $\Q_{29}$
Degree $d$: $8$
Ramification exponent $e$: $2$
Residue field degree $f$: $4$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{29}$
Root number: $-1$
$\card{ \Gal(K/\Q_{ 29 }) }$: $8$
This field is Galois and abelian over $\Q_{29}.$
Visible slopes:None

Intermediate fields

$\Q_{29}(\sqrt{2})$, $\Q_{29}(\sqrt{29})$, $\Q_{29}(\sqrt{29\cdot 2})$, 29.4.0.1, 29.4.2.1, 29.4.2.2

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

Unramified/totally ramified tower

Unramified subfield:29.4.0.1 $\cong \Q_{29}(t)$ where $t$ is a root of \( x^{4} + 2 x^{2} + 15 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + 696 x + 29 \) $\ \in\Q_{29}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Data not computed

Invariants of the Galois closure

Galois group: $C_2\times C_4$ (as 8T2)
Inertia group: Intransitive group isomorphic to $C_2$
Wild inertia group: $C_1$
Unramified degree: $4$
Tame degree: $2$
Wild slopes: None
Galois mean slope: $1/2$
Galois splitting model:Not computed