↑

Properties

Label 16.8.21649597986...2993.1
Degree $16$
Signature $[8, 4]$
Discriminant $43^{4}\cdot 97^{15}$
Root discriminant $186.62$
Ramified primes $43, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9179159021, -7839808132, 3073249780, 4658216687, 1368635976, 128700291, 27735607, 8606523, 18509, 77955, 124718, 10546, -2186, -130, -41, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 41*x^14 - 130*x^13 - 2186*x^12 + 10546*x^11 + 124718*x^10 + 77955*x^9 + 18509*x^8 + 8606523*x^7 + 27735607*x^6 + 128700291*x^5 + 1368635976*x^4 + 4658216687*x^3 + 3073249780*x^2 - 7839808132*x - 9179159021)
 
gp: K = bnfinit(x^16 - 4*x^15 - 41*x^14 - 130*x^13 - 2186*x^12 + 10546*x^11 + 124718*x^10 + 77955*x^9 + 18509*x^8 + 8606523*x^7 + 27735607*x^6 + 128700291*x^5 + 1368635976*x^4 + 4658216687*x^3 + 3073249780*x^2 - 7839808132*x - 9179159021, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 41 x^{14} - 130 x^{13} - 2186 x^{12} + 10546 x^{11} + 124718 x^{10} + 77955 x^{9} + 18509 x^{8} + 8606523 x^{7} + 27735607 x^{6} + 128700291 x^{5} + 1368635976 x^{4} + 4658216687 x^{3} + 3073249780 x^{2} - 7839808132 x - 9179159021 \)

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:  \(2164959798672044689794137876838502993=43^{4}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $186.62$
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:  $43, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{138520095788236766904053035136503347668132915919549944608392024850657675529} a^{15} + \frac{142378976674822602656982179666624406763041611357526408140517415176557352}{451205523740184908482257443441378982632354774982247376574566856191067347} a^{14} - \frac{49464950896409874948693642952696888960340096361954707978352971950405632562}{138520095788236766904053035136503347668132915919549944608392024850657675529} a^{13} - \frac{32161586576284359191341273980644564827716128945019236571305971937303731107}{138520095788236766904053035136503347668132915919549944608392024850657675529} a^{12} - \frac{4334176482240782450825790218755874669091556185783803038470539634153081566}{138520095788236766904053035136503347668132915919549944608392024850657675529} a^{11} + \frac{28839444248161476323251376660988507176926112608647589162707946539490486018}{138520095788236766904053035136503347668132915919549944608392024850657675529} a^{10} + \frac{48545343133289333616136011108251180477115952598622513882823019265384877339}{138520095788236766904053035136503347668132915919549944608392024850657675529} a^{9} + \frac{23633010135914881206785092892241639581396634431519610191172197238643223608}{138520095788236766904053035136503347668132915919549944608392024850657675529} a^{8} + \frac{50245186852584020849600326354629731819145922479753254581657773958828643658}{138520095788236766904053035136503347668132915919549944608392024850657675529} a^{7} + \frac{2156816586269271239411284660650117787005137377425260170990953033654054832}{138520095788236766904053035136503347668132915919549944608392024850657675529} a^{6} + \frac{38058932968855424362890105116273437173283619307073114385944406014567063842}{138520095788236766904053035136503347668132915919549944608392024850657675529} a^{5} + \frac{33237516542026959226688047598901225835890260381027623087011053460501931506}{138520095788236766904053035136503347668132915919549944608392024850657675529} a^{4} - \frac{3434341002421315282574553576111250351524336747745318236228959196558347807}{138520095788236766904053035136503347668132915919549944608392024850657675529} a^{3} - \frac{9748646682109366436231655832758471145003276987988133767049614369114649426}{138520095788236766904053035136503347668132915919549944608392024850657675529} a^{2} + \frac{62832286559522590428612228003302950760426429341596447113012725444824272229}{138520095788236766904053035136503347668132915919549944608392024850657675529} a + \frac{2616804234323650565191330671596346702107980742195330882437837321485908329}{138520095788236766904053035136503347668132915919549944608392024850657675529}$

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:  \( 272048212299 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
$43$43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
97Data not computed