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Properties

Label 16.0.44332440382...5761.5
Degree $16$
Signature $[0, 8]$
Discriminant $41^{14}\cdot 43^{8}$
Root discriminant $169.01$
Ramified primes $41, 43$
Class number $1740$ (GRH)
Class group $[1740]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10143812191, 1158532737, 5798763048, -80045271, 995637614, -141573402, 108110739, -18480176, 5788342, -772973, 151484, -3932, 493, 170, -30, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 30*x^14 + 170*x^13 + 493*x^12 - 3932*x^11 + 151484*x^10 - 772973*x^9 + 5788342*x^8 - 18480176*x^7 + 108110739*x^6 - 141573402*x^5 + 995637614*x^4 - 80045271*x^3 + 5798763048*x^2 + 1158532737*x + 10143812191)
 
gp: K = bnfinit(x^16 - 2*x^15 - 30*x^14 + 170*x^13 + 493*x^12 - 3932*x^11 + 151484*x^10 - 772973*x^9 + 5788342*x^8 - 18480176*x^7 + 108110739*x^6 - 141573402*x^5 + 995637614*x^4 - 80045271*x^3 + 5798763048*x^2 + 1158532737*x + 10143812191, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 30 x^{14} + 170 x^{13} + 493 x^{12} - 3932 x^{11} + 151484 x^{10} - 772973 x^{9} + 5788342 x^{8} - 18480176 x^{7} + 108110739 x^{6} - 141573402 x^{5} + 995637614 x^{4} - 80045271 x^{3} + 5798763048 x^{2} + 1158532737 x + 10143812191 \)

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:  \(443324403828803432927172239238695761=41^{14}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $169.01$
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:  $41, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{86} a^{12} + \frac{6}{43} a^{11} - \frac{16}{43} a^{10} - \frac{9}{86} a^{9} + \frac{1}{43} a^{8} - \frac{17}{86} a^{6} + \frac{2}{43} a^{5} + \frac{8}{43} a^{4} - \frac{1}{2} a^{3} + \frac{2}{43} a^{2} + \frac{6}{43} a + \frac{11}{86}$, $\frac{1}{44978} a^{13} + \frac{63}{22489} a^{12} - \frac{1310}{22489} a^{11} + \frac{10963}{44978} a^{10} + \frac{10797}{22489} a^{9} + \frac{9789}{22489} a^{8} + \frac{14001}{44978} a^{7} - \frac{5310}{22489} a^{6} + \frac{9094}{22489} a^{5} - \frac{6991}{44978} a^{4} - \frac{1417}{22489} a^{3} + \frac{5523}{22489} a^{2} + \frac{7399}{44978} a - \frac{7285}{22489}$, $\frac{1}{3449822180314} a^{14} + \frac{16107749}{3449822180314} a^{13} - \frac{15907249599}{3449822180314} a^{12} - \frac{1095300131941}{3449822180314} a^{11} - \frac{164305768909}{3449822180314} a^{10} + \frac{323551747699}{3449822180314} a^{9} + \frac{1250592645}{3449822180314} a^{8} + \frac{7819450951}{80228422798} a^{7} - \frac{920987759005}{3449822180314} a^{6} - \frac{518909577537}{3449822180314} a^{5} + \frac{823282315451}{3449822180314} a^{4} - \frac{1691087599911}{3449822180314} a^{3} + \frac{1081932876895}{3449822180314} a^{2} - \frac{1610518563287}{3449822180314} a + \frac{1164487442353}{3449822180314}$, $\frac{1}{347723734899841445271611949336494965113597898802189864758} a^{15} + \frac{10403308096187849797829837901063058174681727}{173861867449920722635805974668247482556798949401094932379} a^{14} + \frac{277385426675508538826167742962217821887271011677757}{173861867449920722635805974668247482556798949401094932379} a^{13} + \frac{249058526042729571638465646531140438014917516701447683}{347723734899841445271611949336494965113597898802189864758} a^{12} - \frac{77544283841554456466429578340718095196780772494491893966}{173861867449920722635805974668247482556798949401094932379} a^{11} - \frac{71213633240793346211433311667067672256490064170153929776}{173861867449920722635805974668247482556798949401094932379} a^{10} + \frac{2864978151685638154328200695612100395787593603803850555}{347723734899841445271611949336494965113597898802189864758} a^{9} - \frac{55562347040011813706159599133568172303780610282756293659}{173861867449920722635805974668247482556798949401094932379} a^{8} - \frac{62704784094997591172162893334529186621964779366158586659}{173861867449920722635805974668247482556798949401094932379} a^{7} + \frac{54590381827392871161797268793741965913492694999040677463}{347723734899841445271611949336494965113597898802189864758} a^{6} + \frac{6020290994429228053265785622202867501654241358474631693}{173861867449920722635805974668247482556798949401094932379} a^{5} + \frac{21219397297595324677138712219373709229708734384891750393}{173861867449920722635805974668247482556798949401094932379} a^{4} - \frac{89473989335325357640406256635544193468863271549999885289}{347723734899841445271611949336494965113597898802189864758} a^{3} - \frac{34214248726801064576045377302878896523976999934895994531}{173861867449920722635805974668247482556798949401094932379} a^{2} - \frac{46487173321871924292735073101731382397190395597687347828}{173861867449920722635805974668247482556798949401094932379} a + \frac{82106772022822726182817795062896733636001404744425174141}{173861867449920722635805974668247482556798949401094932379}$

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

Class group and class number

$C_{1740}$, which has order $1740$ (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:  \( 70797623.7718 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-43}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-1763}) \), \(\Q(\sqrt{41}, \sqrt{-43})\), 4.4.68921.1, 4.0.127434929.2, 8.0.16239661129235041.1, 8.4.360100652405969.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$41$41.8.7.1$x^{8} - 41$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.1$x^{8} - 41$$8$$1$$7$$C_8$$[\ ]_{8}$
$43$43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$