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Properties

Label 16.0.22138542742...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 41^{6}\cdot 661^{4}$
Root discriminant $68.25$
Ramified primes $5, 41, 661$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $(C_2\times D_4).C_2^3$ (as 16T322)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![98463142096, 6126502272, -28642020832, -3018651066, 3849869651, 473105398, -305662967, -37075883, 15677672, 1680795, -533858, -45779, 11862, 713, -159, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 159*x^14 + 713*x^13 + 11862*x^12 - 45779*x^11 - 533858*x^10 + 1680795*x^9 + 15677672*x^8 - 37075883*x^7 - 305662967*x^6 + 473105398*x^5 + 3849869651*x^4 - 3018651066*x^3 - 28642020832*x^2 + 6126502272*x + 98463142096)
 
gp: K = bnfinit(x^16 - 5*x^15 - 159*x^14 + 713*x^13 + 11862*x^12 - 45779*x^11 - 533858*x^10 + 1680795*x^9 + 15677672*x^8 - 37075883*x^7 - 305662967*x^6 + 473105398*x^5 + 3849869651*x^4 - 3018651066*x^3 - 28642020832*x^2 + 6126502272*x + 98463142096, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 159 x^{14} + 713 x^{13} + 11862 x^{12} - 45779 x^{11} - 533858 x^{10} + 1680795 x^{9} + 15677672 x^{8} - 37075883 x^{7} - 305662967 x^{6} + 473105398 x^{5} + 3849869651 x^{4} - 3018651066 x^{3} - 28642020832 x^{2} + 6126502272 x + 98463142096 \)

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:  \(221385427428590205586181640625=5^{12}\cdot 41^{6}\cdot 661^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.25$
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:  $5, 41, 661$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{11} + \frac{1}{10} a^{10} + \frac{1}{10} a^{9} + \frac{1}{10} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{110} a^{13} + \frac{1}{22} a^{12} + \frac{7}{110} a^{11} - \frac{6}{55} a^{10} + \frac{3}{110} a^{9} - \frac{39}{110} a^{8} - \frac{13}{55} a^{7} + \frac{4}{11} a^{6} + \frac{9}{22} a^{5} - \frac{6}{55} a^{4} - \frac{53}{110} a^{3} - \frac{6}{55} a^{2} + \frac{7}{55} a + \frac{1}{5}$, $\frac{1}{660} a^{14} - \frac{1}{660} a^{13} + \frac{7}{220} a^{12} + \frac{133}{660} a^{11} + \frac{16}{165} a^{10} - \frac{41}{220} a^{9} - \frac{149}{330} a^{8} - \frac{13}{660} a^{7} + \frac{39}{110} a^{6} - \frac{21}{44} a^{5} - \frac{49}{132} a^{4} - \frac{61}{165} a^{3} - \frac{211}{660} a^{2} - \frac{163}{330} a - \frac{2}{15}$, $\frac{1}{770387812673944087726516413000150834636289614635603520619080} a^{15} - \frac{238817531611756200067043010579098793674917683511902860267}{770387812673944087726516413000150834636289614635603520619080} a^{14} - \frac{1704301698375757977609194837948445068370801698525680310849}{770387812673944087726516413000150834636289614635603520619080} a^{13} + \frac{7302714631753594031920618539806895156610174246535866822535}{154077562534788817545303282600030166927257922927120704123816} a^{12} - \frac{3254873957013718897925570000504991172534484722940185198773}{32099492194747670321938183875006284776512067276483480025795} a^{11} - \frac{107218556809134627410974578749867225309384361455315809473067}{770387812673944087726516413000150834636289614635603520619080} a^{10} + \frac{2255836151786345478796963854134999450952030628793346874063}{38519390633697204386325820650007541731814480731780176030954} a^{9} - \frac{51692854022097079904594550937408709980675207995279828490211}{256795937557981362575505471000050278212096538211867840206360} a^{8} - \frac{25531603453655157129947766743139806648229414300863857464545}{77038781267394408772651641300015083463628961463560352061908} a^{7} + \frac{12715436253636323480603056605460674458469977394627245447723}{256795937557981362575505471000050278212096538211867840206360} a^{6} + \frac{26900143729957692168626300585441508875583180682105929118999}{770387812673944087726516413000150834636289614635603520619080} a^{5} - \frac{2808031111322724311160518432565602854641692342993203914259}{38519390633697204386325820650007541731814480731780176030954} a^{4} - \frac{34261075605920740504864498959849581770056797696112565728163}{256795937557981362575505471000050278212096538211867840206360} a^{3} + \frac{24048119602800749214242475491626889416227125437453283718887}{192596953168486021931629103250037708659072403658900880154770} a^{2} + \frac{33041547282936916504264395439995391865803959646609156994312}{96298476584243010965814551625018854329536201829450440077385} a - \frac{2142777204103992429253358033625404961210332686636014287582}{8754406962203910087801322875001714029957836529950040007035}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( \frac{4779972196359307209920678864091801525719461837}{13222817834505236478777186038930191799737214902262255340} a^{15} - \frac{16866398041886489395519293849706527682785161819}{13222817834505236478777186038930191799737214902262255340} a^{14} - \frac{135623629770906885341132667313367151798892982257}{2644563566901047295755437207786038359947442980452451068} a^{13} + \frac{1777529815600857133976064969729861449856590332849}{13222817834505236478777186038930191799737214902262255340} a^{12} + \frac{3909633868170790858340932659171622340857602829937}{1101901486208769706564765503244182649978101241855187945} a^{11} - \frac{84144266745500808396647923223305520033934002541961}{13222817834505236478777186038930191799737214902262255340} a^{10} - \frac{982127173205071110451834779940980250364874052705031}{6611408917252618239388593019465095899868607451131127670} a^{9} + \frac{661515382823079847244637215868866380934703522722527}{4407605944835078826259062012976730599912404967420751780} a^{8} + \frac{26616547328281251923519081805867397723231974133087409}{6611408917252618239388593019465095899868607451131127670} a^{7} - \frac{4705778974731192017575187754788047643263958969772281}{4407605944835078826259062012976730599912404967420751780} a^{6} - \frac{185194871753989760195838375125822954715009499308774781}{2644563566901047295755437207786038359947442980452451068} a^{5} - \frac{19276817734382801918062495891637594071479998401168837}{661140891725261823938859301946509589986860745113112767} a^{4} + \frac{3150900332375463195261857162524461053246889761428909581}{4407605944835078826259062012976730599912404967420751780} a^{3} + \frac{2520579336749942079859396511381117897301114647667060797}{3305704458626309119694296509732547949934303725565563835} a^{2} - \frac{4319249739283783475328730164608317440429874358198324069}{1322281783450523647877718603893019179973721490226225534} a - \frac{1510983616493470819559078602888563938849152534746806083}{300518587147846283608572409975686177266754884142323985} \) (order $10$)
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:  \( 101264637.982 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4).C_2^3$ (as 16T322):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times D_4).C_2^3$
Character table for $(C_2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.1025.1, 4.4.5125.1, 8.0.26265625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
661Data not computed