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Properties

Label 15.11.9032200559...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{24}\cdot 3^{15}\cdot 5^{6}\cdot 7^{9}\cdot 13^{4}\cdot 456446131^{2}$
Root discriminant $1574.17$
Ramified primes $2, 3, 5, 7, 13, 456446131$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T100

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1456000, 15288000, 47174400, 29720600, -32825520, -32517240, 4954824, 8908788, 204120, -715554, 8208, 15414, -2808, 324, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 324*x^13 - 2808*x^12 + 15414*x^11 + 8208*x^10 - 715554*x^9 + 204120*x^8 + 8908788*x^7 + 4954824*x^6 - 32517240*x^5 - 32825520*x^4 + 29720600*x^3 + 47174400*x^2 + 15288000*x + 1456000)
 
gp: K = bnfinit(x^15 + 324*x^13 - 2808*x^12 + 15414*x^11 + 8208*x^10 - 715554*x^9 + 204120*x^8 + 8908788*x^7 + 4954824*x^6 - 32517240*x^5 - 32825520*x^4 + 29720600*x^3 + 47174400*x^2 + 15288000*x + 1456000, 1)
 

Normalized defining polynomial

\( x^{15} + 324 x^{13} - 2808 x^{12} + 15414 x^{11} + 8208 x^{10} - 715554 x^{9} + 204120 x^{8} + 8908788 x^{7} + 4954824 x^{6} - 32517240 x^{5} - 32825520 x^{4} + 29720600 x^{3} + 47174400 x^{2} + 15288000 x + 1456000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(903220055915539404331852313843496708734976000000=2^{24}\cdot 3^{15}\cdot 5^{6}\cdot 7^{9}\cdot 13^{4}\cdot 456446131^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1574.17$
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:  $2, 3, 5, 7, 13, 456446131$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{20} a^{11} - \frac{1}{20} a^{9} + \frac{1}{10} a^{8} + \frac{1}{5} a^{7} - \frac{1}{10} a^{6} - \frac{1}{5} a^{5} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{12} + \frac{1}{10} a^{10} + \frac{1}{20} a^{9} - \frac{3}{20} a^{8} + \frac{1}{5} a^{7} + \frac{3}{20} a^{6} - \frac{1}{2} a^{5} - \frac{3}{10} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{400} a^{13} + \frac{1}{100} a^{11} - \frac{1}{50} a^{10} + \frac{17}{200} a^{9} - \frac{2}{25} a^{8} - \frac{17}{200} a^{7} - \frac{1}{10} a^{6} - \frac{33}{100} a^{5} + \frac{3}{50} a^{4} - \frac{1}{2} a$, $\frac{1}{41528291911815305038478803115820795630719200} a^{14} + \frac{477314095000349559688137606100421432123}{5191036488976913129809850389477599453839900} a^{13} + \frac{26131455776175410808788667297529135116403}{5191036488976913129809850389477599453839900} a^{12} - \frac{15276421165484056357269251852885044263751}{1297759122244228282452462597369399863459975} a^{11} - \frac{2293556830590357795714010440076585955449129}{20764145955907652519239401557910397815359600} a^{10} - \frac{21209759058924522228592891815064793354568}{1297759122244228282452462597369399863459975} a^{9} - \frac{1862612650275417318620172333625523357051581}{20764145955907652519239401557910397815359600} a^{8} + \frac{19357430611475878036759872695553697670053}{5191036488976913129809850389477599453839900} a^{7} - \frac{1821643257098129271444829535130161637917313}{10382072977953826259619700778955198907679800} a^{6} + \frac{7412979326640311168088768230164908269547}{5191036488976913129809850389477599453839900} a^{5} - \frac{505185592590849163309871372158153253446103}{5191036488976913129809850389477599453839900} a^{4} + \frac{19021257117939858962715274611987439248017}{103820729779538262596197007789551989076798} a^{3} + \frac{371152150547832843104614899511337974548999}{1038207297795382625961970077895519890767980} a^{2} - \frac{10673231045012476674474642913528346064042}{51910364889769131298098503894775994538399} a - \frac{19034994466649961779452185306044104162266}{51910364889769131298098503894775994538399}$

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:  $12$
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:  \( 48708287178600000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T100:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 79 conjugacy class representatives for [1/2.S(5)^3]S(3) are not computed
Character table for [1/2.S(5)^3]S(3) is not computed

Intermediate fields

3.3.756.1

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 R R R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ $15$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$$[\ ]_{3}^{2}$
2.6.11.16$x^{6} + 2 x^{2} + 10$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
2.6.11.15$x^{6} + 2 x^{4} + 2 x^{2} + 10$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
3Data not computed
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.10.9.2$x^{10} + 14$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
456446131Data not computed