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Properties

Label 16.0.22873200555...6849.3
Degree $16$
Signature $[0, 8]$
Discriminant $37^{4}\cdot 73^{14}$
Root discriminant $105.31$
Ramified primes $37, 73$
Class number $89$ (GRH)
Class group $[89]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1859584, 678400, -1660032, 804608, 1216768, -1671056, 1016744, -431096, 138300, -42823, 14527, -4346, 1030, -192, 38, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 38*x^14 - 192*x^13 + 1030*x^12 - 4346*x^11 + 14527*x^10 - 42823*x^9 + 138300*x^8 - 431096*x^7 + 1016744*x^6 - 1671056*x^5 + 1216768*x^4 + 804608*x^3 - 1660032*x^2 + 678400*x + 1859584)
 
gp: K = bnfinit(x^16 - 7*x^15 + 38*x^14 - 192*x^13 + 1030*x^12 - 4346*x^11 + 14527*x^10 - 42823*x^9 + 138300*x^8 - 431096*x^7 + 1016744*x^6 - 1671056*x^5 + 1216768*x^4 + 804608*x^3 - 1660032*x^2 + 678400*x + 1859584, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 38 x^{14} - 192 x^{13} + 1030 x^{12} - 4346 x^{11} + 14527 x^{10} - 42823 x^{9} + 138300 x^{8} - 431096 x^{7} + 1016744 x^{6} - 1671056 x^{5} + 1216768 x^{4} + 804608 x^{3} - 1660032 x^{2} + 678400 x + 1859584 \)

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:  \(228732005557745506375281661426849=37^{4}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.31$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} + \frac{1}{16} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{3}{16} a^{7} + \frac{3}{16} a^{6} + \frac{3}{32} a^{5} - \frac{1}{32} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{384} a^{12} + \frac{1}{384} a^{11} - \frac{5}{192} a^{10} - \frac{1}{16} a^{9} + \frac{19}{192} a^{8} - \frac{29}{192} a^{7} - \frac{11}{128} a^{6} + \frac{11}{128} a^{5} - \frac{17}{96} a^{4} + \frac{1}{3} a^{3} - \frac{13}{48} a^{2} - \frac{1}{4} a + \frac{1}{3}$, $\frac{1}{12288} a^{13} + \frac{1}{4096} a^{12} + \frac{23}{1536} a^{11} - \frac{95}{3072} a^{10} - \frac{293}{6144} a^{9} + \frac{179}{2048} a^{8} - \frac{2549}{12288} a^{7} + \frac{53}{4096} a^{6} - \frac{1105}{6144} a^{5} - \frac{331}{1536} a^{4} + \frac{259}{1536} a^{3} + \frac{209}{768} a^{2} - \frac{19}{192} a - \frac{5}{12}$, $\frac{1}{196608} a^{14} + \frac{5}{196608} a^{13} + \frac{31}{98304} a^{12} - \frac{419}{49152} a^{11} + \frac{885}{32768} a^{10} - \frac{1585}{98304} a^{9} - \frac{3291}{65536} a^{8} + \frac{22453}{196608} a^{7} - \frac{10937}{49152} a^{6} - \frac{1453}{16384} a^{5} + \frac{5197}{24576} a^{4} - \frac{379}{3072} a^{3} - \frac{2485}{6144} a^{2} - \frac{107}{1536} a + \frac{11}{96}$, $\frac{1}{39183685973993340989789442146304} a^{15} + \frac{81733096198030293475275311}{39183685973993340989789442146304} a^{14} + \frac{2039717749820886043685035}{816326791124861270620613378048} a^{13} + \frac{922069369277579640149915765}{1224490186687291905930920067072} a^{12} - \frac{223772170318698701367190953757}{19591842986996670494894721073152} a^{11} - \frac{562357558583716791277771614875}{19591842986996670494894721073152} a^{10} - \frac{302122303677842567371848647845}{39183685973993340989789442146304} a^{9} - \frac{1126069345648793301373458043925}{39183685973993340989789442146304} a^{8} + \frac{488568141325531810284676609255}{19591842986996670494894721073152} a^{7} - \frac{55559415769051945437716047755}{3265307164499445082482453512192} a^{6} + \frac{132459793557985902169519288767}{816326791124861270620613378048} a^{5} + \frac{1786820128095364516126909215}{19910409539630762698063740928} a^{4} - \frac{42992165621153747178738604571}{408163395562430635310306689024} a^{3} + \frac{159336057784251479464736667025}{612245093343645952965460033536} a^{2} - \frac{4881289134568011777459020901}{51020424445303829413788336128} a + \frac{4136874593995521584606245159}{9566329583494468015085313024}$

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

Class group and class number

$C_{89}$, which has order $89$ (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:  \( 18076889725.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.0.11047398519097.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
37Data not computed
73Data not computed