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Properties

Label 16.4.88406959760...5264.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{46}\cdot 3^{14}\cdot 11^{14}\cdot 263^{2}$
Root discriminant $313.80$
Ramified primes $2, 3, 11, 263$
Class number $64$ (GRH)
Class group $[2, 2, 2, 2, 4]$ (GRH)
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![308531367936, 0, 118045509120, 0, 7611413040, 0, -902632896, 0, -38679168, 0, 874632, 0, 41118, 0, 396, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 396*x^14 + 41118*x^12 + 874632*x^10 - 38679168*x^8 - 902632896*x^6 + 7611413040*x^4 + 118045509120*x^2 + 308531367936)
 
gp: K = bnfinit(x^16 + 396*x^14 + 41118*x^12 + 874632*x^10 - 38679168*x^8 - 902632896*x^6 + 7611413040*x^4 + 118045509120*x^2 + 308531367936, 1)
 

Normalized defining polynomial

\( x^{16} + 396 x^{14} + 41118 x^{12} + 874632 x^{10} - 38679168 x^{8} - 902632896 x^{6} + 7611413040 x^{4} + 118045509120 x^{2} + 308531367936 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8840695976077175158703239475879177355264=2^{46}\cdot 3^{14}\cdot 11^{14}\cdot 263^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $313.80$
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:  $2, 3, 11, 263$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{132} a^{8}$, $\frac{1}{132} a^{9}$, $\frac{1}{264} a^{10} - \frac{1}{4} a^{6}$, $\frac{1}{264} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2640} a^{12} + \frac{1}{1320} a^{10} + \frac{1}{1320} a^{8} - \frac{1}{20} a^{6} + \frac{1}{10} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{10560} a^{13} + \frac{1}{880} a^{11} - \frac{3}{1760} a^{9} - \frac{3}{40} a^{7} - \frac{1}{10} a^{5} + \frac{2}{5} a^{3} - \frac{1}{20} a$, $\frac{1}{10185615099874473242657144048328960} a^{14} + \frac{6163522706341284135570439841}{848801258322872770221428670694080} a^{12} - \frac{240429782112262003161149417953}{154327501513249594585714303762560} a^{10} + \frac{865571915331586850520451374107}{424400629161436385110714335347040} a^{8} - \frac{695517809335216548416042859493}{4822734422289049830803571992580} a^{6} - \frac{246040374602545905581742107061}{1607578140763016610267857330860} a^{4} - \frac{2375370920326483716199153389967}{6430312563052066441071429323440} a^{2} - \frac{99568363295064010414731033}{305623220677379583701113561}$, $\frac{1}{40742460399497892970628576193315840} a^{15} + \frac{6163522706341284135570439841}{3395205033291491080885714682776320} a^{13} + \frac{3785584959817184406298785725957}{6790410066582982161771429365552640} a^{11} + \frac{865571915331586850520451374107}{1697602516645745540442857341388160} a^{9} - \frac{950600707453739503058467928819}{9645468844578099661607143985160} a^{7} - \frac{246040374602545905581742107061}{6430312563052066441071429323440} a^{5} - \frac{8805683483378550157270582713407}{25721250252208265764285717293760} a^{3} - \frac{202595791986221797057922297}{611246441354759167402227122} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $64$ (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:  $9$
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:  \( 12586383485500 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{6}) \), 4.4.287496.1, 4.4.1149984.1, \(\Q(\sqrt{6}, \sqrt{22})\), 8.8.21159411204096.1

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 R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.10.6$x^{4} + 6 x^{2} + 3$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.11.12$x^{4} + 26$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.8.25.65$x^{8} + 4 x^{6} + 20 x^{4} + 56$$8$$1$$25$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 3, 7/2, 4, 17/4]^{2}$
$3$3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
11Data not computed
263Data not computed