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Properties

Label 16.0.76573533698...000.53
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}$
Root discriminant $201.96$
Ramified primes $2, 3, 5, 19$
Class number $10862592$ (GRH)
Class group $[2, 4, 8, 8, 21216]$ (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![4488138486, 1226284920, -536880324, 162402720, 398631910, 154887560, 18651344, -5712000, -2500161, -260360, 49484, 17440, 1610, -200, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 24*x^14 - 200*x^13 + 1610*x^12 + 17440*x^11 + 49484*x^10 - 260360*x^9 - 2500161*x^8 - 5712000*x^7 + 18651344*x^6 + 154887560*x^5 + 398631910*x^4 + 162402720*x^3 - 536880324*x^2 + 1226284920*x + 4488138486)
 
gp: K = bnfinit(x^16 - 24*x^14 - 200*x^13 + 1610*x^12 + 17440*x^11 + 49484*x^10 - 260360*x^9 - 2500161*x^8 - 5712000*x^7 + 18651344*x^6 + 154887560*x^5 + 398631910*x^4 + 162402720*x^3 - 536880324*x^2 + 1226284920*x + 4488138486, 1)
 

Normalized defining polynomial

\( x^{16} - 24 x^{14} - 200 x^{13} + 1610 x^{12} + 17440 x^{11} + 49484 x^{10} - 260360 x^{9} - 2500161 x^{8} - 5712000 x^{7} + 18651344 x^{6} + 154887560 x^{5} + 398631910 x^{4} + 162402720 x^{3} - 536880324 x^{2} + 1226284920 x + 4488138486 \)

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:  \(7657353369870241665908736000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $201.96$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4560=2^{4}\cdot 3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4560}(1,·)$, $\chi_{4560}(1027,·)$, $\chi_{4560}(3077,·)$, $\chi_{4560}(1483,·)$, $\chi_{4560}(77,·)$, $\chi_{4560}(3533,·)$, $\chi_{4560}(4369,·)$, $\chi_{4560}(4483,·)$, $\chi_{4560}(533,·)$, $\chi_{4560}(1559,·)$, $\chi_{4560}(1369,·)$, $\chi_{4560}(4559,·)$, $\chi_{4560}(3191,·)$, $\chi_{4560}(3001,·)$, $\chi_{4560}(4027,·)$, $\chi_{4560}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{39} a^{10} - \frac{2}{39} a^{9} + \frac{5}{39} a^{8} - \frac{10}{39} a^{7} - \frac{6}{13} a^{6} + \frac{10}{39} a^{5} - \frac{2}{13} a^{4} + \frac{4}{13} a^{3} + \frac{16}{39} a^{2} - \frac{2}{13} a$, $\frac{1}{39} a^{11} + \frac{1}{39} a^{9} + \frac{1}{39} a^{7} + \frac{1}{3} a^{6} + \frac{14}{39} a^{5} + \frac{1}{39} a^{3} - \frac{1}{3} a^{2} - \frac{4}{13} a$, $\frac{1}{39} a^{12} + \frac{2}{39} a^{9} - \frac{4}{39} a^{8} - \frac{16}{39} a^{7} - \frac{7}{39} a^{6} - \frac{10}{39} a^{5} + \frac{7}{39} a^{4} + \frac{14}{39} a^{3} + \frac{11}{39} a^{2} + \frac{2}{13} a$, $\frac{1}{39} a^{13} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{4}{13} a$, $\frac{1}{2769} a^{14} - \frac{23}{2769} a^{13} + \frac{35}{2769} a^{12} + \frac{7}{2769} a^{11} - \frac{20}{2769} a^{10} + \frac{37}{213} a^{9} - \frac{175}{2769} a^{8} + \frac{466}{2769} a^{7} - \frac{356}{923} a^{6} - \frac{1102}{2769} a^{5} - \frac{4}{923} a^{4} - \frac{1225}{2769} a^{3} + \frac{1364}{2769} a^{2} - \frac{413}{923} a - \frac{23}{71}$, $\frac{1}{2003240170935028251669393717192838557232295229090196622001} a^{15} + \frac{30003441058590038997890358306387154129994618028024770}{667746723645009417223131239064279519077431743030065540667} a^{14} + \frac{412668669342002224798510534745947674363079225514238970}{222582241215003139074377079688093173025810581010021846889} a^{13} - \frac{3459666763676748057590446600860650099261855023934293109}{2003240170935028251669393717192838557232295229090196622001} a^{12} - \frac{14530948381482443233210135881748808885503037999510929159}{2003240170935028251669393717192838557232295229090196622001} a^{11} + \frac{30429035083353116762673021899260693634974099762722385}{2003240170935028251669393717192838557232295229090196622001} a^{10} - \frac{432391997081351391366931088427347054306780660972716265650}{2003240170935028251669393717192838557232295229090196622001} a^{9} - \frac{300278363710426337199323409601700631454202139467698301906}{2003240170935028251669393717192838557232295229090196622001} a^{8} + \frac{159594159187239666212915066090394486653722791092206432319}{667746723645009417223131239064279519077431743030065540667} a^{7} + \frac{61099908730064475961763939155429275827899177101584253266}{667746723645009417223131239064279519077431743030065540667} a^{6} - \frac{797697019267704446689324868475693565737265438524071550535}{2003240170935028251669393717192838557232295229090196622001} a^{5} + \frac{965524733582703047301948798750745008114834703519421285021}{2003240170935028251669393717192838557232295229090196622001} a^{4} - \frac{40152072573099193271670481869794265929079133954469361404}{154095397764232942436107209014833735171715017622322817077} a^{3} + \frac{44504379221040510199656048409537554886004456618733132511}{667746723645009417223131239064279519077431743030065540667} a^{2} + \frac{299569153765613430886226480422977971787838691684901772546}{667746723645009417223131239064279519077431743030065540667} a - \frac{7819900771423436091130976651807922640259294852722593985}{17121710862692549159567467668314859463523890846924757453}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{8}\times C_{8}\times C_{21216}$, which has order $10862592$ (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:  \( 871468.5219091468 \) (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

\(\Q(\sqrt{-285}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-114}) \), \(\Q(\sqrt{-38}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{3}, \sqrt{-95})\), \(\Q(\sqrt{10}, \sqrt{-114})\), \(\Q(\sqrt{30}, \sqrt{-38})\), \(\Q(\sqrt{10}, \sqrt{-38})\), \(\Q(\sqrt{30}, \sqrt{-95})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{-38})\), 4.4.256000.2, 4.0.831744000.1, 4.0.92416000.2, 4.4.2304000.2, 8.0.432373800960000.30, 8.0.2767192326144000000.8, 8.0.2767192326144000000.4, 8.0.8540717056000000.26, 8.0.691798081536000000.15, 8.8.21233664000000.1, 8.0.2767192326144000000.24

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.24.1$x^{8} + 14 x^{4} + 4 x^{2} + 8 x + 22$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.1$x^{8} + 14 x^{4} + 4 x^{2} + 8 x + 22$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$