Properties

Label 2.7.2.21a1.15
Base \(\Q_{2}\)
Degree \(14\)
e \(2\)
f \(7\)
c \(21\)
Galois group $C_2 \wr C_7$ (as 14T29)

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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^7 + x + 1) K.<a> = L.extension(x^2 + (4*t^6 + 4*t)*x + 2)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [3, 6, 5, 4, 8, 4, 0, 2, 6, 0, 4, 4, 0, 0, 1]));
 

$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4 x\right) ( x^{7} + x + 1 ) + 2$ 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 $d$: $14$
Copy content comment:Degree over Qp
 
Copy content sage:K.absolute_degree()
 
Copy content magma:Degree(K);
 
Ramification index $e$:$2$
Copy content comment:Ramification index
 
Copy content sage:K.absolute_e()
 
Copy content magma:RamificationIndex(K);
 
Residue field degree $f$:$7$
Copy content comment:Residue field degree (Inertia degree)
 
Copy content sage:K.absolute_f()
 
Copy content magma:InertiaDegree(K);
 
Discriminant exponent $c$:$21$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{2}(\sqrt{-2})$
Root number: $-i$
$\Aut(K/\Q_{2})$: $C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:$[3]$
Visible Swan slopes:$[2]$
Means:$\langle1\rangle$
Rams:$(2)$
Jump set:$[1, 3]$
Roots of unity:$254 = (2^{ 7 } - 1) \cdot 2$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

2.7.1.0a1.1

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

Canonical tower

Unramified subfield:2.7.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{7} + x + 1 \) Copy content Toggle raw display
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x^{2} + \left(4 t^{6} + 4 t\right) x + 2 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois degree: $896$
Galois group: $C_2\wr C_7$ (as 14T29)
Inertia group: Intransitive group isomorphic to $C_2^7$
Wild inertia group: $C_2^7$
Galois unramified degree: $7$
Galois tame degree: $1$
Galois Artin slopes: $[2, 2, 2, 2, 2, 2, 3]$
Galois Swan slopes: $[1,1,1,1,1,1,2]$
Galois mean slope: $2.484375$
Galois splitting model: $x^{14} - 14 x^{12} - 56 x^{10} + 560 x^{8} + 784 x^{6} - 4480 x^{4} + 1344 x^{2} + 128$ Copy content Toggle raw display