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Properties

Label 16.0.42784533468...000.12
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 5^{12}\cdot 53^{8}$
Root discriminant $194.74$
Ramified primes $2, 5, 53$
Class number $17095680$ (GRH)
Class group $[2, 8, 24, 44520]$ (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![988915277201, 87988803560, 190440765592, 11127338960, 16700763120, 575603880, 891306872, 15174880, 32307694, 177240, 823048, -880, 14560, -40, 168, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 168*x^14 - 40*x^13 + 14560*x^12 - 880*x^11 + 823048*x^10 + 177240*x^9 + 32307694*x^8 + 15174880*x^7 + 891306872*x^6 + 575603880*x^5 + 16700763120*x^4 + 11127338960*x^3 + 190440765592*x^2 + 87988803560*x + 988915277201)
 
gp: K = bnfinit(x^16 + 168*x^14 - 40*x^13 + 14560*x^12 - 880*x^11 + 823048*x^10 + 177240*x^9 + 32307694*x^8 + 15174880*x^7 + 891306872*x^6 + 575603880*x^5 + 16700763120*x^4 + 11127338960*x^3 + 190440765592*x^2 + 87988803560*x + 988915277201, 1)
 

Normalized defining polynomial

\( x^{16} + 168 x^{14} - 40 x^{13} + 14560 x^{12} - 880 x^{11} + 823048 x^{10} + 177240 x^{9} + 32307694 x^{8} + 15174880 x^{7} + 891306872 x^{6} + 575603880 x^{5} + 16700763120 x^{4} + 11127338960 x^{3} + 190440765592 x^{2} + 87988803560 x + 988915277201 \)

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:  \(4278453346814084509597696000000000000=2^{48}\cdot 5^{12}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $194.74$
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:  $2, 5, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4240=2^{4}\cdot 5\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{4240}(1,·)$, $\chi_{4240}(2437,·)$, $\chi_{4240}(2119,·)$, $\chi_{4240}(2121,·)$, $\chi_{4240}(1803,·)$, $\chi_{4240}(4239,·)$, $\chi_{4240}(849,·)$, $\chi_{4240}(3923,·)$, $\chi_{4240}(2969,·)$, $\chi_{4240}(2013,·)$, $\chi_{4240}(4133,·)$, $\chi_{4240}(107,·)$, $\chi_{4240}(2227,·)$, $\chi_{4240}(1271,·)$, $\chi_{4240}(317,·)$, $\chi_{4240}(3391,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{6} a^{6} - \frac{1}{6} a^{4} + \frac{1}{6} a^{2} - \frac{1}{6}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{6} a$, $\frac{1}{12} a^{8} - \frac{1}{12}$, $\frac{1}{24} a^{9} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{1}{12} a^{5} + \frac{1}{12} a^{4} - \frac{1}{12} a^{3} - \frac{1}{12} a^{2} - \frac{11}{24} a - \frac{3}{8}$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{8} + \frac{11}{24} a^{2} - \frac{11}{24}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{1}{12} a^{5} + \frac{1}{12} a^{4} + \frac{3}{8} a^{3} - \frac{1}{12} a^{2} + \frac{1}{12} a - \frac{3}{8}$, $\frac{1}{72} a^{12} + \frac{1}{72} a^{10} - \frac{1}{18} a^{6} + \frac{1}{24} a^{4} + \frac{31}{72} a^{2} - \frac{4}{9}$, $\frac{1}{432} a^{13} + \frac{1}{432} a^{12} - \frac{1}{54} a^{11} + \frac{1}{108} a^{10} - \frac{1}{48} a^{9} - \frac{1}{48} a^{8} + \frac{2}{27} a^{7} + \frac{1}{54} a^{6} - \frac{23}{144} a^{5} - \frac{5}{48} a^{4} - \frac{11}{27} a^{3} - \frac{53}{108} a^{2} - \frac{59}{432} a - \frac{35}{432}$, $\frac{1}{82766827728} a^{14} + \frac{46150769}{82766827728} a^{13} - \frac{127743601}{20691706932} a^{12} + \frac{607955263}{41383413864} a^{11} - \frac{1314368537}{82766827728} a^{10} + \frac{11482735}{9196314192} a^{9} - \frac{256604195}{41383413864} a^{8} + \frac{983747449}{20691706932} a^{7} + \frac{316305983}{82766827728} a^{6} + \frac{5680318957}{27588942576} a^{5} + \frac{472183637}{10345853466} a^{4} - \frac{15442170917}{41383413864} a^{3} - \frac{31619640583}{82766827728} a^{2} + \frac{832981877}{6366679056} a - \frac{18794137819}{41383413864}$, $\frac{1}{102978326222399470061780331237412637429393136} a^{15} - \frac{176342648374283137556526318592985}{34326108740799823353926777079137545809797712} a^{14} - \frac{490676758097875393578494177681987154629}{578529922597749831807754669873104704659512} a^{13} + \frac{30272956143135344920309510616102737439722}{6436145388899966878861270702338289839337071} a^{12} - \frac{630407587844665031809583844242468438090791}{34326108740799823353926777079137545809797712} a^{11} - \frac{1993628414826809943049137207220939274254577}{102978326222399470061780331237412637429393136} a^{10} + \frac{518854453015139162340926921253174593057929}{51489163111199735030890165618706318714696568} a^{9} + \frac{30567022501086775565410108803318384823889}{953503020577772870942410474420487383605492} a^{8} + \frac{55703696368451073586227709534073784386331}{1271337360770363827923213965893983178140656} a^{7} - \frac{4002828055614812328469661802707222266756729}{102978326222399470061780331237412637429393136} a^{6} + \frac{3602193099105315162203299424055581816355449}{51489163111199735030890165618706318714696568} a^{5} + \frac{268858581654024349018656738502543234422947}{4290763592599977919240847134892193226224714} a^{4} + \frac{12084916236648487989210038063609853412778833}{102978326222399470061780331237412637429393136} a^{3} - \frac{1818389936658065031414125341081692174591299}{102978326222399470061780331237412637429393136} a^{2} + \frac{4796912662904314350227818482413457200033405}{17163054370399911676963388539568772904898856} a + \frac{138672447319137678625586358359778710340365}{289264961298874915903877334936552352329756}$

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

Class group and class number

$C_{2}\times C_{8}\times C_{24}\times C_{44520}$, which has order $17095680$ (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:  \( 85299.42553126559 \) (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

\(\Q(\sqrt{-53}) \), \(\Q(\sqrt{-106}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-530}) \), \(\Q(\sqrt{-265}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{-53})\), \(\Q(\sqrt{10}, \sqrt{-53})\), \(\Q(\sqrt{5}, \sqrt{-53})\), \(\Q(\sqrt{10}, \sqrt{-106})\), \(\Q(\sqrt{5}, \sqrt{-106})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-265})\), 4.0.719104000.4, 4.4.256000.2, 4.4.256000.1, 4.0.719104000.2, 8.0.323194101760000.29, 8.0.2068442251264000000.8, 8.0.2068442251264000000.7, 8.0.2068442251264000000.3, 8.0.2068442251264000000.4, 8.0.517110562816000000.5, 8.8.65536000000.1

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
$53$53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$