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Properties

Label 15.15.1631820139...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 5^{18}\cdot 37^{5}\cdot 881^{4}$
Root discriminant $222.59$
Ramified primes $2, 5, 37, 881$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T40

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3524, -88100, 563840, -427285, -2510850, -862375, 1242740, 573440, -214350, -112175, 13476, 8100, -150, -175, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 175*x^13 - 150*x^12 + 8100*x^11 + 13476*x^10 - 112175*x^9 - 214350*x^8 + 573440*x^7 + 1242740*x^6 - 862375*x^5 - 2510850*x^4 - 427285*x^3 + 563840*x^2 - 88100*x + 3524)
 
gp: K = bnfinit(x^15 - 175*x^13 - 150*x^12 + 8100*x^11 + 13476*x^10 - 112175*x^9 - 214350*x^8 + 573440*x^7 + 1242740*x^6 - 862375*x^5 - 2510850*x^4 - 427285*x^3 + 563840*x^2 - 88100*x + 3524, 1)
 

Normalized defining polynomial

\( x^{15} - 175 x^{13} - 150 x^{12} + 8100 x^{11} + 13476 x^{10} - 112175 x^{9} - 214350 x^{8} + 573440 x^{7} + 1242740 x^{6} - 862375 x^{5} - 2510850 x^{4} - 427285 x^{3} + 563840 x^{2} - 88100 x + 3524 \)

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:  \(163182013910625833582031250000000000=2^{10}\cdot 5^{18}\cdot 37^{5}\cdot 881^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $222.59$
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:  $2, 5, 37, 881$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{12} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} + \frac{1}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{44692440892787069673274536294911416} a^{14} - \frac{2680154148111909694556112661782587}{44692440892787069673274536294911416} a^{13} - \frac{195455221786538433683287263834105}{44692440892787069673274536294911416} a^{12} + \frac{628862218478153543889872775967835}{22346220446393534836637268147455708} a^{11} + \frac{181161629668078792011769900993249}{2628967111340415863133796252641848} a^{10} + \frac{17159457693214298243882353454317}{2352233731199319456488133489205864} a^{9} - \frac{8815786167548291493884699128285855}{44692440892787069673274536294911416} a^{8} + \frac{2851294257135821400606816957454005}{22346220446393534836637268147455708} a^{7} - \frac{7491482994816776935327957969364305}{44692440892787069673274536294911416} a^{6} + \frac{626269908851394106222436189668013}{2628967111340415863133796252641848} a^{5} + \frac{12717109850334982797379944259790989}{44692440892787069673274536294911416} a^{4} + \frac{496657087526298404970072036256307}{11173110223196767418318634073727854} a^{3} + \frac{5116355755082721892376868926196805}{11173110223196767418318634073727854} a^{2} + \frac{5306914606240768063273255986954199}{11173110223196767418318634073727854} a - \frac{2009378307997386362452595603877646}{5586555111598383709159317036863927}$

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:  \( 27690515064800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T40:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1500
The 40 conjugacy class representatives for [5^3:2]S(3)
Character table for [5^3:2]S(3) is not computed

Intermediate fields

3.3.148.1

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

Sibling fields

Degree 15 siblings: data not computed
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ R $15$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$5$5.5.6.2$x^{5} + 15 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
5.10.12.14$x^{10} + 10 x^{7} + 10 x^{5} + 100 x^{4} + 50 x^{2} + 25$$5$$2$$12$$D_5^2$$[3/2, 3/2]_{2}^{2}$
37Data not computed
881Data not computed