Properties

Label 2.1.16.72d1.3598
Base \(\Q_{2}\)
Degree \(16\)
e \(16\)
f \(1\)
c \(72\)
Galois group $C_2^5.C_2\wr C_4$ (as 16T1426)

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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^14 + 24*x^12 + 16*x^11 + 16*x^9 + 4*x^8 + 16*x^6 + 8*x^4 + 16*x^2 + 32*x + 10)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [10, 32, 16, 0, 8, 0, 16, 0, 4, 16, 0, 16, 24, 0, 8, 0, 1]));
 

\(x^{16} + 8 x^{14} + 24 x^{12} + 16 x^{11} + 16 x^{9} + 4 x^{8} + 16 x^{6} + 8 x^{4} + 16 x^{2} + 32 x + 10\) 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 $c$:$72$
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:$[3, 4, \frac{19}{4}, \frac{21}{4}]$
Visible Swan slopes:$[2,3,\frac{15}{4},\frac{17}{4}]$
Means:$\langle1, 2, \frac{23}{8}, \frac{57}{16}\rangle$
Rams:$(2, 4, 7, 11)$
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

$\Q_{2}(\sqrt{2\cdot 5})$, 2.1.4.11a1.19, 2.1.8.30a1.115

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: \( x^{16} + 8 x^{14} + 24 x^{12} + 16 x^{11} + 16 x^{9} + 4 x^{8} + 16 x^{6} + 8 x^{4} + 16 x^{2} + 32 x + 10 \) 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:$[57, 46, 32, 16, 0]$

Invariants of the Galois closure

Galois degree: $2048$
Galois group: $C_2^5.C_2\wr C_4$ (as 16T1426)
Inertia group: $C_2^4.C_2\wr C_4$ (as 16T1176)
Wild inertia group: not computed
Galois unramified degree: $2$
Galois tame degree: $1$
Galois Artin slopes: $[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}]$
Galois Swan slopes: $[1,1,2,\frac{5}{2},3,\frac{13}{4},\frac{7}{2},\frac{15}{4},\frac{31}{8},\frac{17}{4}]$
Galois mean slope: $4.958984375$
Galois splitting model: $x^{16} + 8 x^{14} - 80 x^{13} + 256 x^{12} - 960 x^{11} + 3104 x^{10} - 5680 x^{9} + 2644 x^{8} + 15040 x^{7} - 60688 x^{6} + 116800 x^{5} - 131176 x^{4} + 64960 x^{3} + 12736 x^{2} - 33440 x - 5494$ Copy content Toggle raw display