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Properties

Label 7.13.1.0a1.1
Base \(\Q_{7}\)
Degree \(13\)
e \(1\)
f \(13\)
c \(0\)
Galois group $C_{13}$ (as 13T1)

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Defining polynomial

Copy content comment:Define the p-adic field
 
Copy content sage:Prec = 100 # Default precision of 100 Q7 = Qp(7, Prec); x = polygen(QQ) K.<a> = Q7.extension(x^13 + 6*x^2 + 4)
 
Copy content magma:Prec := 100; // Default precision of 100 Q7 := pAdicField(7, Prec); K := LocalField(Q7, Polynomial(Q7, [4, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]));
 

\(x^{13} + 6 x^{2} + 4\) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content magma:DefiningPolynomial(K);
 

Invariants

Base field: $\Q_{7}$
Copy content comment:Base field Qp
 
Copy content sage:K.base()
 
Copy content magma:Q7;
 
Degree $d$: $13$
Copy content comment:Degree over Qp
 
Copy content sage:K.absolute_degree()
 
Copy content magma:Degree(K);
 
Ramification index $e$:$1$
Copy content comment:Ramification index
 
Copy content sage:K.absolute_e()
 
Copy content magma:RamificationIndex(K);
 
Residue field degree $f$:$13$
Copy content comment:Residue field degree (Inertia degree)
 
Copy content sage:K.absolute_f()
 
Copy content magma:InertiaDegree(K);
 
Discriminant exponent $c$:$0$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{7}$
Root number: $1$
$\Aut(K/\Q_{7})$ $=$ $\Gal(K/\Q_{7})$: $C_{13}$
This field is Galois and abelian over $\Q_{7}.$
Visible Artin slopes:$[\ ]$
Visible Swan slopes:$[\ ]$
Means:$\langle\ \rangle$
Rams:$(\ )$
Jump set:undefined
Roots of unity:$96889010406 = (7^{ 13 } - 1)$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$.

Canonical tower

Unramified subfield:7.13.1.0a1.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{13} + 6 x^{2} + 4 \) Copy content Toggle raw display
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x - 7 \) $\ \in\Q_{7}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois degree: $13$
Galois group: $C_{13}$ (as 13T1)
Inertia group: trivial
Wild inertia group: $C_1$
Galois unramified degree: $13$
Galois tame degree: $1$
Galois Artin slopes: $[\ ]$
Galois Swan slopes: $[\ ]$
Galois mean slope: $0.0$
Galois splitting model:$x^{13} - x^{12} - 24 x^{11} + 19 x^{10} + 190 x^{9} - 116 x^{8} - 601 x^{7} + 246 x^{6} + 738 x^{5} - 215 x^{4} - 291 x^{3} + 68 x^{2} + 10 x - 1$