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Properties

Label 16.8.33634732739...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{36}\cdot 5^{8}\cdot 11^{6}\cdot 29^{4}$
Root discriminant $60.66$
Ramified primes $2, 5, 11, 29$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T511)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2411, -25368, -11290, 57500, 513, -5156, -4530, -6400, 4497, -584, -280, 140, -127, 52, -2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 2*x^14 + 52*x^13 - 127*x^12 + 140*x^11 - 280*x^10 - 584*x^9 + 4497*x^8 - 6400*x^7 - 4530*x^6 - 5156*x^5 + 513*x^4 + 57500*x^3 - 11290*x^2 - 25368*x - 2411)
 
gp: K = bnfinit(x^16 - 4*x^15 - 2*x^14 + 52*x^13 - 127*x^12 + 140*x^11 - 280*x^10 - 584*x^9 + 4497*x^8 - 6400*x^7 - 4530*x^6 - 5156*x^5 + 513*x^4 + 57500*x^3 - 11290*x^2 - 25368*x - 2411, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 2 x^{14} + 52 x^{13} - 127 x^{12} + 140 x^{11} - 280 x^{10} - 584 x^{9} + 4497 x^{8} - 6400 x^{7} - 4530 x^{6} - 5156 x^{5} + 513 x^{4} + 57500 x^{3} - 11290 x^{2} - 25368 x - 2411 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33634732739038648729600000000=2^{36}\cdot 5^{8}\cdot 11^{6}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.66$
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:  $2, 5, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1919} a^{14} + \frac{277}{1919} a^{13} + \frac{91}{1919} a^{12} + \frac{6}{19} a^{11} + \frac{17}{1919} a^{10} - \frac{335}{1919} a^{9} + \frac{159}{1919} a^{8} - \frac{470}{1919} a^{7} - \frac{17}{1919} a^{6} + \frac{338}{1919} a^{5} - \frac{288}{1919} a^{4} - \frac{334}{1919} a^{3} + \frac{70}{1919} a^{2} + \frac{917}{1919} a + \frac{959}{1919}$, $\frac{1}{4833754644202334695543953767723777} a^{15} - \frac{46129029378937792712659880890}{4833754644202334695543953767723777} a^{14} - \frac{520377172409055378087602801171152}{4833754644202334695543953767723777} a^{13} - \frac{294047888553368531215843219297409}{4833754644202334695543953767723777} a^{12} - \frac{85502746184617242698647802557044}{254408139168543931344418619353883} a^{11} - \frac{1180323486460133304631079675950414}{4833754644202334695543953767723777} a^{10} + \frac{55394781274165412644150415590184}{254408139168543931344418619353883} a^{9} + \frac{951845152077203366144180039246875}{4833754644202334695543953767723777} a^{8} - \frac{299788479894669961178949057282759}{4833754644202334695543953767723777} a^{7} - \frac{60154493650192253116981534562048}{4833754644202334695543953767723777} a^{6} + \frac{2291646247178077035949743184883419}{4833754644202334695543953767723777} a^{5} - \frac{1136116505312304926230183877010694}{4833754644202334695543953767723777} a^{4} + \frac{617147527027765573716269546129813}{4833754644202334695543953767723777} a^{3} + \frac{505000182492930483055729432708512}{4833754644202334695543953767723777} a^{2} + \frac{1088416007728812142498680354174843}{4833754644202334695543953767723777} a + \frac{344301095707507616680134212001973}{4833754644202334695543953767723777}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 60231330.6439 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T511):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.4400.1, 4.4.127600.1, 4.4.725.1, 8.4.218071040000.8, 8.4.183397744640000.1, 8.8.16281760000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.4.2.1$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}$
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}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$