↑

Properties

Label 16.0.25268810621...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 5^{12}\cdot 197^{4}$
Root discriminant $59.59$
Ramified primes $2, 5, 197$
Class number $464$ (GRH)
Class group $[2, 2, 116]$ (GRH)
Galois group 16T1049

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17271, -7692, 40462, 111532, 155096, 169284, 120690, 38020, 1196, 300, 1450, 252, 104, -12, 30, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 30*x^14 - 12*x^13 + 104*x^12 + 252*x^11 + 1450*x^10 + 300*x^9 + 1196*x^8 + 38020*x^7 + 120690*x^6 + 169284*x^5 + 155096*x^4 + 111532*x^3 + 40462*x^2 - 7692*x + 17271)
 
gp: K = bnfinit(x^16 - 4*x^15 + 30*x^14 - 12*x^13 + 104*x^12 + 252*x^11 + 1450*x^10 + 300*x^9 + 1196*x^8 + 38020*x^7 + 120690*x^6 + 169284*x^5 + 155096*x^4 + 111532*x^3 + 40462*x^2 - 7692*x + 17271, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 30 x^{14} - 12 x^{13} + 104 x^{12} + 252 x^{11} + 1450 x^{10} + 300 x^{9} + 1196 x^{8} + 38020 x^{7} + 120690 x^{6} + 169284 x^{5} + 155096 x^{4} + 111532 x^{3} + 40462 x^{2} - 7692 x + 17271 \)

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:  \(25268810621648896000000000000=2^{36}\cdot 5^{12}\cdot 197^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.59$
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:  $2, 5, 197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{5784891041855327096332947767707320234258} a^{15} - \frac{1369815147661414177457981721013841421049}{5784891041855327096332947767707320234258} a^{14} + \frac{150150905873915543602921723183341945152}{964148506975887849388824627951220039043} a^{13} + \frac{67799574734288786931118198591319555726}{964148506975887849388824627951220039043} a^{12} - \frac{304005747562564085559409407772788320114}{2892445520927663548166473883853660117129} a^{11} - \frac{4975530465537873042572944415098365779}{964148506975887849388824627951220039043} a^{10} + \frac{44048371671069810016596833713304020111}{5784891041855327096332947767707320234258} a^{9} + \frac{271418950138606791806521025954443736239}{1928297013951775698777649255902440078086} a^{8} - \frac{2391281095549950803972911610585426630041}{5784891041855327096332947767707320234258} a^{7} + \frac{752557602546612384124916565347792286961}{5784891041855327096332947767707320234258} a^{6} - \frac{299896721739066266214146549133341400073}{964148506975887849388824627951220039043} a^{5} - \frac{414626460641416195351434853443628857396}{964148506975887849388824627951220039043} a^{4} + \frac{52306546578386121459477140370329181763}{2892445520927663548166473883853660117129} a^{3} + \frac{1012750133147082837496360737289176469103}{2892445520927663548166473883853660117129} a^{2} - \frac{2757199749316736824183134676315041281033}{5784891041855327096332947767707320234258} a + \frac{131296575404439766508678734212334853781}{1928297013951775698777649255902440078086}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{116}$, which has order $464$ (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:  \( 126964.724492 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1049:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 768
The 31 conjugacy class representatives for t16n1049
Character table for t16n1049 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.63040.1, 8.8.99351040000.2

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$197$197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.4.2.1$x^{4} + 985 x^{2} + 349281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
197.4.2.1$x^{4} + 985 x^{2} + 349281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$