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Properties

Label 16.16.1861644105...9936.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{62}\cdot 7^{6}\cdot 193^{4}\cdot 223^{4}$
Root discriminant $438.40$
Ramified primes $2, 7, 193, 223$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $(C_2^2\times C_4).C_2^4$ (as 16T471)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10037906408750643793, 186964374704677232, -650460101056359304, -448750962282272, 13659966875592876, -33098698794736, -137166478126136, 359826968384, 748277249166, -1556101712, -2294565432, 3042144, 3852332, -2160, -3208, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3208*x^14 - 2160*x^13 + 3852332*x^12 + 3042144*x^11 - 2294565432*x^10 - 1556101712*x^9 + 748277249166*x^8 + 359826968384*x^7 - 137166478126136*x^6 - 33098698794736*x^5 + 13659966875592876*x^4 - 448750962282272*x^3 - 650460101056359304*x^2 + 186964374704677232*x + 10037906408750643793)
 
gp: K = bnfinit(x^16 - 3208*x^14 - 2160*x^13 + 3852332*x^12 + 3042144*x^11 - 2294565432*x^10 - 1556101712*x^9 + 748277249166*x^8 + 359826968384*x^7 - 137166478126136*x^6 - 33098698794736*x^5 + 13659966875592876*x^4 - 448750962282272*x^3 - 650460101056359304*x^2 + 186964374704677232*x + 10037906408750643793, 1)
 

Normalized defining polynomial

\( x^{16} - 3208 x^{14} - 2160 x^{13} + 3852332 x^{12} + 3042144 x^{11} - 2294565432 x^{10} - 1556101712 x^{9} + 748277249166 x^{8} + 359826968384 x^{7} - 137166478126136 x^{6} - 33098698794736 x^{5} + 13659966875592876 x^{4} - 448750962282272 x^{3} - 650460101056359304 x^{2} + 186964374704677232 x + 10037906408750643793 \)

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:  \(1861644105089180838863336343231454234279936=2^{62}\cdot 7^{6}\cdot 193^{4}\cdot 223^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $438.40$
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:  $2, 7, 193, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{1100355276} a^{14} - \frac{112289635}{1100355276} a^{13} - \frac{128023775}{1100355276} a^{12} + \frac{31424539}{1100355276} a^{11} + \frac{96375385}{550177638} a^{10} - \frac{15215713}{275088819} a^{9} - \frac{16141789}{91696273} a^{8} - \frac{3864964}{91696273} a^{7} + \frac{6968577}{366785092} a^{6} - \frac{385333993}{1100355276} a^{5} - \frac{1983193}{1100355276} a^{4} + \frac{260008349}{1100355276} a^{3} + \frac{48908655}{183392546} a^{2} - \frac{32843276}{91696273} a - \frac{125451895}{275088819}$, $\frac{1}{4632571409105300052002374854082463956286172113783666561756222489432565722650815491246794057786696079073381265775148} a^{15} - \frac{628591493101682632287864348518688269872255977405801264202958935026937904547798094124624641764347843135189}{2316285704552650026001187427041231978143086056891833280878111244716282861325407745623397028893348039536690632887574} a^{14} - \frac{14245651737396911936443436956961696024110171668929699305647095603280204773459324572588454176793849807135811832031}{1158142852276325013000593713520615989071543028445916640439055622358141430662703872811698514446674019768345316443787} a^{13} + \frac{58467749882480727061952220265833340577625825974983373926464982451970772246533239566486831269247323940502508236641}{4632571409105300052002374854082463956286172113783666561756222489432565722650815491246794057786696079073381265775148} a^{12} + \frac{438190943469543990311328598766426132072688047828083037337750970139783171143551569726749353242746533189202912712651}{4632571409105300052002374854082463956286172113783666561756222489432565722650815491246794057786696079073381265775148} a^{11} - \frac{555654342542310308812109800518993979537594487886044132854777896670929100639998439227878330605724435279813595488099}{2316285704552650026001187427041231978143086056891833280878111244716282861325407745623397028893348039536690632887574} a^{10} + \frac{188754907100450602366834300456291653076342534414134784482043399895649152388749328823042946646246119576081068842245}{772095234850883342000395809013743992714362018963944426959370414905427620441802581874465676297782679845563544295858} a^{9} + \frac{220955812483953664511462282058338681495621092255780403606700473760699158849595370787934539118193747278259633556117}{1544190469701766684000791618027487985428724037927888853918740829810855240883605163748931352595565359691127088591716} a^{8} - \frac{337646207284479836345850522461504554644613696948262754119261655431777376949671531359601178577548110619822277628783}{1544190469701766684000791618027487985428724037927888853918740829810855240883605163748931352595565359691127088591716} a^{7} + \frac{14022325360952249049084008435628781064263146968838098741699211901301741844742422044346815243146320710856714380783}{100708074110984783739182062045270955571438524212688403516439619335490559188061206331452044734493393023334375342938} a^{6} + \frac{480783933940166759234220782527380877599805217801635306700501312832606074298273557082869241105972659103561919836113}{1158142852276325013000593713520615989071543028445916640439055622358141430662703872811698514446674019768345316443787} a^{5} - \frac{976279575334079464628469843419581099112663208880124485074917810176109851584026126292940777496467282658461548684517}{4632571409105300052002374854082463956286172113783666561756222489432565722650815491246794057786696079073381265775148} a^{4} - \frac{612514275233882894187105722307624241450124729677767601657493672051055035283884523467820165402287923847300425236921}{1544190469701766684000791618027487985428724037927888853918740829810855240883605163748931352595565359691127088591716} a^{3} - \frac{26713211555518887205873039648105317305932998473859829229872710604194839492011562496159380352752981513177585236387}{772095234850883342000395809013743992714362018963944426959370414905427620441802581874465676297782679845563544295858} a^{2} - \frac{1093751844197779903126358208649548078485054875980151685951282462132927533580316261795806153101776733398661793765147}{2316285704552650026001187427041231978143086056891833280878111244716282861325407745623397028893348039536690632887574} a + \frac{465155722137129693044500118535801968188470740537259795892586099550088363232079785577445862775274618306473325488319}{1544190469701766684000791618027487985428724037927888853918740829810855240883605163748931352595565359691127088591716}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  $15$
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:  \( 3610704382330000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4).C_2^4$ (as 16T471):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $(C_2^2\times C_4).C_2^4$
Character table for $(C_2^2\times C_4).C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7168.1, \(\Q(\zeta_{16})^+\), 4.4.14336.1, 8.8.3288334336.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
193Data not computed
223Data not computed