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Properties

Label 15.7.96094267600...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{8}\cdot 5^{24}\cdot 229^{5}$
Root discriminant $116.28$
Ramified primes $2, 5, 229$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T51

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, -48320, 126420, -150040, 136600, -156072, 142500, -55090, -12230, 9150, 266, 330, -5, -50, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 50*x^13 - 5*x^12 + 330*x^11 + 266*x^10 + 9150*x^9 - 12230*x^8 - 55090*x^7 + 142500*x^6 - 156072*x^5 + 136600*x^4 - 150040*x^3 + 126420*x^2 - 48320*x + 1024)
 
gp: K = bnfinit(x^15 - 50*x^13 - 5*x^12 + 330*x^11 + 266*x^10 + 9150*x^9 - 12230*x^8 - 55090*x^7 + 142500*x^6 - 156072*x^5 + 136600*x^4 - 150040*x^3 + 126420*x^2 - 48320*x + 1024, 1)
 

Normalized defining polynomial

\( x^{15} - 50 x^{13} - 5 x^{12} + 330 x^{11} + 266 x^{10} + 9150 x^{9} - 12230 x^{8} - 55090 x^{7} + 142500 x^{6} - 156072 x^{5} + 136600 x^{4} - 150040 x^{3} + 126420 x^{2} - 48320 x + 1024 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9609426760086059570312500000000=2^{8}\cdot 5^{24}\cdot 229^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.28$
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:  $2, 5, 229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{68} a^{13} + \frac{3}{17} a^{12} - \frac{5}{34} a^{11} - \frac{13}{68} a^{10} - \frac{5}{34} a^{9} + \frac{1}{34} a^{8} + \frac{7}{34} a^{7} - \frac{15}{34} a^{6} + \frac{5}{34} a^{5} + \frac{4}{17} a^{4} + \frac{6}{17} a^{3} - \frac{7}{17} a^{2} - \frac{2}{17} a - \frac{8}{17}$, $\frac{1}{53189091067588701394456348190825792} a^{14} - \frac{4928932034524208260992754276367}{3324318191724293837153521761926612} a^{13} + \frac{5397518489038298331801227377002679}{26594545533794350697228174095412896} a^{12} + \frac{12351912114693270592647881141718427}{53189091067588701394456348190825792} a^{11} - \frac{231541848675644659510833212841475}{1564385031399667688072245535024288} a^{10} + \frac{1031372472700108866393607312529269}{26594545533794350697228174095412896} a^{9} - \frac{555575990773831353685934896595281}{26594545533794350697228174095412896} a^{8} - \frac{8582024948678232884995120099844995}{26594545533794350697228174095412896} a^{7} - \frac{4450030327617810737812534612786585}{26594545533794350697228174095412896} a^{6} - \frac{2085960759400119432611398011755423}{13297272766897175348614087047706448} a^{5} + \frac{2904181405786918402616002536712707}{6648636383448587674307043523853224} a^{4} - \frac{1277877817623652340446153640609621}{6648636383448587674307043523853224} a^{3} + \frac{1748493770169961434457277650155181}{6648636383448587674307043523853224} a^{2} + \frac{1561753280351738337869951971913157}{13297272766897175348614087047706448} a + \frac{258554291809482716910842101547210}{831079547931073459288380440481653}$

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:  $10$
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:  \( 27870703200.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T51:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3000
The 38 conjugacy class representatives for [1/2.D(5)^3]S(3)
Character table for [1/2.D(5)^3]S(3) is not computed

Intermediate fields

3.3.229.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 $15$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed
229Data not computed