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Properties

Label 15.1.50139512927...9219.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,1259^{7}$
Root discriminant $27.97$
Ramified prime $1259$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![128, 0, 64, 16, -232, 136, 96, -255, 349, -325, 200, -68, 0, 11, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 11*x^13 - 68*x^11 + 200*x^10 - 325*x^9 + 349*x^8 - 255*x^7 + 96*x^6 + 136*x^5 - 232*x^4 + 16*x^3 + 64*x^2 + 128)
 
gp: K = bnfinit(x^15 - 5*x^14 + 11*x^13 - 68*x^11 + 200*x^10 - 325*x^9 + 349*x^8 - 255*x^7 + 96*x^6 + 136*x^5 - 232*x^4 + 16*x^3 + 64*x^2 + 128, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 11 x^{13} - 68 x^{11} + 200 x^{10} - 325 x^{9} + 349 x^{8} - 255 x^{7} + 96 x^{6} + 136 x^{5} - 232 x^{4} + 16 x^{3} + 64 x^{2} + 128 \)

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:  \(-5013951292733215829219=-\,1259^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.97$
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:  $1259$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{4} a^{7} + \frac{3}{16} a^{5} + \frac{1}{16} a^{4} + \frac{5}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{64} a^{12} + \frac{1}{64} a^{11} + \frac{5}{64} a^{10} - \frac{3}{32} a^{9} + \frac{1}{16} a^{8} - \frac{1}{4} a^{7} - \frac{5}{64} a^{6} - \frac{1}{64} a^{5} - \frac{9}{64} a^{4} - \frac{9}{32} a^{3} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{512} a^{13} - \frac{1}{512} a^{12} - \frac{13}{512} a^{11} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{7}{64} a^{8} - \frac{37}{512} a^{7} + \frac{41}{512} a^{6} - \frac{119}{512} a^{5} + \frac{5}{32} a^{4} + \frac{37}{128} a^{3} + \frac{21}{64} a^{2} - \frac{7}{16} a - \frac{3}{16}$, $\frac{1}{65646592} a^{14} - \frac{22287}{65646592} a^{13} + \frac{3119}{1396736} a^{12} - \frac{4971}{698368} a^{11} + \frac{19359}{186496} a^{10} - \frac{160399}{8205824} a^{9} - \frac{5212821}{65646592} a^{8} + \frac{6844975}{65646592} a^{7} - \frac{2185429}{65646592} a^{6} - \frac{149385}{698368} a^{5} + \frac{2149}{16411648} a^{4} - \frac{431579}{4102912} a^{3} - \frac{1295313}{4102912} a^{2} - \frac{910825}{2051456} a + \frac{487085}{1025728}$

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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 487363.237089 \) (assuming GRH)
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

3.1.1259.1, 5.1.1585081.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
1259Data not computed