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Properties

Label 7.2.7.14a8.1
Base \(\Q_{7}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(14\)
Galois group $C_7^2:C_{12}$ (as 14T23)

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Defining polynomial

Copy content comment:Define the p-adic field
 
Copy content sage:Prec = 100 # Default precision of 100 Q7 = Qp(7, Prec); x = polygen(QQ) L.<t> = Q7.extension(x^2 + 6*x + 3) K.<a> = L.extension(x^7 + (7*t + 35)*x + 7)
 
Copy content magma:Prec := 100; // Default precision of 100 Q7 := pAdicField(7, Prec); K := LocalField(Q7, Polynomial(Q7, [2320, 30996, 189105, 673638, 1536003, 2337174, 2411451, 1690632, 803817, 259686, 56889, 8316, 777, 42, 1]));
 

$( x^{2} + 6 x + 3 )^{7} + \left(42 x + 42\right) ( x^{2} + 6 x + 3 ) + 7$ Copy content Toggle raw display

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

Invariants

Base field: $\Q_{7}$
Copy content comment:Base field Qp
 
Copy content sage:K.base()
 
Copy content magma:Q7;
 
Degree $d$: $14$
Copy content comment:Degree over Qp
 
Copy content sage:K.absolute_degree()
 
Copy content magma:Degree(K);
 
Ramification index $e$:$7$
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 $c$:$14$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{7}$
Root number: $-1$
$\Aut(K/\Q_{7})$: $C_1$
This field is not Galois over $\Q_{7}.$
Visible Artin slopes:$[\frac{7}{6}]$
Visible Swan slopes:$[\frac{1}{6}]$
Means:$\langle\frac{1}{7}\rangle$
Rams:$(\frac{1}{6})$
Jump set:undefined
Roots of unity:$48 = (7^{ 2 } - 1)$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

$\Q_{7}(\sqrt{3})$

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

Canonical tower

Unramified subfield:$\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} + 6 x + 3 \) Copy content Toggle raw display
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x^{7} + \left(7 t + 35\right) x + 7 \) $\ \in\Q_{7}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + (4 t + 3)$
Associated inertia:$1$
Indices of inseparability:$[1, 0]$

Invariants of the Galois closure

Galois degree: $588$
Galois group: $C_7^2:C_{12}$ (as 14T23)
Inertia group: Intransitive group isomorphic to $C_7:F_7$
Wild inertia group: $C_7^2$
Galois unramified degree: $2$
Galois tame degree: $6$
Galois Artin slopes: $[\frac{7}{6}, \frac{7}{6}]$
Galois Swan slopes: $[\frac{1}{6},\frac{1}{6}]$
Galois mean slope: $1.1598639455782314$
Galois splitting model: $x^{14} - 4643636887097677602 x^{12} - 1443286778608578361627275785 x^{11} + 6718227867057528796987429463376792693 x^{10} + 3529053853519782968165435040187622080042494119 x^{9} - 2626691794418043870866606803193450499094651729778400594 x^{8} - 1414075916397590878498395147581368637689534854254083246677572468 x^{7} + 415123910212475925089730923014801557701610110581036459187400521656563331 x^{6} + 167024087357848463296702561369362099459818469496992887556269940465746181240958051 x^{5} - 26637862759471051975937392862907589616434828905665337471500468818395123531649302311637312 x^{4} - 2159633478622975679298387120540215721694236528482531790242413817546422066978111440214399356382558 x^{3} + 142799477781558446842542542628476481883297876818076019509301166395430043748711017205499734793224348479844 x^{2} + 8117664833999094473265640050208775005968307328115561535966219327047185991859485757524695309642294557627723598110 x + 92642747175887659868022379286736904675691422821608409863304969812995607176163358769523734616609312284579577622679758397$ Copy content Toggle raw display