Properties

Label 151.2.10.18a1.4
Base \(\Q_{151}\)
Degree \(20\)
e \(10\)
f \(2\)
c \(18\)
Galois group $C_{20}$ (as 20T1)

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Defining polynomial

Copy content comment:Define the p-adic field
 
Copy content sage:Prec = 100 # Default precision of 100 Q151 = Qp(151, Prec); x = polygen(QQ) L.<t> = Q151.extension(x^2 + 149*x + 6) K.<a> = L.extension(x^10 + (19781*t + 14496))
 
Copy content magma:Prec := 100; // Default precision of 100 Q151 := pAdicField(151, Prec); K := LocalField(Q151, Polynomial(Q151, [60480672, 15015786821, 1678112743680, 111144205002240, 4831400493613440, 144038716425848448, 2982948504032080800, 42380902122607330560, 395542177663790122740, 2193009642063025034700, 5525082379989270194053, 365501607010504172450, 10987282712883058965, 196207880197256160, 2301657796321050, 18523497482748, 103553679990, 397034340, 999105, 1490, 1]));
 

$( x^{2} + 149 x + 6 )^{10} + 19781 x + 14496$ Copy content Toggle raw display

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

Invariants

Base field: $\Q_{151}$
Copy content comment:Base field Qp
 
Copy content sage:K.base()
 
Copy content magma:Q151;
 
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 $c$:$18$
Copy content comment:Discriminant exponent
 
Copy content magma:Valuation(Discriminant(K));
 
Discriminant root field:$\Q_{151}(\sqrt{3})$
Root number: $-1$
$\Aut(K/\Q_{151})$ $=$ $\Gal(K/\Q_{151})$: $C_{20}$
This field is Galois and abelian over $\Q_{151}.$
Visible Artin slopes:$[\ ]$
Visible Swan slopes:$[\ ]$
Means:$\langle\ \rangle$
Rams:$(\ )$
Jump set:undefined
Roots of unity:$22800 = (151^{ 2 } - 1)$
Copy content comment:Roots of unity
 
Copy content sage:len(K.roots_of_unity())
 

Intermediate fields

$\Q_{151}(\sqrt{3})$, 151.2.2.2a1.1, 151.1.5.4a1.1, 151.2.5.8a1.2

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

Canonical tower

Unramified subfield:$\Q_{151}(\sqrt{3})$ $\cong \Q_{151}(t)$ where $t$ is a root of \( x^{2} + 149 x + 6 \) Copy content Toggle raw display
Copy content comment:Maximal unramified subextension
 
Copy content sage:K.maximal_unramified_subextension()
 
Relative Eisenstein polynomial: \( x^{10} + 19781 t + 14496 \) $\ \in\Q_{151}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^9 + 10 z^8 + 45 z^7 + 120 z^6 + 59 z^5 + 101 z^4 + 59 z^3 + 120 z^2 + 45 z + 10$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois degree: $20$
Galois group: $C_{20}$ (as 20T1)
Inertia group: Intransitive group isomorphic to $C_{10}$
Wild inertia group: $C_1$
Galois unramified degree: $2$
Galois tame degree: $10$
Galois Artin slopes: $[\ ]$
Galois Swan slopes: $[\ ]$
Galois mean slope: $0.9$
Galois splitting model:not computed