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Properties

Label 15.1.77027586128...1344.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 691^{7}$
Root discriminant $33.56$
Ramified primes $2, 691$
Class number $2$
Class group $[2]$
Galois group $D_{15}$ (as 15T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![972, -3240, 4104, -2232, 1266, 422, -883, 1117, -748, 548, -262, 134, -46, 18, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 + 18*x^13 - 46*x^12 + 134*x^11 - 262*x^10 + 548*x^9 - 748*x^8 + 1117*x^7 - 883*x^6 + 422*x^5 + 1266*x^4 - 2232*x^3 + 4104*x^2 - 3240*x + 972)
 
gp: K = bnfinit(x^15 - 3*x^14 + 18*x^13 - 46*x^12 + 134*x^11 - 262*x^10 + 548*x^9 - 748*x^8 + 1117*x^7 - 883*x^6 + 422*x^5 + 1266*x^4 - 2232*x^3 + 4104*x^2 - 3240*x + 972, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} + 18 x^{13} - 46 x^{12} + 134 x^{11} - 262 x^{10} + 548 x^{9} - 748 x^{8} + 1117 x^{7} - 883 x^{6} + 422 x^{5} + 1266 x^{4} - 2232 x^{3} + 4104 x^{2} - 3240 x + 972 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-77027586128169029561344=-\,2^{10}\cdot 691^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.56$
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:  $2, 691$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} + \frac{1}{3} a$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{36} a^{12} - \frac{1}{36} a^{9} - \frac{1}{36} a^{8} + \frac{2}{9} a^{7} + \frac{2}{9} a^{6} + \frac{17}{36} a^{5} + \frac{5}{18} a^{4} - \frac{1}{9} a^{3} - \frac{5}{18} a^{2} - \frac{1}{3} a$, $\frac{1}{7128} a^{13} - \frac{1}{792} a^{12} + \frac{7}{792} a^{11} + \frac{251}{7128} a^{10} + \frac{437}{7128} a^{9} + \frac{203}{7128} a^{8} - \frac{1039}{7128} a^{7} - \frac{1687}{7128} a^{6} + \frac{1385}{3564} a^{5} + \frac{1513}{3564} a^{4} + \frac{13}{891} a^{3} + \frac{259}{594} a^{2} - \frac{13}{33} a + \frac{1}{66}$, $\frac{1}{35933523912} a^{14} + \frac{13351}{230343102} a^{13} - \frac{5336395}{1996306884} a^{12} + \frac{515561}{74242818} a^{11} + \frac{741067471}{17966761956} a^{10} + \frac{328138090}{4491690489} a^{9} - \frac{356625056}{4491690489} a^{8} + \frac{110551759}{4491690489} a^{7} - \frac{72175715}{2764117224} a^{6} - \frac{1162404212}{4491690489} a^{5} + \frac{1453581595}{17966761956} a^{4} - \frac{395266601}{998153442} a^{3} + \frac{169231111}{998153442} a^{2} + \frac{266489}{30247074} a - \frac{5297423}{110905938}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 817777.771658 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{15}$ (as 15T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 30
The 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.2764.1, 5.1.477481.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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}$
691Data not computed