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Properties

Label 197.4.4.12a1.4
Base \(\Q_{197}\)
Degree \(16\)
e \(4\)
f \(4\)
c \(12\)
Galois group $C_4^2$ (as 16T4)

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Defining polynomial

Copy content comment:Define the p-adic field
 
Copy content sage:Prec = 100 # Default precision of 100 Q197 = Qp(197, Prec); x = polygen(QQ) L.<t> = Q197.extension(x^4 + 16*x^2 + 124*x + 2) K.<a> = L.extension(x^4 + 197)
 
Copy content magma:Prec := 100; // Default precision of 100 Q197 := pAdicField(197, Prec); K := LocalField(Q197, Polynomial(Q197, [213, 3968, 369536, 15348224, 242331936, 122791744, 24020096, 9753344, 3023896, 383904, 109024, 23808, 1544, 496, 64, 0, 1]));
 

$( x^{4} + 16 x^{2} + 124 x + 2 )^{4} + 197$ Copy content Toggle raw display

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

Invariants

Base field: $\Q_{197}$
Copy content comment:Base field Qp
 
Copy content sage:K.base()
 
Copy content magma:Q197;
 
Degree $d$: $16$
Copy content comment:Degree over Qp
 
Copy content sage:K.absolute_degree()
 
Copy content magma:Degree(K);
 
Ramification index $e$:$4$
Copy content comment:Ramification index
 
Copy content sage:K.absolute_e()
 
Copy content magma:RamificationIndex(K);
 
Residue field degree $f$:$4$
Copy content comment:Residue field degree (Inertia degree)
 
Copy content sage:K.absolute_f()
 
Copy content magma:InertiaDegree(K);
 
Discriminant exponent $c$:$12$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{197}$
Root number: $-1$
$\Aut(K/\Q_{197})$ $=$ $\Gal(K/\Q_{197})$: $C_4^2$
This field is Galois and abelian over $\Q_{197}.$
Visible Artin slopes:$[\ ]$
Visible Swan slopes:$[\ ]$
Means:$\langle\ \rangle$
Rams:$(\ )$
Jump set:undefined
Roots of unity:$1506138480 = (197^{ 4 } - 1)$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

$\Q_{197}(\sqrt{2})$, $\Q_{197}(\sqrt{197})$, $\Q_{197}(\sqrt{197\cdot 2})$, 197.4.1.0a1.1, 197.2.2.2a1.2, 197.2.2.2a1.1, 197.1.4.3a1.3, 197.1.4.3a1.1, 197.1.4.3a1.2, 197.1.4.3a1.4, 197.4.2.4a1.2, 197.2.4.6a1.2, 197.2.4.6a1.4

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

Canonical tower

Unramified subfield:197.4.1.0a1.1 $\cong \Q_{197}(t)$ where $t$ is a root of \( x^{4} + 16 x^{2} + 124 x + 2 \) Copy content Toggle raw display
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x^{4} + 197 \) $\ \in\Q_{197}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^3 + 4 z^2 + 6 z + 4$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois degree: $16$
Galois group: $C_4^2$ (as 16T4)
Inertia group: Intransitive group isomorphic to $C_4$
Wild inertia group: $C_1$
Galois unramified degree: $4$
Galois tame degree: $4$
Galois Artin slopes: $[\ ]$
Galois Swan slopes: $[\ ]$
Galois mean slope: $0.75$
Galois splitting model:not computed