↑

Properties

Label 3.17.1.0a1.1
Base \(\Q_{3}\)
Degree \(17\)
e \(1\)
f \(17\)
c \(0\)
Galois group $C_{17}$ (as 17T1)

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

Defining polynomial

Copy content comment:Define the p-adic field
 
Copy content sage:Prec = 100 # Default precision of 100 Q3 = Qp(3, Prec); x = polygen(QQ) K.<a> = Q3.extension(x^17 + 2*x + 1)
 
Copy content magma:Prec := 100; // Default precision of 100 Q3 := pAdicField(3, Prec); K := LocalField(Q3, Polynomial(Q3, [1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]));
 

\(x^{17} + 2 x + 1\) Copy content Toggle raw display

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

Invariants

Base field: $\Q_{3}$
Copy content comment:Base field Qp
 
Copy content sage:K.base()
 
Copy content magma:Q3;
 
Degree $d$: $17$
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 $f$:$17$
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:$\Q_{3}$
Root number: $1$
$\Aut(K/\Q_{3})$ $=$ $\Gal(K/\Q_{3})$: $C_{17}$
This field is Galois and abelian over $\Q_{3}.$
Visible Artin slopes:$[\ ]$
Visible Swan slopes:$[\ ]$
Means:$\langle\ \rangle$
Rams:$(\ )$
Jump set:undefined
Roots of unity:$129140162 = (3^{ 17 } - 1)$
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 $\Q_{ 3 }$.

Canonical tower

Unramified subfield:3.17.1.0a1.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{17} + 2 x + 1 \) Copy content Toggle raw display
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x - 3 \) $\ \in\Q_{3}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois degree: $17$
Galois group: $C_{17}$ (as 17T1)
Inertia group: trivial
Wild inertia group: $C_1$
Galois unramified degree: $17$
Galois tame degree: $1$
Galois Artin slopes: $[\ ]$
Galois Swan slopes: $[\ ]$
Galois mean slope: $0.0$
Galois splitting model:$x^{17} - x^{16} - 48 x^{15} + 105 x^{14} + 763 x^{13} - 2579 x^{12} - 3653 x^{11} + 23311 x^{10} - 11031 x^{9} - 74838 x^{8} + 107759 x^{7} + 50288 x^{6} - 198615 x^{5} + 102976 x^{4} + 58507 x^{3} - 75722 x^{2} + 25763 x - 2837$