↑

Properties

Label 15.1.66912411284...6847.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,1823^{7}$
Root discriminant $33.24$
Ramified prime $1823$
Class number $2$
Class group $[2]$
Galois group $D_{15}$ (as 15T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-200, 152, -254, 1423, -1281, -271, 168, 588, -529, 156, -53, 67, -41, 16, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 16*x^13 - 41*x^12 + 67*x^11 - 53*x^10 + 156*x^9 - 529*x^8 + 588*x^7 + 168*x^6 - 271*x^5 - 1281*x^4 + 1423*x^3 - 254*x^2 + 152*x - 200)
 
gp: K = bnfinit(x^15 - 6*x^14 + 16*x^13 - 41*x^12 + 67*x^11 - 53*x^10 + 156*x^9 - 529*x^8 + 588*x^7 + 168*x^6 - 271*x^5 - 1281*x^4 + 1423*x^3 - 254*x^2 + 152*x - 200, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} + 16 x^{13} - 41 x^{12} + 67 x^{11} - 53 x^{10} + 156 x^{9} - 529 x^{8} + 588 x^{7} + 168 x^{6} - 271 x^{5} - 1281 x^{4} + 1423 x^{3} - 254 x^{2} + 152 x - 200 \)

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:  \(-66912411284534354746847=-\,1823^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.24$
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:  $1823$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{244} a^{13} + \frac{15}{244} a^{12} + \frac{53}{244} a^{11} - \frac{12}{61} a^{10} - \frac{59}{244} a^{9} - \frac{13}{122} a^{8} - \frac{23}{61} a^{7} - \frac{113}{244} a^{6} - \frac{121}{244} a^{5} + \frac{5}{244} a^{4} - \frac{39}{122} a^{3} + \frac{83}{244} a^{2} + \frac{21}{61} a - \frac{23}{61}$, $\frac{1}{2579871483553204304} a^{14} + \frac{4483588933474323}{2579871483553204304} a^{13} - \frac{273338554016127421}{2579871483553204304} a^{12} - \frac{37550221329489411}{1289935741776602152} a^{11} - \frac{17125741509831187}{2579871483553204304} a^{10} + \frac{5920958271558092}{161241967722075269} a^{9} - \frac{60424505597495341}{644967870888301076} a^{8} + \frac{253126968340520043}{2579871483553204304} a^{7} + \frac{410142715848295919}{2579871483553204304} a^{6} - \frac{991163182269872841}{2579871483553204304} a^{5} + \frac{454175958362043}{1781679201348898} a^{4} + \frac{347336504723012743}{2579871483553204304} a^{3} + \frac{359062112228813215}{1289935741776602152} a^{2} + \frac{17540348726609077}{161241967722075269} a - \frac{153626589428279489}{322483935444150538}$

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:  \( 164195.256154 \)
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

3.1.1823.1, 5.1.3323329.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
1823Data not computed