Properties

Label 16.0.560...913.1
Degree $16$
Signature $[0, 8]$
Discriminant $5.600\times 10^{45}$
Root discriminant $723.21$
Ramified primes $17, 89$
Class number $13655584$ (GRH)
Class group $[2, 2, 3413896]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 94*x^14 + 2302*x^13 + 26159*x^12 - 397975*x^11 + 6284898*x^10 - 24995672*x^9 + 142093700*x^8 - 83458080*x^7 + 7911836541*x^6 - 60914865824*x^5 + 494959575758*x^4 - 1703233593999*x^3 + 5636562271845*x^2 - 6761439893945*x + 19724305333097)
 
gp: K = bnfinit(x^16 - 3*x^15 + 94*x^14 + 2302*x^13 + 26159*x^12 - 397975*x^11 + 6284898*x^10 - 24995672*x^9 + 142093700*x^8 - 83458080*x^7 + 7911836541*x^6 - 60914865824*x^5 + 494959575758*x^4 - 1703233593999*x^3 + 5636562271845*x^2 - 6761439893945*x + 19724305333097, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19724305333097, -6761439893945, 5636562271845, -1703233593999, 494959575758, -60914865824, 7911836541, -83458080, 142093700, -24995672, 6284898, -397975, 26159, 2302, 94, -3, 1]);
 

\( x^{16} - 3 x^{15} + 94 x^{14} + 2302 x^{13} + 26159 x^{12} - 397975 x^{11} + 6284898 x^{10} - 24995672 x^{9} + 142093700 x^{8} - 83458080 x^{7} + 7911836541 x^{6} - 60914865824 x^{5} + 494959575758 x^{4} - 1703233593999 x^{3} + 5636562271845 x^{2} - 6761439893945 x + 19724305333097 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(5600075906386661768959437042812533816731958913\)\(\medspace = 17^{15}\cdot 89^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $723.21$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $17, 89$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1867176554733441482505045549562908774628951339214396775177640220443888295196550154912567390086991} a^{15} - \frac{109315889252526172147254217106454663822350434244017655456214434581969469852283395239762221580604}{1867176554733441482505045549562908774628951339214396775177640220443888295196550154912567390086991} a^{14} - \frac{778472422645740030944150550009794543502550210091039546379142219764938226777293043792450267473783}{1867176554733441482505045549562908774628951339214396775177640220443888295196550154912567390086991} a^{13} - \frac{643803176403802902273398646505724667312041194880751356002492688183678018956458076530472098019825}{1867176554733441482505045549562908774628951339214396775177640220443888295196550154912567390086991} a^{12} + \frac{626053386694812152537093049689121043494084552412737284397159151540958888851719143770231366611729}{1867176554733441482505045549562908774628951339214396775177640220443888295196550154912567390086991} a^{11} + \frac{794040466771596065583998959099660300072663712949040854493591973505944982517413383315929793365495}{1867176554733441482505045549562908774628951339214396775177640220443888295196550154912567390086991} a^{10} + \frac{417447020739055823860395747398673554110220458379130380944363074244954213357015869963445840120749}{1867176554733441482505045549562908774628951339214396775177640220443888295196550154912567390086991} a^{9} - \frac{416468056910891805338207036003572599365144355080918018324631677097598768172347569438299289114734}{1867176554733441482505045549562908774628951339214396775177640220443888295196550154912567390086991} a^{8} + \frac{340035586320431133295154228123125166682692309732024647825400445013654815850424505137179535535976}{1867176554733441482505045549562908774628951339214396775177640220443888295196550154912567390086991} a^{7} + \frac{5054319847947057516576079182179290066128986306001462488869231363575717483264610402453865075408}{1867176554733441482505045549562908774628951339214396775177640220443888295196550154912567390086991} a^{6} - \frac{457524753504957680516211815344998682571246200525656831287312641294084562870976672169151582617195}{1867176554733441482505045549562908774628951339214396775177640220443888295196550154912567390086991} a^{5} + \frac{533372912690474968237580980629607242288576045241548048553058801093391567129701578659989382295647}{1867176554733441482505045549562908774628951339214396775177640220443888295196550154912567390086991} a^{4} + \frac{476242161879031080680737248748767363736009988351410588854148006785243007307064021342839726535867}{1867176554733441482505045549562908774628951339214396775177640220443888295196550154912567390086991} a^{3} - \frac{374302836242552562035942349119779065649995167120029270266615248958904760263508521453232855303443}{1867176554733441482505045549562908774628951339214396775177640220443888295196550154912567390086991} a^{2} - \frac{666497063699153112567174309548625195735769234343992048359732852256843459873524099713290613898974}{1867176554733441482505045549562908774628951339214396775177640220443888295196550154912567390086991} a - \frac{200059831872983962292040472796132259543329635998711615367293478155976164518079232562353945442}{31647060249719347161102466941744216519134768461260962291146444414303191444009324659535040509949}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{3413896}$, which has order $13655584$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 921838016.564 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 921838016.564 \cdot 13655584}{2\sqrt{5600075906386661768959437042812533816731958913}}\approx 0.204304008115$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.38915873.1, 8.8.203930643438763634753.1

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
17Data not computed
$89$89.8.7.4$x^{8} - 64881$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.4$x^{8} - 64881$$8$$1$$7$$C_8$$[\ ]_{8}$