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Properties

Label 16.0.35363757399...6096.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{64}\cdot 61^{8}$
Root discriminant $124.96$
Ramified primes $2, 61$
Class number $1491750$ (GRH)
Class group $[15, 15, 6630]$ (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![25558736641, 0, 10935000000, 0, 3827250000, 0, 510300000, 0, 33412500, 0, 1188000, 0, 23400, 0, 240, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 240*x^14 + 23400*x^12 + 1188000*x^10 + 33412500*x^8 + 510300000*x^6 + 3827250000*x^4 + 10935000000*x^2 + 25558736641)
 
gp: K = bnfinit(x^16 + 240*x^14 + 23400*x^12 + 1188000*x^10 + 33412500*x^8 + 510300000*x^6 + 3827250000*x^4 + 10935000000*x^2 + 25558736641, 1)
 

Normalized defining polynomial

\( x^{16} + 240 x^{14} + 23400 x^{12} + 1188000 x^{10} + 33412500 x^{8} + 510300000 x^{6} + 3827250000 x^{4} + 10935000000 x^{2} + 25558736641 \)

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:  \(3536375739919375385601751537156096=2^{64}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $124.96$
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:  $2, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1952=2^{5}\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{1952}(1,·)$, $\chi_{1952}(1219,·)$, $\chi_{1952}(1221,·)$, $\chi_{1952}(975,·)$, $\chi_{1952}(977,·)$, $\chi_{1952}(731,·)$, $\chi_{1952}(733,·)$, $\chi_{1952}(1951,·)$, $\chi_{1952}(487,·)$, $\chi_{1952}(489,·)$, $\chi_{1952}(1707,·)$, $\chi_{1952}(1709,·)$, $\chi_{1952}(243,·)$, $\chi_{1952}(245,·)$, $\chi_{1952}(1463,·)$, $\chi_{1952}(1465,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{15841} a^{8} + \frac{120}{15841} a^{6} + \frac{4500}{15841} a^{4} + \frac{6477}{15841} a^{2} + \frac{6204}{15841}$, $\frac{1}{2532516511} a^{9} + \frac{660427251}{2532516511} a^{7} + \frac{966907458}{2532516511} a^{5} + \frac{1149461119}{2532516511} a^{3} - \frac{243153146}{2532516511} a$, $\frac{1}{2532516511} a^{10} + \frac{150}{2532516511} a^{8} + \frac{223667179}{2532516511} a^{6} - \frac{18660854}{361788073} a^{4} - \frac{409099244}{2532516511} a^{2} + \frac{2014}{15841}$, $\frac{1}{2532516511} a^{11} - \frac{72276542}{2532516511} a^{7} - \frac{813303551}{2532516511} a^{5} - \frac{88163478}{361788073} a^{3} - \frac{1192795571}{2532516511} a$, $\frac{1}{2532516511} a^{12} - \frac{14850}{2532516511} a^{8} + \frac{260549956}{2532516511} a^{6} + \frac{398356246}{2532516511} a^{4} + \frac{863145489}{2532516511} a^{2} + \frac{351}{15841}$, $\frac{1}{2532516511} a^{13} - \frac{831219797}{2532516511} a^{7} - \frac{394509824}{2532516511} a^{5} + \frac{1199478499}{2532516511} a^{3} + \frac{600441307}{2532516511} a$, $\frac{1}{2532516511} a^{14} - \frac{1628}{81694081} a^{8} + \frac{577665727}{2532516511} a^{6} + \frac{30147492}{81694081} a^{4} - \frac{629002}{34692007} a^{2} + \frac{2320}{15841}$, $\frac{1}{2532516511} a^{15} + \frac{81481132}{361788073} a^{7} - \frac{1108239}{11670583} a^{5} + \frac{1134635580}{2532516511} a^{3} - \frac{1039575813}{2532516511} a$

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

Class group and class number

$C_{15}\times C_{15}\times C_{6630}$, which has order $1491750$ (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

\(\Q(\sqrt{-122}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-61}) \), \(\Q(\sqrt{2}, \sqrt{-61})\), \(\Q(\zeta_{16})^+\), 4.0.7620608.2, 8.0.232294665158656.11, \(\Q(\zeta_{32})^+\), 8.0.29733717140307968.6

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
2Data not computed
61Data not computed