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Properties

Label 16.0.13344870400...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{14}\cdot 19^{4}$
Root discriminant $24.15$
Ramified primes $2, 5, 19$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2\times C_2^3.C_4$ (as 16T99)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1681, 5658, 4432, -2506, -89, 3618, -3600, -1378, 1954, -1874, 1580, -664, 256, -78, 18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 18*x^14 - 78*x^13 + 256*x^12 - 664*x^11 + 1580*x^10 - 1874*x^9 + 1954*x^8 - 1378*x^7 - 3600*x^6 + 3618*x^5 - 89*x^4 - 2506*x^3 + 4432*x^2 + 5658*x + 1681)
 
gp: K = bnfinit(x^16 - 4*x^15 + 18*x^14 - 78*x^13 + 256*x^12 - 664*x^11 + 1580*x^10 - 1874*x^9 + 1954*x^8 - 1378*x^7 - 3600*x^6 + 3618*x^5 - 89*x^4 - 2506*x^3 + 4432*x^2 + 5658*x + 1681, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 18 x^{14} - 78 x^{13} + 256 x^{12} - 664 x^{11} + 1580 x^{10} - 1874 x^{9} + 1954 x^{8} - 1378 x^{7} - 3600 x^{6} + 3618 x^{5} - 89 x^{4} - 2506 x^{3} + 4432 x^{2} + 5658 x + 1681 \)

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:  \(13344870400000000000000=2^{24}\cdot 5^{14}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.15$
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:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} - \frac{3}{11} a^{12} - \frac{5}{11} a^{11} + \frac{1}{11} a^{9} + \frac{3}{11} a^{8} + \frac{1}{11} a^{7} + \frac{4}{11} a^{6} + \frac{5}{11} a^{5} + \frac{4}{11} a^{2} + \frac{3}{11} a + \frac{2}{11}$, $\frac{1}{11} a^{14} - \frac{3}{11} a^{12} - \frac{4}{11} a^{11} + \frac{1}{11} a^{10} - \frac{5}{11} a^{9} - \frac{1}{11} a^{8} - \frac{4}{11} a^{7} - \frac{5}{11} a^{6} + \frac{4}{11} a^{5} + \frac{4}{11} a^{3} + \frac{4}{11} a^{2} - \frac{5}{11}$, $\frac{1}{53761556375033425489631361713071} a^{15} + \frac{2329118095078578287124934976682}{53761556375033425489631361713071} a^{14} + \frac{237729086798537515773094964542}{53761556375033425489631361713071} a^{13} - \frac{515019221647765130056275805304}{4887414215912129589966487428461} a^{12} - \frac{16229958783315454725327861881456}{53761556375033425489631361713071} a^{11} + \frac{6539058814960712957446208866731}{53761556375033425489631361713071} a^{10} + \frac{1576879770187992055821597832803}{4887414215912129589966487428461} a^{9} - \frac{14591972609723207936465756477734}{53761556375033425489631361713071} a^{8} - \frac{6622577261738805014080105564864}{53761556375033425489631361713071} a^{7} + \frac{21757792305204584010377619608890}{53761556375033425489631361713071} a^{6} + \frac{20317710428584548127171201698232}{53761556375033425489631361713071} a^{5} - \frac{15985694829672072298484094927827}{53761556375033425489631361713071} a^{4} - \frac{312494650330654256419610763962}{4887414215912129589966487428461} a^{3} - \frac{5818234731715382058606926782972}{53761556375033425489631361713071} a^{2} - \frac{12560915466810877207749819039209}{53761556375033425489631361713071} a - \frac{52819798131380790744513974562}{1311257472561790865600764919831}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{59162391136029444622593864880}{53761556375033425489631361713071} a^{15} - \frac{262086708774237669068312987924}{53761556375033425489631361713071} a^{14} + \frac{1167781928564663806335430793875}{53761556375033425489631361713071} a^{13} - \frac{462101491971895493519597014380}{4887414215912129589966487428461} a^{12} + \frac{17176282165280160906818359389273}{53761556375033425489631361713071} a^{11} - \frac{46003932229593402866589130622668}{53761556375033425489631361713071} a^{10} + \frac{10106670243900329341394339664923}{4887414215912129589966487428461} a^{9} - \frac{153553846164501328043636065092076}{53761556375033425489631361713071} a^{8} + \frac{169751538691975711035208108544369}{53761556375033425489631361713071} a^{7} - \frac{144658921778841370097550585077891}{53761556375033425489631361713071} a^{6} - \frac{160832136102961943528791507104298}{53761556375033425489631361713071} a^{5} + \frac{287683733174569650100306614727302}{53761556375033425489631361713071} a^{4} - \frac{8433475979231294465638580398560}{4887414215912129589966487428461} a^{3} - \frac{115292229998716712617591921659693}{53761556375033425489631361713071} a^{2} + \frac{298392789493962175986333164970975}{53761556375033425489631361713071} a + \frac{5546788651355150726313373342390}{1311257472561790865600764919831} \) (order $20$)
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:  \( 58305.1656546 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^3.C_4$ (as 16T99):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2\times C_2^3.C_4$
Character table for $C_2\times C_2^3.C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), \(\Q(i, \sqrt{5})\), 8.0.115520000000.1, 8.8.115520000000.1, \(\Q(\zeta_{20})\)

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$