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Properties

Label 16.0.15434889281...0944.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 257^{14}$
Root discriminant $1027.50$
Ramified primes $2, 257$
Class number $10601103360$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 4, 4, 8, 16, 80880]$ (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![17868678762496, 0, 0, 0, -521459496960, 0, 0, 0, 3810894945, 0, 0, 0, 123360, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 123360*x^12 + 3810894945*x^8 - 521459496960*x^4 + 17868678762496)
 
gp: K = bnfinit(x^16 + 123360*x^12 + 3810894945*x^8 - 521459496960*x^4 + 17868678762496, 1)
 

Normalized defining polynomial

\( x^{16} + 123360 x^{12} + 3810894945 x^{8} - 521459496960 x^{4} + 17868678762496 \)

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:  \(1543488928107008169638324168125095939257028050944=2^{48}\cdot 257^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1027.50$
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:  $2, 257$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4112=2^{4}\cdot 257\)
Dirichlet character group:    $\lbrace$$\chi_{4112}(1,·)$, $\chi_{4112}(835,·)$, $\chi_{4112}(261,·)$, $\chi_{4112}(513,·)$, $\chi_{4112}(1803,·)$, $\chi_{4112}(3851,·)$, $\chi_{4112}(3277,·)$, $\chi_{4112}(4111,·)$, $\chi_{4112}(707,·)$, $\chi_{4112}(1815,·)$, $\chi_{4112}(2329,·)$, $\chi_{4112}(3599,·)$, $\chi_{4112}(2309,·)$, $\chi_{4112}(3405,·)$, $\chi_{4112}(1783,·)$, $\chi_{4112}(2297,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{61937} a^{8} - \frac{1}{241} a^{4} - \frac{60}{241}$, $\frac{1}{123874} a^{9} + \frac{120}{241} a^{5} + \frac{181}{482} a$, $\frac{1}{495496} a^{10} + \frac{30}{241} a^{6} - \frac{783}{1928} a^{2}$, $\frac{1}{254684944} a^{11} - \frac{4082}{61937} a^{7} + \frac{41633}{990992} a^{3}$, $\frac{1}{15206281409590336} a^{12} - \frac{19061}{142231755178} a^{8} + \frac{566861410533}{4551416165696} a^{4} + \frac{1798153185}{3597301201}$, $\frac{1}{60825125638361344} a^{13} + \frac{2277333}{568927020712} a^{9} + \frac{547975866277}{18205664662784} a^{5} + \frac{902559525}{14389204804} a$, $\frac{1}{31264114578117730816} a^{14} + \frac{68872759}{292428488645968} a^{10} - \frac{3145028275413083}{9357711636670976} a^{6} - \frac{5332747205}{14389204804} a^{2}$, $\frac{1}{125056458312470923264} a^{15} + \frac{2277333}{1169713954583872} a^{11} - \frac{4805747495108699}{37430846546683904} a^{7} - \frac{13366668703391}{29584205077024} a^{3}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{8}\times C_{16}\times C_{80880}$, which has order $10601103360$ (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:  \( \frac{15}{245512075328} a^{12} + \frac{4453}{590173258} a^{8} + \frac{17166315}{73484608} a^{4} - \frac{238298640}{14926561} \) (order $4$)
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:  \( 638360561.5229604 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{257}) \), \(\Q(\sqrt{-257}) \), \(\Q(i, \sqrt{257})\), 4.4.1086373952.2, 4.0.1086373952.2, 8.0.18883333817345572864.18, 8.0.310593074627699982467072.6, 8.8.310593074627699982467072.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.24.10$x^{8} + 16$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.10$x^{8} + 16$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
257Data not computed