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sage:Prec = 100 # Default precision of 100
Q2 = Qp(2, Prec); x = polygen(QQ)
K.<a> = Q2.extension(x^8 + 2*x^7 + 6)
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magma:Prec := 100; // Default precision of 100
Q2 := pAdicField(2, Prec);
K := LocalField(Q2, Polynomial(Q2, [6, 0, 0, 0, 0, 0, 0, 2, 1]));
\(x^{8} + 2 x^{7} + 6\)
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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 $c$: | $14$ |
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magma:Valuation(Discriminant(K));
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| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$:
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$C_2$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[2, 2, 2]$ |
| Visible Swan slopes: | $[1,1,1]$ |
| Means: | $\langle\frac{1}{2}, \frac{3}{4}, \frac{7}{8}\rangle$ |
| Rams: | $(1, 1, 1)$ |
| Jump set: | $[1, 3, 7, 16]$ |
| 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.
