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Properties

Label 16.0.14591662792...0704.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{54}\cdot 3^{4}$
Root discriminant $13.65$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_4 \times D_4$ (as 16T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -64, 224, -480, 768, -992, 992, -736, 364, -80, 24, -64, 100, -72, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 72*x^13 + 100*x^12 - 64*x^11 + 24*x^10 - 80*x^9 + 364*x^8 - 736*x^7 + 992*x^6 - 992*x^5 + 768*x^4 - 480*x^3 + 224*x^2 - 64*x + 8)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 72*x^13 + 100*x^12 - 64*x^11 + 24*x^10 - 80*x^9 + 364*x^8 - 736*x^7 + 992*x^6 - 992*x^5 + 768*x^4 - 480*x^3 + 224*x^2 - 64*x + 8, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 72 x^{13} + 100 x^{12} - 64 x^{11} + 24 x^{10} - 80 x^{9} + 364 x^{8} - 736 x^{7} + 992 x^{6} - 992 x^{5} + 768 x^{4} - 480 x^{3} + 224 x^{2} - 64 x + 8 \)

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:  \(1459166279268040704=2^{54}\cdot 3^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.65$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{4} a^{14}$, $\frac{1}{1044662852} a^{15} - \frac{84863933}{1044662852} a^{14} + \frac{69891819}{1044662852} a^{13} - \frac{29236081}{522331426} a^{12} - \frac{26168120}{261165713} a^{11} - \frac{31264975}{261165713} a^{10} - \frac{77885105}{522331426} a^{9} + \frac{44596207}{522331426} a^{8} + \frac{115076141}{522331426} a^{7} + \frac{60884639}{261165713} a^{6} + \frac{79718424}{261165713} a^{5} - \frac{7502358}{261165713} a^{4} + \frac{108352561}{261165713} a^{3} - \frac{115827114}{261165713} a^{2} - \frac{2008868}{11355031} a - \frac{87886627}{261165713}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{16824955281}{1044662852} a^{15} + \frac{62058648451}{522331426} a^{14} - \frac{460964862237}{1044662852} a^{13} + \frac{230867325747}{261165713} a^{12} - \frac{276147718659}{261165713} a^{11} + \frac{192020410249}{522331426} a^{10} - \frac{39991488069}{261165713} a^{9} + \frac{623177491583}{522331426} a^{8} - \frac{1336717326054}{261165713} a^{7} + \frac{4521878080371}{522331426} a^{6} - \frac{2756633223735}{261165713} a^{5} + \frac{2444569955959}{261165713} a^{4} - \frac{1695566598064}{261165713} a^{3} + \frac{953034164163}{261165713} a^{2} - \frac{14885088856}{11355031} a + \frac{53027319449}{261165713} \) (order $16$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3502.04946069 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_4$ (as 16T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_4 \times D_4$
Character table for $C_4 \times D_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), 4.0.2048.2, \(\Q(\zeta_{16})^+\), 4.0.3072.2, \(\Q(\zeta_{8})\), 4.0.3072.1, 8.0.37748736.1, 8.0.150994944.1, \(\Q(\zeta_{16})\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 siblings: data not computed

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$