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sage:Prec = 100 # Default precision of 100
Q2 = Qp(2, Prec); x = polygen(QQ)
K.<a> = Q2.extension(x^16 + 16*x^15 + 8*x^14 + 8*x^10 + 16*x^9 + 4*x^8 + 32*x^7 + 32*x^5 + 16*x^4 + 32*x^3 + 16*x^2 + 32*x + 26)
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magma:Prec := 100; // Default precision of 100
Q2 := pAdicField(2, Prec);
K := LocalField(Q2, Polynomial(Q2, [26, 32, 16, 32, 16, 32, 0, 32, 4, 16, 8, 0, 0, 0, 8, 16, 1]));
\(x^{16} + 16 x^{15} + 8 x^{14} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 32 x^{7} + 32 x^{5} + 16 x^{4} + 32 x^{3} + 16 x^{2} + 32 x + 26\)
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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 $c$: | $72$ |
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magma:Valuation(Discriminant(K));
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| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $-i$ |
| $\Aut(K/\Q_{2})$:
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$C_2$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[3, 4, \frac{17}{4}, \frac{11}{2}]$ |
| Visible Swan slopes: | $[2,3,\frac{13}{4},\frac{9}{2}]$ |
| Means: | $\langle1, 2, \frac{21}{8}, \frac{57}{16}\rangle$ |
| Rams: | $(2, 4, 5, 15)$ |
| Jump set: | $[1, 3, 7, 15, 31]$ |
| 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.
| Residual polynomials: | $z^8 + 1$,$z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$,$1$,$1$ |
| Indices of inseparability: | $[57, 42, 32, 16, 0]$ |
