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Properties

Label 16.0.24041492054...7797.2
Degree $16$
Signature $[0, 8]$
Discriminant $11^{10}\cdot 53^{11}$
Root discriminant $68.60$
Ramified primes $11, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10161, 40872, 14387, -96554, 127119, -40402, 13315, 2519, 3305, 347, -784, 250, -10, 0, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 4*x^14 - 10*x^12 + 250*x^11 - 784*x^10 + 347*x^9 + 3305*x^8 + 2519*x^7 + 13315*x^6 - 40402*x^5 + 127119*x^4 - 96554*x^3 + 14387*x^2 + 40872*x + 10161)
 
gp: K = bnfinit(x^16 - x^15 + 4*x^14 - 10*x^12 + 250*x^11 - 784*x^10 + 347*x^9 + 3305*x^8 + 2519*x^7 + 13315*x^6 - 40402*x^5 + 127119*x^4 - 96554*x^3 + 14387*x^2 + 40872*x + 10161, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 4 x^{14} - 10 x^{12} + 250 x^{11} - 784 x^{10} + 347 x^{9} + 3305 x^{8} + 2519 x^{7} + 13315 x^{6} - 40402 x^{5} + 127119 x^{4} - 96554 x^{3} + 14387 x^{2} + 40872 x + 10161 \)

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:  \(240414920542051180813313277797=11^{10}\cdot 53^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.60$
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:  $11, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{35} a^{13} - \frac{11}{35} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{13}{35} a^{9} + \frac{2}{35} a^{8} - \frac{3}{7} a^{7} - \frac{12}{35} a^{6} + \frac{16}{35} a^{5} + \frac{4}{35} a^{4} + \frac{4}{35} a^{3} + \frac{1}{35} a^{2} - \frac{9}{35} a - \frac{17}{35}$, $\frac{1}{1785} a^{14} + \frac{1}{85} a^{13} - \frac{233}{1785} a^{12} - \frac{1}{15} a^{11} + \frac{271}{595} a^{10} + \frac{286}{1785} a^{9} - \frac{41}{85} a^{8} + \frac{383}{1785} a^{7} - \frac{368}{1785} a^{6} + \frac{277}{595} a^{5} + \frac{832}{1785} a^{4} - \frac{202}{595} a^{3} + \frac{31}{595} a^{2} + \frac{10}{21} a + \frac{87}{595}$, $\frac{1}{385020435408283559525986002947966985} a^{15} - \frac{34266801009257222352660534800807}{385020435408283559525986002947966985} a^{14} - \frac{1479109021125213651900547044405347}{385020435408283559525986002947966985} a^{13} + \frac{187024622869788931417632735293851391}{385020435408283559525986002947966985} a^{12} + \frac{144314454792269408241672099469471186}{385020435408283559525986002947966985} a^{11} + \frac{187265117687071845969644859625714954}{385020435408283559525986002947966985} a^{10} - \frac{74653840401257055744149768263728491}{385020435408283559525986002947966985} a^{9} + \frac{81243553045804312965052434214139429}{385020435408283559525986002947966985} a^{8} - \frac{22251464124709821040172076591966919}{128340145136094519841995334315988995} a^{7} - \frac{964509142112933060415004169886982}{3235465843767088735512487419730815} a^{6} + \frac{164638835316045978435543151661304853}{385020435408283559525986002947966985} a^{5} + \frac{43771610601776459710072447684796}{3235465843767088735512487419730815} a^{4} + \frac{21493549685290871919349357157268294}{128340145136094519841995334315988995} a^{3} + \frac{23317326043336574891876145788638253}{77004087081656711905197200589593397} a^{2} - \frac{23649635961067223397043708913582737}{77004087081656711905197200589593397} a - \frac{4433293232230843145244164640998152}{128340145136094519841995334315988995}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 133049380.535 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{53}) \), 4.2.30899.1, 8.0.6122800213013.1

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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.8.7.1$x^{8} + 33$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
53Data not computed