↑

Properties

Label 15.15.1327997606...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 5^{6}\cdot 17^{2}\cdot 37^{5}\cdot 4441^{2}\cdot 1449121^{2}$
Root discriminant $298.46$
Ramified primes $2, 5, 17, 37, 4441, 1449121$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T60

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1080, 297900, -17391330, 272475995, -2895510, -68813252, 792140, 7054867, -53828, -375108, 1462, 10897, -14, -164, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 164*x^13 - 14*x^12 + 10897*x^11 + 1462*x^10 - 375108*x^9 - 53828*x^8 + 7054867*x^7 + 792140*x^6 - 68813252*x^5 - 2895510*x^4 + 272475995*x^3 - 17391330*x^2 + 297900*x - 1080)
 
gp: K = bnfinit(x^15 - 164*x^13 - 14*x^12 + 10897*x^11 + 1462*x^10 - 375108*x^9 - 53828*x^8 + 7054867*x^7 + 792140*x^6 - 68813252*x^5 - 2895510*x^4 + 272475995*x^3 - 17391330*x^2 + 297900*x - 1080, 1)
 

Normalized defining polynomial

\( x^{15} - 164 x^{13} - 14 x^{12} + 10897 x^{11} + 1462 x^{10} - 375108 x^{9} - 53828 x^{8} + 7054867 x^{7} + 792140 x^{6} - 68813252 x^{5} - 2895510 x^{4} + 272475995 x^{3} - 17391330 x^{2} + 297900 x - 1080 \)

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:  \(13279976064852192427125304206928000000=2^{10}\cdot 5^{6}\cdot 17^{2}\cdot 37^{5}\cdot 4441^{2}\cdot 1449121^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $298.46$
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:  $2, 5, 17, 37, 4441, 1449121$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{4} + \frac{3}{16} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{96} a^{8} - \frac{1}{96} a^{7} + \frac{1}{24} a^{6} - \frac{5}{48} a^{5} - \frac{11}{96} a^{4} - \frac{1}{96} a^{3} + \frac{7}{48} a^{2} - \frac{5}{24} a + \frac{1}{4}$, $\frac{1}{192} a^{9} + \frac{1}{64} a^{7} - \frac{1}{32} a^{6} + \frac{1}{64} a^{5} + \frac{1}{16} a^{4} - \frac{11}{192} a^{3} - \frac{5}{32} a^{2} + \frac{1}{48} a + \frac{1}{8}$, $\frac{1}{4608} a^{10} + \frac{1}{4608} a^{9} - \frac{13}{4608} a^{8} - \frac{35}{4608} a^{7} - \frac{19}{4608} a^{6} + \frac{559}{4608} a^{5} - \frac{23}{512} a^{4} - \frac{73}{4608} a^{3} + \frac{139}{2304} a^{2} + \frac{85}{384} a + \frac{11}{64}$, $\frac{1}{36864} a^{11} - \frac{1}{9216} a^{10} - \frac{11}{6144} a^{9} + \frac{5}{6144} a^{8} - \frac{95}{3072} a^{7} + \frac{349}{6144} a^{6} - \frac{2149}{18432} a^{5} + \frac{1921}{18432} a^{4} + \frac{1171}{36864} a^{3} - \frac{1769}{18432} a^{2} - \frac{189}{1024} a - \frac{7}{512}$, $\frac{1}{884736} a^{12} - \frac{1}{98304} a^{11} + \frac{41}{442368} a^{10} - \frac{83}{110592} a^{9} - \frac{1093}{442368} a^{8} + \frac{1849}{442368} a^{7} - \frac{451}{55296} a^{6} - \frac{3907}{221184} a^{5} - \frac{5375}{98304} a^{4} + \frac{53071}{884736} a^{3} + \frac{4089}{16384} a^{2} + \frac{3041}{8192} a + \frac{1121}{4096}$, $\frac{1}{21233664} a^{13} + \frac{1}{10616832} a^{12} - \frac{17}{21233664} a^{11} + \frac{119}{10616832} a^{10} + \frac{4471}{10616832} a^{9} + \frac{13345}{5308416} a^{8} - \frac{17597}{1179648} a^{7} - \frac{9413}{196608} a^{6} + \frac{867205}{21233664} a^{5} - \frac{1087399}{10616832} a^{4} - \frac{909589}{21233664} a^{3} + \frac{7775}{1179648} a^{2} - \frac{275897}{589824} a - \frac{42965}{98304}$, $\frac{1}{509607936} a^{14} - \frac{11}{509607936} a^{13} - \frac{43}{509607936} a^{12} + \frac{17}{18874368} a^{11} + \frac{731}{63700992} a^{10} - \frac{31433}{254803968} a^{9} + \frac{158209}{254803968} a^{8} - \frac{196357}{28311552} a^{7} - \frac{17767439}{509607936} a^{6} - \frac{19523333}{169869312} a^{5} - \frac{61553183}{509607936} a^{4} - \frac{90222401}{509607936} a^{3} - \frac{149791}{1048576} a^{2} - \frac{6550681}{14155776} a + \frac{214481}{2359296}$

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:  \( 279965726842000 \) (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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ R ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{3}^{2}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.5.6.2$x^{5} + 15 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.10.0.1$x^{10} - x + 7$$1$$10$$0$$C_{10}$$[\ ]^{10}$
37Data not computed
4441Data not computed
1449121Data not computed