Defining polynomial
\(x^{13} + 78 x^{3} + 13\) |
Invariants
Base field: | $\Q_{13}$ |
Degree $d$: | $13$ |
Ramification exponent $e$: | $13$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $15$ |
Discriminant root field: | $\Q_{13}(\sqrt{13\cdot 2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 13 }) }$: | $1$ |
This field is not Galois over $\Q_{13}.$ | |
Visible slopes: | $[5/4]$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 13 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{13}$ |
Relative Eisenstein polynomial: | \( x^{13} + 78 x^{3} + 13 \) |
Ramification polygon
Residual polynomials: | $z^{3} + 8$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[3, 0]$ |
Invariants of the Galois closure
Galois group: | $C_{13}:C_4$ (as 13T4) |
Inertia group: | $C_{13}:C_4$ (as 13T4) |
Wild inertia group: | $C_{13}$ |
Unramified degree: | $1$ |
Tame degree: | $4$ |
Wild slopes: | $[5/4]$ |
Galois mean slope: | $63/52$ |
Galois splitting model: | Not computed |