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Properties

Label 16.0.88848937405...8553.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 73^{13}$
Root discriminant $74.44$
Ramified primes $3, 73$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7274677, 2209756, 1279138, 823765, -1969087, -1549502, -115337, 341342, 190159, 16035, -10904, 1245, 1726, 151, -41, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 41*x^14 + 151*x^13 + 1726*x^12 + 1245*x^11 - 10904*x^10 + 16035*x^9 + 190159*x^8 + 341342*x^7 - 115337*x^6 - 1549502*x^5 - 1969087*x^4 + 823765*x^3 + 1279138*x^2 + 2209756*x + 7274677)
 
gp: K = bnfinit(x^16 - 3*x^15 - 41*x^14 + 151*x^13 + 1726*x^12 + 1245*x^11 - 10904*x^10 + 16035*x^9 + 190159*x^8 + 341342*x^7 - 115337*x^6 - 1549502*x^5 - 1969087*x^4 + 823765*x^3 + 1279138*x^2 + 2209756*x + 7274677, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 41 x^{14} + 151 x^{13} + 1726 x^{12} + 1245 x^{11} - 10904 x^{10} + 16035 x^{9} + 190159 x^{8} + 341342 x^{7} - 115337 x^{6} - 1549502 x^{5} - 1969087 x^{4} + 823765 x^{3} + 1279138 x^{2} + 2209756 x + 7274677 \)

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:  \(888489374058862559333388518553=3^{12}\cdot 73^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.44$
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:  $3, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{73281642747697193787200847432243051336506705321572078} a^{15} - \frac{8344707612578263025474487186279356754683701635689669}{36640821373848596893600423716121525668253352660786039} a^{14} + \frac{7355255063879048808347520439901894389435755243661215}{73281642747697193787200847432243051336506705321572078} a^{13} + \frac{93329459495823157271598690853987753200633116660414}{2818524721065276684123109516624732743711796358522003} a^{12} - \frac{770585290893292640480295507084478531540692469303343}{5234403053406942413371489102303075095464764665826577} a^{11} + \frac{8200105235492320813422199126447219991866455927515426}{36640821373848596893600423716121525668253352660786039} a^{10} - \frac{2623759061608475740718470792613108017443313851594202}{36640821373848596893600423716121525668253352660786039} a^{9} - \frac{15781301210250611423615535047008808625360957100912453}{73281642747697193787200847432243051336506705321572078} a^{8} - \frac{2096300701918246686662100861236728425991100727031406}{36640821373848596893600423716121525668253352660786039} a^{7} + \frac{14235922608718514312730010736721893382971496235022371}{36640821373848596893600423716121525668253352660786039} a^{6} + \frac{312911290232862955323595715539372087371809575624842}{1181961979801567641729045926326500828008172666476969} a^{5} + \frac{3665511400210713434729983229412615095686367429598090}{36640821373848596893600423716121525668253352660786039} a^{4} + \frac{36429378032703579816353535476810297049902995888194793}{73281642747697193787200847432243051336506705321572078} a^{3} + \frac{2547591975585049882544609063648872103278707354219136}{5234403053406942413371489102303075095464764665826577} a^{2} + \frac{184023849442980761541121371680570690209167775322865}{402646388723610954874729930946390391958828051217429} a + \frac{487878075372787101762152146483177553841303269417675}{2363923959603135283458091852653001656016345332953938}$

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

Class group and class number

$C_{4}$, 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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{383193680935110737903130285}{226874586663317278495723029519958} a^{15} + \frac{2122539536700367523704458897}{226874586663317278495723029519958} a^{14} + \frac{10623218160707013721924415209}{226874586663317278495723029519958} a^{13} - \frac{43144861001028287079392297706}{113437293331658639247861514759979} a^{12} - \frac{225573719161968420261824902251}{113437293331658639247861514759979} a^{11} + \frac{726946916764553446613736639089}{226874586663317278495723029519958} a^{10} + \frac{2716878065348317920096276097169}{226874586663317278495723029519958} a^{9} - \frac{13232343007499669411633323870019}{226874586663317278495723029519958} a^{8} - \frac{20471090070574333253425978927344}{113437293331658639247861514759979} a^{7} - \frac{9062760119135104151154610160044}{113437293331658639247861514759979} a^{6} + \frac{127363770252001588496193268689475}{226874586663317278495723029519958} a^{5} + \frac{326764542020309378165986440876779}{226874586663317278495723029519958} a^{4} - \frac{83931809793270766264191556070777}{226874586663317278495723029519958} a^{3} - \frac{142504817766679494047003403392487}{113437293331658639247861514759979} a^{2} + \frac{100752231544348269579606696117207}{113437293331658639247861514759979} a - \frac{568076104555547188344362018135045}{113437293331658639247861514759979} \) (order $6$)
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:  \( 322311080.328 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.657.1, 8.0.167918799033.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ $16$ $16$ $16$

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
3Data not computed
$73$73.8.7.5$x^{8} + 365$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.6.4$x^{8} + 2336 x^{4} + 7092899$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$