Defining polynomial
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$( x^{3} + 3 x + 3 )^{5} + \left(50 x^{2} + 50 x + 100\right) ( x^{3} + 3 x + 3 ) + 5$
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Invariants
| Base field: | $\Q_{5}$ |
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| Degree $d$: | $15$ |
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| Ramification index $e$: | $5$ |
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| Residue field degree $f$: | $3$ |
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| Discriminant exponent $c$: | $27$ |
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| Discriminant root field: | $\Q_{5}(\sqrt{5})$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{5})$: | $C_1$ | |
| This field is not Galois over $\Q_{5}.$ | ||
| Visible Artin slopes: | $[\frac{9}{4}]$ | |
| Visible Swan slopes: | $[\frac{5}{4}]$ | |
| Means: | $\langle1\rangle$ | |
| Rams: | $(\frac{5}{4})$ | |
| Jump set: | undefined | |
| Roots of unity: | $124 = (5^{ 3 } - 1)$ |
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Intermediate fields
| 5.3.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 5.3.1.0a1.1 $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{3} + 3 x + 3 \)
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| Relative Eisenstein polynomial: |
\( x^{5} + \left(100 t^{2} + 25 t + 75\right) x + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + (4 t + 3)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[5, 0]$ |
Invariants of the Galois closure
| Galois degree: | $1500$ |
| Galois group: | $C_5^3:C_{12}$ (as 15T38) |
| Inertia group: | Intransitive group isomorphic to $C_5^3:C_4$ |
| Wild inertia group: | $C_5^3$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\frac{5}{4}, \frac{5}{4}, \frac{9}{4}]$ |
| Galois Swan slopes: | $[\frac{1}{4},\frac{1}{4},\frac{5}{4}]$ |
| Galois mean slope: | $2.046$ |
| Galois splitting model: |
$x^{15} - 19475 x^{13} + 141414125 x^{11} - 211371400 x^{10} - 498384981250 x^{9} + 1299084364500 x^{8} + 902874792015625 x^{7} - 1904419730622500 x^{6} - 816415675837290100 x^{5} - 567856011751843750 x^{4} + 331565059777803938125 x^{3} + 1461116604969779062500 x^{2} - 36380117201061149262500 x - 187537551183568627847000$
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