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Properties

Label 16.0.81606169470...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{10}\cdot 41^{6}$
Root discriminant $74.04$
Ramified primes $2, 5, 41$
Class number $11664$ (GRH)
Class group $[2, 2, 2, 1458]$ (GRH)
Galois group 16T869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![672400, 0, 2689600, 0, 3171760, 0, 1515360, 0, 340044, 0, 39920, 0, 2516, 0, 80, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 80*x^14 + 2516*x^12 + 39920*x^10 + 340044*x^8 + 1515360*x^6 + 3171760*x^4 + 2689600*x^2 + 672400)
 
gp: K = bnfinit(x^16 + 80*x^14 + 2516*x^12 + 39920*x^10 + 340044*x^8 + 1515360*x^6 + 3171760*x^4 + 2689600*x^2 + 672400, 1)
 

Normalized defining polynomial

\( x^{16} + 80 x^{14} + 2516 x^{12} + 39920 x^{10} + 340044 x^{8} + 1515360 x^{6} + 3171760 x^{4} + 2689600 x^{2} + 672400 \)

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:  \(816061694707436093440000000000=2^{44}\cdot 5^{10}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.04$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{16} a^{8} - \frac{1}{4} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4}$, $\frac{1}{16} a^{9} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} - \frac{1}{4} a$, $\frac{1}{48} a^{10} - \frac{1}{48} a^{8} + \frac{1}{24} a^{6} + \frac{5}{24} a^{4} - \frac{5}{12} a^{2} + \frac{1}{12}$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{9} + \frac{1}{24} a^{7} + \frac{5}{24} a^{5} - \frac{5}{12} a^{3} + \frac{1}{12} a$, $\frac{1}{413280} a^{12} - \frac{11}{13776} a^{10} + \frac{53}{12915} a^{8} - \frac{47}{2296} a^{6} - \frac{647}{51660} a^{4} - \frac{5}{252} a^{2} - \frac{37}{252}$, $\frac{1}{413280} a^{13} - \frac{11}{13776} a^{11} + \frac{53}{12915} a^{9} - \frac{47}{2296} a^{7} - \frac{647}{51660} a^{5} - \frac{5}{252} a^{3} - \frac{37}{252} a$, $\frac{1}{26102351520} a^{14} + \frac{311}{3262793940} a^{12} - \frac{1541677}{815698485} a^{10} + \frac{152298217}{6525587880} a^{8} - \frac{66819527}{3262793940} a^{6} - \frac{430318141}{3262793940} a^{4} - \frac{1915919}{5305356} a^{2} + \frac{2243587}{7958034}$, $\frac{1}{26102351520} a^{15} + \frac{311}{3262793940} a^{13} - \frac{1541677}{815698485} a^{11} + \frac{152298217}{6525587880} a^{9} - \frac{66819527}{3262793940} a^{7} - \frac{430318141}{3262793940} a^{5} - \frac{1915919}{5305356} a^{3} + \frac{2243587}{7958034} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{1458}$, which has order $11664$ (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:  \( 19190.6159055 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n869
Character table for t16n869 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), 4.4.65600.2, 4.4.2624.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.4303360000.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.22.64$x^{8} + 4 x^{6} + 6 x^{4} + 28$$4$$2$$22$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 7/2, 4]^{2}$
2.8.22.63$x^{8} + 2 x^{4} + 8 x^{2} + 60$$4$$2$$22$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 7/2, 4]^{2}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$41$41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$