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Properties

Label 15.1.80263207751...2991.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,1871^{7}$
Root discriminant $33.65$
Ramified prime $1871$
Class number $1$
Class group Trivial
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![-19, -81, -69, 704, -944, 1017, -580, 207, -127, -84, 29, 38, -11, 9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + 9*x^13 - 11*x^12 + 38*x^11 + 29*x^10 - 84*x^9 - 127*x^8 + 207*x^7 - 580*x^6 + 1017*x^5 - 944*x^4 + 704*x^3 - 69*x^2 - 81*x - 19)
 
gp: K = bnfinit(x^15 - x^14 + 9*x^13 - 11*x^12 + 38*x^11 + 29*x^10 - 84*x^9 - 127*x^8 + 207*x^7 - 580*x^6 + 1017*x^5 - 944*x^4 + 704*x^3 - 69*x^2 - 81*x - 19, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} + 9 x^{13} - 11 x^{12} + 38 x^{11} + 29 x^{10} - 84 x^{9} - 127 x^{8} + 207 x^{7} - 580 x^{6} + 1017 x^{5} - 944 x^{4} + 704 x^{3} - 69 x^{2} - 81 x - 19 \)

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:  \(-80263207751705459602991=-\,1871^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.65$
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:  $1871$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{979} a^{13} + \frac{241}{979} a^{12} + \frac{258}{979} a^{11} - \frac{40}{979} a^{10} - \frac{70}{979} a^{9} + \frac{464}{979} a^{8} - \frac{79}{979} a^{7} + \frac{459}{979} a^{6} - \frac{148}{979} a^{5} - \frac{447}{979} a^{4} - \frac{283}{979} a^{3} - \frac{454}{979} a^{2} - \frac{309}{979} a + \frac{229}{979}$, $\frac{1}{2015483992184381411} a^{14} - \frac{58360025282675}{2015483992184381411} a^{13} - \frac{113915560995622515}{287926284597768773} a^{12} - \frac{680914275786945232}{2015483992184381411} a^{11} - \frac{648392000620344378}{2015483992184381411} a^{10} + \frac{42902839498257875}{183225817471307401} a^{9} - \frac{688403368860433150}{2015483992184381411} a^{8} - \frac{488426463687566430}{2015483992184381411} a^{7} - \frac{21399392151514571}{49158146150838571} a^{6} - \frac{8490922650572482}{287926284597768773} a^{5} + \frac{914718425808341786}{2015483992184381411} a^{4} + \frac{507494341735822863}{2015483992184381411} a^{3} + \frac{403783029421965548}{2015483992184381411} a^{2} + \frac{57994032889318055}{287926284597768773} a + \frac{592208333310769185}{2015483992184381411}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( 198116.498535 \)
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

3.1.1871.1, 5.1.3500641.1

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 $15$ $15$ $15$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ $15$ $15$ $15$

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
1871Data not computed