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Properties

Label 15.15.2483989775...3125.1
Degree $15$
Signature $[15, 0]$
Discriminant $5^{13}\cdot 257^{5}\cdot 883^{2}\cdot 15257253491^{2}$
Root discriminant $1444.36$
Ramified primes $5, 257, 883, 15257253491$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T97

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![402871723939, 693107493395, 365694137360, 16640248995, -35890644050, -5695992303, 1303942895, 278723315, -22494140, -6191100, 184558, 71510, -580, -420, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 420*x^13 - 580*x^12 + 71510*x^11 + 184558*x^10 - 6191100*x^9 - 22494140*x^8 + 278723315*x^7 + 1303942895*x^6 - 5695992303*x^5 - 35890644050*x^4 + 16640248995*x^3 + 365694137360*x^2 + 693107493395*x + 402871723939)
 
gp: K = bnfinit(x^15 - 420*x^13 - 580*x^12 + 71510*x^11 + 184558*x^10 - 6191100*x^9 - 22494140*x^8 + 278723315*x^7 + 1303942895*x^6 - 5695992303*x^5 - 35890644050*x^4 + 16640248995*x^3 + 365694137360*x^2 + 693107493395*x + 402871723939, 1)
 

Normalized defining polynomial

\( x^{15} - 420 x^{13} - 580 x^{12} + 71510 x^{11} + 184558 x^{10} - 6191100 x^{9} - 22494140 x^{8} + 278723315 x^{7} + 1303942895 x^{6} - 5695992303 x^{5} - 35890644050 x^{4} + 16640248995 x^{3} + 365694137360 x^{2} + 693107493395 x + 402871723939 \)

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:  \(248398977558713794925562108830363540177001953125=5^{13}\cdot 257^{5}\cdot 883^{2}\cdot 15257253491^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1444.36$
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:  $5, 257, 883, 15257253491$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{95} a^{13} + \frac{7}{95} a^{12} - \frac{42}{95} a^{11} + \frac{4}{95} a^{10} - \frac{8}{19} a^{9} - \frac{36}{95} a^{8} + \frac{33}{95} a^{7} + \frac{32}{95} a^{6} + \frac{26}{95} a^{5} + \frac{8}{19} a^{4} + \frac{26}{95} a^{3} + \frac{27}{95} a^{2} - \frac{22}{95} a + \frac{9}{95}$, $\frac{1}{173466587123158486530790704272142873975} a^{14} + \frac{179198229188564828495460907154213666}{173466587123158486530790704272142873975} a^{13} + \frac{61281476026902118747942370808335839036}{173466587123158486530790704272142873975} a^{12} + \frac{10633763369933231961957785545803919753}{24780941017594069504398672038877553425} a^{11} - \frac{53710867860727476732924500826232033979}{173466587123158486530790704272142873975} a^{10} + \frac{3363746772626623405975203167572209789}{8260313672531356501466224012959184475} a^{9} + \frac{3881223224210515837050068306045237293}{57822195707719495510263568090714291325} a^{8} + \frac{67071415307809553069364601483919497474}{173466587123158486530790704272142873975} a^{7} + \frac{79026842199333099014015081195323943674}{173466587123158486530790704272142873975} a^{6} - \frac{2625144618634229912797107726086231171}{173466587123158486530790704272142873975} a^{5} + \frac{4578599907798057704687833720979097386}{173466587123158486530790704272142873975} a^{4} - \frac{78490345790120118412671081511356888049}{173466587123158486530790704272142873975} a^{3} + \frac{70657565590106301381360940049974968811}{173466587123158486530790704272142873975} a^{2} - \frac{72025429336533598342132353377403253664}{173466587123158486530790704272142873975} a - \frac{51241239088116318733541588187036277754}{173466587123158486530790704272142873975}$

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:  \( 12239607085800000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T97:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 70 conjugacy class representatives for [A(5)^3:2]S(3) are not computed
Character table for [A(5)^3:2]S(3) is not computed

Intermediate fields

3.3.257.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $15$ $15$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.10.10.11$x^{10} + 5 x^{6} + 10 x^{5} + 25 x^{2} + 25 x + 25$$5$$2$$10$$(C_5^2 : C_4) : C_2$$[5/4, 5/4]_{4}^{2}$
257Data not computed
883Data not computed
15257253491Data not computed