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Properties

Label 16.0.15673665258...6729.5
Degree $16$
Signature $[0, 8]$
Discriminant $3^{10}\cdot 61^{12}$
Root discriminant $43.37$
Ramified primes $3, 61$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $C_2^4.D_4$ (as 16T330)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![50625, 73575, 83325, 47622, 15196, 3843, 2385, -1011, 879, -462, 388, -114, 69, -39, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 15*x^14 - 39*x^13 + 69*x^12 - 114*x^11 + 388*x^10 - 462*x^9 + 879*x^8 - 1011*x^7 + 2385*x^6 + 3843*x^5 + 15196*x^4 + 47622*x^3 + 83325*x^2 + 73575*x + 50625)
 
gp: K = bnfinit(x^16 - 6*x^15 + 15*x^14 - 39*x^13 + 69*x^12 - 114*x^11 + 388*x^10 - 462*x^9 + 879*x^8 - 1011*x^7 + 2385*x^6 + 3843*x^5 + 15196*x^4 + 47622*x^3 + 83325*x^2 + 73575*x + 50625, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 15 x^{14} - 39 x^{13} + 69 x^{12} - 114 x^{11} + 388 x^{10} - 462 x^{9} + 879 x^{8} - 1011 x^{7} + 2385 x^{6} + 3843 x^{5} + 15196 x^{4} + 47622 x^{3} + 83325 x^{2} + 73575 x + 50625 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(156736652583298165062696729=3^{10}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.37$
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:  $3, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{15} a^{9} + \frac{2}{15} a^{8} - \frac{1}{15} a^{7} - \frac{1}{15} a^{6} - \frac{7}{15} a^{5} - \frac{4}{15} a^{4} - \frac{2}{15} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{15} a^{10} + \frac{1}{15} a^{7} + \frac{1}{3} a^{6} + \frac{2}{5} a^{4} + \frac{1}{3} a^{3} + \frac{2}{5} a$, $\frac{1}{45} a^{11} + \frac{1}{45} a^{10} + \frac{1}{45} a^{9} - \frac{7}{45} a^{8} + \frac{1}{9} a^{7} - \frac{16}{45} a^{6} + \frac{4}{45} a^{5} + \frac{22}{45} a^{4} + \frac{13}{45} a^{3} - \frac{2}{5} a^{2} + \frac{1}{15} a$, $\frac{1}{45} a^{12} + \frac{1}{45} a^{9} - \frac{4}{45} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{19}{45} a^{3} + \frac{1}{3} a$, $\frac{1}{135} a^{13} + \frac{4}{135} a^{10} - \frac{1}{45} a^{9} - \frac{7}{45} a^{8} + \frac{2}{135} a^{7} - \frac{1}{5} a^{6} - \frac{1}{15} a^{5} + \frac{11}{135} a^{4} - \frac{1}{15} a^{3} + \frac{14}{45} a^{2} - \frac{2}{15} a$, $\frac{1}{2025} a^{14} + \frac{2}{2025} a^{13} + \frac{2}{675} a^{12} + \frac{4}{2025} a^{11} + \frac{41}{2025} a^{10} - \frac{7}{675} a^{9} + \frac{2}{81} a^{8} + \frac{148}{2025} a^{7} - \frac{44}{675} a^{6} - \frac{682}{2025} a^{5} - \frac{401}{2025} a^{4} + \frac{2}{5} a^{3} - \frac{158}{675} a^{2} + \frac{1}{5} a$, $\frac{1}{677922352552517633976825} a^{15} + \frac{5738365969005935569}{27116894102100705359073} a^{14} + \frac{1532688652373539570222}{677922352552517633976825} a^{13} + \frac{2466742545945940715917}{677922352552517633976825} a^{12} - \frac{4320125019830658740957}{677922352552517633976825} a^{11} - \frac{6680900019542297312618}{677922352552517633976825} a^{10} - \frac{1094085355505910658093}{677922352552517633976825} a^{9} - \frac{20309893142539818010987}{677922352552517633976825} a^{8} - \frac{2020460249788261233733}{677922352552517633976825} a^{7} - \frac{169818707578502330676883}{677922352552517633976825} a^{6} - \frac{27108946996502885837557}{677922352552517633976825} a^{5} - \frac{286940756830496935938988}{677922352552517633976825} a^{4} + \frac{830403843999136335388}{2073157041445008054975} a^{3} + \frac{14535598616219340252401}{225974117517505877992275} a^{2} - \frac{834116394563104452614}{5021647055944575066495} a - \frac{82201956161115123649}{334776470396305004433}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( 4498736.68408 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T330):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{61}) \), 4.2.11163.1, 8.0.68412300381.1, 8.2.1391050107747.2, 8.2.205236901143.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
61Data not computed