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Properties

Label 16.0.58189572765...5321.5
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 61^{10}$
Root discriminant $47.08$
Ramified primes $13, 61$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2.SD_{16}$ (as 16T163)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2197, -14196, 43670, -86543, 127502, -144991, 123062, -73079, 27762, -4714, -1271, 1165, -308, 25, 5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 5*x^14 + 25*x^13 - 308*x^12 + 1165*x^11 - 1271*x^10 - 4714*x^9 + 27762*x^8 - 73079*x^7 + 123062*x^6 - 144991*x^5 + 127502*x^4 - 86543*x^3 + 43670*x^2 - 14196*x + 2197)
 
gp: K = bnfinit(x^16 - 4*x^15 + 5*x^14 + 25*x^13 - 308*x^12 + 1165*x^11 - 1271*x^10 - 4714*x^9 + 27762*x^8 - 73079*x^7 + 123062*x^6 - 144991*x^5 + 127502*x^4 - 86543*x^3 + 43670*x^2 - 14196*x + 2197, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 5 x^{14} + 25 x^{13} - 308 x^{12} + 1165 x^{11} - 1271 x^{10} - 4714 x^{9} + 27762 x^{8} - 73079 x^{7} + 123062 x^{6} - 144991 x^{5} + 127502 x^{4} - 86543 x^{3} + 43670 x^{2} - 14196 x + 2197 \)

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:  \(581895727651002533052085321=13^{8}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.08$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13} a^{10} + \frac{5}{13} a^{8} + \frac{4}{13} a^{7} - \frac{1}{13} a^{4} - \frac{4}{13} a^{3} - \frac{1}{13} a^{2} + \frac{5}{13} a$, $\frac{1}{13} a^{11} + \frac{5}{13} a^{9} + \frac{4}{13} a^{8} - \frac{1}{13} a^{5} - \frac{4}{13} a^{4} - \frac{1}{13} a^{3} + \frac{5}{13} a^{2}$, $\frac{1}{39} a^{12} + \frac{1}{39} a^{11} + \frac{1}{39} a^{10} - \frac{17}{39} a^{9} - \frac{1}{13} a^{8} + \frac{10}{39} a^{7} + \frac{4}{13} a^{6} - \frac{5}{39} a^{5} + \frac{4}{13} a^{4} - \frac{2}{13} a^{3} + \frac{3}{13} a^{2} + \frac{19}{39} a + \frac{1}{3}$, $\frac{1}{39} a^{13} + \frac{14}{39} a^{9} - \frac{14}{39} a^{8} - \frac{4}{39} a^{7} - \frac{17}{39} a^{6} + \frac{17}{39} a^{5} + \frac{1}{13} a^{4} - \frac{6}{13} a^{3} - \frac{8}{39} a^{2} + \frac{2}{13} a - \frac{1}{3}$, $\frac{1}{39} a^{14} - \frac{1}{39} a^{10} - \frac{14}{39} a^{9} - \frac{1}{39} a^{8} + \frac{1}{39} a^{7} + \frac{17}{39} a^{6} + \frac{1}{13} a^{5} - \frac{1}{13} a^{4} + \frac{1}{3} a^{3} - \frac{6}{13} a^{2} - \frac{10}{39} a$, $\frac{1}{35096399360001685114964061} a^{15} + \frac{142167647849128414790311}{35096399360001685114964061} a^{14} - \frac{436667990723017194529717}{35096399360001685114964061} a^{13} - \frac{67738436282219819858243}{11698799786667228371654687} a^{12} + \frac{867298790623071960304115}{35096399360001685114964061} a^{11} + \frac{6139149499755177937152}{11698799786667228371654687} a^{10} - \frac{5908995218096065845398816}{35096399360001685114964061} a^{9} + \frac{13882036527510198080136977}{35096399360001685114964061} a^{8} - \frac{2019791317945769769344807}{35096399360001685114964061} a^{7} - \frac{179225653160217929109224}{35096399360001685114964061} a^{6} - \frac{12255504944771431217171720}{35096399360001685114964061} a^{5} - \frac{263764617342867234573251}{2699723027692437316535697} a^{4} + \frac{1970118366514130819461036}{35096399360001685114964061} a^{3} - \frac{9182899965331335790338515}{35096399360001685114964061} a^{2} - \frac{2595472329544798102694344}{35096399360001685114964061} a - \frac{1137604440742087053499187}{2699723027692437316535697}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 2807721.54944 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.10309.1, 8.4.24122514952861.2, 8.0.6482804341.1, 8.4.395451064801.2

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
$61$61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$