↑

Properties

Label 16.4.17139616816...3125.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{8}\cdot 19^{5}\cdot 29^{6}\cdot 31^{3}$
Root discriminant $37.77$
Ramified primes $5, 19, 29, 31$
Class number $4$
Class group $[2, 2]$
Galois group 16T1477

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-21059, 85873, -115415, 112080, -88705, 56450, -31412, 14121, -5274, 1565, -307, -23, 52, -33, 10, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 10*x^14 - 33*x^13 + 52*x^12 - 23*x^11 - 307*x^10 + 1565*x^9 - 5274*x^8 + 14121*x^7 - 31412*x^6 + 56450*x^5 - 88705*x^4 + 112080*x^3 - 115415*x^2 + 85873*x - 21059)
 
gp: K = bnfinit(x^16 - 3*x^15 + 10*x^14 - 33*x^13 + 52*x^12 - 23*x^11 - 307*x^10 + 1565*x^9 - 5274*x^8 + 14121*x^7 - 31412*x^6 + 56450*x^5 - 88705*x^4 + 112080*x^3 - 115415*x^2 + 85873*x - 21059, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 10 x^{14} - 33 x^{13} + 52 x^{12} - 23 x^{11} - 307 x^{10} + 1565 x^{9} - 5274 x^{8} + 14121 x^{7} - 31412 x^{6} + 56450 x^{5} - 88705 x^{4} + 112080 x^{3} - 115415 x^{2} + 85873 x - 21059 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17139616816488152808203125=5^{8}\cdot 19^{5}\cdot 29^{6}\cdot 31^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.77$
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:  $5, 19, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1313057861121147769330120508658349} a^{15} + \frac{393738438430473686143133792120349}{1313057861121147769330120508658349} a^{14} + \frac{36293993366511636776946281739769}{1313057861121147769330120508658349} a^{13} - \frac{493337771466568627833332699027628}{1313057861121147769330120508658349} a^{12} - \frac{549195381322137121881431094485466}{1313057861121147769330120508658349} a^{11} + \frac{187196966160856447358664837337238}{1313057861121147769330120508658349} a^{10} + \frac{2107314460309476615434186245250}{1313057861121147769330120508658349} a^{9} - \frac{531768111134312189581084241321943}{1313057861121147769330120508658349} a^{8} + \frac{394249813913553500295411540227537}{1313057861121147769330120508658349} a^{7} + \frac{306931743637858392666323628492292}{1313057861121147769330120508658349} a^{6} + \frac{70088038224571816005960422780255}{1313057861121147769330120508658349} a^{5} - \frac{463235037025079268961971130546826}{1313057861121147769330120508658349} a^{4} - \frac{227918193978787088967375341961340}{1313057861121147769330120508658349} a^{3} - \frac{172350371354772207241701120751073}{1313057861121147769330120508658349} a^{2} + \frac{217710407934140309041250219235222}{1313057861121147769330120508658349} a + \frac{638446624764116886430157895152601}{1313057861121147769330120508658349}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 349774.101138 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1477:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1477 are not computed
Character table for t16n1477 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.309593125.1

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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ $16$ R ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
5Data not computed
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.8.0.1$x^{8} - x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$