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Properties

Label 16.0.64866971506...6736.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 37^{8}$
Root discriminant $84.29$
Ramified primes $2, 3, 37$
Class number $106080$ (GRH)
Class group $[106080]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![350398078, -17519784, 168754196, -1933320, 37553914, 175584, 5163148, 38592, 483207, 1848, 32324, -24, 1558, 0, 52, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 52*x^14 + 1558*x^12 - 24*x^11 + 32324*x^10 + 1848*x^9 + 483207*x^8 + 38592*x^7 + 5163148*x^6 + 175584*x^5 + 37553914*x^4 - 1933320*x^3 + 168754196*x^2 - 17519784*x + 350398078)
 
gp: K = bnfinit(x^16 + 52*x^14 + 1558*x^12 - 24*x^11 + 32324*x^10 + 1848*x^9 + 483207*x^8 + 38592*x^7 + 5163148*x^6 + 175584*x^5 + 37553914*x^4 - 1933320*x^3 + 168754196*x^2 - 17519784*x + 350398078, 1)
 

Normalized defining polynomial

\( x^{16} + 52 x^{14} + 1558 x^{12} - 24 x^{11} + 32324 x^{10} + 1848 x^{9} + 483207 x^{8} + 38592 x^{7} + 5163148 x^{6} + 175584 x^{5} + 37553914 x^{4} - 1933320 x^{3} + 168754196 x^{2} - 17519784 x + 350398078 \)

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:  \(6486697150600796010022736756736=2^{48}\cdot 3^{8}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.29$
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:  $2, 3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1776=2^{4}\cdot 3\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{1776}(1,·)$, $\chi_{1776}(1703,·)$, $\chi_{1776}(1553,·)$, $\chi_{1776}(1109,·)$, $\chi_{1776}(665,·)$, $\chi_{1776}(1627,·)$, $\chi_{1776}(221,·)$, $\chi_{1776}(1183,·)$, $\chi_{1776}(739,·)$, $\chi_{1776}(295,·)$, $\chi_{1776}(1259,·)$, $\chi_{1776}(815,·)$, $\chi_{1776}(371,·)$, $\chi_{1776}(1333,·)$, $\chi_{1776}(889,·)$, $\chi_{1776}(445,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{131547407} a^{14} + \frac{11894826}{131547407} a^{13} - \frac{43382590}{131547407} a^{12} - \frac{64406050}{131547407} a^{11} + \frac{20294361}{131547407} a^{10} + \frac{30643768}{131547407} a^{9} - \frac{49252126}{131547407} a^{8} + \frac{54405816}{131547407} a^{7} + \frac{48379834}{131547407} a^{6} + \frac{39156889}{131547407} a^{5} + \frac{11735333}{131547407} a^{4} + \frac{44904034}{131547407} a^{3} + \frac{6215745}{131547407} a^{2} - \frac{21919196}{131547407} a - \frac{12312998}{131547407}$, $\frac{1}{315210357105884732516865942349726793903} a^{15} - \frac{267286306559537504299819477233}{315210357105884732516865942349726793903} a^{14} + \frac{4785021134891026754547370153553877570}{18541785712110866618639173079395693759} a^{13} + \frac{278341852698575843008980209700588059}{18541785712110866618639173079395693759} a^{12} + \frac{13558414040094980777548057609948550138}{45030051015126390359552277478532399129} a^{11} + \frac{1028370614397804838840378121776048653}{18541785712110866618639173079395693759} a^{10} - \frac{85310041041341626956938042482449264331}{315210357105884732516865942349726793903} a^{9} - \frac{101605814603122108855624770883486116727}{315210357105884732516865942349726793903} a^{8} - \frac{83131442427603773774209992809110922005}{315210357105884732516865942349726793903} a^{7} + \frac{133282575596779884312912803091112570372}{315210357105884732516865942349726793903} a^{6} + \frac{131398478552386365086145260890315946589}{315210357105884732516865942349726793903} a^{5} + \frac{7030434032581049252092153180046662695}{45030051015126390359552277478532399129} a^{4} - \frac{105762580362026568651884053442723173249}{315210357105884732516865942349726793903} a^{3} + \frac{21304172755150310142513374227191384889}{45030051015126390359552277478532399129} a^{2} - \frac{131511173620329203113689138454826961382}{315210357105884732516865942349726793903} a + \frac{56935299604779599193849299376934263888}{315210357105884732516865942349726793903}$

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

Class group and class number

$C_{106080}$, which has order $106080$ (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:  \( 11964.310642723332 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-222}) \), \(\Q(\sqrt{-37}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-111}) \), \(\Q(\sqrt{-74}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}, \sqrt{-37})\), \(\Q(\sqrt{2}, \sqrt{-111})\), \(\Q(\sqrt{3}, \sqrt{-74})\), \(\Q(\sqrt{2}, \sqrt{-37})\), \(\Q(\sqrt{3}, \sqrt{-37})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{6}, \sqrt{-74})\), 4.0.2803712.2, 4.4.18432.1, \(\Q(\zeta_{16})^+\), 4.0.25233408.2, 8.0.9948826238976.5, 8.0.636724879294464.73, 8.0.636724879294464.58, 8.0.31443203915776.12, 8.0.2546899517177856.49, 8.0.2546899517177856.41, \(\Q(\zeta_{48})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$37$37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$