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Properties

Label 15.7.46147275726...2416.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{12}\cdot 3^{20}\cdot 17^{2}\cdot 3343733^{2}$
Root discriminant $81.46$
Ramified primes $2, 3, 17, 3343733$
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![-10313, 36987, -49518, -28655, 101934, -5610, -65914, 6408, 17799, 125, -1944, -177, 195, -15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 - 15*x^13 + 195*x^12 - 177*x^11 - 1944*x^10 + 125*x^9 + 17799*x^8 + 6408*x^7 - 65914*x^6 - 5610*x^5 + 101934*x^4 - 28655*x^3 - 49518*x^2 + 36987*x - 10313)
 
gp: K = bnfinit(x^15 - 6*x^14 - 15*x^13 + 195*x^12 - 177*x^11 - 1944*x^10 + 125*x^9 + 17799*x^8 + 6408*x^7 - 65914*x^6 - 5610*x^5 + 101934*x^4 - 28655*x^3 - 49518*x^2 + 36987*x - 10313, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} - 15 x^{13} + 195 x^{12} - 177 x^{11} - 1944 x^{10} + 125 x^{9} + 17799 x^{8} + 6408 x^{7} - 65914 x^{6} - 5610 x^{5} + 101934 x^{4} - 28655 x^{3} - 49518 x^{2} + 36987 x - 10313 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(46147275726319772173643452416=2^{12}\cdot 3^{20}\cdot 17^{2}\cdot 3343733^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.46$
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:  $2, 3, 17, 3343733$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{4} + \frac{3}{8} a^{3} + \frac{1}{8} a^{2} - \frac{3}{8} a - \frac{3}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{6748519705752712771539719576} a^{14} - \frac{57687662051303865511786591}{3374259852876356385769859788} a^{13} - \frac{24928730776682143393886931}{3374259852876356385769859788} a^{12} - \frac{139992192397915168659054741}{6748519705752712771539719576} a^{11} - \frac{8318700189758705018700198}{843564963219089096442464947} a^{10} - \frac{4938993344453734438406135}{3374259852876356385769859788} a^{9} + \frac{1133969918463546255397486459}{6748519705752712771539719576} a^{8} + \frac{863383405312484557272296049}{6748519705752712771539719576} a^{7} - \frac{400941344851688978121541111}{6748519705752712771539719576} a^{6} - \frac{843551196745092140930699683}{6748519705752712771539719576} a^{5} + \frac{590727105052776965663285057}{1687129926438178192884929894} a^{4} + \frac{1672095461468469832047215157}{6748519705752712771539719576} a^{3} + \frac{837891947712118826020024035}{6748519705752712771539719576} a^{2} + \frac{677123798120556753316692013}{1687129926438178192884929894} a - \frac{2557550027450659471470868729}{6748519705752712771539719576}$

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:  $10$
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:  \( 574507211.585 \) (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

\(\Q(\zeta_{9})^+\)

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 R $15$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ $15$ R ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{7}$ $15$

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$$[\ ]^{3}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
3Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.5.0.1$x^{5} - x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
17.5.0.1$x^{5} - x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
3343733Data not computed