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Properties

Label 15.3.14335961191...6704.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{10}\cdot 3^{20}\cdot 47^{6}\cdot 193^{2}$
Root discriminant $64.63$
Ramified primes $2, 3, 47, 193$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T80

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6176, -46320, 138960, -208744, 158154, -51003, 4328, -2547, 1512, -776, 60, -45, -2, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 3*x^13 - 2*x^12 - 45*x^11 + 60*x^10 - 776*x^9 + 1512*x^8 - 2547*x^7 + 4328*x^6 - 51003*x^5 + 158154*x^4 - 208744*x^3 + 138960*x^2 - 46320*x + 6176)
 
gp: K = bnfinit(x^15 + 3*x^13 - 2*x^12 - 45*x^11 + 60*x^10 - 776*x^9 + 1512*x^8 - 2547*x^7 + 4328*x^6 - 51003*x^5 + 158154*x^4 - 208744*x^3 + 138960*x^2 - 46320*x + 6176, 1)
 

Normalized defining polynomial

\( x^{15} + 3 x^{13} - 2 x^{12} - 45 x^{11} + 60 x^{10} - 776 x^{9} + 1512 x^{8} - 2547 x^{7} + 4328 x^{6} - 51003 x^{5} + 158154 x^{4} - 208744 x^{3} + 138960 x^{2} - 46320 x + 6176 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1433596119184117362506056704=2^{10}\cdot 3^{20}\cdot 47^{6}\cdot 193^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.63$
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:  $2, 3, 47, 193$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{3} - \frac{2}{9}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{4} - \frac{2}{9} a$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{5} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{9} - \frac{1}{9} a^{3} + \frac{2}{27}$, $\frac{1}{27} a^{10} - \frac{1}{9} a^{4} + \frac{2}{27} a$, $\frac{1}{216} a^{11} + \frac{1}{72} a^{9} + \frac{1}{36} a^{8} + \frac{1}{72} a^{7} - \frac{1}{18} a^{6} - \frac{1}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{72} a^{3} - \frac{8}{27} a^{2} + \frac{7}{72} a + \frac{11}{36}$, $\frac{1}{5184} a^{12} - \frac{1}{864} a^{11} + \frac{1}{192} a^{10} + \frac{23}{1296} a^{9} + \frac{31}{576} a^{8} - \frac{1}{96} a^{7} - \frac{5}{216} a^{6} + \frac{1}{8} a^{5} + \frac{23}{192} a^{4} + \frac{61}{2592} a^{3} - \frac{145}{1728} a^{2} + \frac{5}{48} a - \frac{287}{1296}$, $\frac{1}{41472} a^{13} - \frac{1}{10368} a^{12} + \frac{5}{13824} a^{11} - \frac{23}{20736} a^{10} + \frac{79}{41472} a^{9} - \frac{1}{576} a^{8} - \frac{67}{3456} a^{7} + \frac{17}{1728} a^{6} - \frac{619}{4608} a^{5} + \frac{1637}{10368} a^{4} - \frac{4223}{41472} a^{3} - \frac{1303}{6912} a^{2} + \frac{175}{10368} a - \frac{383}{5184}$, $\frac{1}{331776} a^{14} - \frac{1}{165888} a^{13} + \frac{7}{331776} a^{12} - \frac{1}{20736} a^{11} - \frac{13}{331776} a^{10} + \frac{43}{165888} a^{9} - \frac{79}{27648} a^{8} + \frac{71}{6912} a^{7} - \frac{3121}{110592} a^{6} + \frac{11527}{165888} a^{5} + \frac{13481}{331776} a^{4} + \frac{2575}{41472} a^{3} + \frac{10229}{41472} a^{2} - \frac{1057}{2592} a + \frac{193}{20736}$

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:  $8$
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:  \( 37740429.2419 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T80:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 38880
The 48 conjugacy class representatives for [1/2.S(3)^5]D(5)
Character table for [1/2.S(3)^5]D(5) is not computed

Intermediate fields

5.1.2209.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 siblings: data not computed
Degree 45 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ R $15$ $15$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.1$x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
3Data not computed
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
193Data not computed