↑

Properties

Label 16.0.26581578912...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 101^{6}\cdot 156506561$
Root discriminant $122.75$
Ramified primes $2, 5, 101, 156506561$
Class number $294912$ (GRH)
Class group $[2, 8, 18432]$ (GRH)
Galois group 16T1643

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![156506561, 0, 154910568, 0, 60637212, 0, 12428760, 0, 1480966, 0, 106200, 0, 4508, 0, 104, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 104*x^14 + 4508*x^12 + 106200*x^10 + 1480966*x^8 + 12428760*x^6 + 60637212*x^4 + 154910568*x^2 + 156506561)
 
gp: K = bnfinit(x^16 + 104*x^14 + 4508*x^12 + 106200*x^10 + 1480966*x^8 + 12428760*x^6 + 60637212*x^4 + 154910568*x^2 + 156506561, 1)
 

Normalized defining polynomial

\( x^{16} + 104 x^{14} + 4508 x^{12} + 106200 x^{10} + 1480966 x^{8} + 12428760 x^{6} + 60637212 x^{4} + 154910568 x^{2} + 156506561 \)

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:  \(2658157891244233490576000000000000=2^{16}\cdot 5^{12}\cdot 101^{6}\cdot 156506561\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.75$
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:  $2, 5, 101, 156506561$
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 \(a\))

$1$, $a$, $\frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{4} - \frac{1}{8} a^{2} - \frac{3}{16}$, $\frac{1}{16} a^{5} - \frac{1}{8} a^{3} - \frac{3}{16} a$, $\frac{1}{64} a^{6} - \frac{1}{64} a^{4} - \frac{5}{64} a^{2} - \frac{3}{64}$, $\frac{1}{64} a^{7} - \frac{1}{64} a^{5} - \frac{5}{64} a^{3} - \frac{3}{64} a$, $\frac{1}{256} a^{8} - \frac{3}{128} a^{4} - \frac{1}{32} a^{2} - \frac{3}{256}$, $\frac{1}{256} a^{9} - \frac{3}{128} a^{5} - \frac{1}{32} a^{3} - \frac{3}{256} a$, $\frac{1}{1024} a^{10} + \frac{1}{1024} a^{8} - \frac{3}{512} a^{6} - \frac{7}{512} a^{4} - \frac{11}{1024} a^{2} - \frac{3}{1024}$, $\frac{1}{1024} a^{11} + \frac{1}{1024} a^{9} - \frac{3}{512} a^{7} - \frac{7}{512} a^{5} - \frac{11}{1024} a^{3} - \frac{3}{1024} a$, $\frac{1}{4096} a^{12} - \frac{1}{2048} a^{10} + \frac{7}{4096} a^{8} + \frac{1}{1024} a^{6} - \frac{65}{4096} a^{4} - \frac{49}{2048} a^{2} - \frac{39}{4096}$, $\frac{1}{4096} a^{13} - \frac{1}{2048} a^{11} + \frac{7}{4096} a^{9} + \frac{1}{1024} a^{7} - \frac{65}{4096} a^{5} - \frac{49}{2048} a^{3} - \frac{39}{4096} a$, $\frac{1}{1163264} a^{14} + \frac{107}{1163264} a^{12} + \frac{285}{1163264} a^{10} - \frac{2001}{1163264} a^{8} - \frac{1837}{1163264} a^{6} + \frac{27217}{1163264} a^{4} - \frac{38961}{1163264} a^{2} - \frac{65835}{1163264}$, $\frac{1}{1163264} a^{15} + \frac{107}{1163264} a^{13} + \frac{285}{1163264} a^{11} - \frac{2001}{1163264} a^{9} - \frac{1837}{1163264} a^{7} + \frac{27217}{1163264} a^{5} - \frac{38961}{1163264} a^{3} - \frac{65835}{1163264} a$

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

Class group and class number

$C_{2}\times C_{8}\times C_{18432}$, which has order $294912$ (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:  \( 38120.6275869 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1643:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 64 conjugacy class representatives for t16n1643 are not computed
Character table for t16n1643 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.16098453125.1

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 $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
156506561Data not computed