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Properties

Label 2.19.1.0a1.1
Base \(\Q_{2}\)
Degree \(19\)
e \(1\)
f \(19\)
c \(0\)
Galois group $C_{19}$ (as 19T1)

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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^19 + x^5 + x^2 + x + 1)
 
Copy content magma:Prec := 100; // Default precision of 100 Q2 := pAdicField(2, Prec); K := LocalField(Q2, Polynomial(Q2, [1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]));
 

\(x^{19} + x^{5} + x^{2} + x + 1\) 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 $d$: $19$
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 $f$:$19$
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:$\Q_{2}$
Root number: $1$
$\Aut(K/\Q_{2})$ $=$ $\Gal(K/\Q_{2})$: $C_{19}$
This field is Galois and abelian over $\Q_{2}.$
Visible Artin slopes:$[\ ]$
Visible Swan slopes:$[\ ]$
Means:$\langle\ \rangle$
Rams:$(\ )$
Jump set:$[1]$
Roots of unity:$1048574 = (2^{ 19 } - 1) \cdot 2$
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 $\Q_{ 2 }$.

Canonical tower

Unramified subfield:2.19.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{19} + x^{5} + x^{2} + x + 1 \) Copy content Toggle raw display
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x - 2 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois degree: $19$
Galois group: $C_{19}$ (as 19T1)
Inertia group: trivial
Wild inertia group: $C_1$
Galois unramified degree: $19$
Galois tame degree: $1$
Galois Artin slopes: $[\ ]$
Galois Swan slopes: $[\ ]$
Galois mean slope: $0.0$
Galois splitting model:$x^{19} - x^{18} - 90 x^{17} + 57 x^{16} + 3044 x^{15} - 1124 x^{14} - 51184 x^{13} + 4822 x^{12} + 474003 x^{11} + 90110 x^{10} - 2465084 x^{9} - 1153239 x^{8} + 6854098 x^{7} + 5023125 x^{6} - 8711114 x^{5} - 8950277 x^{4} + 2600136 x^{3} + 5125792 x^{2} + 1553447 x + 117649$