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Properties

Label 2.3.6.24a2.7
Base \(\Q_{2}\)
Degree \(18\)
e \(6\)
f \(3\)
c \(24\)
Galois group $C_2^5.(A_4\times S_4)$ (as 18T544)

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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 + (2*t^2 + 2*t + 2)*x^5 + 2*t*x^3 + 6)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [3, 10, 33, 62, 103, 158, 194, 218, 232, 210, 177, 140, 95, 60, 35, 16, 8, 2, 1]));
 

$( x^{3} + x + 1 )^{6} + \left(2 x^{2} + 2 x\right) ( x^{3} + x + 1 )^{5} + 2 x ( x^{3} + x + 1 )^{3} + 2$ 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 $d$: $18$
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 $c$:$24$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{2}(\sqrt{-5})$
Root number: $-i$
$\Aut(K/\Q_{2})$: $C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:$[2]$
Visible Swan slopes:$[1]$
Means:$\langle\frac{1}{2}\rangle$
Rams:$(3)$
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: \( x^{6} + \left(2 t^{2} + 2 t + 2\right) x^{5} + 2 t x^{3} + 6 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^4 + z^2 + 1$,$z + (t^2 + t + 1)$
Associated inertia:$2$,$1$
Indices of inseparability:$[3, 0]$

Invariants of the Galois closure

Galois degree: $9216$
Galois group: $C_2^5.(A_4\times S_4)$ (as 18T544)
Inertia group: Intransitive group isomorphic to $C_2^3\wr C_3$
Wild inertia group: $C_2^9$
Galois unramified degree: $6$
Galois tame degree: $3$
Galois Artin slopes: $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2]$
Galois Swan slopes: $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1]$
Galois mean slope: $1.9153645833333333$
Galois splitting model: $x^{18} - 4 x^{17} + 51 x^{16} - 146 x^{15} - 27 x^{14} + 22974 x^{13} + 46702 x^{12} - 987192 x^{11} + 914436 x^{10} + 28334512 x^{9} + 1403583 x^{8} - 132753006 x^{7} + 4794669 x^{6} - 2537817756 x^{5} - 239584757 x^{4} + 1265684030 x^{3} - 1997599554 x^{2} - 2545162020 x - 686881313$ Copy content Toggle raw display