↑

Properties

Label 16.0.77825707884...0000.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{10}\cdot 41^{6}$
Root discriminant $31.13$
Ramified primes $2, 5, 41$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T869

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3231, -822, 5227, -4324, 6498, -5146, 3094, -2462, 1183, -654, 486, -258, 148, -68, 23, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 23*x^14 - 68*x^13 + 148*x^12 - 258*x^11 + 486*x^10 - 654*x^9 + 1183*x^8 - 2462*x^7 + 3094*x^6 - 5146*x^5 + 6498*x^4 - 4324*x^3 + 5227*x^2 - 822*x + 3231)
 
gp: K = bnfinit(x^16 - 6*x^15 + 23*x^14 - 68*x^13 + 148*x^12 - 258*x^11 + 486*x^10 - 654*x^9 + 1183*x^8 - 2462*x^7 + 3094*x^6 - 5146*x^5 + 6498*x^4 - 4324*x^3 + 5227*x^2 - 822*x + 3231, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 23 x^{14} - 68 x^{13} + 148 x^{12} - 258 x^{11} + 486 x^{10} - 654 x^{9} + 1183 x^{8} - 2462 x^{7} + 3094 x^{6} - 5146 x^{5} + 6498 x^{4} - 4324 x^{3} + 5227 x^{2} - 822 x + 3231 \)

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:  \(778257078845440000000000=2^{24}\cdot 5^{10}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.13$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{30} a^{12} + \frac{1}{5} a^{10} - \frac{4}{15} a^{9} - \frac{1}{10} a^{8} + \frac{7}{15} a^{7} - \frac{7}{30} a^{6} + \frac{1}{15} a^{5} + \frac{1}{10} a^{4} + \frac{1}{3} a^{3} - \frac{1}{15} a^{2} - \frac{4}{15} a - \frac{3}{10}$, $\frac{1}{30} a^{13} - \frac{2}{15} a^{11} + \frac{1}{15} a^{10} - \frac{13}{30} a^{9} + \frac{7}{15} a^{8} - \frac{7}{30} a^{7} + \frac{2}{5} a^{6} + \frac{1}{10} a^{5} - \frac{1}{3} a^{4} - \frac{2}{5} a^{3} - \frac{4}{15} a^{2} + \frac{11}{30} a$, $\frac{1}{4590} a^{14} + \frac{49}{4590} a^{13} - \frac{2}{459} a^{12} - \frac{64}{765} a^{11} + \frac{1199}{4590} a^{10} - \frac{209}{918} a^{9} - \frac{2003}{4590} a^{8} - \frac{19}{102} a^{7} - \frac{1237}{4590} a^{6} + \frac{47}{102} a^{5} + \frac{62}{153} a^{4} - \frac{1058}{2295} a^{3} + \frac{1121}{4590} a^{2} - \frac{131}{1530} a - \frac{67}{255}$, $\frac{1}{26588268802778060414790} a^{15} - \frac{226110067845720143}{26588268802778060414790} a^{14} - \frac{239311422692787141641}{26588268802778060414790} a^{13} - \frac{114430679418287423033}{8862756267592686804930} a^{12} - \frac{510255203761052857771}{26588268802778060414790} a^{11} - \frac{2475416306662499500219}{26588268802778060414790} a^{10} + \frac{2416631055075335802458}{13294134401389030207395} a^{9} + \frac{9837306821298530513}{1477126044598781134155} a^{8} + \frac{3198822179985845802457}{13294134401389030207395} a^{7} - \frac{130537528696588788434}{1477126044598781134155} a^{6} + \frac{3753878226749015180837}{8862756267592686804930} a^{5} - \frac{4732285984407276064831}{26588268802778060414790} a^{4} - \frac{8986065313871407232191}{26588268802778060414790} a^{3} - \frac{2312043748795762593617}{8862756267592686804930} a^{2} + \frac{135010092650693479781}{984750696399187422770} a - \frac{68287445549927188669}{984750696399187422770}$

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

Class group and class number

$C_{4}$, 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:  \( -1 \) (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:  \( 109907.73837 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n869
Character table for t16n869 is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), 4.0.65600.2, 4.0.65600.5, \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.4303360000.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$