↑

Properties

Label 16.0.28643813255...2081.2
Degree $16$
Signature $[0, 8]$
Discriminant $11^{4}\cdot 89^{14}$
Root discriminant $92.48$
Ramified primes $11, 89$
Class number $226$ (GRH)
Class group $[226]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![50324192, -91102296, 60329180, -10586538, -438001, -743188, 1007165, 112522, 22417, -41278, -5620, 716, 879, 2, -21, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 21*x^14 + 2*x^13 + 879*x^12 + 716*x^11 - 5620*x^10 - 41278*x^9 + 22417*x^8 + 112522*x^7 + 1007165*x^6 - 743188*x^5 - 438001*x^4 - 10586538*x^3 + 60329180*x^2 - 91102296*x + 50324192)
 
gp: K = bnfinit(x^16 - 4*x^15 - 21*x^14 + 2*x^13 + 879*x^12 + 716*x^11 - 5620*x^10 - 41278*x^9 + 22417*x^8 + 112522*x^7 + 1007165*x^6 - 743188*x^5 - 438001*x^4 - 10586538*x^3 + 60329180*x^2 - 91102296*x + 50324192, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 21 x^{14} + 2 x^{13} + 879 x^{12} + 716 x^{11} - 5620 x^{10} - 41278 x^{9} + 22417 x^{8} + 112522 x^{7} + 1007165 x^{6} - 743188 x^{5} - 438001 x^{4} - 10586538 x^{3} + 60329180 x^{2} - 91102296 x + 50324192 \)

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:  \(28643813255402702732772283462081=11^{4}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.48$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{4} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{7}{32} a^{5} - \frac{1}{8} a^{4} - \frac{11}{32} a^{3} + \frac{3}{8} a^{2} - \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{3}{64} a^{9} - \frac{1}{16} a^{8} + \frac{1}{16} a^{7} - \frac{5}{64} a^{6} - \frac{11}{64} a^{5} + \frac{9}{64} a^{4} + \frac{7}{64} a^{3} + \frac{1}{4} a^{2} - \frac{7}{16} a + \frac{1}{4}$, $\frac{1}{256} a^{13} - \frac{1}{128} a^{11} + \frac{3}{64} a^{10} + \frac{9}{256} a^{9} - \frac{1}{8} a^{8} - \frac{1}{256} a^{7} + \frac{1}{16} a^{6} + \frac{31}{128} a^{5} - \frac{1}{8} a^{4} - \frac{89}{256} a^{3} - \frac{27}{64} a^{2} + \frac{5}{64} a - \frac{7}{16}$, $\frac{1}{4352} a^{14} - \frac{1}{1088} a^{13} - \frac{9}{2176} a^{12} - \frac{11}{1088} a^{11} + \frac{9}{4352} a^{10} - \frac{41}{1088} a^{9} + \frac{15}{256} a^{8} + \frac{21}{1088} a^{7} - \frac{377}{2176} a^{6} + \frac{93}{544} a^{5} + \frac{951}{4352} a^{4} + \frac{107}{544} a^{3} + \frac{353}{1088} a^{2} - \frac{15}{68} a + \frac{15}{68}$, $\frac{1}{40923134491728353642745067655450585495775232} a^{15} - \frac{357593984246815276205437995507357108211}{40923134491728353642745067655450585495775232} a^{14} + \frac{5829112205606415645418055698673174626337}{5115391811466044205343133456931323186971904} a^{13} - \frac{28077458062811852694797673618281518502123}{20461567245864176821372533827725292747887616} a^{12} + \frac{565055968901264979357618839021799139878265}{40923134491728353642745067655450585495775232} a^{11} - \frac{378973938687930481734091195038574611537195}{40923134491728353642745067655450585495775232} a^{10} + \frac{137859811660792791749803655404197968125233}{40923134491728353642745067655450585495775232} a^{9} - \frac{2395376806401506728051872522360059836237725}{40923134491728353642745067655450585495775232} a^{8} - \frac{132120774908066400517787644458088761493383}{10230783622932088410686266913862646373943808} a^{7} + \frac{3806210000141456763104756274229285163890695}{20461567245864176821372533827725292747887616} a^{6} + \frac{4141943385920340108316410987248201684862955}{40923134491728353642745067655450585495775232} a^{5} - \frac{655206120562637161381231023091990750803961}{40923134491728353642745067655450585495775232} a^{4} - \frac{8817060879140581524124952436907488083359229}{20461567245864176821372533827725292747887616} a^{3} - \frac{1598087058837446748141243284766810733529897}{10230783622932088410686266913862646373943808} a^{2} - \frac{826159206771068750689655731362527208966769}{5115391811466044205343133456931323186971904} a + \frac{166536792494355444767606913287456639371}{4518897359952335870444464184568306702272}$

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

Class group and class number

$C_{226}$, which has order $226$ (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:  \( 1882188039.33 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 4.2.87131.1, 4.2.7754659.1, 8.0.44231334895529.1, 8.4.5351991522359009.1, 8.4.60134736206281.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$11.2.0.1$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}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$