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Properties

Label 15.1.50558739784...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 3^{20}\cdot 5^{18}\cdot 1459^{4}$
Root discriminant $602.91$
Ramified primes $2, 3, 5, 1459$
Class number $15$ (GRH)
Class group $[15]$ (GRH)
Galois group 15T40

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9378358624, 10724525400, 5043879720, 981089960, -73095900, -29685606, 7999460, 794220, -455490, -36275, 2880, 75, -30, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 15*x^13 - 30*x^12 + 75*x^11 + 2880*x^10 - 36275*x^9 - 455490*x^8 + 794220*x^7 + 7999460*x^6 - 29685606*x^5 - 73095900*x^4 + 981089960*x^3 + 5043879720*x^2 + 10724525400*x + 9378358624)
 
gp: K = bnfinit(x^15 - 15*x^13 - 30*x^12 + 75*x^11 + 2880*x^10 - 36275*x^9 - 455490*x^8 + 794220*x^7 + 7999460*x^6 - 29685606*x^5 - 73095900*x^4 + 981089960*x^3 + 5043879720*x^2 + 10724525400*x + 9378358624, 1)
 

Normalized defining polynomial

\( x^{15} - 15 x^{13} - 30 x^{12} + 75 x^{11} + 2880 x^{10} - 36275 x^{9} - 455490 x^{8} + 794220 x^{7} + 7999460 x^{6} - 29685606 x^{5} - 73095900 x^{4} + 981089960 x^{3} + 5043879720 x^{2} + 10724525400 x + 9378358624 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-505587397847240362501152000000000000000000=-\,2^{23}\cdot 3^{20}\cdot 5^{18}\cdot 1459^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $602.91$
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:  $2, 3, 5, 1459$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4}$, $\frac{1}{232} a^{13} - \frac{11}{232} a^{12} + \frac{1}{116} a^{11} - \frac{5}{116} a^{10} - \frac{23}{232} a^{9} - \frac{3}{232} a^{8} + \frac{11}{116} a^{7} + \frac{27}{116} a^{6} + \frac{51}{116} a^{5} + \frac{45}{116} a^{4} + \frac{9}{29} a^{3} + \frac{6}{29} a^{2} + \frac{14}{29} a + \frac{1}{29}$, $\frac{1}{1727940255898401644064098811719586633598238021575592608} a^{14} - \frac{3623775933345639144548097813792743792117449515388743}{1727940255898401644064098811719586633598238021575592608} a^{13} - \frac{31356880527597643423026414321667672181207934607526911}{863970127949200822032049405859793316799119010787796304} a^{12} - \frac{7042210769134081889793161201332058133964141655699459}{431985063974600411016024702929896658399559505393898152} a^{11} + \frac{69198445515308058347528147902855385387848563994208935}{1727940255898401644064098811719586633598238021575592608} a^{10} - \frac{203424244187300526726741746159440082706038401182060601}{1727940255898401644064098811719586633598238021575592608} a^{9} - \frac{1447797677628758465206891052732994738409156465289357}{14896036688779324517793955273444712358605500185996488} a^{8} - \frac{72209588145321615928293779166420895175915076312637483}{863970127949200822032049405859793316799119010787796304} a^{7} - \frac{214997810527252832341639966971189188027026187767720489}{863970127949200822032049405859793316799119010787796304} a^{6} - \frac{99845229466112591568446356849642612412304557222516947}{863970127949200822032049405859793316799119010787796304} a^{5} - \frac{91851197080080520264075985525591265589755750384712775}{431985063974600411016024702929896658399559505393898152} a^{4} + \frac{97451755560155787787247901522835941685539578218717171}{215992531987300205508012351464948329199779752696949076} a^{3} - \frac{777154246617495349522554086208633751213075157114033}{3724009172194831129448488818361178089651375046499122} a^{2} + \frac{100600984274871558548308496156226008055914564849503793}{215992531987300205508012351464948329199779752696949076} a + \frac{5470298890105622659306762068058021726754503122590205}{53998132996825051377003087866237082299944938174237269}$

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

Class group and class number

$C_{15}$, which has order $15$ (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:  $7$
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:  \( 2356551602180000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T40:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1500
The 40 conjugacy class representatives for [5^3:2]S(3)
Character table for [5^3:2]S(3) is not computed

Intermediate fields

3.1.648.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
3Data not computed
$5$5.5.6.2$x^{5} + 15 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
5.10.12.13$x^{10} + 15 x^{7} + 10 x^{5} + 100 x^{4} + 75 x^{2} + 25$$5$$2$$12$$D_5^2$$[3/2, 3/2]_{2}^{2}$
1459Data not computed