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Properties

Label 15.7.47320369987...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{18}\cdot 5^{4}\cdot 13^{10}\cdot 2131^{2}\cdot 2309^{4}\cdot 127399^{2}$
Root discriminant $2049.60$
Ramified primes $2, 5, 13, 2131, 2309, 127399$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T98

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9457664000, -49652736000, -112427980800, -143254054400, -112357048320, -55720224640, -17240494976, -3119550496, -257694912, 9892440, 3920352, 229632, -8208, -1026, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 1026*x^13 - 8208*x^12 + 229632*x^11 + 3920352*x^10 + 9892440*x^9 - 257694912*x^8 - 3119550496*x^7 - 17240494976*x^6 - 55720224640*x^5 - 112357048320*x^4 - 143254054400*x^3 - 112427980800*x^2 - 49652736000*x - 9457664000)
 
gp: K = bnfinit(x^15 - 1026*x^13 - 8208*x^12 + 229632*x^11 + 3920352*x^10 + 9892440*x^9 - 257694912*x^8 - 3119550496*x^7 - 17240494976*x^6 - 55720224640*x^5 - 112357048320*x^4 - 143254054400*x^3 - 112427980800*x^2 - 49652736000*x - 9457664000, 1)
 

Normalized defining polynomial

\( x^{15} - 1026 x^{13} - 8208 x^{12} + 229632 x^{11} + 3920352 x^{10} + 9892440 x^{9} - 257694912 x^{8} - 3119550496 x^{7} - 17240494976 x^{6} - 55720224640 x^{5} - 112357048320 x^{4} - 143254054400 x^{3} - 112427980800 x^{2} - 49652736000 x - 9457664000 \)

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:  \(47320369987690955076753433023568430483460751360000=2^{18}\cdot 5^{4}\cdot 13^{10}\cdot 2131^{2}\cdot 2309^{4}\cdot 127399^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2049.60$
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:  $2, 5, 13, 2131, 2309, 127399$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{6} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{400} a^{7} - \frac{1}{200} a^{6} - \frac{1}{40} a^{5} + \frac{7}{100} a^{4} + \frac{3}{25} a^{3} + \frac{1}{50} a^{2} + \frac{1}{10} a - \frac{1}{5}$, $\frac{1}{4000} a^{8} - \frac{7}{2000} a^{6} + \frac{1}{500} a^{5} - \frac{31}{250} a^{4} - \frac{31}{250} a^{3} + \frac{7}{500} a^{2} + \frac{1}{5} a - \frac{11}{25}$, $\frac{1}{16000} a^{9} + \frac{3}{8000} a^{7} - \frac{1}{500} a^{6} - \frac{3}{500} a^{5} + \frac{1}{250} a^{4} - \frac{273}{2000} a^{3} - \frac{1}{25} a^{2} - \frac{31}{100} a + \frac{2}{5}$, $\frac{1}{32000} a^{10} - \frac{1}{16000} a^{8} - \frac{1}{1000} a^{7} + \frac{1}{2000} a^{6} + \frac{223}{4000} a^{4} - \frac{73}{500} a^{3} - \frac{169}{1000} a^{2} + \frac{11}{25}$, $\frac{1}{640000} a^{11} + \frac{1}{80000} a^{10} + \frac{9}{320000} a^{9} - \frac{1}{10000} a^{8} - \frac{7}{16000} a^{7} - \frac{249}{20000} a^{6} - \frac{433}{80000} a^{5} - \frac{263}{5000} a^{4} + \frac{1053}{10000} a^{3} - \frac{221}{1000} a^{2} - \frac{113}{1000} a - \frac{3}{50}$, $\frac{1}{1280000} a^{12} - \frac{3}{640000} a^{10} + \frac{1}{40000} a^{9} + \frac{19}{160000} a^{8} + \frac{13}{20000} a^{7} + \frac{211}{32000} a^{6} - \frac{453}{20000} a^{5} + \frac{13}{625} a^{4} - \frac{59}{1250} a^{3} - \frac{443}{2000} a^{2} + \frac{24}{125} a + \frac{12}{25}$, $\frac{1}{25600000000} a^{13} - \frac{231}{640000000} a^{12} + \frac{5367}{12800000000} a^{11} + \frac{11601}{800000000} a^{10} - \frac{15019}{800000000} a^{9} + \frac{38231}{800000000} a^{8} - \frac{515473}{640000000} a^{7} + \frac{1203893}{100000000} a^{6} + \frac{14196387}{800000000} a^{5} - \frac{20025577}{200000000} a^{4} - \frac{8167559}{40000000} a^{3} + \frac{778151}{5000000} a^{2} + \frac{941409}{2000000} a + \frac{93167}{500000}$, $\frac{1}{3276800000000} a^{14} - \frac{11}{819200000000} a^{13} + \frac{59847}{1638400000000} a^{12} - \frac{254863}{409600000000} a^{11} + \frac{735277}{102400000000} a^{10} - \frac{2376493}{102400000000} a^{9} - \frac{6808261}{409600000000} a^{8} + \frac{52549009}{102400000000} a^{7} - \frac{188673389}{102400000000} a^{6} - \frac{37923983}{3200000000} a^{5} + \frac{2632556113}{25600000000} a^{4} - \frac{268123839}{1280000000} a^{3} - \frac{49829763}{1280000000} a^{2} - \frac{9970771}{32000000} a - \frac{89067}{16000000}$

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:  \( 138611432489000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T98:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 71 conjugacy class representatives for [1/2.S(5)^3]3 are not computed
Character table for [1/2.S(5)^3]3 is not computed

Intermediate fields

3.3.169.1

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 $15$ R ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ $15$ $15$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ $15$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.4$x^{6} + 4 x^{2} + 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13Data not computed
2131Data not computed
2309Data not computed
127399Data not computed