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Properties

Label 15.1.72498065296...5264.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{14}\cdot 461^{7}$
Root discriminant $33.42$
Ramified primes $2, 461$
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![-256, 208, -336, 244, -472, 172, -308, 349, -258, 284, -214, 126, -70, 24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 24*x^13 - 70*x^12 + 126*x^11 - 214*x^10 + 284*x^9 - 258*x^8 + 349*x^7 - 308*x^6 + 172*x^5 - 472*x^4 + 244*x^3 - 336*x^2 + 208*x - 256)
 
gp: K = bnfinit(x^15 - 6*x^14 + 24*x^13 - 70*x^12 + 126*x^11 - 214*x^10 + 284*x^9 - 258*x^8 + 349*x^7 - 308*x^6 + 172*x^5 - 472*x^4 + 244*x^3 - 336*x^2 + 208*x - 256, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} + 24 x^{13} - 70 x^{12} + 126 x^{11} - 214 x^{10} + 284 x^{9} - 258 x^{8} + 349 x^{7} - 308 x^{6} + 172 x^{5} - 472 x^{4} + 244 x^{3} - 336 x^{2} + 208 x - 256 \)

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:  \(-72498065296628233355264=-\,2^{14}\cdot 461^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.42$
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:  $2, 461$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{1}{16} a^{7} - \frac{3}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{12} - \frac{1}{64} a^{11} - \frac{1}{128} a^{10} - \frac{1}{64} a^{9} - \frac{5}{128} a^{8} - \frac{7}{64} a^{7} + \frac{13}{128} a^{6} - \frac{15}{64} a^{5} - \frac{3}{32} a^{4} + \frac{5}{16} a^{3} + \frac{1}{32} a^{2} - \frac{7}{16} a$, $\frac{1}{7424} a^{13} - \frac{25}{7424} a^{12} - \frac{51}{7424} a^{11} + \frac{149}{7424} a^{10} - \frac{407}{7424} a^{9} + \frac{389}{7424} a^{8} + \frac{751}{7424} a^{7} - \frac{73}{7424} a^{6} - \frac{685}{3712} a^{5} - \frac{265}{1856} a^{4} - \frac{325}{1856} a^{3} + \frac{427}{1856} a^{2} + \frac{321}{928} a - \frac{27}{58}$, $\frac{1}{29146624} a^{14} - \frac{385}{29146624} a^{13} + \frac{106679}{29146624} a^{12} + \frac{626697}{29146624} a^{11} - \frac{521121}{29146624} a^{10} - \frac{110883}{2242048} a^{9} - \frac{594531}{29146624} a^{8} - \frac{1258173}{29146624} a^{7} - \frac{323419}{3643328} a^{6} - \frac{241863}{1821664} a^{5} - \frac{161459}{7286656} a^{4} + \frac{1462679}{7286656} a^{3} + \frac{87315}{1821664} a^{2} - \frac{850131}{1821664} a - \frac{10017}{113854}$

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:  \( 29224114.7546 \) (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

3.1.1844.1, 5.1.3400336.2

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
461Data not computed