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Defining polynomial

Copy content comment:Define the p-adic field
 
Copy content sage:Prec = 100 # Default precision of 100 Q5 = Qp(5, Prec); x = polygen(QQ) K.<a> = Q5.extension(x^20 + 10*x^8 + 5)
 
Copy content magma:Prec := 100; // Default precision of 100 Q5 := pAdicField(5, Prec); K := LocalField(Q5, Polynomial(Q5, [5, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]));
 

\(x^{20} + 10 x^{8} + 5\) Copy content Toggle raw display

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

Invariants

Base field: $\Q_{5}$
Copy content comment:Base field Qp
 
Copy content sage:K.base()
 
Copy content magma:Q5;
 
Degree $d$: $20$
Copy content comment:Degree over Qp
 
Copy content sage:K.absolute_degree()
 
Copy content magma:Degree(K);
 
Ramification index $e$:$20$
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 $c$:$27$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{5}(\sqrt{5})$
Root number: $-1$
$\Aut(K/\Q_{5})$: $C_4$
This field is not Galois over $\Q_{5}.$
Visible Artin slopes:$[\frac{3}{2}]$
Visible Swan slopes:$[\frac{1}{2}]$
Means:$\langle\frac{2}{5}\rangle$
Rams:$(2)$
Jump set:$[1, 13]$
Roots of unity:$20 = (5 - 1) \cdot 5$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

$\Q_{5}(\sqrt{5})$, 5.1.4.3a1.1, 5.1.5.6a1.1, 5.1.10.13a2.1

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

Canonical tower

Unramified subfield:$\Q_{5}$
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x^{20} + 10 x^{8} + 5 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{15} + 4 z^{10} + z^5 + 4$,$4 z^4 + 4$
Associated inertia:$1$,$2$
Indices of inseparability:$[8, 0]$

Invariants of the Galois closure

Galois degree: $40$
Galois group: $C_4\times D_5$ (as 20T6)
Inertia group: $C_5:C_4$ (as 20T2)
Wild inertia group: $C_5$
Galois unramified degree: $2$
Galois tame degree: $4$
Galois Artin slopes: $[\frac{3}{2}]$
Galois Swan slopes: $[\frac{1}{2}]$
Galois mean slope: not computed
Galois splitting model:$x^{20} - 10 x^{19} + 40 x^{18} - 75 x^{17} + 45 x^{16} + 48 x^{15} - 5 x^{14} - 295 x^{13} + 535 x^{12} - 285 x^{11} - 261 x^{10} + 375 x^{9} + 50 x^{8} - 275 x^{7} + 65 x^{6} + 92 x^{5} - 35 x^{4} - 25 x^{3} + 10 x^{2} + 5 x + 1$