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Properties

Label 15.5.29055909763...0000.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{23}\cdot 3^{15}\cdot 5^{6}\cdot 37^{2}\cdot 181^{2}\cdot 257^{4}\cdot 281^{2}$
Root discriminant $498.37$
Ramified primes $2, 3, 5, 37, 181, 257, 281$
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![-2056000, -21588000, -82188600, -149548300, -164593080, -115536600, -50109174, -10844463, 717984, 942597, 162330, 9756, -108, 189, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 189*x^13 - 108*x^12 + 9756*x^11 + 162330*x^10 + 942597*x^9 + 717984*x^8 - 10844463*x^7 - 50109174*x^6 - 115536600*x^5 - 164593080*x^4 - 149548300*x^3 - 82188600*x^2 - 21588000*x - 2056000)
 
gp: K = bnfinit(x^15 + 189*x^13 - 108*x^12 + 9756*x^11 + 162330*x^10 + 942597*x^9 + 717984*x^8 - 10844463*x^7 - 50109174*x^6 - 115536600*x^5 - 164593080*x^4 - 149548300*x^3 - 82188600*x^2 - 21588000*x - 2056000, 1)
 

Normalized defining polynomial

\( x^{15} + 189 x^{13} - 108 x^{12} + 9756 x^{11} + 162330 x^{10} + 942597 x^{9} + 717984 x^{8} - 10844463 x^{7} - 50109174 x^{6} - 115536600 x^{5} - 164593080 x^{4} - 149548300 x^{3} - 82188600 x^{2} - 21588000 x - 2056000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-29055909763442875483084887483088896000000=-\,2^{23}\cdot 3^{15}\cdot 5^{6}\cdot 37^{2}\cdot 181^{2}\cdot 257^{4}\cdot 281^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $498.37$
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:  $2, 3, 5, 37, 181, 257, 281$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{20} a^{11} - \frac{1}{20} a^{9} + \frac{1}{10} a^{8} + \frac{1}{20} a^{7} - \frac{1}{4} a^{6} + \frac{7}{20} a^{5} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{40} a^{12} - \frac{1}{40} a^{10} + \frac{1}{20} a^{9} - \frac{1}{10} a^{8} - \frac{3}{40} a^{6} + \frac{7}{20} a^{5} - \frac{3}{40} a^{4} + \frac{2}{5} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{400} a^{13} - \frac{1}{400} a^{11} - \frac{1}{50} a^{10} + \frac{23}{200} a^{9} - \frac{1}{8} a^{8} - \frac{43}{400} a^{7} + \frac{21}{100} a^{6} - \frac{193}{400} a^{5} - \frac{67}{200} a^{4} - \frac{3}{40} a^{3} + \frac{9}{20} a^{2} - \frac{1}{2} a$, $\frac{1}{28744843781758500478612412496273950506622941600} a^{14} - \frac{391598882901479771497487217172686919254691}{359310547271981255982655156203424381332786770} a^{13} + \frac{293305982840802376210283754604797630620071819}{28744843781758500478612412496273950506622941600} a^{12} - \frac{174136982439220453518308421933018924969966907}{7186210945439625119653103124068487626655735400} a^{11} + \frac{68338174814398250962274902318889943611793591}{1105570914683019249177400480625921173331651600} a^{10} + \frac{81827240295117982112623346197893832219137061}{2874484378175850047861241249627395050662294160} a^{9} + \frac{205039038246105323378634605486807346839365577}{28744843781758500478612412496273950506622941600} a^{8} + \frac{121678505911207711035646815907836683611613831}{7186210945439625119653103124068487626655735400} a^{7} - \frac{3662672399814816328124584837935362983709431273}{28744843781758500478612412496273950506622941600} a^{6} + \frac{5457205890706247694909066471949195116842368693}{14372421890879250239306206248136975253311470800} a^{5} + \frac{476347697341403172981016584951319350670403033}{2874484378175850047861241249627395050662294160} a^{4} - \frac{5233422970242641120099528528727160511588505}{287448437817585004786124124962739505066229416} a^{3} - \frac{44428429151226539311397831161487284840246189}{143724218908792502393062062481369752533114708} a^{2} + \frac{2837219388880802720217400634591663233690972}{35931054727198125598265515620342438133278677} a + \frac{12145251594592306970313502659174523678398696}{35931054727198125598265515620342438133278677}$

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:  $9$
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:  \( 8400622176900000 \) (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

3.1.216.1

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 $15$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$0$Trivial$[\ ]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.4.10.4$x^{4} + 6 x^{2} - 1$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{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$$[\ ]_{2}^{3}$
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.8.0.1$x^{8} - x + 18$$1$$8$$0$$C_8$$[\ ]^{8}$
181Data not computed
257Data not computed
281Data not computed