Base \(\Q_{19}\)
Degree \(15\)
e \(15\)
f \(1\)
c \(14\)
Galois group $D_5\times C_3$ (as 15T3)

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Defining polynomial

\(x^{15} + 76\) Copy content Toggle raw display


Base field: $\Q_{19}$
Degree $d$: $15$
Ramification exponent $e$: $15$
Residue field degree $f$: $1$
Discriminant exponent $c$: $14$
Discriminant root field: $\Q_{19}$
Root number: $1$
$\card{ \Aut(K/\Q_{ 19 }) }$: $3$
This field is not Galois over $\Q_{19}.$
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:$\Q_{19}$
Relative Eisenstein polynomial: \( x^{15} + 76 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{14} + 15z^{13} + 10z^{12} + 18z^{11} + 16z^{10} + z^{9} + 8z^{8} + 13z^{7} + 13z^{6} + 8z^{5} + z^{4} + 16z^{3} + 18z^{2} + 10z + 15$
Associated inertia:$2$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_3\times D_5$ (as 15T3)
Inertia group:$C_{15}$ (as 15T1)
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$15$
Wild slopes:None
Galois mean slope:$14/15$
Galois splitting model:Not computed