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Properties

Label 16.4.46370927160...3125.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 11^{6}\cdot 101^{7}$
Root discriminant $61.89$
Ramified primes $5, 11, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1574

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![512111, -510826, 22908, 526104, -19483, 287265, -91833, -5923, 10060, -19593, 7262, -2730, 797, -171, 48, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 48*x^14 - 171*x^13 + 797*x^12 - 2730*x^11 + 7262*x^10 - 19593*x^9 + 10060*x^8 - 5923*x^7 - 91833*x^6 + 287265*x^5 - 19483*x^4 + 526104*x^3 + 22908*x^2 - 510826*x + 512111)
 
gp: K = bnfinit(x^16 - 6*x^15 + 48*x^14 - 171*x^13 + 797*x^12 - 2730*x^11 + 7262*x^10 - 19593*x^9 + 10060*x^8 - 5923*x^7 - 91833*x^6 + 287265*x^5 - 19483*x^4 + 526104*x^3 + 22908*x^2 - 510826*x + 512111, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 48 x^{14} - 171 x^{13} + 797 x^{12} - 2730 x^{11} + 7262 x^{10} - 19593 x^{9} + 10060 x^{8} - 5923 x^{7} - 91833 x^{6} + 287265 x^{5} - 19483 x^{4} + 526104 x^{3} + 22908 x^{2} - 510826 x + 512111 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(46370927160987469302001953125=5^{12}\cdot 11^{6}\cdot 101^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.89$
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:  $5, 11, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} + \frac{3}{11} a^{11} + \frac{2}{11} a^{10} - \frac{4}{11} a^{9} + \frac{5}{11} a^{8} - \frac{1}{11} a^{7} - \frac{3}{11} a^{6} - \frac{5}{11} a^{5} - \frac{2}{11} a^{4} - \frac{5}{11} a^{3} - \frac{1}{11} a^{2} - \frac{5}{11} a - \frac{3}{11}$, $\frac{1}{33} a^{13} - \frac{1}{33} a^{12} + \frac{4}{11} a^{11} - \frac{4}{11} a^{10} - \frac{4}{11} a^{9} + \frac{4}{11} a^{8} - \frac{10}{33} a^{7} + \frac{7}{33} a^{6} + \frac{7}{33} a^{5} - \frac{8}{33} a^{4} - \frac{1}{11} a^{3} + \frac{10}{33} a^{2} + \frac{2}{11} a - \frac{10}{33}$, $\frac{1}{33} a^{14} - \frac{1}{33} a^{12} - \frac{1}{11} a^{11} - \frac{5}{11} a^{10} + \frac{5}{11} a^{9} + \frac{8}{33} a^{8} + \frac{3}{11} a^{7} - \frac{16}{33} a^{6} - \frac{7}{33} a^{5} + \frac{13}{33} a^{4} + \frac{1}{33} a^{3} - \frac{5}{33} a^{2} - \frac{10}{33} a - \frac{7}{33}$, $\frac{1}{1943835516394169898601759614144122555404377759} a^{15} + \frac{9077082655203268733607178295305336077036380}{1943835516394169898601759614144122555404377759} a^{14} + \frac{971085436574388478965867127426474202341286}{176712319672197263509250874013102050491307069} a^{13} - \frac{3674786892658170329421163599310512142546847}{176712319672197263509250874013102050491307069} a^{12} - \frac{34955785419588528802033704424307745820371103}{647945172131389966200586538048040851801459253} a^{11} - \frac{15829590035709688766545681551915646566766862}{215981724043796655400195512682680283933819751} a^{10} - \frac{153257096420187580460093053318970874059955439}{1943835516394169898601759614144122555404377759} a^{9} + \frac{946868972035434616192145398116067325522359054}{1943835516394169898601759614144122555404377759} a^{8} + \frac{184286428063667344460679134650733810067496}{58904106557399087836416958004367350163769023} a^{7} - \frac{58842023704185799126131279603872476491222685}{1943835516394169898601759614144122555404377759} a^{6} + \frac{363926332256653752770189643303981890189553553}{1943835516394169898601759614144122555404377759} a^{5} - \frac{74170325361047844743205231021349498111383595}{1943835516394169898601759614144122555404377759} a^{4} + \frac{95709500637887927883436313724693280953732995}{215981724043796655400195512682680283933819751} a^{3} - \frac{11223413577421682786948780098964566350300313}{215981724043796655400195512682680283933819751} a^{2} - \frac{3802898197857774495225094487449108118813471}{647945172131389966200586538048040851801459253} a + \frac{492616865014368432232988192539465870727221307}{1943835516394169898601759614144122555404377759}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 80676981.1247 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1574:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 88 conjugacy class representatives for t16n1574 are not computed
Character table for t16n1574 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.4.1947912828125.6

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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R $16$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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}$
$11$11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
11.8.0.1$x^{8} + x^{2} - 2 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
$101$101.4.3.3$x^{4} + 202$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$
101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$