↑

Properties

Label 16.0.39137690856...1609.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{14}\cdot 67^{12}$
Root discriminant $61.24$
Ramified primes $3, 67$
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![1577536, 1863904, 1069344, 180536, -53896, 33990, 32545, 3274, 5097, 1696, -1775, -840, 89, 92, 9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 9*x^14 + 92*x^13 + 89*x^12 - 840*x^11 - 1775*x^10 + 1696*x^9 + 5097*x^8 + 3274*x^7 + 32545*x^6 + 33990*x^5 - 53896*x^4 + 180536*x^3 + 1069344*x^2 + 1863904*x + 1577536)
 
gp: K = bnfinit(x^16 - 2*x^15 + 9*x^14 + 92*x^13 + 89*x^12 - 840*x^11 - 1775*x^10 + 1696*x^9 + 5097*x^8 + 3274*x^7 + 32545*x^6 + 33990*x^5 - 53896*x^4 + 180536*x^3 + 1069344*x^2 + 1863904*x + 1577536, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 9 x^{14} + 92 x^{13} + 89 x^{12} - 840 x^{11} - 1775 x^{10} + 1696 x^{9} + 5097 x^{8} + 3274 x^{7} + 32545 x^{6} + 33990 x^{5} - 53896 x^{4} + 180536 x^{3} + 1069344 x^{2} + 1863904 x + 1577536 \)

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:  \(39137690856572912103863781609=3^{14}\cdot 67^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.24$
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:  $3, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{1}{12} a^{6} - \frac{1}{12} a^{5} + \frac{1}{12} a^{4} - \frac{1}{12} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{9} + \frac{1}{4} a^{3} + \frac{1}{3}$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{24} a^{4} + \frac{1}{24} a^{3} + \frac{5}{24} a^{2} + \frac{1}{12} a + \frac{1}{6}$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{10} + \frac{1}{48} a^{9} - \frac{1}{24} a^{8} + \frac{1}{24} a^{7} + \frac{5}{24} a^{6} + \frac{5}{48} a^{5} + \frac{7}{48} a^{4} - \frac{19}{48} a^{3} - \frac{1}{12} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{240} a^{12} + \frac{1}{120} a^{11} + \frac{1}{120} a^{10} + \frac{1}{240} a^{9} - \frac{1}{60} a^{8} + \frac{7}{60} a^{7} - \frac{5}{48} a^{6} + \frac{29}{120} a^{5} - \frac{11}{120} a^{4} - \frac{1}{240} a^{3} - \frac{7}{30} a^{2} + \frac{3}{10} a + \frac{1}{10}$, $\frac{1}{720} a^{13} + \frac{1}{720} a^{12} - \frac{11}{720} a^{10} + \frac{5}{144} a^{9} - \frac{1}{40} a^{8} + \frac{3}{80} a^{7} - \frac{19}{240} a^{6} + \frac{1}{12} a^{5} - \frac{149}{720} a^{4} - \frac{61}{144} a^{3} - \frac{47}{120} a^{2} - \frac{47}{180} a - \frac{14}{45}$, $\frac{1}{1176735096000} a^{14} - \frac{4240019}{147091887000} a^{13} - \frac{7168459}{235347019200} a^{12} - \frac{46124539}{23534701920} a^{11} + \frac{2971794869}{1176735096000} a^{10} + \frac{2321625301}{117673509600} a^{9} - \frac{1200144239}{130748344000} a^{8} - \frac{23642534249}{196122516000} a^{7} + \frac{5109471967}{392245032000} a^{6} + \frac{125374037}{5883675480} a^{5} + \frac{80718696041}{1176735096000} a^{4} - \frac{7190252279}{29418377400} a^{3} + \frac{21458833817}{58836754800} a^{2} - \frac{10105339157}{36772971750} a + \frac{104522257}{468445500}$, $\frac{1}{2624687521225209360000} a^{15} + \frac{5837487}{36453993350350130000} a^{14} + \frac{291015063949697773}{2624687521225209360000} a^{13} - \frac{16978064012314733}{29163194680280104000} a^{12} + \frac{4809328357616057923}{874895840408403120000} a^{11} - \frac{26999368054010952043}{1312343760612604680000} a^{10} - \frac{26676154035742961191}{2624687521225209360000} a^{9} - \frac{250642589011358017}{87489584040840312000} a^{8} + \frac{10783795476471163433}{291631946802801040000} a^{7} - \frac{53601045574776309821}{656171880306302340000} a^{6} - \frac{18841815654184777851}{291631946802801040000} a^{5} + \frac{61366543361866918187}{328085940153151170000} a^{4} - \frac{62567792936160167}{43744792020420156000} a^{3} + \frac{4622607638870426071}{27340495012762597500} a^{2} + \frac{1336196543059481797}{6561718803063023400} a + \frac{112022437803475703}{261214920504101250}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1609572213}{175092908435000} a^{15} - \frac{2376528189}{87546454217500} a^{14} + \frac{7959231647}{87546454217500} a^{13} + \frac{180966811283}{210111490122000} a^{12} - \frac{169329087031}{350185816870000} a^{11} - \frac{7854021098423}{1050557450610000} a^{10} - \frac{1060649145181}{131319681326250} a^{9} + \frac{1193533582901}{35018581687000} a^{8} - \frac{126069117499}{175092908435000} a^{7} + \frac{6262145363621}{350185816870000} a^{6} + \frac{109680979056641}{350185816870000} a^{5} - \frac{42015144052649}{350185816870000} a^{4} - \frac{31215081398773}{52527872530500} a^{3} + \frac{128026268853727}{43773227108750} a^{2} + \frac{74659010029823}{13131968132625} a + \frac{4207123793566}{418215545625} \) (order $6$)
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:  \( 1128505968.29 \) (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

\(\Q(\sqrt{-67}) \), \(\Q(\sqrt{201}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-67})\), 4.2.121203.2 x2, 4.0.1809.1 x2, 8.0.14690167209.1, 8.2.197832481803603.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
3Data not computed
$67$67.8.6.2$x^{8} + 1541 x^{4} + 646416$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
67.8.6.2$x^{8} + 1541 x^{4} + 646416$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$