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Properties

Label 16.0.50301776770...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{4}\cdot 13^{4}$
Root discriminant $19.67$
Ramified primes $2, 3, 5, 13$
Class number $1$
Class group Trivial
Galois group $C_2\times Q_8:C_2^2$ (as 16T69)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![324, -1296, 2592, -1512, 1152, -1944, 2088, -924, 613, -668, 488, -180, 86, -60, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 60*x^13 + 86*x^12 - 180*x^11 + 488*x^10 - 668*x^9 + 613*x^8 - 924*x^7 + 2088*x^6 - 1944*x^5 + 1152*x^4 - 1512*x^3 + 2592*x^2 - 1296*x + 324)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 60*x^13 + 86*x^12 - 180*x^11 + 488*x^10 - 668*x^9 + 613*x^8 - 924*x^7 + 2088*x^6 - 1944*x^5 + 1152*x^4 - 1512*x^3 + 2592*x^2 - 1296*x + 324, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 60 x^{13} + 86 x^{12} - 180 x^{11} + 488 x^{10} - 668 x^{9} + 613 x^{8} - 924 x^{7} + 2088 x^{6} - 1944 x^{5} + 1152 x^{4} - 1512 x^{3} + 2592 x^{2} - 1296 x + 324 \)

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:  \(503017767708917760000=2^{32}\cdot 3^{8}\cdot 5^{4}\cdot 13^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.67$
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:  $2, 3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{12} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{1}{4} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{12} a^{9} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{12} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{10} + \frac{1}{12} a^{6} - \frac{1}{6} a^{2}$, $\frac{1}{168} a^{11} + \frac{1}{168} a^{10} - \frac{1}{28} a^{9} + \frac{1}{84} a^{8} + \frac{29}{168} a^{7} - \frac{23}{168} a^{6} - \frac{41}{84} a^{5} + \frac{1}{28} a^{4} - \frac{17}{84} a^{3} - \frac{5}{28} a^{2} + \frac{1}{7}$, $\frac{1}{1512} a^{12} + \frac{1}{378} a^{11} + \frac{53}{1512} a^{10} + \frac{1}{28} a^{9} - \frac{7}{216} a^{8} + \frac{17}{126} a^{7} - \frac{151}{1512} a^{6} + \frac{83}{756} a^{5} - \frac{179}{378} a^{4} - \frac{2}{63} a^{3} - \frac{11}{28} a^{2} + \frac{1}{14} a + \frac{8}{21}$, $\frac{1}{1512} a^{13} + \frac{1}{1512} a^{11} + \frac{29}{756} a^{10} - \frac{7}{216} a^{9} - \frac{25}{756} a^{8} + \frac{257}{1512} a^{7} + \frac{169}{756} a^{6} - \frac{29}{63} a^{5} - \frac{23}{756} a^{4} - \frac{115}{252} a^{3} - \frac{10}{21} a^{2} + \frac{2}{21} a + \frac{17}{42}$, $\frac{1}{4536} a^{14} + \frac{1}{4536} a^{13} - \frac{1}{4536} a^{12} - \frac{1}{378} a^{11} - \frac{17}{2268} a^{10} + \frac{19}{504} a^{9} + \frac{179}{4536} a^{8} + \frac{157}{1134} a^{7} + \frac{109}{648} a^{6} - \frac{50}{189} a^{5} + \frac{179}{756} a^{4} + \frac{23}{126} a^{3} + \frac{125}{252} a^{2} + \frac{5}{42} a - \frac{5}{42}$, $\frac{1}{2089816848} a^{15} - \frac{111773}{2089816848} a^{14} + \frac{401525}{2089816848} a^{13} - \frac{73957}{232201872} a^{12} + \frac{80327}{298545264} a^{11} - \frac{15815143}{696605616} a^{10} + \frac{42354743}{2089816848} a^{9} - \frac{82846391}{2089816848} a^{8} + \frac{75734357}{1044908424} a^{7} + \frac{10055}{74376} a^{6} - \frac{10947077}{24878772} a^{5} - \frac{10829533}{24878772} a^{4} + \frac{12132439}{29025234} a^{3} - \frac{143848}{691077} a^{2} - \frac{69205}{19350156} a + \frac{4770209}{19350156}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{1321511}{522454212} a^{15} + \frac{712847}{38700312} a^{14} - \frac{69712381}{1044908424} a^{13} + \frac{102743705}{1044908424} a^{12} - \frac{17507071}{130613553} a^{11} + \frac{28111795}{74636316} a^{10} - \frac{1079361959}{1044908424} a^{9} + \frac{1086306919}{1044908424} a^{8} - \frac{146273707}{174151404} a^{7} + \frac{10581217}{4685688} a^{6} - \frac{410983043}{87075702} a^{5} + \frac{122367787}{43537851} a^{4} - \frac{2668132}{1612513} a^{3} + \frac{258302603}{58050468} a^{2} - \frac{18430969}{3225026} a + \frac{9234445}{4837539} \) (order $24$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 121122.735511 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times Q_8:C_2^2$ (as 16T69):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 34 conjugacy class representatives for $C_2\times Q_8:C_2^2$
Character table for $C_2\times Q_8:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{24})\), 8.4.22428057600.1, 8.4.22428057600.2

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 R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$