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Properties

Label 16.0.22873200555...6849.2
Degree $16$
Signature $[0, 8]$
Discriminant $37^{4}\cdot 73^{14}$
Root discriminant $105.31$
Ramified primes $37, 73$
Class number $89$ (GRH)
Class group $[89]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![430336, -2368160, 6571656, -9803772, 10291030, -7439391, 3943200, -1545333, 498951, -145668, 38313, -7501, 1532, -331, 37, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 37*x^14 - 331*x^13 + 1532*x^12 - 7501*x^11 + 38313*x^10 - 145668*x^9 + 498951*x^8 - 1545333*x^7 + 3943200*x^6 - 7439391*x^5 + 10291030*x^4 - 9803772*x^3 + 6571656*x^2 - 2368160*x + 430336)
 
gp: K = bnfinit(x^16 - 4*x^15 + 37*x^14 - 331*x^13 + 1532*x^12 - 7501*x^11 + 38313*x^10 - 145668*x^9 + 498951*x^8 - 1545333*x^7 + 3943200*x^6 - 7439391*x^5 + 10291030*x^4 - 9803772*x^3 + 6571656*x^2 - 2368160*x + 430336, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 37 x^{14} - 331 x^{13} + 1532 x^{12} - 7501 x^{11} + 38313 x^{10} - 145668 x^{9} + 498951 x^{8} - 1545333 x^{7} + 3943200 x^{6} - 7439391 x^{5} + 10291030 x^{4} - 9803772 x^{3} + 6571656 x^{2} - 2368160 x + 430336 \)

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:  \(228732005557745506375281661426849=37^{4}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.31$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{96} a^{12} + \frac{1}{48} a^{11} - \frac{5}{96} a^{9} + \frac{1}{16} a^{8} - \frac{1}{24} a^{7} - \frac{17}{96} a^{6} - \frac{3}{16} a^{5} - \frac{5}{24} a^{4} + \frac{29}{96} a^{3} + \frac{17}{48} a^{2} - \frac{5}{12} a - \frac{1}{3}$, $\frac{1}{192} a^{13} + \frac{1}{24} a^{11} + \frac{7}{192} a^{10} - \frac{1}{24} a^{9} + \frac{1}{24} a^{8} + \frac{5}{64} a^{7} + \frac{5}{24} a^{6} + \frac{1}{48} a^{5} + \frac{11}{64} a^{4} + \frac{1}{8} a^{3} - \frac{3}{16} a^{2} + \frac{1}{3}$, $\frac{1}{113664} a^{14} + \frac{199}{113664} a^{13} - \frac{1}{296} a^{12} - \frac{3049}{113664} a^{11} + \frac{953}{113664} a^{10} + \frac{341}{14208} a^{9} + \frac{11519}{113664} a^{8} - \frac{7477}{37888} a^{7} - \frac{205}{9472} a^{6} + \frac{21541}{113664} a^{5} - \frac{14993}{113664} a^{4} - \frac{6085}{28416} a^{3} - \frac{13853}{28416} a^{2} - \frac{283}{2368} a - \frac{191}{888}$, $\frac{1}{1727840860462287931804904210109966336} a^{15} + \frac{2047735967017819183701176999317}{1727840860462287931804904210109966336} a^{14} + \frac{651437491480111593577212471730379}{287973476743714655300817368351661056} a^{13} - \frac{138919398773227635883666342782773}{46698401634115890048781194867836928} a^{12} + \frac{12771129717980990571698842470769641}{575946953487429310601634736703322112} a^{11} + \frac{28591010291802962598992730693130739}{863920430231143965902452105054983168} a^{10} + \frac{35940455987650543857106346254377583}{1727840860462287931804904210109966336} a^{9} + \frac{37821515482190554973392787170307505}{575946953487429310601634736703322112} a^{8} + \frac{199466865356827324477904905053782441}{863920430231143965902452105054983168} a^{7} - \frac{138526505802827067226926182534618081}{575946953487429310601634736703322112} a^{6} - \frac{134135340208241391533779692435089867}{1727840860462287931804904210109966336} a^{5} + \frac{151908024996884582252437931753033711}{863920430231143965902452105054983168} a^{4} - \frac{50267549551687346247140730238112755}{431960215115571982951226052527491584} a^{3} + \frac{89097107846981847902144067874096627}{215980107557785991475613026263745792} a^{2} + \frac{6910790355655488342552295484910159}{17998342296482165956301085521978816} a + \frac{23986101707358108226394968848847}{164618984419044200819827001725416}$

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:  \( 20533345115.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

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

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
37Data not computed
73Data not computed