Properties

Label 5.2.10.20a3.2
Base \(\Q_{5}\)
Degree \(20\)
e \(10\)
f \(2\)
c \(20\)
Galois group $C_5^2:\OD_{16}$ (as 20T107)

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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) L.<t> = Q5.extension(x^2 + 4*x + 2) K.<a> = L.extension(x^10 + (20*t + 5)*x + 5)
 
Copy content magma:Prec := 100; // Default precision of 100 Q5 := pAdicField(5, Prec); K := LocalField(Q5, Polynomial(Q5, [1029, 20490, 189460, 1075205, 4189440, 11882496, 25390080, 41748480, 53527680, 53941760, 42904960, 26970880, 13381920, 5218560, 1586880, 371328, 65460, 8400, 740, 40, 1]));
 

$( x^{2} + 4 x + 2 )^{10} + 5 x ( x^{2} + 4 x + 2 ) + 5$ 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 $e$:$10$
Copy content comment:Ramification index
 
Copy content sage:K.absolute_e()
 
Copy content magma:RamificationIndex(K);
 
Residue field degree $f$:$2$
Copy content comment:Residue field degree (Inertia degree)
 
Copy content sage:K.absolute_f()
 
Copy content magma:InertiaDegree(K);
 
Discriminant exponent $c$:$20$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{5}$
Root number: $1$
$\Aut(K/\Q_{5})$: $C_1$
This field is not Galois over $\Q_{5}.$
Visible Artin slopes:$[\frac{9}{8}]$
Visible Swan slopes:$[\frac{1}{8}]$
Means:$\langle\frac{1}{10}\rangle$
Rams:$(\frac{1}{4})$
Jump set:undefined
Roots of unity:$24 = (5^{ 2 } - 1)$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

$\Q_{5}(\sqrt{2})$, $\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5\cdot 2})$, 5.2.2.2a1.2

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

Canonical tower

Unramified subfield:$\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} + 4 x + 2 \) Copy content Toggle raw display
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x^{10} + \left(20 t + 5\right) x + 5 \) $\ \in\Q_{5}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^5 + 2$,$2 z + t$
Associated inertia:$1$,$1$
Indices of inseparability:$[1, 0]$

Invariants of the Galois closure

Galois degree: $400$
Galois group: $C_5^2:\OD_{16}$ (as 20T107)
Inertia group: Intransitive group isomorphic to $C_5^2:C_8$
Wild inertia group: $C_5^2$
Galois unramified degree: $2$
Galois tame degree: $8$
Galois Artin slopes: not computed
Galois Swan slopes: not computed
Galois mean slope: not computed
Galois splitting model: $x^{20} + 352830 x^{18} - 4218060 x^{17} + 47154323955 x^{16} - 791063046252 x^{15} + 2984460294660200 x^{14} - 43120598007534720 x^{13} + 91938842206127448415 x^{12} - 547733130795410297920 x^{11} + 1360705096955545410501614 x^{10} - 4435031111938124581414700 x^{9} + 10347298524118653617459360785 x^{8} - 34179710250256145046855856620 x^{7} + 41159663184685131194408196587860 x^{6} - 154629645274942129300260075284488 x^{5} + 80528565150310879384866797286884800 x^{4} - 251796253573501789899691823964253160 x^{3} + 63890892005639586544510396028070856120 x^{2} - 117721419619697979372523158708109640960 x + 8357636003163370514676239692349531807036$ Copy content Toggle raw display