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sage:Prec = 100 # Default precision of 100
Q2 = Qp(2, Prec); x = polygen(QQ)
K.<a> = Q2.extension(x^8 + 16*x^7 + 8*x^6 + 16*x^3 + 8*x^2 + 2)
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magma:Prec := 100; // Default precision of 100
Q2 := pAdicField(2, Prec);
K := LocalField(Q2, Polynomial(Q2, [2, 0, 8, 16, 0, 0, 8, 16, 1]));
\(x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{3} + 8 x^{2} + 2\)
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sage:K.defining_polynomial()
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magma:DefiningPolynomial(K);
| Base field: | $\Q_{2}$ |
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sage:K.base()
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magma:Q2;
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| Degree $d$: | $8$ |
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sage:K.absolute_degree()
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magma:Degree(K);
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| Ramification index $e$: | $8$ |
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sage:K.absolute_e()
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magma:RamificationIndex(K);
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| Residue field degree $f$: | $1$ |
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sage:K.absolute_f()
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magma:InertiaDegree(K);
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| Discriminant exponent $c$: | $31$ |
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magma:Valuation(Discriminant(K));
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| Discriminant root field: | $\Q_{2}(\sqrt{2})$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$:
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$C_2$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[3, 4, 5]$ |
| Visible Swan slopes: | $[2,3,4]$ |
| Means: | $\langle1, 2, 3\rangle$ |
| Rams: | $(2, 4, 8)$ |
| Jump set: | $[1, 3, 7, 15]$ |
| Roots of unity: | $2$ |
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sage:len(K.roots_of_unity())
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Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| Galois degree: |
$128$
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| Galois group: |
$C_2\wr D_4$ (as 8T35)
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| Inertia group: |
$C_2\wr C_4$ (as 8T27)
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| Wild inertia group: |
$C_2\wr C_4$
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| Galois unramified degree: |
$2$
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| Galois tame degree: |
$1$
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| Galois Artin slopes: |
$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, 5]$
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| Galois Swan slopes: |
$[1,2,\frac{5}{2},3,\frac{13}{4},4]$
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| Galois mean slope: |
$4.40625$
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| Galois splitting model: | $x^{8} + 8 x^{4} - 8 x^{2} + 2$ |
