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Properties

Label 16.0.11463919801...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 5^{12}\cdot 41^{2}\cdot 101^{4}$
Root discriminant $56.72$
Ramified primes $2, 5, 41, 101$
Class number $1040$ (GRH)
Class group $[2, 2, 2, 130]$ (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13011916, -11583064, 16315160, -10858344, 8233280, -4249040, 2222292, -904216, 352922, -113188, 33828, -8352, 1948, -340, 64, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 64*x^14 - 340*x^13 + 1948*x^12 - 8352*x^11 + 33828*x^10 - 113188*x^9 + 352922*x^8 - 904216*x^7 + 2222292*x^6 - 4249040*x^5 + 8233280*x^4 - 10858344*x^3 + 16315160*x^2 - 11583064*x + 13011916)
 
gp: K = bnfinit(x^16 - 6*x^15 + 64*x^14 - 340*x^13 + 1948*x^12 - 8352*x^11 + 33828*x^10 - 113188*x^9 + 352922*x^8 - 904216*x^7 + 2222292*x^6 - 4249040*x^5 + 8233280*x^4 - 10858344*x^3 + 16315160*x^2 - 11583064*x + 13011916, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 64 x^{14} - 340 x^{13} + 1948 x^{12} - 8352 x^{11} + 33828 x^{10} - 113188 x^{9} + 352922 x^{8} - 904216 x^{7} + 2222292 x^{6} - 4249040 x^{5} + 8233280 x^{4} - 10858344 x^{3} + 16315160 x^{2} - 11583064 x + 13011916 \)

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:  \(11463919801532416000000000000=2^{28}\cdot 5^{12}\cdot 41^{2}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.72$
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:  $2, 5, 41, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{10} a^{12} + \frac{1}{5} a^{11} + \frac{1}{10} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{13} + \frac{1}{5} a^{11} - \frac{1}{10} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{10} a^{14} + \frac{1}{10} a^{11} + \frac{1}{5} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{8554756310787689511841782503769563396810} a^{15} + \frac{12331385662441189700244402879180420007}{1710951262157537902368356500753912679362} a^{14} - \frac{3661906677122511152402830424408717813}{1710951262157537902368356500753912679362} a^{13} - \frac{212656912775836663180987969873721988911}{8554756310787689511841782503769563396810} a^{12} - \frac{54738059415788780891517596794576285636}{4277378155393844755920891251884781698405} a^{11} + \frac{78753066457518612080847201339702272647}{8554756310787689511841782503769563396810} a^{10} + \frac{1741999150533964423108858874006160512023}{8554756310787689511841782503769563396810} a^{9} + \frac{514871809951660709481997740109939880856}{4277378155393844755920891251884781698405} a^{8} - \frac{1384327346980162444506210025420797001942}{4277378155393844755920891251884781698405} a^{7} - \frac{1955557501066641586307187121457827868863}{4277378155393844755920891251884781698405} a^{6} + \frac{368351693869503201812577387764952927847}{855475631078768951184178250376956339681} a^{5} + \frac{181783985616629632930502951712436209899}{4277378155393844755920891251884781698405} a^{4} - \frac{860738358558444047917024761383589700928}{4277378155393844755920891251884781698405} a^{3} - \frac{1624069229136489461694683774563511115486}{4277378155393844755920891251884781698405} a^{2} + \frac{936387056296916722343299401966685460971}{4277378155393844755920891251884781698405} a - \frac{1292802345765666930596340836978889964376}{4277378155393844755920891251884781698405}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{130}$, which has order $1040$ (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:  \( -1 \) (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:  \( 20635.0035043 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.6464000000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$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}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
101Data not computed