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Properties

Label 15.15.5721702580...4641.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{20}\cdot 71^{12}$
Root discriminant $130.97$
Ramified primes $3, 71$
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![8621, -17184, -414456, -1198347, -1101825, 87771, 624775, 200511, -90183, -48026, 2367, 3387, 90, -96, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 96*x^13 + 90*x^12 + 3387*x^11 + 2367*x^10 - 48026*x^9 - 90183*x^8 + 200511*x^7 + 624775*x^6 + 87771*x^5 - 1101825*x^4 - 1198347*x^3 - 414456*x^2 - 17184*x + 8621)
 
gp: K = bnfinit(x^15 - 3*x^14 - 96*x^13 + 90*x^12 + 3387*x^11 + 2367*x^10 - 48026*x^9 - 90183*x^8 + 200511*x^7 + 624775*x^6 + 87771*x^5 - 1101825*x^4 - 1198347*x^3 - 414456*x^2 - 17184*x + 8621, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 96 x^{13} + 90 x^{12} + 3387 x^{11} + 2367 x^{10} - 48026 x^{9} - 90183 x^{8} + 200511 x^{7} + 624775 x^{6} + 87771 x^{5} - 1101825 x^{4} - 1198347 x^{3} - 414456 x^{2} - 17184 x + 8621 \)

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:  \(57217025805425309006858635774641=3^{20}\cdot 71^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.97$
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:  $3, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(639=3^{2}\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{639}(1,·)$, $\chi_{639}(451,·)$, $\chi_{639}(196,·)$, $\chi_{639}(214,·)$, $\chi_{639}(199,·)$, $\chi_{639}(427,·)$, $\chi_{639}(76,·)$, $\chi_{639}(289,·)$, $\chi_{639}(622,·)$, $\chi_{639}(625,·)$, $\chi_{639}(238,·)$, $\chi_{639}(502,·)$, $\chi_{639}(409,·)$, $\chi_{639}(25,·)$, $\chi_{639}(412,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{37} a^{11} + \frac{8}{37} a^{10} - \frac{11}{37} a^{9} - \frac{2}{37} a^{8} + \frac{5}{37} a^{7} - \frac{16}{37} a^{6} - \frac{9}{37} a^{5} - \frac{10}{37} a^{4} + \frac{9}{37} a^{3} - \frac{15}{37} a$, $\frac{1}{851} a^{12} - \frac{2}{851} a^{11} + \frac{242}{851} a^{10} + \frac{256}{851} a^{9} - \frac{86}{851} a^{8} - \frac{399}{851} a^{7} - \frac{367}{851} a^{6} + \frac{376}{851} a^{5} - \frac{2}{851} a^{4} + \frac{243}{851} a^{3} + \frac{281}{851} a^{2} - \frac{220}{851} a + \frac{11}{23}$, $\frac{1}{31487} a^{13} + \frac{14}{31487} a^{12} + \frac{49}{31487} a^{11} + \frac{6244}{31487} a^{10} - \frac{14643}{31487} a^{9} - \frac{12516}{31487} a^{8} - \frac{748}{31487} a^{7} + \frac{11547}{31487} a^{6} - \frac{196}{31487} a^{5} + \frac{15437}{31487} a^{4} + \frac{6124}{31487} a^{3} + \frac{2574}{31487} a^{2} - \frac{1549}{31487} a - \frac{422}{851}$, $\frac{1}{21090257424948843289102541} a^{14} + \frac{20830916630868465049}{21090257424948843289102541} a^{13} - \frac{10442178626567078766976}{21090257424948843289102541} a^{12} - \frac{34272685903801701072657}{21090257424948843289102541} a^{11} + \frac{1476617342197588494718810}{21090257424948843289102541} a^{10} - \frac{5558747935039766824979279}{21090257424948843289102541} a^{9} - \frac{1474146959994119417661836}{21090257424948843289102541} a^{8} - \frac{310126491337335212303726}{21090257424948843289102541} a^{7} - \frac{6471220492686949910617315}{21090257424948843289102541} a^{6} + \frac{194128873173831865234938}{916967714128210577787067} a^{5} - \frac{1870692539447552370757051}{21090257424948843289102541} a^{4} + \frac{5267604599517193427046059}{21090257424948843289102541} a^{3} + \frac{141901536320789334408579}{916967714128210577787067} a^{2} - \frac{2876347458460177147245409}{21090257424948843289102541} a - \frac{257829004931613475726492}{570006957431049818624393}$

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:  \( 94930923317.51544 \) (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

\(\Q(\zeta_{9})^+\), 5.5.25411681.1

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 $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ $15$ $15$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ $15$ $15$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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}$$[2]^{5}$
$71$71.5.4.1$x^{5} - 71$$5$$1$$4$$C_5$$[\ ]_{5}$
71.5.4.1$x^{5} - 71$$5$$1$$4$$C_5$$[\ ]_{5}$
71.5.4.1$x^{5} - 71$$5$$1$$4$$C_5$$[\ ]_{5}$