↑

Properties

Label 16.0.20515766860...2529.1
Degree $16$
Signature $[0, 8]$
Discriminant $41^{2}\cdot 73^{14}$
Root discriminant $67.92$
Ramified primes $41, 73$
Class number $89$ (GRH)
Class group $[89]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![741392, -1825008, 2665644, -1785584, 669739, -137199, 76085, -20450, -9503, 7077, 552, -909, 91, 30, -1, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - x^14 + 30*x^13 + 91*x^12 - 909*x^11 + 552*x^10 + 7077*x^9 - 9503*x^8 - 20450*x^7 + 76085*x^6 - 137199*x^5 + 669739*x^4 - 1785584*x^3 + 2665644*x^2 - 1825008*x + 741392)
 
gp: K = bnfinit(x^16 - 5*x^15 - x^14 + 30*x^13 + 91*x^12 - 909*x^11 + 552*x^10 + 7077*x^9 - 9503*x^8 - 20450*x^7 + 76085*x^6 - 137199*x^5 + 669739*x^4 - 1785584*x^3 + 2665644*x^2 - 1825008*x + 741392, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - x^{14} + 30 x^{13} + 91 x^{12} - 909 x^{11} + 552 x^{10} + 7077 x^{9} - 9503 x^{8} - 20450 x^{7} + 76085 x^{6} - 137199 x^{5} + 669739 x^{4} - 1785584 x^{3} + 2665644 x^{2} - 1825008 x + 741392 \)

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:  \(205157668600814015560481982529=41^{2}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.92$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{16} a^{5} + \frac{1}{8} a^{4} - \frac{5}{16} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{8} a^{7} + \frac{3}{16} a^{6} + \frac{1}{8} a^{5} - \frac{1}{16} a^{4} + \frac{1}{4} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{9} - \frac{3}{16} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{96} a^{14} - \frac{1}{96} a^{13} + \frac{1}{48} a^{12} + \frac{1}{96} a^{11} + \frac{1}{96} a^{10} - \frac{1}{24} a^{9} - \frac{3}{32} a^{8} + \frac{17}{96} a^{7} + \frac{3}{16} a^{6} + \frac{1}{32} a^{5} - \frac{1}{96} a^{4} + \frac{1}{3} a^{3} - \frac{7}{24} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{46480592225887976823361731272360086656} a^{15} + \frac{2253448571404379537874777629484805}{5810074028235997102920216409045010832} a^{14} - \frac{1130707841035286980535924848804496233}{46480592225887976823361731272360086656} a^{13} + \frac{1047390144814784934939891547497796753}{46480592225887976823361731272360086656} a^{12} - \frac{96153436046197927858729668483911329}{5810074028235997102920216409045010832} a^{11} + \frac{2506861816068908703166890428337863219}{46480592225887976823361731272360086656} a^{10} - \frac{365696429286953003428766910154397019}{15493530741962658941120577090786695552} a^{9} + \frac{57040194726573452414021380338544309}{5810074028235997102920216409045010832} a^{8} - \frac{23195526488540635272386327088223137}{142142483871217054505693367805382528} a^{7} - \frac{3058980781317823645788552578662369079}{15493530741962658941120577090786695552} a^{6} + \frac{742632495629708555268923892137182961}{11620148056471994205840432818090021664} a^{5} - \frac{52998204086454463907576019522499019}{522253845234696368801817205307416704} a^{4} - \frac{2672971079993657626013651485646613895}{11620148056471994205840432818090021664} a^{3} - \frac{917943641477453762283391906591916803}{11620148056471994205840432818090021664} a^{2} + \frac{256286212727759404748618488767408013}{726259253529499637865027051130626354} a + \frac{13344955599061218445275304417235301}{968345671372666183820036068174168472}$

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

Class group and class number

$C_{89}$, which has order $89$ (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:  \( 7728102.44392 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.0.11047398519097.1, 8.4.452943339282977.1, 8.4.6204703277849.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73Data not computed