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sage:Prec = 100 # Default precision of 100
Q2 = Qp(2, Prec); x = polygen(QQ)
K.<a> = Q2.extension(x^16 + 4*x^14 + 8*x^13 + 4*x^12 + 8*x^11 + 26*x^8 + 16*x^7 + 4*x^4 + 2)
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magma:Prec := 100; // Default precision of 100
Q2 := pAdicField(2, Prec);
K := LocalField(Q2, Polynomial(Q2, [2, 0, 0, 0, 4, 0, 0, 16, 26, 0, 0, 8, 4, 8, 4, 0, 1]));
\(x^{16} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 26 x^{8} + 16 x^{7} + 4 x^{4} + 2\)
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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 $d$: | $16$ |
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sage:K.absolute_degree()
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magma:Degree(K);
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| Ramification index $e$: | $16$ |
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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 $c$: | $58$ |
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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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$C_4$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[2, 3, \frac{7}{2}, \frac{9}{2}]$ |
| Visible Swan slopes: | $[1,2,\frac{5}{2},\frac{7}{2}]$ |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}, \frac{15}{8}, \frac{43}{16}\rangle$ |
| Rams: | $(1, 3, 5, 13)$ |
| Jump set: | $[1, 7, 23, 39, 55]$ |
| Roots of unity: | $8 = 2^{ 3 }$ |
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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.
| Residual polynomials: | $z^8 + 1$,$z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$,$1$,$1$ |
| Indices of inseparability: | $[43, 30, 20, 8, 0]$ |
