↑

Properties

Label 16.0.14764424308...881.43
Degree $16$
Signature $[0, 8]$
Discriminant $37^{12}\cdot 157^{12}$
Root discriminant $665.39$
Ramified primes $37, 157$
Class number $326861312$ (GRH)
Class group $[2, 2, 2, 2, 2, 3196, 3196]$ (GRH)
Galois group $Q_{16}$ (as 16T14)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5423234506172099, 3629213187441572, 709619381151297, 13345645225409, 1622671062847, 2895427880223, 234290252130, -16644451521, 2476254439, 527271324, 9678642, 477002, 344716, -929, 1245, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 1245*x^14 - 929*x^13 + 344716*x^12 + 477002*x^11 + 9678642*x^10 + 527271324*x^9 + 2476254439*x^8 - 16644451521*x^7 + 234290252130*x^6 + 2895427880223*x^5 + 1622671062847*x^4 + 13345645225409*x^3 + 709619381151297*x^2 + 3629213187441572*x + 5423234506172099)
 
gp: K = bnfinit(x^16 - 3*x^15 + 1245*x^14 - 929*x^13 + 344716*x^12 + 477002*x^11 + 9678642*x^10 + 527271324*x^9 + 2476254439*x^8 - 16644451521*x^7 + 234290252130*x^6 + 2895427880223*x^5 + 1622671062847*x^4 + 13345645225409*x^3 + 709619381151297*x^2 + 3629213187441572*x + 5423234506172099, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 1245 x^{14} - 929 x^{13} + 344716 x^{12} + 477002 x^{11} + 9678642 x^{10} + 527271324 x^{9} + 2476254439 x^{8} - 16644451521 x^{7} + 234290252130 x^{6} + 2895427880223 x^{5} + 1622671062847 x^{4} + 13345645225409 x^{3} + 709619381151297 x^{2} + 3629213187441572 x + 5423234506172099 \)

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:  \(1476442430876140116899060702773874183614562881=37^{12}\cdot 157^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $665.39$
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:  $37, 157$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{63458345249222692163559694149646556317029514979012717687660392300507830842954516615674754252663350401780207644831229} a^{15} + \frac{9633186553354356584756146475855335920986373654906496292396313624961582401947605234819528836880139036688268219233182}{63458345249222692163559694149646556317029514979012717687660392300507830842954516615674754252663350401780207644831229} a^{14} + \frac{7952581121733879808093813715052279379476775551933497316006402918060598728112216623598609025576794630718651721627482}{21152781749740897387853231383215518772343171659670905895886797433502610280984838871891584750887783467260069214943743} a^{13} - \frac{964220987542939336793003937526960063707776096793495779888925904942364304533430513858982431581357263450987785965243}{21152781749740897387853231383215518772343171659670905895886797433502610280984838871891584750887783467260069214943743} a^{12} + \frac{1719280971198718221917307755012697141454790255965696598630488229655087372259331870307902747580945008396506373461136}{63458345249222692163559694149646556317029514979012717687660392300507830842954516615674754252663350401780207644831229} a^{11} - \frac{18927689688060528016976846408052185241212582485597901966647061426858948342565922014336553172329651781483579042701606}{63458345249222692163559694149646556317029514979012717687660392300507830842954516615674754252663350401780207644831229} a^{10} + \frac{9093470043071474692400128504904061233776140815802338927678806423166644797225977967092258540485718166077679446057292}{21152781749740897387853231383215518772343171659670905895886797433502610280984838871891584750887783467260069214943743} a^{9} + \frac{26283598001256714393839972350684337163772309142790690722438137843487715610797129420874807522351213244841796810892639}{63458345249222692163559694149646556317029514979012717687660392300507830842954516615674754252663350401780207644831229} a^{8} + \frac{3992267106772178090304089655028048381394078547424440998652430750523425634032434421866407825597020583033760253849341}{21152781749740897387853231383215518772343171659670905895886797433502610280984838871891584750887783467260069214943743} a^{7} - \frac{4799345276576686461371560710947706467600342209861157272646622951836373907557334029285752729346150468912617684682347}{21152781749740897387853231383215518772343171659670905895886797433502610280984838871891584750887783467260069214943743} a^{6} + \frac{57413732811617632306584024614761801122065160550328341584549023732815468231020293004478009249071744819523429741882}{21152781749740897387853231383215518772343171659670905895886797433502610280984838871891584750887783467260069214943743} a^{5} - \frac{2022929038299260471775191713040626142596772228965060279759496570208603724882602028999460532129569989672634732526902}{21152781749740897387853231383215518772343171659670905895886797433502610280984838871891584750887783467260069214943743} a^{4} + \frac{2125722619653377695081108702185560663845221365784667778496091725736580364037666258007255396421339251553102614290894}{63458345249222692163559694149646556317029514979012717687660392300507830842954516615674754252663350401780207644831229} a^{3} - \frac{11669807764797443178556477726323686332355828467950162557891575345315518373245368671690713663552235755080147604648776}{63458345249222692163559694149646556317029514979012717687660392300507830842954516615674754252663350401780207644831229} a^{2} + \frac{6016326105250816779294105813391040574269325651341872958661043491186961819894963495023683456445856672130662704274564}{63458345249222692163559694149646556317029514979012717687660392300507830842954516615674754252663350401780207644831229} a + \frac{18282524771832567024092190405237449401519187180977525747362502169017777644179971895662590334565832456837870257432122}{63458345249222692163559694149646556317029514979012717687660392300507830842954516615674754252663350401780207644831229}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{3196}\times C_{3196}$, which has order $326861312$ (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:  \( 11997899.8215 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_{16}$ (as 16T14):

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 $Q_{16}$
Character table for $Q_{16}$

Intermediate fields

\(\Q(\sqrt{5809}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{157}) \), \(\Q(\sqrt{37}, \sqrt{157})\), 4.4.214933.1 x2, 4.4.912013.1 x2, 8.8.1138689997959361.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
$157$157.4.3.1$x^{4} - 157$$4$$1$$3$$C_4$$[\ ]_{4}$
157.4.3.1$x^{4} - 157$$4$$1$$3$$C_4$$[\ ]_{4}$
157.4.3.1$x^{4} - 157$$4$$1$$3$$C_4$$[\ ]_{4}$
157.4.3.1$x^{4} - 157$$4$$1$$3$$C_4$$[\ ]_{4}$