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Properties

Label 16.8.13854815685...5057.4
Degree $16$
Signature $[8, 4]$
Discriminant $41^{15}\cdot 73^{15}$
Root discriminant $1814.89$
Ramified primes $41, 73$
Class number $48$ (GRH)
Class group $[48]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20617776290452519, 28165196463819167, 7605509253401021, -1546925722655262, -238409158886151, 40524218769108, -404880849047, -1048567964, 15068632831, -1114369992, -10993465, -145121, -246687, 6208, 94, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 94*x^14 + 6208*x^13 - 246687*x^12 - 145121*x^11 - 10993465*x^10 - 1114369992*x^9 + 15068632831*x^8 - 1048567964*x^7 - 404880849047*x^6 + 40524218769108*x^5 - 238409158886151*x^4 - 1546925722655262*x^3 + 7605509253401021*x^2 + 28165196463819167*x + 20617776290452519)
 
gp: K = bnfinit(x^16 - x^15 + 94*x^14 + 6208*x^13 - 246687*x^12 - 145121*x^11 - 10993465*x^10 - 1114369992*x^9 + 15068632831*x^8 - 1048567964*x^7 - 404880849047*x^6 + 40524218769108*x^5 - 238409158886151*x^4 - 1546925722655262*x^3 + 7605509253401021*x^2 + 28165196463819167*x + 20617776290452519, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 94 x^{14} + 6208 x^{13} - 246687 x^{12} - 145121 x^{11} - 10993465 x^{10} - 1114369992 x^{9} + 15068632831 x^{8} - 1048567964 x^{7} - 404880849047 x^{6} + 40524218769108 x^{5} - 238409158886151 x^{4} - 1546925722655262 x^{3} + 7605509253401021 x^{2} + 28165196463819167 x + 20617776290452519 \)

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:  \(13854815685101981032101054249860279864869741101695057=41^{15}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1814.89$
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:  $41, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{329} a^{14} + \frac{102}{329} a^{13} - \frac{121}{329} a^{12} + \frac{50}{329} a^{11} - \frac{57}{329} a^{10} - \frac{90}{329} a^{9} + \frac{156}{329} a^{8} - \frac{65}{329} a^{7} - \frac{88}{329} a^{6} - \frac{29}{329} a^{5} - \frac{6}{329} a^{4} + \frac{153}{329} a^{3} + \frac{137}{329} a^{2} - \frac{16}{329} a + \frac{111}{329}$, $\frac{1}{1789969517951911236583178498447296960316435923202806389094193207718936379920142618285143864475703023980501346368189119} a^{15} + \frac{2154473693511957125704084449845757217873860067649316987138371976632734892935258421857191561108771916020518022518688}{1789969517951911236583178498447296960316435923202806389094193207718936379920142618285143864475703023980501346368189119} a^{14} - \frac{795127055883799704678513490872904001884233263198011141115040871498464709018690452154402582896544924831437937910804776}{1789969517951911236583178498447296960316435923202806389094193207718936379920142618285143864475703023980501346368189119} a^{13} + \frac{542146018670907586021812557819669619636844716631733531504511212026267703110792755967051018172420906142008756418542210}{1789969517951911236583178498447296960316435923202806389094193207718936379920142618285143864475703023980501346368189119} a^{12} - \frac{494561952202547497760412046916351256292096034807213074299832788728955210756462428681991358554236942704379757617374280}{1789969517951911236583178498447296960316435923202806389094193207718936379920142618285143864475703023980501346368189119} a^{11} - \frac{691805837946008291406226395861731033665113758854187032716244015336316321680071266091076066782036918610939371012355705}{1789969517951911236583178498447296960316435923202806389094193207718936379920142618285143864475703023980501346368189119} a^{10} - \frac{98481937192753788183572080233647675006100323521851193207976771805171959410339098129019886273757782311309904718692002}{255709931135987319511882642635328137188062274743258055584884743959848054274306088326449123496529003425785906624027017} a^{9} - \frac{812380180145173775455282319017679324899377224727335531874067317266976127112135365039522603848453102103226107192587388}{1789969517951911236583178498447296960316435923202806389094193207718936379920142618285143864475703023980501346368189119} a^{8} + \frac{126945177115517610936265370378559478914206394790165219170161581352589185489617868087014773067771994577435869363285481}{255709931135987319511882642635328137188062274743258055584884743959848054274306088326449123496529003425785906624027017} a^{7} + \frac{79294115294179972850424774057228504832223998119565638078939857952529168339757708260957556039974271057923464068218262}{255709931135987319511882642635328137188062274743258055584884743959848054274306088326449123496529003425785906624027017} a^{6} - \frac{351323779586169879301984418829695345022792332958160492452369570315122737760167185408041581861024005378182701246658407}{1789969517951911236583178498447296960316435923202806389094193207718936379920142618285143864475703023980501346368189119} a^{5} - \frac{259519894230959062535035036912170579039975935965807440590269571861821251402281273414826439772349207702746728669289024}{1789969517951911236583178498447296960316435923202806389094193207718936379920142618285143864475703023980501346368189119} a^{4} - \frac{833819853821176642775134980700770191443421657678841429503656966498539570355681261350806909870665199046257616510380391}{1789969517951911236583178498447296960316435923202806389094193207718936379920142618285143864475703023980501346368189119} a^{3} - \frac{858383118382873876009971635283922788711523114564086236026743556301434663071713528539338548128405827412317338709718785}{1789969517951911236583178498447296960316435923202806389094193207718936379920142618285143864475703023980501346368189119} a^{2} - \frac{789443092253196996695390376390987581096027372440829570236169895511856949180836270047742131375004137411530926135955735}{1789969517951911236583178498447296960316435923202806389094193207718936379920142618285143864475703023980501346368189119} a + \frac{624341134908758441383582433475744563087026527840250194935564842018440892126521109097468625278235342470040046861699872}{1789969517951911236583178498447296960316435923202806389094193207718936379920142618285143864475703023980501346368189119}$

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

Class group and class number

$C_{48}$, 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:  $11$
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:  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:  \( 8256970769730000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{2993}) \), 4.4.26811440657.2, 8.8.2151528076860770946805457.3

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

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ R $16$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ $16$

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
41Data not computed
73Data not computed