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Properties

Label 16.0.57268745716...5629.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{8}\cdot 149^{7}$
Root discriminant $54.31$
Ramified primes $37, 149$
Class number $47$ (GRH)
Class group $[47]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![50919, -57305, -55032, 26263, 57643, 40365, 16030, 323, -1443, -660, 2, 157, 69, -5, -5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 5*x^14 - 5*x^13 + 69*x^12 + 157*x^11 + 2*x^10 - 660*x^9 - 1443*x^8 + 323*x^7 + 16030*x^6 + 40365*x^5 + 57643*x^4 + 26263*x^3 - 55032*x^2 - 57305*x + 50919)
 
gp: K = bnfinit(x^16 - 3*x^15 - 5*x^14 - 5*x^13 + 69*x^12 + 157*x^11 + 2*x^10 - 660*x^9 - 1443*x^8 + 323*x^7 + 16030*x^6 + 40365*x^5 + 57643*x^4 + 26263*x^3 - 55032*x^2 - 57305*x + 50919, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 5 x^{14} - 5 x^{13} + 69 x^{12} + 157 x^{11} + 2 x^{10} - 660 x^{9} - 1443 x^{8} + 323 x^{7} + 16030 x^{6} + 40365 x^{5} + 57643 x^{4} + 26263 x^{3} - 55032 x^{2} - 57305 x + 50919 \)

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:  \(5726874571603625384731365629=37^{8}\cdot 149^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.31$
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:  $37, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{4}{9} a^{6} - \frac{1}{9} a^{5} + \frac{4}{9} a^{4} + \frac{4}{9} a^{2} - \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{45} a^{12} - \frac{1}{45} a^{11} - \frac{2}{45} a^{10} - \frac{1}{9} a^{9} - \frac{1}{45} a^{8} + \frac{1}{15} a^{7} - \frac{4}{9} a^{6} + \frac{2}{15} a^{5} - \frac{8}{45} a^{4} + \frac{4}{45} a^{3} - \frac{2}{5} a^{2} + \frac{11}{45} a - \frac{2}{15}$, $\frac{1}{135} a^{13} + \frac{7}{135} a^{11} - \frac{4}{45} a^{10} - \frac{7}{45} a^{9} - \frac{2}{15} a^{8} - \frac{7}{135} a^{7} - \frac{8}{45} a^{6} + \frac{11}{45} a^{5} + \frac{2}{45} a^{4} - \frac{14}{135} a^{3} - \frac{19}{45} a^{2} - \frac{7}{27} a - \frac{22}{45}$, $\frac{1}{405} a^{14} + \frac{1}{405} a^{13} + \frac{4}{405} a^{12} - \frac{2}{405} a^{11} - \frac{1}{15} a^{10} + \frac{22}{135} a^{9} - \frac{67}{405} a^{8} + \frac{10}{81} a^{7} + \frac{38}{135} a^{6} + \frac{67}{135} a^{5} - \frac{29}{405} a^{4} + \frac{142}{405} a^{3} - \frac{128}{405} a^{2} + \frac{91}{405} a - \frac{16}{135}$, $\frac{1}{12703149707345046005504447835} a^{15} - \frac{338721076925284325848943}{747244100432061529735555755} a^{14} - \frac{15890431875919101633753298}{12703149707345046005504447835} a^{13} + \frac{116392822297416339831786398}{12703149707345046005504447835} a^{12} + \frac{641786823834612871835831644}{12703149707345046005504447835} a^{11} - \frac{34519413334942090042066097}{4234383235781682001834815945} a^{10} + \frac{46116496312726884449361821}{12703149707345046005504447835} a^{9} - \frac{1994025344770459171863251399}{12703149707345046005504447835} a^{8} + \frac{928452311325880386435246653}{12703149707345046005504447835} a^{7} + \frac{52756658149764615820735229}{470487026197964666870535105} a^{6} + \frac{5484462095628206595504917362}{12703149707345046005504447835} a^{5} + \frac{3068245420262477332654131791}{12703149707345046005504447835} a^{4} - \frac{122061135183662464599441575}{2540629941469009201100889567} a^{3} + \frac{833718314031412772744539}{747244100432061529735555755} a^{2} + \frac{237848859948174748159415788}{12703149707345046005504447835} a + \frac{132077698404273274746742006}{384943930525607454712255995}$

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

Class group and class number

$C_{47}$, which has order $47$ (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:  \( 1665869.40914 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{37}) \), 4.4.203981.1, 8.8.6199629005789.1

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 $16$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ $16$

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
$37$37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$149$$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$