Base \(\Q_{11}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(12\)
Galois group $C_{15}$ (as 15T1)

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Defining polynomial

\(x^{15} - 396 x^{10} + 98978 x^{5} + 74617191\) Copy content Toggle raw display


Base field: $\Q_{11}$
Degree $d$: $15$
Ramification exponent $e$: $5$
Residue field degree $f$: $3$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{11}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 11 }) }$: $15$
This field is Galois and abelian over $\Q_{11}.$
Visible slopes:None

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: $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{3} + 2 x + 9 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{5} + 99 t^{2} + 22 t \) $\ \in\Q_{11}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{4} + 5z^{3} + 10z^{2} + 10z + 5$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_{15}$ (as 15T1)
Inertia group:Intransitive group isomorphic to $C_5$
Wild inertia group:$C_1$
Unramified degree:$3$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:Not computed