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Properties

Label 16.8.69858727014...5808.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{71}\cdot 7^{8}\cdot 150513919^{4}$
Root discriminant $6349.85$
Ramified primes $2, 7, 150513919$
Class number Not computed
Class group Not computed
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![83142230203097022619741298888816802, 0, 11574363150647086989170162447488, 0, -3384507326332862097584407792, 0, -7267228884846036391424, 0, 13611700064197971142, 0, 36863389517120, 0, -7200731392, 0, -8752, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8752*x^14 - 7200731392*x^12 + 36863389517120*x^10 + 13611700064197971142*x^8 - 7267228884846036391424*x^6 - 3384507326332862097584407792*x^4 + 11574363150647086989170162447488*x^2 + 83142230203097022619741298888816802)
 
gp: K = bnfinit(x^16 - 8752*x^14 - 7200731392*x^12 + 36863389517120*x^10 + 13611700064197971142*x^8 - 7267228884846036391424*x^6 - 3384507326332862097584407792*x^4 + 11574363150647086989170162447488*x^2 + 83142230203097022619741298888816802, 1)
 

Normalized defining polynomial

\( x^{16} - 8752 x^{14} - 7200731392 x^{12} + 36863389517120 x^{10} + 13611700064197971142 x^{8} - 7267228884846036391424 x^{6} - 3384507326332862097584407792 x^{4} + 11574363150647086989170162447488 x^{2} + 83142230203097022619741298888816802 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6985872701411915091439460916010822936571645898168438313975808=2^{71}\cdot 7^{8}\cdot 150513919^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $6349.85$
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:  $2, 7, 150513919$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{150513919} a^{10} - \frac{8752}{150513919} a^{8} + \frac{23936720}{150513919} a^{6} - \frac{27982603}{150513919} a^{4} - \frac{7092852}{150513919} a^{2}$, $\frac{1}{150513919} a^{11} - \frac{8752}{150513919} a^{9} + \frac{23936720}{150513919} a^{7} - \frac{27982603}{150513919} a^{5} - \frac{7092852}{150513919} a^{3}$, $\frac{1}{22654439812738561} a^{12} - \frac{8752}{22654439812738561} a^{10} - \frac{7200731392}{22654439812738561} a^{8} + \frac{36863389517120}{22654439812738561} a^{6} - \frac{3618263257904019}{22654439812738561} a^{4} - \frac{11660562}{150513919} a^{2}$, $\frac{1}{203889958314647049} a^{13} + \frac{301019086}{203889958314647049} a^{11} + \frac{90615117454584676}{203889958314647049} a^{9} - \frac{38066397165548642}{203889958314647049} a^{7} - \frac{11565415184448298}{67963319438215683} a^{5} - \frac{326874104}{1354625271} a^{3} + \frac{4}{9} a$, $\frac{1}{620778721258813329697221965581988507871407249743883384358926246101865651761712152308187304940522807} a^{14} + \frac{1738986580553839148603407500652983721874075755093253569925474633612613233762741644}{620778721258813329697221965581988507871407249743883384358926246101865651761712152308187304940522807} a^{12} + \frac{1617993435932621939224001249800881978226348818867156893340485964599841763894416650179621051}{620778721258813329697221965581988507871407249743883384358926246101865651761712152308187304940522807} a^{10} - \frac{115453339269484567932621524661259749278788915316840586606514285906623619073738884056833718261195122}{620778721258813329697221965581988507871407249743883384358926246101865651761712152308187304940522807} a^{8} - \frac{37552971582499924994663213574156830362173999032661867895306524647762459451539588688821074833997273}{206926240419604443232407321860662835957135749914627794786308748700621883920570717436062434980174269} a^{6} - \frac{792701599144542335241372547348142859547059789648469067459575506053213275053637139700333842}{4124394111742006596062534027713333993193063093014562888093600473600489080094394142432682953} a^{4} + \frac{843477600727071347976428639646645665769019985892879115438480255885369257080343529}{27402077755626086621680045602382687233019712236810224096906283289324817062894988087} a^{2} - \frac{2470356328871116982356379859738417405005221585292328173925506807967278861}{20228529868926449860745323126621847314569781296225518368397752480231720897}$, $\frac{1}{620778721258813329697221965581988507871407249743883384358926246101865651761712152308187304940522807} a^{15} - \frac{145076525069648571410486149339578194273494202974949900834284093664337011839880411}{68975413473201481077469107286887611985711916638209264928769582900207294640190239145354144993391423} a^{13} + \frac{233829352699781537051890968020014143367765798973954919516980328073664662431848000877294851}{206926240419604443232407321860662835957135749914627794786308748700621883920570717436062434980174269} a^{11} + \frac{25492419056521468767131359696258741223638503930455464538897709245436169797432067737650386661249313}{68975413473201481077469107286887611985711916638209264928769582900207294640190239145354144993391423} a^{9} + \frac{3240904697712264718184572929963156259198637409717387888137766328314393654664694765837487210122387}{620778721258813329697221965581988507871407249743883384358926246101865651761712152308187304940522807} a^{7} - \frac{90847559629098627795796641552269636271839907833410619934554766893628534027646679512283924}{4124394111742006596062534027713333993193063093014562888093600473600489080094394142432682953} a^{5} + \frac{828406686318849120900942323205737839282647043619172668227115279274987859740588313}{3044675306180676291297782844709187470335523581867802677434031476591646340321665343} a^{3} + \frac{78909442384792196462519196895463479927801912213496638276659201129453094736}{182056768820338048746707908139596625831128031666029665315579772322085488073} a$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{7}) \), 4.4.25088.1, 8.8.322256764928.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ $16$ R $16$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
7Data not computed
150513919Data not computed