Defining polynomial
\(x^{8} + 2784 x^{7} + 2906616 x^{6} + 1348864734 x^{5} + 234834277018 x^{4} + 41857830864 x^{3} + 492109772617 x^{2} + 3561769809750 x + 616658760166\) |
Invariants
Base field: | $\Q_{29}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $4$ |
Discriminant root field: | $\Q_{29}$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 29 }) }$: | $8$ |
This field is Galois and abelian over $\Q_{29}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{29}(\sqrt{2})$, $\Q_{29}(\sqrt{29})$, $\Q_{29}(\sqrt{29\cdot 2})$, 29.4.0.1, 29.4.2.1, 29.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 29.4.0.1 $\cong \Q_{29}(t)$ where $t$ is a root of \( x^{4} + 2 x^{2} + 15 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{2} + 696 x + 29 \) $\ \in\Q_{29}(t)[x]$ |
Ramification polygon
Data not computedInvariants of the Galois closure
Galois group: | $C_2\times C_4$ (as 8T2) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $4$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | Not computed |