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Properties

Label 16.8.30417192973...5625.2
Degree $16$
Signature $[8, 4]$
Discriminant $5^{12}\cdot 11^{3}\cdot 101^{6}\cdot 4451^{3}$
Root discriminant $142.95$
Ramified primes $5, 11, 101, 4451$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1643

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5972069905, -10291908080, 7219747165, -339713070, -377718884, 345170910, -51262919, -24295974, 2077146, 474192, 74882, -8447, -126, 165, -94, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 94*x^14 + 165*x^13 - 126*x^12 - 8447*x^11 + 74882*x^10 + 474192*x^9 + 2077146*x^8 - 24295974*x^7 - 51262919*x^6 + 345170910*x^5 - 377718884*x^4 - 339713070*x^3 + 7219747165*x^2 - 10291908080*x + 5972069905)
 
gp: K = bnfinit(x^16 - x^15 - 94*x^14 + 165*x^13 - 126*x^12 - 8447*x^11 + 74882*x^10 + 474192*x^9 + 2077146*x^8 - 24295974*x^7 - 51262919*x^6 + 345170910*x^5 - 377718884*x^4 - 339713070*x^3 + 7219747165*x^2 - 10291908080*x + 5972069905, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 94 x^{14} + 165 x^{13} - 126 x^{12} - 8447 x^{11} + 74882 x^{10} + 474192 x^{9} + 2077146 x^{8} - 24295974 x^{7} - 51262919 x^{6} + 345170910 x^{5} - 377718884 x^{4} - 339713070 x^{3} + 7219747165 x^{2} - 10291908080 x + 5972069905 \)

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:  \(30417192973888736937325752197265625=5^{12}\cdot 11^{3}\cdot 101^{6}\cdot 4451^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $142.95$
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:  $5, 11, 101, 4451$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4}$, $\frac{1}{1243541281931939259739718052740244876398898027536435380337962692057529572755} a^{15} - \frac{102505343835484075741205164225354361898638947327087911274464456924761778197}{1243541281931939259739718052740244876398898027536435380337962692057529572755} a^{14} + \frac{59159454506284518399508105989592948270148021637082508927343951919432752976}{1243541281931939259739718052740244876398898027536435380337962692057529572755} a^{13} + \frac{46695861425806486382580452394804044212496778021856936641362053544964286768}{1243541281931939259739718052740244876398898027536435380337962692057529572755} a^{12} + \frac{400234338650469045447527027278505640328096303453610471946055579980182420168}{1243541281931939259739718052740244876398898027536435380337962692057529572755} a^{11} + \frac{35323096836793912568906897471548671752694230269104078769406450089933487010}{248708256386387851947943610548048975279779605507287076067592538411505914551} a^{10} - \frac{50509578725177867893085692649016440143347188800595204453092004049939669624}{248708256386387851947943610548048975279779605507287076067592538411505914551} a^{9} + \frac{304571084514123756642633061179241961930768031303240509202598162324581548348}{1243541281931939259739718052740244876398898027536435380337962692057529572755} a^{8} + \frac{252426235614580955972841538760045010078227498307636968781971006882372219744}{1243541281931939259739718052740244876398898027536435380337962692057529572755} a^{7} + \frac{63599977024109038710498242279312166035893183562201037252163828607953255328}{248708256386387851947943610548048975279779605507287076067592538411505914551} a^{6} - \frac{348291019232599915586044821464795042589785881464490630983111278210179581357}{1243541281931939259739718052740244876398898027536435380337962692057529572755} a^{5} - \frac{56147593691105319994019363155879295444897473557153287006433617141307824126}{1243541281931939259739718052740244876398898027536435380337962692057529572755} a^{4} - \frac{21203455901495970670698424688916987561975807264709950085810672804853470551}{248708256386387851947943610548048975279779605507287076067592538411505914551} a^{3} - \frac{72831419565183135369282388569364738568115863481079953957046277921567619291}{248708256386387851947943610548048975279779605507287076067592538411505914551} a^{2} - \frac{91648266122414794735837882638847266570134535475266886599277400188687713220}{248708256386387851947943610548048975279779605507287076067592538411505914551} a + \frac{104006241120676878078247618297626920243637071632857063493082767999982791863}{248708256386387851947943610548048975279779605507287076067592538411505914551}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 236820451265 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1643:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 64 conjugacy class representatives for t16n1643 are not computed
Character table for t16n1643 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.16098453125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R $16$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$101$101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
4451Data not computed