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Properties

Label 15.15.1923356492...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 3^{26}\cdot 5^{11}\cdot 73^{6}$
Root discriminant $193.02$
Ramified primes $2, 3, 5, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T76

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4500000, -20700000, 12600000, 41190000, -25140000, -28350000, 3254000, 4185000, -162000, -271500, 3600, 9000, -30, -150, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 150*x^13 - 30*x^12 + 9000*x^11 + 3600*x^10 - 271500*x^9 - 162000*x^8 + 4185000*x^7 + 3254000*x^6 - 28350000*x^5 - 25140000*x^4 + 41190000*x^3 + 12600000*x^2 - 20700000*x + 4500000)
 
gp: K = bnfinit(x^15 - 150*x^13 - 30*x^12 + 9000*x^11 + 3600*x^10 - 271500*x^9 - 162000*x^8 + 4185000*x^7 + 3254000*x^6 - 28350000*x^5 - 25140000*x^4 + 41190000*x^3 + 12600000*x^2 - 20700000*x + 4500000, 1)
 

Normalized defining polynomial

\( x^{15} - 150 x^{13} - 30 x^{12} + 9000 x^{11} + 3600 x^{10} - 271500 x^{9} - 162000 x^{8} + 4185000 x^{7} + 3254000 x^{6} - 28350000 x^{5} - 25140000 x^{4} + 41190000 x^{3} + 12600000 x^{2} - 20700000 x + 4500000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(19233564923030865637054050000000000=2^{10}\cdot 3^{26}\cdot 5^{11}\cdot 73^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $193.02$
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:  $2, 3, 5, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{10} a^{3}$, $\frac{1}{10} a^{4}$, $\frac{1}{10} a^{5}$, $\frac{1}{100} a^{6}$, $\frac{1}{100} a^{7}$, $\frac{1}{500} a^{8} + \frac{1}{50} a^{5} + \frac{2}{5} a^{2}$, $\frac{1}{1000} a^{9}$, $\frac{1}{1000} a^{10}$, $\frac{1}{5000} a^{11} + \frac{1}{50} a^{5} - \frac{2}{5} a^{2}$, $\frac{1}{30000} a^{12} - \frac{1}{30} a^{3}$, $\frac{1}{30000} a^{13} - \frac{1}{30} a^{4}$, $\frac{1}{616050000} a^{14} + \frac{43}{13690000} a^{13} + \frac{149}{13690000} a^{12} + \frac{23}{855625} a^{11} + \frac{1}{34225} a^{10} - \frac{5}{10952} a^{9} - \frac{1727}{2053500} a^{8} + \frac{82}{34225} a^{7} + \frac{197}{68450} a^{6} + \frac{10546}{308025} a^{5} + \frac{24}{6845} a^{4} + \frac{71}{6845} a^{3} + \frac{9673}{20535} a^{2} - \frac{683}{1369} a - \frac{566}{1369}$

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:  $14$
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:  \( 6526396541930 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T76:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 29160
The 48 conjugacy class representatives for [3^5:2]A(5)
Character table for [3^5:2]A(5) is not computed

Intermediate fields

5.5.10791225.1

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

Sibling fields

Degree 15 sibling: data not computed
Degree 30 siblings: data not computed
Degree 45 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 ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
2Data not computed
$3$3.6.7.5$x^{6} + 6 x^{2} + 3$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
3.9.19.45$x^{9} + 3 x^{3} + 18 x^{2} + 6$$9$$1$$19$$((C_3^3:C_3):C_2):C_2$$[3/2, 2, 5/2, 8/3]_{2}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.12.10.2$x^{12} + 15 x^{6} + 100$$6$$2$$10$$C_6\times S_3$$[\ ]_{6}^{6}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$
73.6.4.2$x^{6} - 73 x^{3} + 58619$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$