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Properties

Label 15.3.60500833661...2241.3
Degree $15$
Signature $[3, 6]$
Discriminant $3^{20}\cdot 1609^{6}$
Root discriminant $82.94$
Ramified primes $3, 1609$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $\GL(2,4):C_2$ (as 15T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-89027, -2825475, -6780969, -6229693, -2115414, 400995, 498495, 72636, -43227, -13977, 1680, 1062, -18, -45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 45*x^13 - 18*x^12 + 1062*x^11 + 1680*x^10 - 13977*x^9 - 43227*x^8 + 72636*x^7 + 498495*x^6 + 400995*x^5 - 2115414*x^4 - 6229693*x^3 - 6780969*x^2 - 2825475*x - 89027)
 
gp: K = bnfinit(x^15 - 45*x^13 - 18*x^12 + 1062*x^11 + 1680*x^10 - 13977*x^9 - 43227*x^8 + 72636*x^7 + 498495*x^6 + 400995*x^5 - 2115414*x^4 - 6229693*x^3 - 6780969*x^2 - 2825475*x - 89027, 1)
 

Normalized defining polynomial

\( x^{15} - 45 x^{13} - 18 x^{12} + 1062 x^{11} + 1680 x^{10} - 13977 x^{9} - 43227 x^{8} + 72636 x^{7} + 498495 x^{6} + 400995 x^{5} - 2115414 x^{4} - 6229693 x^{3} - 6780969 x^{2} - 2825475 x - 89027 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(60500833661891746134479882241=3^{20}\cdot 1609^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.94$
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:  $3, 1609$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} + \frac{3}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{797710646419637991257116500753019109104} a^{14} + \frac{22105568453764443940859555854129156239}{797710646419637991257116500753019109104} a^{13} + \frac{12136535604011221042343655829290219097}{199427661604909497814279125188254777276} a^{12} - \frac{16502232675916336597936839336757395767}{398855323209818995628558250376509554552} a^{11} + \frac{52959344066238508112008862740675396823}{199427661604909497814279125188254777276} a^{10} + \frac{52213956561597042649194806225589529711}{199427661604909497814279125188254777276} a^{9} - \frac{130860444246762176466613046311024552861}{797710646419637991257116500753019109104} a^{8} - \frac{17396274040133761022732046461839353003}{398855323209818995628558250376509554552} a^{7} - \frac{3450183437206944177085023922725234683}{398855323209818995628558250376509554552} a^{6} - \frac{217348245771545478235991168644052239483}{797710646419637991257116500753019109104} a^{5} + \frac{135459770649068257696997640912743824147}{398855323209818995628558250376509554552} a^{4} - \frac{79210341560949679025836510047022401031}{199427661604909497814279125188254777276} a^{3} - \frac{341884772196397762239563602593839027761}{797710646419637991257116500753019109104} a^{2} - \frac{44859774191391790458818442217943241213}{99713830802454748907139562594127388638} a + \frac{182603755905877159560033506296906618405}{797710646419637991257116500753019109104}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $8$
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:  \( 95269546.8655 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_5$ (as 15T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 12 conjugacy class representatives for $\GL(2,4):C_2$
Character table for $\GL(2,4):C_2$

Intermediate fields

3.3.130329.1, 5.1.1609.1

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

Sibling fields

Degree 15 siblings: data not computed
Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 45 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ R $15$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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
3Data not computed
1609Data not computed