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Properties

Label 2.1.12.26b1.63
Base \(\Q_{2}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(26\)
Galois group $C_2^2\wr S_3$ (as 12T139)

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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^12 + 4*x^11 + 2*x^10 + 2*x^6 + 4*x^3 + 6)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [6, 0, 0, 4, 0, 0, 2, 0, 0, 0, 2, 4, 1]));
 

\(x^{12} + 4 x^{11} + 2 x^{10} + 2 x^{6} + 4 x^{3} + 6\) 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 $d$: $12$
Copy content comment:Degree over Qp
 
Copy content sage:K.absolute_degree()
 
Copy content magma:Degree(K);
 
Ramification index $e$:$12$
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 $c$:$26$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{2}$
Root number: $-1$
$\Aut(K/\Q_{2})$: $C_2^2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:$[2, 3]$
Visible Swan slopes:$[1,2]$
Means:$\langle\frac{1}{2}, \frac{5}{4}\rangle$
Rams:$(3, 9)$
Jump set:$[3, 11, 23]$
Roots of unity:$2$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

2.1.3.2a1.1, 2.1.6.8a1.4, 2.1.6.11a1.12, 2.1.6.11a1.15

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: \( x^{12} + 4 x^{11} + 2 x^{10} + 2 x^{6} + 4 x^{3} + 6 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^8 + z^4 + 1$,$z^2 + 1$,$z + 1$
Associated inertia:$2$,$1$,$1$
Indices of inseparability:$[15, 6, 0]$

Invariants of the Galois closure

Galois degree: $384$
Galois group: $C_2^2\wr S_3$ (as 12T139)
Inertia group: $C_2^2\wr C_3$ (as 12T90)
Wild inertia group: $C_2^6$
Galois unramified degree: $2$
Galois tame degree: $3$
Galois Artin slopes: $[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}, 3]$
Galois Swan slopes: $[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3},2]$
Galois mean slope: $2.6979166666666665$
Galois splitting model:$x^{12} - 4 x^{11} + 14 x^{10} - 24 x^{9} + 35 x^{8} - 60 x^{7} + 154 x^{6} - 408 x^{5} + 743 x^{4} - 964 x^{3} + 964 x^{2} - 652 x + 419$