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Properties

Label 15.15.2231784480...4761.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{20}\cdot 401^{7}\cdot 19593331^{2}$
Root discriminant $665.65$
Ramified primes $3, 401, 19593331$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T45

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-44516048032, -177750698832, -246875970600, -152139299988, -37671758364, 1781931375, 2496923998, 250121274, -51480954, -8513738, 437550, 106245, -1318, -564, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 564*x^13 - 1318*x^12 + 106245*x^11 + 437550*x^10 - 8513738*x^9 - 51480954*x^8 + 250121274*x^7 + 2496923998*x^6 + 1781931375*x^5 - 37671758364*x^4 - 152139299988*x^3 - 246875970600*x^2 - 177750698832*x - 44516048032)
 
gp: K = bnfinit(x^15 - 564*x^13 - 1318*x^12 + 106245*x^11 + 437550*x^10 - 8513738*x^9 - 51480954*x^8 + 250121274*x^7 + 2496923998*x^6 + 1781931375*x^5 - 37671758364*x^4 - 152139299988*x^3 - 246875970600*x^2 - 177750698832*x - 44516048032, 1)
 

Normalized defining polynomial

\( x^{15} - 564 x^{13} - 1318 x^{12} + 106245 x^{11} + 437550 x^{10} - 8513738 x^{9} - 51480954 x^{8} + 250121274 x^{7} + 2496923998 x^{6} + 1781931375 x^{5} - 37671758364 x^{4} - 152139299988 x^{3} - 246875970600 x^{2} - 177750698832 x - 44516048032 \)

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:  \(2231784480979129609810320595530550951774761=3^{20}\cdot 401^{7}\cdot 19593331^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $665.65$
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:  $3, 401, 19593331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{10} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{177946531088217032930927569037646403166594090780544239630091632} a^{14} + \frac{1543429463133703546912612072255605729812580765027941653593985}{44486632772054258232731892259411600791648522695136059907522908} a^{13} - \frac{3359900032454051207408471939305795607240819374092800797654337}{44486632772054258232731892259411600791648522695136059907522908} a^{12} + \frac{8854948843637985375234465670258876578536509125776966745217053}{88973265544108516465463784518823201583297045390272119815045816} a^{11} + \frac{3292497155836643052134457038856137574334654851058431970577477}{177946531088217032930927569037646403166594090780544239630091632} a^{10} + \frac{19235117282736368897856529403254974157526038849071362026960477}{88973265544108516465463784518823201583297045390272119815045816} a^{9} + \frac{6941270345657088881825940914568861314150928528088421413676867}{88973265544108516465463784518823201583297045390272119815045816} a^{8} + \frac{17926980777827499610715501234687956036225797708784972590372719}{88973265544108516465463784518823201583297045390272119815045816} a^{7} - \frac{15183931096901655643019247791331362539935600141201188882877179}{88973265544108516465463784518823201583297045390272119815045816} a^{6} + \frac{25981977045838856045760911578028553898326917937927683561885923}{88973265544108516465463784518823201583297045390272119815045816} a^{5} + \frac{18920973748260140110998454768414761972520599807148252014293927}{177946531088217032930927569037646403166594090780544239630091632} a^{4} + \frac{253342150210844522595879838097017605341524310229147930398403}{11121658193013564558182973064852900197912130673784014976880727} a^{3} - \frac{19544380990919743931198704214554605105394176459364266749868271}{44486632772054258232731892259411600791648522695136059907522908} a^{2} - \frac{5355328523971455892418245251887706926694102944477154755605339}{11121658193013564558182973064852900197912130673784014976880727} a - \frac{1077849787239094577471586211530584454117432041432469709439655}{11121658193013564558182973064852900197912130673784014976880727}$

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:  \( 35897750262200000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T45:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2430
The 45 conjugacy class representatives for 1/2[3^5:2]D(5)
Character table for 1/2[3^5:2]D(5) is not computed

Intermediate fields

5.5.160801.1

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 30 siblings: data not computed
Degree 45 siblings: 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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.6.8.8$x^{6} + 3 x^{5} + 63$$3$$2$$8$$S_3\times C_3$$[2, 2]^{2}$
3.6.8.10$x^{6} + 6 x^{5} + 36$$3$$2$$8$$S_3\times C_3$$[2, 2]^{2}$
401Data not computed
19593331Data not computed