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Properties

Label 15.15.2946961580...0928.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 37^{5}\cdot 20371966273189^{2}$
Root discriminant $314.74$
Ramified primes $2, 37, 20371966273189$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T60

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![262144, -8192000, -10485760, 70144000, 5849088, -25716480, -1186880, 3704800, 113792, -259640, -5424, 9150, 121, -155, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 155*x^13 + 121*x^12 + 9150*x^11 - 5424*x^10 - 259640*x^9 + 113792*x^8 + 3704800*x^7 - 1186880*x^6 - 25716480*x^5 + 5849088*x^4 + 70144000*x^3 - 10485760*x^2 - 8192000*x + 262144)
 
gp: K = bnfinit(x^15 - x^14 - 155*x^13 + 121*x^12 + 9150*x^11 - 5424*x^10 - 259640*x^9 + 113792*x^8 + 3704800*x^7 - 1186880*x^6 - 25716480*x^5 + 5849088*x^4 + 70144000*x^3 - 10485760*x^2 - 8192000*x + 262144, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 155 x^{13} + 121 x^{12} + 9150 x^{11} - 5424 x^{10} - 259640 x^{9} + 113792 x^{8} + 3704800 x^{7} - 1186880 x^{6} - 25716480 x^{5} + 5849088 x^{4} + 70144000 x^{3} - 10485760 x^{2} - 8192000 x + 262144 \)

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:  \(29469615804756686585980269796501500928=2^{10}\cdot 37^{5}\cdot 20371966273189^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $314.74$
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:  $2, 37, 20371966273189$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{5} - \frac{1}{12} a^{4} - \frac{1}{4} a^{3} + \frac{5}{12} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{24} a^{6} - \frac{1}{24} a^{5} - \frac{1}{8} a^{4} + \frac{5}{24} a^{3} - \frac{5}{12} a^{2} - \frac{1}{6} a$, $\frac{1}{144} a^{7} - \frac{1}{144} a^{6} + \frac{5}{144} a^{5} + \frac{11}{48} a^{4} - \frac{5}{72} a^{3} + \frac{1}{6} a^{2} - \frac{2}{9} a + \frac{1}{9}$, $\frac{1}{576} a^{8} + \frac{1}{576} a^{7} + \frac{1}{192} a^{6} - \frac{5}{576} a^{5} + \frac{13}{72} a^{4} - \frac{35}{144} a^{3} - \frac{5}{36} a^{2} + \frac{1}{4} a - \frac{1}{9}$, $\frac{1}{3456} a^{9} - \frac{1}{3456} a^{8} + \frac{1}{3456} a^{7} + \frac{37}{3456} a^{6} - \frac{5}{576} a^{5} + \frac{7}{32} a^{4} + \frac{91}{432} a^{3} - \frac{77}{216} a^{2} + \frac{49}{108} a - \frac{5}{27}$, $\frac{1}{6912} a^{10} - \frac{1}{6912} a^{9} + \frac{1}{6912} a^{8} - \frac{11}{6912} a^{7} + \frac{1}{384} a^{6} - \frac{5}{576} a^{5} - \frac{35}{864} a^{4} + \frac{61}{432} a^{3} - \frac{77}{216} a^{2} + \frac{25}{54} a + \frac{2}{9}$, $\frac{1}{248832} a^{11} - \frac{1}{248832} a^{10} - \frac{1}{82944} a^{9} - \frac{31}{248832} a^{8} - \frac{53}{124416} a^{7} + \frac{481}{31104} a^{6} - \frac{713}{31104} a^{5} - \frac{583}{7776} a^{4} - \frac{197}{2592} a^{3} + \frac{481}{3888} a^{2} - \frac{104}{243} a + \frac{61}{243}$, $\frac{1}{995328} a^{12} - \frac{1}{995328} a^{11} + \frac{23}{331776} a^{10} - \frac{103}{995328} a^{9} - \frac{17}{497664} a^{8} - \frac{25}{62208} a^{7} + \frac{2473}{124416} a^{6} + \frac{343}{15552} a^{5} + \frac{2077}{10368} a^{4} - \frac{149}{15552} a^{3} - \frac{623}{3888} a^{2} + \frac{475}{972} a + \frac{2}{9}$, $\frac{1}{71663616} a^{13} + \frac{35}{71663616} a^{12} + \frac{65}{71663616} a^{11} - \frac{35}{884736} a^{10} - \frac{2495}{35831808} a^{9} - \frac{205}{2985984} a^{8} - \frac{253}{2985984} a^{7} + \frac{20545}{2239488} a^{6} + \frac{45743}{2239488} a^{5} - \frac{5297}{41472} a^{4} - \frac{1913}{17496} a^{3} - \frac{6967}{23328} a^{2} + \frac{598}{2187} a - \frac{2101}{4374}$, $\frac{1}{286654464} a^{14} - \frac{1}{286654464} a^{13} - \frac{43}{286654464} a^{12} - \frac{7}{3538944} a^{11} - \frac{7913}{143327232} a^{10} - \frac{61}{746496} a^{9} + \frac{647}{11943936} a^{8} - \frac{373}{279936} a^{7} - \frac{179365}{8957952} a^{6} + \frac{2885}{165888} a^{5} + \frac{130267}{1119744} a^{4} + \frac{21521}{93312} a^{3} - \frac{21055}{69984} a^{2} - \frac{7087}{17496} a + \frac{119}{486}$

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:  \( 234947681045000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T60:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6000
The 40 conjugacy class representatives for [D(5)^3]S(3)=D(5)wrS(3)
Character table for [D(5)^3]S(3)=D(5)wrS(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 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$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$$[\ ]_{3}^{2}$
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.10.5.1$x^{10} - 2738 x^{6} + 1874161 x^{2} - 11719128733$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
20371966273189Data not computed