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Properties

Label 16.0.26268600651...000.56
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 17^{12}$
Root discriminant $387.90$
Ramified primes $2, 3, 5, 17$
Class number $267264000$ (GRH)
Class group $[2, 2, 2, 2, 20, 120, 6960]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89663714721, 0, -409240944, 0, 3865158504, 0, -687937848, 0, 14583259, 0, 2873344, 0, 71756, 0, 488, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 488*x^14 + 71756*x^12 + 2873344*x^10 + 14583259*x^8 - 687937848*x^6 + 3865158504*x^4 - 409240944*x^2 + 89663714721)
 
gp: K = bnfinit(x^16 + 488*x^14 + 71756*x^12 + 2873344*x^10 + 14583259*x^8 - 687937848*x^6 + 3865158504*x^4 - 409240944*x^2 + 89663714721, 1)
 

Normalized defining polynomial

\( x^{16} + 488 x^{14} + 71756 x^{12} + 2873344 x^{10} + 14583259 x^{8} - 687937848 x^{6} + 3865158504 x^{4} - 409240944 x^{2} + 89663714721 \)

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:  \(262686006513622818702917369856000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $387.90$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4080=2^{4}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{4080}(1,·)$, $\chi_{4080}(2821,·)$, $\chi_{4080}(781,·)$, $\chi_{4080}(3469,·)$, $\chi_{4080}(3107,·)$, $\chi_{4080}(1429,·)$, $\chi_{4080}(2903,·)$, $\chi_{4080}(2843,·)$, $\chi_{4080}(863,·)$, $\chi_{4080}(2209,·)$, $\chi_{4080}(803,·)$, $\chi_{4080}(3047,·)$, $\chi_{4080}(169,·)$, $\chi_{4080}(1067,·)$, $\chi_{4080}(1007,·)$, $\chi_{4080}(2041,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{7} a^{5} + \frac{1}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{7} a^{6} + \frac{1}{7} a^{4} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{147} a^{10} + \frac{2}{147} a^{8} - \frac{4}{147} a^{6} - \frac{5}{147} a^{4} + \frac{43}{147} a^{2}$, $\frac{1}{147} a^{11} + \frac{2}{147} a^{9} - \frac{4}{147} a^{7} - \frac{5}{147} a^{5} + \frac{43}{147} a^{3}$, $\frac{1}{122157} a^{12} - \frac{130}{122157} a^{10} - \frac{4090}{122157} a^{8} + \frac{4807}{122157} a^{6} + \frac{59377}{122157} a^{4} - \frac{3061}{40719} a^{2} + \frac{60}{277}$, $\frac{1}{1741836663} a^{13} - \frac{3580}{248833809} a^{11} - \frac{16963969}{1741836663} a^{9} + \frac{24780241}{1741836663} a^{7} + \frac{403630}{17957079} a^{5} + \frac{21686039}{580612221} a^{3} - \frac{118219}{3949743} a$, $\frac{1}{12243135909242409821005601637441} a^{14} - \frac{968073466248122845944973}{1749019415606058545857943091063} a^{12} - \frac{21228345633681173223463425010}{12243135909242409821005601637441} a^{10} - \frac{343139508448424834607554886020}{12243135909242409821005601637441} a^{8} - \frac{3209325305353825908145588235}{126217895971571235268098985953} a^{6} + \frac{40882405473601063450065441295}{453449478120089252629837097683} a^{4} + \frac{6045392976725517422442542510}{27762212946127913426316557001} a^{2} - \frac{525212188712787816671306}{1946995788353174375925139}$, $\frac{1}{36729407727727229463016804912323} a^{15} - \frac{7716631303260917947270}{36729407727727229463016804912323} a^{13} - \frac{24281136625699683697472660464}{36729407727727229463016804912323} a^{11} + \frac{335519241233437806049054152712}{5247058246818175637573829273189} a^{9} + \frac{597994464032046916415236779577}{36729407727727229463016804912323} a^{7} - \frac{63031459993074641395901219459}{4081045303080803273668533879147} a^{5} + \frac{2020677989814285859117911215656}{4081045303080803273668533879147} a^{3} - \frac{2183780963563069644356178949}{27762212946127913426316557001} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{20}\times C_{120}\times C_{6960}$, which has order $267264000$ (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:  \( 6183243.81077213 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{170}) \), 4.4.591872.2, \(\Q(\sqrt{2}, \sqrt{85})\), 4.4.51200.1, 4.0.88434000.2, 4.0.353736000.4, 4.0.11319552000.5, 4.0.11319552000.7, 8.8.218945290240000.6, 8.0.2002066523136000000.24, 8.0.128132257480704000000.3

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 ${\href{/LocalNumberField/7.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$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}$
5Data not computed
$17$17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$