↑

Properties

Label 16.0.24389423094...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 43^{12}$
Root discriminant $68.66$
Ramified primes $5, 43$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12313081, 88567160, 206802536, 133312655, 11437842, -14067960, -2205112, 973820, 115330, -77990, -16878, 1520, 1312, 125, -46, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 46*x^14 + 125*x^13 + 1312*x^12 + 1520*x^11 - 16878*x^10 - 77990*x^9 + 115330*x^8 + 973820*x^7 - 2205112*x^6 - 14067960*x^5 + 11437842*x^4 + 133312655*x^3 + 206802536*x^2 + 88567160*x + 12313081)
 
gp: K = bnfinit(x^16 - 5*x^15 - 46*x^14 + 125*x^13 + 1312*x^12 + 1520*x^11 - 16878*x^10 - 77990*x^9 + 115330*x^8 + 973820*x^7 - 2205112*x^6 - 14067960*x^5 + 11437842*x^4 + 133312655*x^3 + 206802536*x^2 + 88567160*x + 12313081, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 46 x^{14} + 125 x^{13} + 1312 x^{12} + 1520 x^{11} - 16878 x^{10} - 77990 x^{9} + 115330 x^{8} + 973820 x^{7} - 2205112 x^{6} - 14067960 x^{5} + 11437842 x^{4} + 133312655 x^{3} + 206802536 x^{2} + 88567160 x + 12313081 \)

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:  \(243894230940323342291259765625=5^{14}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.66$
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:  $5, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11} a^{9} + \frac{2}{11} a^{8} - \frac{3}{11} a^{7} + \frac{2}{11} a^{6} + \frac{3}{11} a^{5} + \frac{3}{11} a^{4} - \frac{4}{11} a^{3} + \frac{4}{11} a^{2} + \frac{3}{11} a$, $\frac{1}{11} a^{10} + \frac{4}{11} a^{8} - \frac{3}{11} a^{7} - \frac{1}{11} a^{6} - \frac{3}{11} a^{5} + \frac{1}{11} a^{4} + \frac{1}{11} a^{3} - \frac{5}{11} a^{2} + \frac{5}{11} a$, $\frac{1}{11} a^{11} - \frac{1}{11} a$, $\frac{1}{11} a^{12} - \frac{1}{11} a^{2}$, $\frac{1}{2299} a^{13} - \frac{90}{2299} a^{12} - \frac{2}{2299} a^{11} + \frac{8}{2299} a^{10} + \frac{81}{2299} a^{9} - \frac{510}{2299} a^{8} + \frac{294}{2299} a^{7} + \frac{75}{209} a^{6} - \frac{232}{2299} a^{5} + \frac{1032}{2299} a^{4} - \frac{823}{2299} a^{3} - \frac{24}{209} a^{2} + \frac{769}{2299} a - \frac{6}{19}$, $\frac{1}{12899689} a^{14} - \frac{2524}{12899689} a^{13} - \frac{3527}{12899689} a^{12} - \frac{353350}{12899689} a^{11} + \frac{520456}{12899689} a^{10} - \frac{153774}{12899689} a^{9} - \frac{3893287}{12899689} a^{8} - \frac{366995}{12899689} a^{7} + \frac{1870967}{12899689} a^{6} - \frac{6393771}{12899689} a^{5} + \frac{5651247}{12899689} a^{4} + \frac{6096183}{12899689} a^{3} - \frac{4993594}{12899689} a^{2} + \frac{2606398}{12899689} a + \frac{31514}{106609}$, $\frac{1}{235927484147341172106625598144604521995661659} a^{15} - \frac{2092330758953219326283246064862979224}{235927484147341172106625598144604521995661659} a^{14} + \frac{20581032999601240455510054542900787894475}{235927484147341172106625598144604521995661659} a^{13} - \frac{662410784695388804060378967798581036743994}{21447953104303742918784145285873138363241969} a^{12} - \frac{5265186599768769586375825267445369545045349}{235927484147341172106625598144604521995661659} a^{11} - \frac{3311734829118701436790971522525351563524182}{235927484147341172106625598144604521995661659} a^{10} - \frac{108778142401702423465285831830790323743970}{8135430487839350762297434418779466275712471} a^{9} - \frac{23312438809616617472367778123312349665893915}{235927484147341172106625598144604521995661659} a^{8} + \frac{116133386087446378123793755119853730659182810}{235927484147341172106625598144604521995661659} a^{7} - \frac{1058293095258215053342506882501913693461390}{8135430487839350762297434418779466275712471} a^{6} + \frac{19834864982250017731018819948533029194826557}{235927484147341172106625598144604521995661659} a^{5} + \frac{82215256769235966629973401457859541299743118}{235927484147341172106625598144604521995661659} a^{4} + \frac{44665671192274596655399527622808904444475715}{235927484147341172106625598144604521995661659} a^{3} - \frac{98964052825328975187878967784433128894248043}{235927484147341172106625598144604521995661659} a^{2} + \frac{5391445371732409372484312407428498893106452}{21447953104303742918784145285873138363241969} a + \frac{462425821730483570992142596039128758913}{6112269337219647454768921426581116660941}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( -\frac{31441485324216947426343466419334100115}{235927484147341172106625598144604521995661659} a^{15} + \frac{184591114402795987283018775367952479842}{235927484147341172106625598144604521995661659} a^{14} + \frac{1256004282352759375245263120751608407879}{235927484147341172106625598144604521995661659} a^{13} - \frac{432656785575521600549130276334272140275}{21447953104303742918784145285873138363241969} a^{12} - \frac{36594798994684740122728173868027060569673}{235927484147341172106625598144604521995661659} a^{11} - \frac{22499800837012370012624685792956877673211}{235927484147341172106625598144604521995661659} a^{10} + \frac{18296471418928654632199234476218084943482}{8135430487839350762297434418779466275712471} a^{9} + \frac{2029335856125408007545439728367251186558038}{235927484147341172106625598144604521995661659} a^{8} - \frac{4877600428191850719504013405825786513993895}{235927484147341172106625598144604521995661659} a^{7} - \frac{879567804268895112861929284408941621325645}{8135430487839350762297434418779466275712471} a^{6} + \frac{84050169395624331372506840058208351303988252}{235927484147341172106625598144604521995661659} a^{5} + \frac{361117355653314344643003666761746857767183379}{235927484147341172106625598144604521995661659} a^{4} - \frac{570999628678779247735933760401749407414761622}{235927484147341172106625598144604521995661659} a^{3} - \frac{3592839333808971647552496540293073913937804983}{235927484147341172106625598144604521995661659} a^{2} - \frac{390957472851074964680214026201786478006301367}{21447953104303742918784145285873138363241969} a - \frac{27373201149140124573127765319437389765584}{6112269337219647454768921426581116660941} \) (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:  \( 286364980.697 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-215}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{5}, \sqrt{-43})\), 4.4.231125.1, \(\Q(\zeta_{5})\), 8.0.53418765625.1, 8.4.493856488203125.1 x2

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
43Data not computed