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Properties

Label 16.0.64905889088...0000.9
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 5^{12}\cdot 7^{8}$
Root discriminant $129.80$
Ramified primes $2, 5, 7$
Class number $1427456$ (GRH)
Class group $[2, 4, 4, 8, 5576]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14412002500, 0, 16470860000, 0, 5529503000, 0, 739508000, 0, 47899950, 0, 1646400, 0, 30380, 0, 280, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 280*x^14 + 30380*x^12 + 1646400*x^10 + 47899950*x^8 + 739508000*x^6 + 5529503000*x^4 + 16470860000*x^2 + 14412002500)
 
gp: K = bnfinit(x^16 + 280*x^14 + 30380*x^12 + 1646400*x^10 + 47899950*x^8 + 739508000*x^6 + 5529503000*x^4 + 16470860000*x^2 + 14412002500, 1)
 

Normalized defining polynomial

\( x^{16} + 280 x^{14} + 30380 x^{12} + 1646400 x^{10} + 47899950 x^{8} + 739508000 x^{6} + 5529503000 x^{4} + 16470860000 x^{2} + 14412002500 \)

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:  \(6490588908866265677824000000000000=2^{62}\cdot 5^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.80$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1120=2^{5}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1120}(1,·)$, $\chi_{1120}(643,·)$, $\chi_{1120}(449,·)$, $\chi_{1120}(841,·)$, $\chi_{1120}(587,·)$, $\chi_{1120}(83,·)$, $\chi_{1120}(281,·)$, $\chi_{1120}(729,·)$, $\chi_{1120}(923,·)$, $\chi_{1120}(867,·)$, $\chi_{1120}(561,·)$, $\chi_{1120}(169,·)$, $\chi_{1120}(363,·)$, $\chi_{1120}(27,·)$, $\chi_{1120}(1009,·)$, $\chi_{1120}(307,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{245} a^{4}$, $\frac{1}{245} a^{5}$, $\frac{1}{1715} a^{6}$, $\frac{1}{1715} a^{7}$, $\frac{1}{120050} a^{8}$, $\frac{1}{120050} a^{9}$, $\frac{1}{840350} a^{10}$, $\frac{1}{840350} a^{11}$, $\frac{1}{29412250} a^{12}$, $\frac{1}{29412250} a^{13}$, $\frac{1}{44677207750} a^{14} - \frac{38}{3191229125} a^{12} + \frac{3}{26050850} a^{10} + \frac{43}{26050850} a^{8} - \frac{13}{372155} a^{6} + \frac{1}{53165} a^{4} + \frac{78}{1519} a^{2} - \frac{71}{217}$, $\frac{1}{44677207750} a^{15} - \frac{38}{3191229125} a^{13} + \frac{3}{26050850} a^{11} + \frac{43}{26050850} a^{9} - \frac{13}{372155} a^{7} + \frac{1}{53165} a^{5} + \frac{78}{1519} a^{3} - \frac{71}{217} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{8}\times C_{5576}$, which has order $1427456$ (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:  \( 12198.951274811623 \) (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

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.51200.1, \(\Q(\zeta_{16})^+\), 8.8.2621440000.1, 8.0.80564191232000000.78, 8.0.80564191232000000.92

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
5Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$