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Properties

Label 15.5.43014554344...0000.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{24}\cdot 5^{6}\cdot 37^{5}\cdot 683^{2}\cdot 225223^{2}$
Root discriminant $237.45$
Ramified primes $2, 5, 37, 683, 225223$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T102

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16000, 56000, 23200, -206200, 64080, 179240, -118584, -4484, 14904, 10602, -3400, -2132, -432, -54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 54*x^13 - 432*x^12 - 2132*x^11 - 3400*x^10 + 10602*x^9 + 14904*x^8 - 4484*x^7 - 118584*x^6 + 179240*x^5 + 64080*x^4 - 206200*x^3 + 23200*x^2 + 56000*x - 16000)
 
gp: K = bnfinit(x^15 - 54*x^13 - 432*x^12 - 2132*x^11 - 3400*x^10 + 10602*x^9 + 14904*x^8 - 4484*x^7 - 118584*x^6 + 179240*x^5 + 64080*x^4 - 206200*x^3 + 23200*x^2 + 56000*x - 16000, 1)
 

Normalized defining polynomial

\( x^{15} - 54 x^{13} - 432 x^{12} - 2132 x^{11} - 3400 x^{10} + 10602 x^{9} + 14904 x^{8} - 4484 x^{7} - 118584 x^{6} + 179240 x^{5} + 64080 x^{4} - 206200 x^{3} + 23200 x^{2} + 56000 x - 16000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-430145543444459125122090139648000000=-\,2^{24}\cdot 5^{6}\cdot 37^{5}\cdot 683^{2}\cdot 225223^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $237.45$
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:  $2, 5, 37, 683, 225223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{20} a^{11} + \frac{1}{20} a^{9} - \frac{1}{10} a^{8} - \frac{1}{10} a^{7} + \frac{1}{10} a^{5} + \frac{1}{5} a^{4} + \frac{3}{10} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{40} a^{12} - \frac{1}{10} a^{10} - \frac{1}{20} a^{9} + \frac{1}{5} a^{8} + \frac{1}{20} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{10} a^{3}$, $\frac{1}{400} a^{13} + \frac{3}{200} a^{11} - \frac{2}{25} a^{10} + \frac{7}{100} a^{9} + \frac{1}{5} a^{8} + \frac{41}{200} a^{7} - \frac{6}{25} a^{6} + \frac{9}{100} a^{5} + \frac{7}{50} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{1868421490007687814626029498400} a^{14} - \frac{556892251757168320302463079}{934210745003843907313014749200} a^{13} + \frac{1460411413513645579376558313}{934210745003843907313014749200} a^{12} - \frac{990243985983377486944265049}{93421074500384390731301474920} a^{11} + \frac{43423075263451896641296529251}{467105372501921953656507374600} a^{10} - \frac{23592563823144333881246364773}{233552686250960976828253687300} a^{9} - \frac{66254262609306839231002243299}{934210745003843907313014749200} a^{8} - \frac{85004118920270396948244141813}{467105372501921953656507374600} a^{7} + \frac{82863201501267910996207146611}{467105372501921953656507374600} a^{6} - \frac{35398229817453975582363010697}{116776343125480488414126843650} a^{5} - \frac{116510575807100698529924823941}{233552686250960976828253687300} a^{4} + \frac{2579223933343354105202211667}{23355268625096097682825368730} a^{3} - \frac{4998755168406385891310025503}{46710537250192195365650737460} a^{2} + \frac{300681363340575334028670065}{4671053725019219536565073746} a - \frac{687192631346884727752855182}{2335526862509609768282536873}$

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:  $9$
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:  \( 14859928654700 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T102:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10368000
The 140 conjugacy class representatives for [S(5)^3]S(3)=S(5)wrS(3) are not computed
Character table for [S(5)^3]S(3)=S(5)wrS(3) is not computed

Intermediate fields

3.3.148.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: 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 R ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.11.5$x^{6} + 6$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.14$x^{6} + 2 x^{4} + 10$$6$$1$$11$$S_4\times C_2$$[4/3, 4/3, 3]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37Data not computed
683Data not computed
225223Data not computed