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Properties

Label 16.4.17328553034...8849.1
Degree $16$
Signature $[4, 6]$
Discriminant $61^{12}\cdot 97^{14}$
Root discriminant $1195.15$
Ramified primes $61, 97$
Class number $48$ (GRH)
Class group $[2, 2, 12]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-37606158695886896, 34234229555680560, -13908684943930888, 3183147219370626, -404015089637913, 23198289208957, -34767420017, -13822223688, -3792716246, -90797059, -27287937, 3884251, 79228, 2816, -479, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 479*x^14 + 2816*x^13 + 79228*x^12 + 3884251*x^11 - 27287937*x^10 - 90797059*x^9 - 3792716246*x^8 - 13822223688*x^7 - 34767420017*x^6 + 23198289208957*x^5 - 404015089637913*x^4 + 3183147219370626*x^3 - 13908684943930888*x^2 + 34234229555680560*x - 37606158695886896)
 
gp: K = bnfinit(x^16 - 3*x^15 - 479*x^14 + 2816*x^13 + 79228*x^12 + 3884251*x^11 - 27287937*x^10 - 90797059*x^9 - 3792716246*x^8 - 13822223688*x^7 - 34767420017*x^6 + 23198289208957*x^5 - 404015089637913*x^4 + 3183147219370626*x^3 - 13908684943930888*x^2 + 34234229555680560*x - 37606158695886896, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 479 x^{14} + 2816 x^{13} + 79228 x^{12} + 3884251 x^{11} - 27287937 x^{10} - 90797059 x^{9} - 3792716246 x^{8} - 13822223688 x^{7} - 34767420017 x^{6} + 23198289208957 x^{5} - 404015089637913 x^{4} + 3183147219370626 x^{3} - 13908684943930888 x^{2} + 34234229555680560 x - 37606158695886896 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17328553034618181867273005889005225191371184588849=61^{12}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1195.15$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{471755547440276847719003429624628489529374382755487312687323674218124344709797068116464145381324742860091939305044969256} a^{15} + \frac{13660381605280068452804536122153737213575613838071985276338868551821956691579518392401292910476337480046286507362084917}{471755547440276847719003429624628489529374382755487312687323674218124344709797068116464145381324742860091939305044969256} a^{14} + \frac{64611466171526658146531907304006021649913151267407486175960777013924524499891916112501081234527442630951700325414473797}{471755547440276847719003429624628489529374382755487312687323674218124344709797068116464145381324742860091939305044969256} a^{13} - \frac{9246699420851930227065488872270039695168849021503996123871000393023352494714656865994621477864398783875367652890052024}{58969443430034605964875428703078561191171797844435914085915459277265543088724633514558018172665592857511492413130621157} a^{12} + \frac{7133478229935284712203986243784273322212551147524021558976376727084504886936959120297687690286085207399977586433063083}{58969443430034605964875428703078561191171797844435914085915459277265543088724633514558018172665592857511492413130621157} a^{11} - \frac{129632428927000132380506953110833190400766443054470742766947609123225715387653156076690801091147401846762077029410237781}{471755547440276847719003429624628489529374382755487312687323674218124344709797068116464145381324742860091939305044969256} a^{10} + \frac{24161044220137233010294223581819447835495852314672302093848267417961467110881541767312876827169203802558000964883095}{1050680506548500774429851736357747192715755863597967288835910187568205667505115964624641749178896977416685833641525544} a^{9} + \frac{172403296109817614212429164928862302997718729802655682677674481087209204468213459843712547586488076659614564557063549777}{471755547440276847719003429624628489529374382755487312687323674218124344709797068116464145381324742860091939305044969256} a^{8} + \frac{14832262458049124917085215567410579742146431491281154925962790204929609763250983102230943356460244815721848474434250481}{235877773720138423859501714812314244764687191377743656343661837109062172354898534058232072690662371430045969652522484628} a^{7} - \frac{11065342537421680797446074254684235001876054762769545079797984241064635654129922616798587325496306995984510221999108631}{117938886860069211929750857406157122382343595688871828171830918554531086177449267029116036345331185715022984826261242314} a^{6} + \frac{192316409446021059624501697175314635243293002492764085034024092152441423949133862598384577477859236038779484000065829399}{471755547440276847719003429624628489529374382755487312687323674218124344709797068116464145381324742860091939305044969256} a^{5} - \frac{62588444143054065623668863859830386496218875060337385228598292239288472048824711605562233078668751138664631037962973019}{471755547440276847719003429624628489529374382755487312687323674218124344709797068116464145381324742860091939305044969256} a^{4} - \frac{5272553674087563783837997105229546773809085136852310834991510072432820391498514670770502875903387105367440567890496213}{471755547440276847719003429624628489529374382755487312687323674218124344709797068116464145381324742860091939305044969256} a^{3} - \frac{20902430003267595852177258996737053365238124977577726239680296507164142040473963075943410342560648459365004614548333563}{235877773720138423859501714812314244764687191377743656343661837109062172354898534058232072690662371430045969652522484628} a^{2} + \frac{30909114816280191946259279949662552331860425747878686651587667166885919971284312943323691696139797094944886613155627963}{117938886860069211929750857406157122382343595688871828171830918554531086177449267029116036345331185715022984826261242314} a + \frac{4663495220210942710807659099810635183150644619130527182096350559115338362104360120087216769987110229358573049258333055}{58969443430034605964875428703078561191171797844435914085915459277265543088724633514558018172665592857511492413130621157}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{12}$, which has order $48$ (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:  $9$
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:  \( 44650517371300000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.1118720199956720578033.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
61Data not computed
97Data not computed