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Properties

Label 2.1.16.58n1.825
Base \(\Q_{2}\)
Degree \(16\)
e \(16\)
f \(1\)
c \(58\)
Galois group $C_2^4:Q_8$ (as 16T354)

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Defining polynomial

Copy content comment:Define the p-adic field
 
Copy content sage:Prec = 100 # Default precision of 100 Q2 = Qp(2, Prec); x = polygen(QQ) K.<a> = Q2.extension(x^16 + 8*x^15 + 8*x^14 + 8*x^13 + 4*x^12 + 8*x^11 + 8*x^10 + 2*x^8 + 8*x^6 + 20*x^4 + 16*x^3 + 8*x^2 + 16*x + 14)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [14, 16, 8, 16, 20, 0, 8, 0, 2, 0, 8, 8, 4, 8, 8, 8, 1]));
 

\(x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 8 x^{10} + 2 x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 14\) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content magma:DefiningPolynomial(K);
 

Invariants

Base field: $\Q_{2}$
Copy content comment:Base field Qp
 
Copy content sage:K.base()
 
Copy content magma:Q2;
 
Degree $d$: $16$
Copy content comment:Degree over Qp
 
Copy content sage:K.absolute_degree()
 
Copy content magma:Degree(K);
 
Ramification index $e$:$16$
Copy content comment:Ramification index
 
Copy content sage:K.absolute_e()
 
Copy content magma:RamificationIndex(K);
 
Residue field degree $f$:$1$
Copy content comment:Residue field degree (Inertia degree)
 
Copy content sage:K.absolute_f()
 
Copy content magma:InertiaDegree(K);
 
Discriminant exponent $c$:$58$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{2}$
Root number: $1$
$\Aut(K/\Q_{2})$: $C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:$[2, 3, 4, \frac{17}{4}]$
Visible Swan slopes:$[1,2,3,\frac{13}{4}]$
Means:$\langle\frac{1}{2}, \frac{5}{4}, \frac{17}{8}, \frac{43}{16}\rangle$
Rams:$(1, 3, 7, 9)$
Jump set:$[1, 2, 4, 8, 32]$
Roots of unity:$2$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

$\Q_{2}(\sqrt{-5})$, $\Q_{2}(\sqrt{-2\cdot 5})$, $\Q_{2}(\sqrt{2})$, 2.1.4.8b1.6, 2.1.8.24c1.56

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Canonical tower

Unramified subfield:$\Q_{2}$
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 8 x^{10} + 2 x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 14 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^8 + 1$,$z^4 + 1$,$z^2 + 1$,$z + 1$
Associated inertia:$1$,$1$,$1$,$1$
Indices of inseparability:$[43, 34, 20, 8, 0]$

Invariants of the Galois closure

Galois degree: $128$
Galois group: $C_2^4:Q_8$ (as 16T354)
Inertia group: $C_2^2.C_4^2$ (as 16T77)
Wild inertia group: $C_2^2.C_4^2$
Galois unramified degree: $2$
Galois tame degree: $1$
Galois Artin slopes: $[2, 3, 3, \frac{7}{2}, 4, \frac{17}{4}]$
Galois Swan slopes: $[1,2,2,\frac{5}{2},3,\frac{13}{4}]$
Galois mean slope: $3.875$
Galois splitting model:$x^{16} - 8 x^{14} - 56 x^{12} + 40 x^{10} + 274 x^{8} - 8 x^{6} - 248 x^{4} + 40 x^{2} + 1$