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Properties

Label 16.8.13854815685...5057.2
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![11036826128683, 7482245749632, -8192417846781, 1324048078030, -420732794356, -291033786670, 99309516517, -1617435648, -39325421, 300555100, -64876517, 2599284, -433123, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 433123*x^12 + 2599284*x^11 - 64876517*x^10 + 300555100*x^9 - 39325421*x^8 - 1617435648*x^7 + 99309516517*x^6 - 291033786670*x^5 - 420732794356*x^4 + 1324048078030*x^3 - 8192417846781*x^2 + 7482245749632*x + 11036826128683)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 433123*x^12 + 2599284*x^11 - 64876517*x^10 + 300555100*x^9 - 39325421*x^8 - 1617435648*x^7 + 99309516517*x^6 - 291033786670*x^5 - 420732794356*x^4 + 1324048078030*x^3 - 8192417846781*x^2 + 7482245749632*x + 11036826128683, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 433123 x^{12} + 2599284 x^{11} - 64876517 x^{10} + 300555100 x^{9} - 39325421 x^{8} - 1617435648 x^{7} + 99309516517 x^{6} - 291033786670 x^{5} - 420732794356 x^{4} + 1324048078030 x^{3} - 8192417846781 x^{2} + 7482245749632 x + 11036826128683 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{11178576285121553636506496093396995194727603876598} a^{14} - \frac{7}{11178576285121553636506496093396995194727603876598} a^{13} + \frac{1672245240108862996163132375139845963624440399161}{11178576285121553636506496093396995194727603876598} a^{12} + \frac{1145104844468375659527701842557919412980961481723}{11178576285121553636506496093396995194727603876598} a^{11} + \frac{2755835533264338466651755632586015172417771546549}{11178576285121553636506496093396995194727603876598} a^{10} - \frac{55723456185102999831970184017514225837862916077}{11178576285121553636506496093396995194727603876598} a^{9} + \frac{772652130794901313613445256888323284736572891370}{5589288142560776818253248046698497597363801938299} a^{8} - \frac{4429299305218013891617516776261382574802433211857}{11178576285121553636506496093396995194727603876598} a^{7} - \frac{2441198799861398120691259664440251872697216706133}{11178576285121553636506496093396995194727603876598} a^{6} + \frac{3288806391644745946732195013186724221754003487163}{11178576285121553636506496093396995194727603876598} a^{5} + \frac{246322290238258299914037195369393893490285861048}{5589288142560776818253248046698497597363801938299} a^{4} - \frac{3361529398648761584612860191423977858628373375415}{11178576285121553636506496093396995194727603876598} a^{3} - \frac{4216371194057615984176125336824484585785316344829}{11178576285121553636506496093396995194727603876598} a^{2} + \frac{3604181302418250284799982384981671990520308134885}{11178576285121553636506496093396995194727603876598} a + \frac{3801776767258079598639875236647958740427375895265}{11178576285121553636506496093396995194727603876598}$, $\frac{1}{187593982179073315008912864100595558871018552294173562674} a^{15} + \frac{4195387}{93796991089536657504456432050297779435509276147086781337} a^{14} + \frac{739840974246360225304609690713164245086154380778396243}{5070107626461440946186834164880961050568068980923609802} a^{13} - \frac{23204823247776142314833800425694852365526586907823170559}{187593982179073315008912864100595558871018552294173562674} a^{12} - \frac{20186698377401272884638888486757794790691344072817454151}{187593982179073315008912864100595558871018552294173562674} a^{11} + \frac{16906618867163079939446693450423671790915377100148326086}{93796991089536657504456432050297779435509276147086781337} a^{10} - \frac{383688388681567886769196932101551183181835731725000028}{93796991089536657504456432050297779435509276147086781337} a^{9} + \frac{11611154001544884857190190356060721250138507103236068422}{93796991089536657504456432050297779435509276147086781337} a^{8} + \frac{950529840589429810519655875364337915837936492430408272}{2535053813230720473093417082440480525284034490461804901} a^{7} + \frac{74226686018008802732112121535383405890014934264293335433}{187593982179073315008912864100595558871018552294173562674} a^{6} - \frac{11489454804251680265106294302819750117780369907660778741}{187593982179073315008912864100595558871018552294173562674} a^{5} - \frac{37683810870187800548128473641355906911335749719090080336}{93796991089536657504456432050297779435509276147086781337} a^{4} - \frac{33813051857976047421717577026174086070252385360561765641}{187593982179073315008912864100595558871018552294173562674} a^{3} - \frac{79920960854916131569622562007806121924870245901821088035}{187593982179073315008912864100595558871018552294173562674} a^{2} + \frac{34131806835199793213990348905788760942033055280197460523}{93796991089536657504456432050297779435509276147086781337} a - \frac{20621604193589074800810967087865106562463585835954418419}{93796991089536657504456432050297779435509276147086781337}$

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:  \( 1924563711160000000 \) (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, 8.8.2151528076860770946805457.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ R $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ $16$

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