↑

Properties

Label 16.0.69964294870...8144.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 79^{8}$
Root discriminant $130.41$
Ramified primes $2, 79$
Class number $3732480$ (GRH)
Class group $[4, 432, 2160]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53156361278, -16228594896, 18507232360, -4749263824, 2865222448, -618835120, 257684608, -46483984, 14717225, -2173192, 545756, -63272, 12790, -1064, 172, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717225*x^8 - 46483984*x^7 + 257684608*x^6 - 618835120*x^5 + 2865222448*x^4 - 4749263824*x^3 + 18507232360*x^2 - 16228594896*x + 53156361278)
 
gp: K = bnfinit(x^16 - 8*x^15 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717225*x^8 - 46483984*x^7 + 257684608*x^6 - 618835120*x^5 + 2865222448*x^4 - 4749263824*x^3 + 18507232360*x^2 - 16228594896*x + 53156361278, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 172 x^{14} - 1064 x^{13} + 12790 x^{12} - 63272 x^{11} + 545756 x^{10} - 2173192 x^{9} + 14717225 x^{8} - 46483984 x^{7} + 257684608 x^{6} - 618835120 x^{5} + 2865222448 x^{4} - 4749263824 x^{3} + 18507232360 x^{2} - 16228594896 x + 53156361278 \)

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:  \(6996429487079101204569997541638144=2^{62}\cdot 79^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.41$
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:  $2, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2528=2^{5}\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{2528}(1,·)$, $\chi_{2528}(2053,·)$, $\chi_{2528}(2369,·)$, $\chi_{2528}(1737,·)$, $\chi_{2528}(1421,·)$, $\chi_{2528}(1105,·)$, $\chi_{2528}(789,·)$, $\chi_{2528}(473,·)$, $\chi_{2528}(157,·)$, $\chi_{2528}(2213,·)$, $\chi_{2528}(1897,·)$, $\chi_{2528}(1581,·)$, $\chi_{2528}(1265,·)$, $\chi_{2528}(949,·)$, $\chi_{2528}(633,·)$, $\chi_{2528}(317,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{527} a^{13} + \frac{257}{527} a^{12} - \frac{57}{527} a^{11} - \frac{142}{527} a^{10} - \frac{241}{527} a^{9} + \frac{66}{527} a^{8} - \frac{226}{527} a^{7} + \frac{159}{527} a^{6} + \frac{183}{527} a^{5} - \frac{230}{527} a^{4} + \frac{98}{527} a^{3} - \frac{2}{17} a^{2} + \frac{162}{527} a + \frac{16}{527}$, $\frac{1}{105453458350254733715201} a^{14} - \frac{7}{105453458350254733715201} a^{13} - \frac{12343801230949783352698}{105453458350254733715201} a^{12} - \frac{31390650964556033598922}{105453458350254733715201} a^{11} - \frac{1484511337845133848677}{6203144608838513747953} a^{10} - \frac{25458312234128038685841}{105453458350254733715201} a^{9} + \frac{36434260330411764240372}{105453458350254733715201} a^{8} - \frac{682712716850839015}{3283824567939922577} a^{7} + \frac{36739591995566052594204}{105453458350254733715201} a^{6} - \frac{6019606293620541595272}{105453458350254733715201} a^{5} + \frac{48744642483317873203726}{105453458350254733715201} a^{4} + \frac{28867136505963724280289}{105453458350254733715201} a^{3} + \frac{34688559645481986267417}{105453458350254733715201} a^{2} + \frac{42352284332365999078136}{105453458350254733715201} a - \frac{21426352294159790143891}{105453458350254733715201}$, $\frac{1}{11366933729032308001895230991} a^{15} + \frac{53888}{11366933729032308001895230991} a^{14} + \frac{938738338311290727367476}{11366933729032308001895230991} a^{13} + \frac{19502294653986290126583423}{44229314120748280163016463} a^{12} + \frac{13758513674274154936085403}{117184883804456783524693103} a^{11} + \frac{1953961038189120240576674574}{11366933729032308001895230991} a^{10} - \frac{4481184046773136974145782432}{11366933729032308001895230991} a^{9} - \frac{3318429125260457247147256265}{11366933729032308001895230991} a^{8} - \frac{1639991331876758352898043144}{11366933729032308001895230991} a^{7} - \frac{1478586435063965019251238523}{11366933729032308001895230991} a^{6} - \frac{2554563301943423431846016976}{11366933729032308001895230991} a^{5} + \frac{3009035380443258385728466985}{11366933729032308001895230991} a^{4} - \frac{4977132995214556220834662741}{11366933729032308001895230991} a^{3} - \frac{3705866131492560525916118579}{11366933729032308001895230991} a^{2} + \frac{3562662272610434369934697225}{11366933729032308001895230991} a + \frac{146262424865437822186455292}{668643160531312235405601823}$

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

Class group and class number

$C_{4}\times C_{432}\times C_{2160}$, which has order $3732480$ (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:  \( 15753.94986242651 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

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 $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-158}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-79}) \), \(\Q(\sqrt{2}, \sqrt{-79})\), \(\Q(\zeta_{16})^+\), 4.0.12781568.9, 8.0.163368480538624.33, \(\Q(\zeta_{32})^+\), 8.0.83644662035775488.51

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
$79$79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$