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Properties

Label 15.15.6644769363...4064.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{12}\cdot 7^{10}\cdot 13^{4}\cdot 83^{2}\cdot 293^{2}\cdot 18439^{2}$
Root discriminant $179.82$
Ramified primes $2, 7, 13, 83, 293, 18439$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T50

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3877, 54733, -110047, -197917, 402515, 182005, -367383, -25813, 124473, -15393, -13381, 2481, 473, -97, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 97*x^13 + 473*x^12 + 2481*x^11 - 13381*x^10 - 15393*x^9 + 124473*x^8 - 25813*x^7 - 367383*x^6 + 182005*x^5 + 402515*x^4 - 197917*x^3 - 110047*x^2 + 54733*x - 3877)
 
gp: K = bnfinit(x^15 - 5*x^14 - 97*x^13 + 473*x^12 + 2481*x^11 - 13381*x^10 - 15393*x^9 + 124473*x^8 - 25813*x^7 - 367383*x^6 + 182005*x^5 + 402515*x^4 - 197917*x^3 - 110047*x^2 + 54733*x - 3877, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 97 x^{13} + 473 x^{12} + 2481 x^{11} - 13381 x^{10} - 15393 x^{9} + 124473 x^{8} - 25813 x^{7} - 367383 x^{6} + 182005 x^{5} + 402515 x^{4} - 197917 x^{3} - 110047 x^{2} + 54733 x - 3877 \)

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:  \(6644769363943959400399066678104064=2^{12}\cdot 7^{10}\cdot 13^{4}\cdot 83^{2}\cdot 293^{2}\cdot 18439^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $179.82$
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:  $2, 7, 13, 83, 293, 18439$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{5} + \frac{1}{16} a^{4} - \frac{1}{16} a^{3} + \frac{1}{16} a^{2} - \frac{7}{16} a + \frac{7}{16}$, $\frac{1}{32} a^{8} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{3}{16} a^{2} - \frac{3}{8} a - \frac{5}{32}$, $\frac{1}{128} a^{9} + \frac{1}{128} a^{8} + \frac{1}{64} a^{7} + \frac{3}{64} a^{6} + \frac{3}{32} a^{5} + \frac{1}{16} a^{4} + \frac{11}{64} a^{3} - \frac{15}{64} a^{2} - \frac{37}{128} a + \frac{15}{128}$, $\frac{1}{256} a^{10} + \frac{1}{256} a^{8} + \frac{1}{64} a^{7} + \frac{3}{128} a^{6} + \frac{7}{64} a^{5} - \frac{9}{128} a^{4} - \frac{13}{64} a^{3} - \frac{7}{256} a^{2} + \frac{5}{64} a + \frac{17}{256}$, $\frac{1}{1024} a^{11} - \frac{1}{1024} a^{10} + \frac{1}{1024} a^{9} - \frac{13}{1024} a^{8} + \frac{1}{512} a^{7} - \frac{5}{512} a^{6} - \frac{23}{512} a^{5} + \frac{15}{512} a^{4} - \frac{83}{1024} a^{3} + \frac{59}{1024} a^{2} + \frac{381}{1024} a - \frac{321}{1024}$, $\frac{1}{2048} a^{12} + \frac{1}{512} a^{9} + \frac{5}{2048} a^{8} + \frac{3}{256} a^{7} + \frac{5}{256} a^{6} - \frac{5}{128} a^{5} - \frac{181}{2048} a^{4} - \frac{23}{256} a^{3} + \frac{59}{256} a^{2} + \frac{59}{512} a - \frac{337}{2048}$, $\frac{1}{4096} a^{13} - \frac{1}{4096} a^{12} + \frac{1}{1024} a^{10} + \frac{1}{4096} a^{9} + \frac{19}{4096} a^{8} + \frac{1}{256} a^{7} - \frac{15}{512} a^{6} - \frac{101}{4096} a^{5} - \frac{3}{4096} a^{4} - \frac{23}{256} a^{3} + \frac{197}{1024} a^{2} - \frac{1597}{4096} a + \frac{1361}{4096}$, $\frac{1}{6124476039725056} a^{14} + \frac{161539701643}{3062238019862528} a^{13} + \frac{1015687584865}{6124476039725056} a^{12} + \frac{430575834593}{1531119009931264} a^{11} + \frac{5931141082653}{6124476039725056} a^{10} + \frac{395949108441}{235556770758656} a^{9} + \frac{94132518842205}{6124476039725056} a^{8} - \frac{10877841989337}{765559504965632} a^{7} + \frac{205589010908947}{6124476039725056} a^{6} - \frac{182611587599307}{3062238019862528} a^{5} + \frac{132891919643123}{6124476039725056} a^{4} - \frac{116961664726575}{1531119009931264} a^{3} + \frac{755249521620111}{6124476039725056} a^{2} - \frac{57658088878789}{3062238019862528} a + \frac{2948815738374287}{6124476039725056}$

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:  \( 2157836539700 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T50:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3000
The 32 conjugacy class representatives for [D(5)^3]3=D(5)wr3
Character table for [D(5)^3]3=D(5)wr3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 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 $15$ $15$ R $15$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ $15$ $15$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
7Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.5.0.1$x^{5} - 2 x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
293Data not computed
18439Data not computed