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Properties

Label 15.1.32192049904...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{18}\cdot 3^{12}\cdot 5^{17}\cdot 13^{13}$
Root discriminant $316.60$
Ramified primes $2, 3, 5, 13$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-501418879, 169553865, -238258865, 56139395, -19232005, -636307, 1814005, -790005, -51485, 6615, -1927, 265, -15, -5, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 5*x^13 - 15*x^12 + 265*x^11 - 1927*x^10 + 6615*x^9 - 51485*x^8 - 790005*x^7 + 1814005*x^6 - 636307*x^5 - 19232005*x^4 + 56139395*x^3 - 238258865*x^2 + 169553865*x - 501418879)
 
gp: K = bnfinit(x^15 - 5*x^14 - 5*x^13 - 15*x^12 + 265*x^11 - 1927*x^10 + 6615*x^9 - 51485*x^8 - 790005*x^7 + 1814005*x^6 - 636307*x^5 - 19232005*x^4 + 56139395*x^3 - 238258865*x^2 + 169553865*x - 501418879, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 5 x^{13} - 15 x^{12} + 265 x^{11} - 1927 x^{10} + 6615 x^{9} - 51485 x^{8} - 790005 x^{7} + 1814005 x^{6} - 636307 x^{5} - 19232005 x^{4} + 56139395 x^{3} - 238258865 x^{2} + 169553865 x - 501418879 \)

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:  \(-32192049904498705314600000000000000000=-\,2^{18}\cdot 3^{12}\cdot 5^{17}\cdot 13^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $316.60$
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:  $2, 3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{10} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a + \frac{1}{10}$, $\frac{1}{30} a^{6} + \frac{1}{30} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{3}{10} a - \frac{7}{15}$, $\frac{1}{30} a^{7} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{7}{15} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{30} a^{8} + \frac{1}{3} a^{4} - \frac{7}{15} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{300} a^{9} - \frac{1}{300} a^{8} - \frac{1}{75} a^{7} + \frac{1}{75} a^{6} - \frac{1}{75} a^{5} + \frac{53}{150} a^{4} + \frac{12}{25} a^{3} - \frac{31}{75} a^{2} + \frac{33}{100} a - \frac{49}{300}$, $\frac{1}{3900} a^{10} + \frac{1}{975} a^{9} + \frac{61}{3900} a^{8} + \frac{9}{650} a^{7} + \frac{1}{650} a^{6} + \frac{53}{1950} a^{5} + \frac{49}{150} a^{4} - \frac{3}{50} a^{3} + \frac{73}{300} a^{2} + \frac{61}{150} a + \frac{5}{12}$, $\frac{1}{3900} a^{11} + \frac{1}{650} a^{9} - \frac{7}{1300} a^{8} - \frac{9}{650} a^{7} + \frac{14}{975} a^{6} - \frac{17}{1950} a^{5} - \frac{13}{50} a^{4} - \frac{9}{100} a^{3} + \frac{13}{75} a^{2} - \frac{1}{3} a - \frac{133}{300}$, $\frac{1}{117000} a^{12} + \frac{1}{58500} a^{11} + \frac{1}{19500} a^{10} + \frac{4}{2925} a^{9} + \frac{233}{23400} a^{8} + \frac{2}{125} a^{7} + \frac{7}{2250} a^{6} - \frac{439}{9750} a^{5} + \frac{53}{600} a^{4} + \frac{323}{900} a^{3} - \frac{79}{4500} a^{2} - \frac{73}{750} a + \frac{3427}{9000}$, $\frac{1}{117000} a^{13} + \frac{1}{58500} a^{11} - \frac{1}{58500} a^{10} - \frac{29}{23400} a^{9} - \frac{709}{58500} a^{8} + \frac{89}{5850} a^{7} - \frac{82}{14625} a^{6} + \frac{877}{39000} a^{5} - \frac{106}{225} a^{4} + \frac{159}{500} a^{3} - \frac{239}{900} a^{2} + \frac{4009}{9000} a - \frac{967}{4500}$, $\frac{1}{209347933940892428194040010561000} a^{14} - \frac{38082353711340288837122833}{17445661161741035682836667546750} a^{13} - \frac{288443925965088037362676811}{104673966970446214097020005280500} a^{12} + \frac{11146109792973449873586711299}{104673966970446214097020005280500} a^{11} - \frac{8958970109236577379915572227}{209347933940892428194040010561000} a^{10} + \frac{56263819699602213717504621521}{104673966970446214097020005280500} a^{9} - \frac{713947183874937190405739262091}{104673966970446214097020005280500} a^{8} + \frac{529314195906593779613703758179}{52336983485223107048510002640250} a^{7} - \frac{355620203726205255740485564843}{69782644646964142731346670187000} a^{6} - \frac{2184574506459882665248563937561}{52336983485223107048510002640250} a^{5} + \frac{929412186395302520795540693647}{2683947871037082412744102699500} a^{4} - \frac{2563057866753559001499345620461}{8051843613111247238232308098500} a^{3} - \frac{61768035820231047081873900149}{16103687226222494476464616197000} a^{2} + \frac{2168083585123220152461768197723}{8051843613111247238232308098500} a + \frac{1085967612145489307742103107961}{2683947871037082412744102699500}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  \( 1023778341600.2203 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.1300.1, 5.1.115672050000.14

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

Sibling fields

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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.14.5$x^{10} - 2 x^{5} - 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.15.17.3$x^{15} - 5 x^{13} + 10 x^{12} + 10 x^{11} + 10 x^{9} - 5 x^{8} - 10 x^{6} + 10 x^{5} + 5 x^{3} + 10$$15$$1$$17$$F_5 \times S_3$$[5/4]_{12}^{2}$
$13$13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
13.10.9.1$x^{10} - 13$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$