Properties

Label 16.0.543...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5.438\times 10^{37}$
Root discriminant $228.28$
Ramified primes $5, 73$
Class number $39204$ (GRH)
Class group $[99, 396]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 59*x^14 - 355*x^13 + 2757*x^12 + 38575*x^11 + 76998*x^10 - 1276065*x^9 + 40270*x^8 + 15027260*x^7 + 51825752*x^6 - 512018190*x^5 + 1751927332*x^4 - 4935660000*x^3 + 10347212754*x^2 - 11733017535*x + 5267375811)
 
gp: K = bnfinit(x^16 - 5*x^15 - 59*x^14 - 355*x^13 + 2757*x^12 + 38575*x^11 + 76998*x^10 - 1276065*x^9 + 40270*x^8 + 15027260*x^7 + 51825752*x^6 - 512018190*x^5 + 1751927332*x^4 - 4935660000*x^3 + 10347212754*x^2 - 11733017535*x + 5267375811, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5267375811, -11733017535, 10347212754, -4935660000, 1751927332, -512018190, 51825752, 15027260, 40270, -1276065, 76998, 38575, 2757, -355, -59, -5, 1]);
 

\( x^{16} - 5 x^{15} - 59 x^{14} - 355 x^{13} + 2757 x^{12} + 38575 x^{11} + 76998 x^{10} - 1276065 x^{9} + 40270 x^{8} + 15027260 x^{7} + 51825752 x^{6} - 512018190 x^{5} + 1751927332 x^{4} - 4935660000 x^{3} + 10347212754 x^{2} - 11733017535 x + 5267375811 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(54377966460580450275755400738525390625\)\(\medspace = 5^{14}\cdot 73^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $228.28$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 73$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{3} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{3} a^{4} + \frac{1}{9} a^{3} + \frac{2}{9} a^{2}$, $\frac{1}{27} a^{11} + \frac{1}{27} a^{10} + \frac{1}{27} a^{9} + \frac{1}{27} a^{7} - \frac{1}{27} a^{6} - \frac{1}{9} a^{5} - \frac{8}{27} a^{4} - \frac{1}{3} a^{3} - \frac{1}{27} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{81} a^{12} + \frac{1}{81} a^{11} - \frac{2}{81} a^{10} + \frac{7}{81} a^{8} + \frac{2}{81} a^{7} + \frac{1}{81} a^{5} - \frac{1}{9} a^{4} - \frac{40}{81} a^{3} + \frac{10}{27} a^{2} + \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{729} a^{13} - \frac{1}{243} a^{12} + \frac{1}{243} a^{11} + \frac{17}{729} a^{10} - \frac{11}{729} a^{9} - \frac{53}{729} a^{8} - \frac{26}{729} a^{7} - \frac{116}{729} a^{6} + \frac{41}{729} a^{5} + \frac{32}{729} a^{4} + \frac{82}{729} a^{3} - \frac{103}{243} a^{2} + \frac{17}{81} a + \frac{7}{27}$, $\frac{1}{1113183} a^{14} - \frac{343}{1113183} a^{13} - \frac{568}{371061} a^{12} + \frac{10607}{1113183} a^{11} + \frac{54905}{1113183} a^{10} - \frac{6061}{371061} a^{9} - \frac{15161}{371061} a^{8} + \frac{40186}{371061} a^{7} - \frac{19892}{1113183} a^{6} - \frac{39646}{371061} a^{5} - \frac{526201}{1113183} a^{4} + \frac{85832}{1113183} a^{3} + \frac{52081}{371061} a^{2} - \frac{1115}{123687} a + \frac{28}{81}$, $\frac{1}{632559759308874602754566767941769434038938314648704417884971} a^{15} - \frac{236481567198962454740098391753813991270039630411288208}{632559759308874602754566767941769434038938314648704417884971} a^{14} - \frac{3149851982251597450155155972715459188505942144215561753}{9167532743606878300790822723793759913607801661575426346159} a^{13} - \frac{2286729354482377800633586164248511000071728695912163009804}{632559759308874602754566767941769434038938314648704417884971} a^{12} - \frac{10099407180601230182025649986526664570672171390088184296124}{632559759308874602754566767941769434038938314648704417884971} a^{11} + \frac{325494425914714084193399710132503274168626186597095037995}{70284417700986066972729640882418826004326479405411601987219} a^{10} - \frac{1278344786963888834133582826299587924626512006056175209965}{210853253102958200918188922647256478012979438216234805961657} a^{9} - \frac{7088883706057661843983040808882734393156650422816828539500}{210853253102958200918188922647256478012979438216234805961657} a^{8} - \frac{13041103176489697538662675400722412221876708897774082971766}{632559759308874602754566767941769434038938314648704417884971} a^{7} - \frac{5801709462036138362654241352579793261933088880168605255101}{210853253102958200918188922647256478012979438216234805961657} a^{6} + \frac{45688611813402123205364790963963003807011248118673835689332}{632559759308874602754566767941769434038938314648704417884971} a^{5} + \frac{3481523290672644715701098366017161731180404339885884861182}{632559759308874602754566767941769434038938314648704417884971} a^{4} + \frac{91372946242210131035338288448108842485650072311313484033682}{210853253102958200918188922647256478012979438216234805961657} a^{3} - \frac{13792424794994450880812838523962019444924697485260656793791}{70284417700986066972729640882418826004326479405411601987219} a^{2} - \frac{6474311303908502623385081190204762740993561009877246470485}{23428139233662022324243213627472942001442159801803867329073} a - \frac{572603908925068854433664181936694635051035400476587066}{5114197606125741611928228253104767954909879895613155933}$

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

Class group and class number

$C_{99}\times C_{396}$, which has order $39204$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1182290911.59 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 1182290911.59 \cdot 39204}{2\sqrt{54377966460580450275755400738525390625}}\approx 7.63400080824$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.9725425.1, 8.8.172615601860890625.1

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ R $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ $16$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
73Data not computed