Properties

Label 5.2.10.20a7.1
Base \(\Q_{5}\)
Degree \(20\)
e \(10\)
f \(2\)
c \(20\)
Galois group $C_5^2:\OD_{16}$ (as 20T115)

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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 + (15*t + 5)*x + 5)
 
Copy content magma:Prec := 100; // Default precision of 100 Q5 := pAdicField(5, Prec); K := LocalField(Q5, Polynomial(Q5, [1069, 20580, 189500, 1075210, 4189440, 11882496, 25390080, 41748480, 53527680, 53941760, 42904960, 26970880, 13381920, 5218560, 1586880, 371328, 65460, 8400, 740, 40, 1]));
 

$( x^{2} + 4 x + 2 )^{10} + \left(10 x + 20\right) ( x^{2} + 4 x + 2 ) + 5$ 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: $1$
$\Aut(K/\Q_{5})$: $C_2$
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, 5.1.10.10a1.3

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: \( x^{10} + \left(15 t + 5\right) x + 5 \) $\ \in\Q_{5}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^5 + 2$,$2 z + (3 t + 1)$
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 20T115)
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: $x^{20} + 865575010 x^{18} - 24884980 x^{17} + 256113858597817305 x^{16} + 7948377664711652 x^{15} + 31711128920868753905384740 x^{14} - 20000337957633058711333680 x^{13} + 2039736151082075999289926921781400 x^{12} - 11348324557227138103093068987042680 x^{11} + 77185317738691745317039131809179289270360 x^{10} - 838993049396563243548448434315228102351280 x^{9} + 1816892689659575402377334752059417040738503233160 x^{8} - 26143511068523818247751089613306940989280711906800 x^{7} + 26776526327970922236981451918945019388287994738699181200 x^{6} - 412434482772585311079911522229538252505305896259105687616 x^{5} + 236967121330834698693681613636114619946217467666716729425577680 x^{4} - 3296085796668753311900336655931119525214381455485275522434711520 x^{3} + 1125574659211001261491142522522715580136771678682254274975905636805440 x^{2} - 10884651012181132417737823014975147376388840729987057805431461743872640 x + 2156076475410504564756848931566998580467677590915953022486985974533455867024$ Copy content Toggle raw display