Properties

Label 2.1.14.26a1.45
Base \(\Q_{2}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(26\)
Galois group $C_2\wr C_7:C_3$ (as 14T44)

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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^14 + 2*x^13 + 4*x^9 + 4*x^7 + 4*x^3 + 2)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [2, 0, 0, 4, 0, 0, 0, 4, 0, 4, 0, 0, 0, 2, 1]));
 

\(x^{14} + 2 x^{13} + 4 x^{9} + 4 x^{7} + 4 x^{3} + 2\) 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 $d$: $14$
Copy content comment:Degree over Qp
 
Copy content sage:K.absolute_degree()
 
Copy content magma:Degree(K);
 
Ramification index $e$:$14$
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:$\Q_{2}(\sqrt{5})$
Root number: $1$
$\Aut(K/\Q_{2})$: $C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:$[\frac{20}{7}]$
Visible Swan slopes:$[\frac{13}{7}]$
Means:$\langle\frac{13}{14}\rangle$
Rams:$(13)$
Jump set:$[7, 21]$
Roots of unity:$2$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

2.1.7.6a1.1

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: \( x^{14} + 2 x^{13} + 4 x^{9} + 4 x^{7} + 4 x^{3} + 2 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1$,$z + 1$
Associated inertia:$3$,$1$
Indices of inseparability:$[13, 0]$

Invariants of the Galois closure

Galois degree: $2688$
Galois group: $C_2\wr C_7:C_3$ (as 14T44)
Inertia group: $C_2^3:F_8$ (as 14T21)
Wild inertia group: $C_2^6$
Galois unramified degree: $6$
Galois tame degree: $7$
Galois Artin slopes: $[\frac{18}{7}, \frac{18}{7}, \frac{18}{7}, \frac{20}{7}, \frac{20}{7}, \frac{20}{7}]$
Galois Swan slopes: $[\frac{11}{7},\frac{11}{7},\frac{11}{7},\frac{13}{7},\frac{13}{7},\frac{13}{7}]$
Galois mean slope: $2.794642857142857$
Galois splitting model: $x^{14} - 21 x^{12} - 217 x^{10} + 245 x^{8} + 945 x^{6} - 945 x^{4} - 189 x^{2} + 243$ Copy content Toggle raw display