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Properties

Label 2.1.16.48h1.1218
Base \(\Q_{2}\)
Degree \(16\)
e \(16\)
f \(1\)
c \(48\)
Galois group $C_2^6.(D_4\times S_4)$ (as 16T1757)

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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 + 4*x^6 + 2*x^4 + 8*x^3 + 4*x^2 + 8*x + 2)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [2, 8, 4, 8, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 8, 1]));
 

\(x^{16} + 8 x^{15} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 2\) 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 $c$:$48$
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:$[\frac{4}{3}, \frac{4}{3}, 3, 4]$
Visible Swan slopes:$[\frac{1}{3},\frac{1}{3},2,3]$
Means:$\langle\frac{1}{6}, \frac{1}{4}, \frac{9}{8}, \frac{33}{16}\rangle$
Rams:$(\frac{1}{3}, \frac{1}{3}, 7, 15)$
Jump set:$[1, 2, 5, 13, 29]$
Roots of unity:$2$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

2.1.4.4a1.1, 2.1.8.16b1.9

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: \( x^{16} + 8 x^{15} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 2 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^4 + 1$,$z^2 + 1$,$z + 1$
Associated inertia:$1$,$1$,$1$
Indices of inseparability:$[33, 18, 4, 4, 0]$

Invariants of the Galois closure

Galois degree: $12288$
Galois group: $C_2^6.(D_4\times S_4)$ (as 16T1757)
Inertia group: not computed
Wild inertia group: not computed
Galois unramified degree: not computed
Galois tame degree: not computed
Galois Artin slopes: not computed
Galois Swan slopes: not computed
Galois mean slope: not computed
Galois splitting model:not computed