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Properties

Label 2.3.6.30a1.14
Base \(\Q_{2}\)
Degree \(18\)
e \(6\)
f \(3\)
c \(30\)
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*x^5 + 4*x^4 + (4*t^2 + 4*t)*x^3 + (4*t^2 + 4*t)*x + 2)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [9, 40, 75, 100, 167, 200, 188, 220, 192, 136, 135, 80, 49, 40, 15, 8, 6, 0, 1]));
 

$( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + 4 x ( x^{3} + x + 1 ) + 2$ 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 $c$:$30$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{2}(\sqrt{-1})$
Root number: $-i$
$\Aut(K/\Q_{2})$: $C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:$[\frac{8}{3}]$
Visible Swan slopes:$[\frac{5}{3}]$
Means:$\langle\frac{5}{6}\rangle$
Rams:$(5)$
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: \( x^{6} + 2 x^{5} + 4 x^{4} + \left(4 t^{2} + 4 t\right) x^{3} + \left(4 t^{2} + 4 t\right) x + 2 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^4 + z^2 + 1$,$z + t^2$
Associated inertia:$2$,$1$
Indices of inseparability:$[5, 0]$

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: $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2, \frac{8}{3}, \frac{8}{3}]$
Galois Swan slopes: $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3},\frac{5}{3}]$
Galois mean slope: $2.4778645833333335$
Galois splitting model: $x^{18} - 201 x^{16} - 405513 x^{14} + 127996740 x^{12} - 5795689086 x^{10} - 759043749519 x^{8} + 7015312213569 x^{6} + 840182491490265 x^{4} + 3265225350952950 x^{2} - 98345477832273375$ Copy content Toggle raw display