↑

Properties

Label 16.16.2165234002...9441.1
Degree $16$
Signature $[16, 0]$
Discriminant $13^{8}\cdot 61^{12}$
Root discriminant $78.70$
Ramified primes $13, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $QD_{16}$ (as 16T12)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-17848727, -31380089, 2446090, 30871725, 9737766, -10614120, -5130166, 1550091, 1008688, -95986, -94158, 1965, 4506, 25, -107, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 107*x^14 + 25*x^13 + 4506*x^12 + 1965*x^11 - 94158*x^10 - 95986*x^9 + 1008688*x^8 + 1550091*x^7 - 5130166*x^6 - 10614120*x^5 + 9737766*x^4 + 30871725*x^3 + 2446090*x^2 - 31380089*x - 17848727)
 
gp: K = bnfinit(x^16 - x^15 - 107*x^14 + 25*x^13 + 4506*x^12 + 1965*x^11 - 94158*x^10 - 95986*x^9 + 1008688*x^8 + 1550091*x^7 - 5130166*x^6 - 10614120*x^5 + 9737766*x^4 + 30871725*x^3 + 2446090*x^2 - 31380089*x - 17848727, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 107 x^{14} + 25 x^{13} + 4506 x^{12} + 1965 x^{11} - 94158 x^{10} - 95986 x^{9} + 1008688 x^{8} + 1550091 x^{7} - 5130166 x^{6} - 10614120 x^{5} + 9737766 x^{4} + 30871725 x^{3} + 2446090 x^{2} - 31380089 x - 17848727 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2165234002589380425486809479441=13^{8}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.70$
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 Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{2}{13} a^{11} - \frac{1}{13} a^{9} - \frac{3}{13} a^{7} - \frac{1}{13} a^{6} + \frac{3}{13} a^{5} - \frac{1}{13} a^{4} + \frac{4}{13} a^{3} - \frac{4}{13} a^{2}$, $\frac{1}{13} a^{13} - \frac{4}{13} a^{11} - \frac{1}{13} a^{10} + \frac{2}{13} a^{9} - \frac{3}{13} a^{8} + \frac{5}{13} a^{7} + \frac{5}{13} a^{6} + \frac{6}{13} a^{5} + \frac{6}{13} a^{4} + \frac{1}{13} a^{3} - \frac{5}{13} a^{2}$, $\frac{1}{3705} a^{14} - \frac{56}{3705} a^{13} + \frac{7}{3705} a^{12} - \frac{704}{3705} a^{11} - \frac{492}{1235} a^{10} - \frac{776}{3705} a^{9} - \frac{73}{195} a^{8} - \frac{358}{741} a^{7} - \frac{44}{741} a^{6} + \frac{74}{195} a^{5} + \frac{1604}{3705} a^{4} - \frac{127}{1235} a^{3} - \frac{778}{3705} a^{2} + \frac{67}{285} a - \frac{124}{285}$, $\frac{1}{5563026630817922769289782613913907105} a^{15} + \frac{4399986900234628672699191313906}{5563026630817922769289782613913907105} a^{14} + \frac{26024746498236072645145615751645810}{1112605326163584553857956522782781421} a^{13} + \frac{35801422150787234807692193718942668}{1112605326163584553857956522782781421} a^{12} + \frac{763385610014631205663718057672629867}{1854342210272640923096594204637969035} a^{11} + \frac{1397912908238316107215386654226445617}{5563026630817922769289782613913907105} a^{10} - \frac{1596230899656166924714664934284921734}{5563026630817922769289782613913907105} a^{9} + \frac{63616248739533714192346062756841426}{5563026630817922769289782613913907105} a^{8} - \frac{440584192784644159011234721237859}{6583463468423577241763056347827109} a^{7} - \frac{966763896148453815937924884930498589}{5563026630817922769289782613913907105} a^{6} + \frac{220292330456259443980817640924067541}{5563026630817922769289782613913907105} a^{5} - \frac{818076660960054093288467070074395586}{1854342210272640923096594204637969035} a^{4} - \frac{25701977930535171167986813064130962}{85585025089506504142919732521752417} a^{3} - \frac{9199952987422274365000283827517251}{58558175061241292308313501199093759} a^{2} + \frac{7819308697541431875068698220295274}{85585025089506504142919732521752417} a - \frac{59998407760140530874890855379211181}{142641708482510840238199554202920695}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 15423219542.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{793}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{61})\), 4.4.48373.1 x2, 4.4.10309.1 x2, 8.8.395451064801.1, 8.8.113190262471117.1 x4

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$