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Properties

Label 2.1.16.62f1.74
Base \(\Q_{2}\)
Degree \(16\)
e \(16\)
f \(1\)
c \(62\)
Galois group $C_2^6.(D_4\times S_4)$ (as 16T1756)

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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^12 + 8*x^10 + 16*x^5 + 10)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [10, 0, 0, 0, 0, 16, 0, 0, 0, 0, 8, 0, 4, 0, 0, 8, 1]));
 

\(x^{16} + 8 x^{15} + 4 x^{12} + 8 x^{10} + 16 x^{5} + 10\) 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 $c$:$62$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{2}$
Root number: $1$
$\Aut(K/\Q_{2})$: $C_1$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:$[3, \frac{7}{2}, \frac{13}{3}, \frac{13}{3}]$
Visible Swan slopes:$[2,\frac{5}{2},\frac{10}{3},\frac{10}{3}]$
Means:$\langle1, \frac{7}{4}, \frac{61}{24}, \frac{47}{16}\rangle$
Rams:$(2, 3, \frac{19}{3}, \frac{19}{3})$
Jump set:$[1, 3, 7, 15, 31]$
Roots of unity:$2$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

$\Q_{2}(\sqrt{2\cdot 5})$, 2.1.4.10a1.3

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: \( x^{16} + 8 x^{15} + 4 x^{12} + 8 x^{10} + 16 x^{5} + 10 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^8 + 1$,$z^4 + 1$,$z + 1$
Associated inertia:$1$,$1$,$1$
Indices of inseparability:$[47, 42, 28, 16, 0]$

Invariants of the Galois closure

Galois degree: $12288$
Galois group: $C_2^6.(D_4\times S_4)$ (as 16T1756)
Inertia group: $C_2^6.(D_4\times A_4)$ (as 16T1655)
Wild inertia group: not computed
Galois unramified degree: $2$
Galois tame degree: $3$
Galois Artin slopes: $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, \frac{7}{2}, \frac{43}{12}, \frac{43}{12}, \frac{13}{3}, \frac{13}{3}]$
Galois Swan slopes: $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},\frac{5}{2},\frac{31}{12},\frac{31}{12},\frac{10}{3},\frac{10}{3}]$
Galois mean slope: $4.123372395833333$
Galois splitting model: $x^{16} - 24 x^{14} - 64 x^{13} - 268 x^{12} - 624 x^{11} + 7880 x^{10} + 48976 x^{9} + 69066 x^{8} - 196768 x^{7} - 815192 x^{6} - 808800 x^{5} + 964828 x^{4} + 3329008 x^{3} + 3673896 x^{2} + 1900080 x + 381665$ Copy content Toggle raw display