Properties

Label 89.12.10.1
Base \(\Q_{89}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $D_6$ (as 12T3)

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Defining polynomial

\(x^{12} + 492 x^{11} + 100878 x^{10} + 11034740 x^{9} + 679393095 x^{8} + 22343681112 x^{7} + 308081214422 x^{6} + 67031087124 x^{5} + 6123506385 x^{4} + 1278059380 x^{3} + 60258731628 x^{2} + 1975336395024 x + 26984212327944\) Copy content Toggle raw display

Invariants

Base field: $\Q_{89}$
Degree $d$: $12$
Ramification exponent $e$: $6$
Residue field degree $f$: $2$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{89}$
Root number: $-1$
$\card{ \Gal(K/\Q_{ 89 }) }$: $12$
This field is Galois over $\Q_{89}.$
Visible slopes:None

Intermediate fields

$\Q_{89}(\sqrt{3})$, $\Q_{89}(\sqrt{89})$, $\Q_{89}(\sqrt{89\cdot 3})$, 89.3.2.1 x3, 89.4.2.1, 89.6.4.1, 89.6.5.1 x3, 89.6.5.2 x3

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{89}(\sqrt{3})$ $\cong \Q_{89}(t)$ where $t$ is a root of \( x^{2} + 82 x + 3 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{6} + 89 \) $\ \in\Q_{89}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{5} + 6z^{4} + 15z^{3} + 20z^{2} + 15z + 6$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$D_6$ (as 12T3)
Inertia group:Intransitive group isomorphic to $C_6$
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:Not computed