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Properties

Label 15.15.9255142598...0721.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{20}\cdot 61^{12}$
Root discriminant $115.99$
Ramified primes $3, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{15}$ (as 15T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![109169, -213990, -477888, 516547, 575889, -455961, -273581, 170109, 60063, -29200, -5847, 2391, 228, -84, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 84*x^13 + 228*x^12 + 2391*x^11 - 5847*x^10 - 29200*x^9 + 60063*x^8 + 170109*x^7 - 273581*x^6 - 455961*x^5 + 575889*x^4 + 516547*x^3 - 477888*x^2 - 213990*x + 109169)
 
gp: K = bnfinit(x^15 - 3*x^14 - 84*x^13 + 228*x^12 + 2391*x^11 - 5847*x^10 - 29200*x^9 + 60063*x^8 + 170109*x^7 - 273581*x^6 - 455961*x^5 + 575889*x^4 + 516547*x^3 - 477888*x^2 - 213990*x + 109169, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 84 x^{13} + 228 x^{12} + 2391 x^{11} - 5847 x^{10} - 29200 x^{9} + 60063 x^{8} + 170109 x^{7} - 273581 x^{6} - 455961 x^{5} + 575889 x^{4} + 516547 x^{3} - 477888 x^{2} - 213990 x + 109169 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9255142598391173348787179150721=3^{20}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.99$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(549=3^{2}\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{549}(1,·)$, $\chi_{549}(34,·)$, $\chi_{549}(325,·)$, $\chi_{549}(70,·)$, $\chi_{549}(424,·)$, $\chi_{549}(142,·)$, $\chi_{549}(367,·)$, $\chi_{549}(400,·)$, $\chi_{549}(241,·)$, $\chi_{549}(436,·)$, $\chi_{549}(184,·)$, $\chi_{549}(217,·)$, $\chi_{549}(58,·)$, $\chi_{549}(508,·)$, $\chi_{549}(253,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{29} a^{12} - \frac{7}{29} a^{11} - \frac{10}{29} a^{10} + \frac{12}{29} a^{9} - \frac{10}{29} a^{7} - \frac{6}{29} a^{6} + \frac{3}{29} a^{5} + \frac{4}{29} a^{4} - \frac{8}{29} a^{3} + \frac{14}{29} a^{2} - \frac{8}{29} a + \frac{2}{29}$, $\frac{1}{29} a^{13} - \frac{1}{29} a^{11} - \frac{3}{29} a^{9} - \frac{10}{29} a^{8} + \frac{11}{29} a^{7} - \frac{10}{29} a^{6} - \frac{4}{29} a^{5} - \frac{9}{29} a^{4} - \frac{13}{29} a^{3} + \frac{3}{29} a^{2} + \frac{4}{29} a + \frac{14}{29}$, $\frac{1}{279885047705981975174083855442723} a^{14} - \frac{1114093766211005207960386707049}{279885047705981975174083855442723} a^{13} - \frac{2264619948963748081898231344751}{279885047705981975174083855442723} a^{12} + \frac{18406777271079361868085624749454}{279885047705981975174083855442723} a^{11} + \frac{101776508439870288086581625645202}{279885047705981975174083855442723} a^{10} + \frac{130584391123979835976490041863614}{279885047705981975174083855442723} a^{9} - \frac{99217073501828183921287281337379}{279885047705981975174083855442723} a^{8} + \frac{50944189905642610926344247432532}{279885047705981975174083855442723} a^{7} - \frac{73276694165533537297391095946461}{279885047705981975174083855442723} a^{6} + \frac{74701407859288418068255260456527}{279885047705981975174083855442723} a^{5} - \frac{136343801615147023814959054022485}{279885047705981975174083855442723} a^{4} - \frac{135701821685613249550641451465647}{279885047705981975174083855442723} a^{3} + \frac{86921326955350936766455653943535}{279885047705981975174083855442723} a^{2} - \frac{16009486796283040792385013249904}{279885047705981975174083855442723} a - \frac{118930034601059602970783342522263}{279885047705981975174083855442723}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $14$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 23247159013.898647 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{15}$ (as 15T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 15
The 15 conjugacy class representatives for $C_{15}$
Character table for $C_{15}$

Intermediate fields

\(\Q(\zeta_{9})^+\), 5.5.13845841.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $15$ R $15$ $15$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.15.20.65$x^{15} + 78 x^{14} + 39 x^{13} + 49 x^{12} + 24 x^{11} + 36 x^{10} + 2 x^{9} + 54 x^{8} + 69 x^{7} + 47 x^{6} + 18 x^{5} + 15 x^{4} + 36 x^{3} + 36 x^{2} + 63 x + 73$$3$$5$$20$$C_{15}$$[2]^{5}$
$61$61.15.12.1$x^{15} + 3050 x^{10} + 1856779 x^{5} + 22698100000$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$