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Properties

Label 2.3.6.33a1.36
Base \(\Q_{2}\)
Degree \(18\)
e \(6\)
f \(3\)
c \(33\)
Galois group $C_2^4:(A_4\times D_6)$ (as 18T366)

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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) L.<t> = Q2.extension(x^3 + x + 1) K.<a> = L.extension(x^6 + 4*t*x^5 + (4*t^2 + 4*t + 4)*x^3 + 10)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [15, 22, 43, 74, 125, 162, 200, 238, 228, 224, 199, 140, 115, 70, 35, 26, 6, 4, 1]));
 

$( x^{3} + x + 1 )^{6} + 4 x^{2} ( x^{3} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + 10$ 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 $d$: $18$
Copy content comment:Degree over Qp
 
Copy content sage:K.absolute_degree()
 
Copy content magma:Degree(K);
 
Ramification index $e$:$6$
Copy content comment:Ramification index
 
Copy content sage:K.absolute_e()
 
Copy content magma:RamificationIndex(K);
 
Residue field degree $f$:$3$
Copy content comment:Residue field degree (Inertia degree)
 
Copy content sage:K.absolute_f()
 
Copy content magma:InertiaDegree(K);
 
Discriminant exponent $c$:$33$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{2}(\sqrt{2\cdot 5})$
Root number: $-1$
$\Aut(K/\Q_{2})$: $C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:$[3]$
Visible Swan slopes:$[2]$
Means:$\langle1\rangle$
Rams:$(6)$
Jump set:$[3, 9]$
Roots of unity:$14 = (2^{ 3 } - 1) \cdot 2$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

2.3.1.0a1.1, 2.1.3.2a1.1, 2.3.3.6a1.1

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

Canonical tower

Unramified subfield:2.3.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} + x + 1 \) Copy content Toggle raw display
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x^{6} + 4 t x^{5} + \left(4 t^{2} + 4 t + 4\right) x^{3} + 10 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois degree: $2304$
Galois group: $C_2^4:(A_4\times D_6)$ (as 18T366)
Inertia group: Intransitive group isomorphic to $C_2^5:A_4$
Wild inertia group: $C_2^7$
Galois unramified degree: $6$
Galois tame degree: $3$
Galois Artin slopes: $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 3]$
Galois Swan slopes: $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,2]$
Galois mean slope: $2.4114583333333335$
Galois splitting model: $x^{18} + 40 x^{16} + 1780 x^{14} + 18122 x^{12} + 6596776 x^{10} + 729159864 x^{8} - 3260067956 x^{6} - 616163795760 x^{4} + 2473153994016 x^{2} + 179921953064664$ Copy content Toggle raw display