Properties

Label 7.12.11.2
Base \(\Q_{7}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(11\)
Galois group $D_4 \times C_3$ (as 12T14)

Related objects

Downloads

Learn more

Defining polynomial

\(x^{12} + 7\) Copy content Toggle raw display

Invariants

Base field: $\Q_{7}$
Degree $d$: $12$
Ramification exponent $e$: $12$
Residue field degree $f$: $1$
Discriminant exponent $c$: $11$
Discriminant root field: $\Q_{7}(\sqrt{7})$
Root number: $-i$
$\card{ \Aut(K/\Q_{ 7 }) }$: $6$
This field is not Galois over $\Q_{7}.$
Visible slopes:None

Intermediate fields

$\Q_{7}(\sqrt{7\cdot 3})$, 7.3.2.2, 7.4.3.1, 7.6.5.5

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}$
Relative Eisenstein polynomial: \( x^{12} + 7 \) Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_3\times D_4$ (as 12T14)
Inertia group:$C_{12}$ (as 12T1)
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$12$
Wild slopes:None
Galois mean slope:$11/12$
Galois splitting model:$x^{12} - 4 x^{11} + 5 x^{10} + 3 x^{9} - 11 x^{8} - 3 x^{7} + 35 x^{6} - 47 x^{5} + 27 x^{4} - 4 x^{3} - x^{2} - x + 1$