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Properties

Label 16.0.11497853914...5776.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 53^{8}$
Root discriminant $100.88$
Ramified primes $2, 3, 53$
Class number $532440$ (GRH)
Class group $[3, 3, 59160]$ (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![3821989918, -85763016, 1466638388, -6750888, 259368858, 589920, 27955244, 83712, 2027175, 2520, 102884, -24, 3638, 0, 84, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 84*x^14 + 3638*x^12 - 24*x^11 + 102884*x^10 + 2520*x^9 + 2027175*x^8 + 83712*x^7 + 27955244*x^6 + 589920*x^5 + 259368858*x^4 - 6750888*x^3 + 1466638388*x^2 - 85763016*x + 3821989918)
 
gp: K = bnfinit(x^16 + 84*x^14 + 3638*x^12 - 24*x^11 + 102884*x^10 + 2520*x^9 + 2027175*x^8 + 83712*x^7 + 27955244*x^6 + 589920*x^5 + 259368858*x^4 - 6750888*x^3 + 1466638388*x^2 - 85763016*x + 3821989918, 1)
 

Normalized defining polynomial

\( x^{16} + 84 x^{14} + 3638 x^{12} - 24 x^{11} + 102884 x^{10} + 2520 x^{9} + 2027175 x^{8} + 83712 x^{7} + 27955244 x^{6} + 589920 x^{5} + 259368858 x^{4} - 6750888 x^{3} + 1466638388 x^{2} - 85763016 x + 3821989918 \)

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:  \(114978539144999765882759100235776=2^{48}\cdot 3^{8}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.88$
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:  $2, 3, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2544=2^{4}\cdot 3\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{2544}(1,·)$, $\chi_{2544}(2119,·)$, $\chi_{2544}(1483,·)$, $\chi_{2544}(847,·)$, $\chi_{2544}(211,·)$, $\chi_{2544}(953,·)$, $\chi_{2544}(2015,·)$, $\chi_{2544}(1379,·)$, $\chi_{2544}(743,·)$, $\chi_{2544}(107,·)$, $\chi_{2544}(317,·)$, $\chi_{2544}(2225,·)$, $\chi_{2544}(1909,·)$, $\chi_{2544}(1273,·)$, $\chi_{2544}(637,·)$, $\chi_{2544}(1589,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{493689551} a^{14} + \frac{23633264}{493689551} a^{13} + \frac{99788415}{493689551} a^{12} - \frac{168158517}{493689551} a^{11} - \frac{238961293}{493689551} a^{10} - \frac{158749987}{493689551} a^{9} + \frac{119706262}{493689551} a^{8} - \frac{156807623}{493689551} a^{7} + \frac{3821905}{493689551} a^{6} + \frac{194186134}{493689551} a^{5} - \frac{168128692}{493689551} a^{4} - \frac{156883760}{493689551} a^{3} + \frac{1600862}{493689551} a^{2} + \frac{239552152}{493689551} a - \frac{57851348}{493689551}$, $\frac{1}{646428445194350978617895347040412420106031} a^{15} + \frac{502449166498982388559313733955507}{646428445194350978617895347040412420106031} a^{14} - \frac{82094104434962968617443926402801532381481}{646428445194350978617895347040412420106031} a^{13} + \frac{191790390694198765344954469720455483799973}{646428445194350978617895347040412420106031} a^{12} - \frac{250810578045440373405004600927400952346413}{646428445194350978617895347040412420106031} a^{11} + \frac{21937726809193178684384428304993078683784}{646428445194350978617895347040412420106031} a^{10} - \frac{35141694942295821866715369299666922251432}{92346920742050139802556478148630345729433} a^{9} + \frac{33375140582177114113296385885267404118774}{92346920742050139802556478148630345729433} a^{8} - \frac{83316129543902168336645089304056040749486}{646428445194350978617895347040412420106031} a^{7} + \frac{148222021313486235855178091759431124115116}{646428445194350978617895347040412420106031} a^{6} + \frac{244116599833987859493706355457111810135388}{646428445194350978617895347040412420106031} a^{5} - \frac{240850057665848883429263945578136930490366}{646428445194350978617895347040412420106031} a^{4} + \frac{20881254827309966762523272261308317217443}{646428445194350978617895347040412420106031} a^{3} + \frac{194384122631049107195265973668890613080016}{646428445194350978617895347040412420106031} a^{2} + \frac{223996742439800043151133558194188968599711}{646428445194350978617895347040412420106031} a + \frac{48825079741273201109760775794805100857881}{646428445194350978617895347040412420106031}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{59160}$, which has order $532440$ (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

\(\Q(\sqrt{-53}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-159}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-106}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-318}) \), \(\Q(\sqrt{3}, \sqrt{-53})\), \(\Q(\sqrt{2}, \sqrt{-53})\), \(\Q(\sqrt{6}, \sqrt{-53})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-106})\), \(\Q(\sqrt{2}, \sqrt{-159})\), \(\Q(\sqrt{6}, \sqrt{-106})\), 4.4.18432.1, 4.0.51775488.2, \(\Q(\zeta_{16})^+\), 4.0.5752832.2, 8.0.41885955588096.13, 8.0.10722804630552576.17, 8.0.132380304080896.14, \(\Q(\zeta_{48})^+\), 8.0.10722804630552576.23, 8.0.2680701157638144.44, 8.0.2680701157638144.60

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ 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
$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}$
$53$53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$