Properties

Label 16.8.925...409.1
Degree $16$
Signature $[8, 4]$
Discriminant $9.253\times 10^{34}$
Root discriminant $153.25$
Ramified primes $3, 89$
Class number $1$ (GRH)
Class group trivial (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 - 6*x^15 - 61*x^14 + 129*x^13 + 1709*x^12 + 11091*x^11 - 31783*x^10 - 454524*x^9 + 25606*x^8 + 6052602*x^7 + 10610353*x^6 - 30220476*x^5 - 125101939*x^4 - 49470471*x^3 + 344285185*x^2 + 698844519*x + 585525649)
 
gp: K = bnfinit(x^16 - 6*x^15 - 61*x^14 + 129*x^13 + 1709*x^12 + 11091*x^11 - 31783*x^10 - 454524*x^9 + 25606*x^8 + 6052602*x^7 + 10610353*x^6 - 30220476*x^5 - 125101939*x^4 - 49470471*x^3 + 344285185*x^2 + 698844519*x + 585525649, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![585525649, 698844519, 344285185, -49470471, -125101939, -30220476, 10610353, 6052602, 25606, -454524, -31783, 11091, 1709, 129, -61, -6, 1]);
 

\( x^{16} - 6 x^{15} - 61 x^{14} + 129 x^{13} + 1709 x^{12} + 11091 x^{11} - 31783 x^{10} - 454524 x^{9} + 25606 x^{8} + 6052602 x^{7} + 10610353 x^{6} - 30220476 x^{5} - 125101939 x^{4} - 49470471 x^{3} + 344285185 x^{2} + 698844519 x + 585525649 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(92534813992455271438265413802888409\)\(\medspace = 3^{12}\cdot 89^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $153.25$
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:  $3, 89$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{44} a^{14} - \frac{1}{22} a^{13} + \frac{1}{11} a^{12} - \frac{1}{44} a^{11} - \frac{5}{44} a^{10} - \frac{1}{22} a^{9} - \frac{7}{22} a^{8} - \frac{2}{11} a^{7} - \frac{1}{2} a^{6} - \frac{3}{22} a^{5} - \frac{7}{44} a^{4} - \frac{1}{11} a^{3} - \frac{4}{11} a^{2} - \frac{21}{44} a + \frac{15}{44}$, $\frac{1}{14778648639472525184217036248228920345763033586959589745192} a^{15} - \frac{46193200481253395680838627182852436895388545852805128963}{14778648639472525184217036248228920345763033586959589745192} a^{14} + \frac{857202871310539449786536483203145125946444529152509685217}{7389324319736262592108518124114460172881516793479794872596} a^{13} + \frac{523698950667321257970688240108389407270740926610808062943}{14778648639472525184217036248228920345763033586959589745192} a^{12} + \frac{1668998219306720625550460996372801287796150954190474095401}{7389324319736262592108518124114460172881516793479794872596} a^{11} - \frac{3341080362749660067094982877480217238761364727245328527603}{14778648639472525184217036248228920345763033586959589745192} a^{10} + \frac{340497217787180601030590392932824606193405805783560061375}{1847331079934065648027129531028615043220379198369948718149} a^{9} + \frac{797387362515800090219546211819173990614079334125738920539}{1847331079934065648027129531028615043220379198369948718149} a^{8} + \frac{496635495844707960930237521841780716495046787280635488177}{7389324319736262592108518124114460172881516793479794872596} a^{7} - \frac{983109731268835980101679583858624587588762217722551268005}{3694662159868131296054259062057230086440758396739897436298} a^{6} + \frac{492145186850098164862971673731029155012554426929284243277}{14778648639472525184217036248228920345763033586959589745192} a^{5} - \frac{6432692643524417816144376724931998889809172767888115365545}{14778648639472525184217036248228920345763033586959589745192} a^{4} - \frac{1456953610177799229989833543372888311385931842112109228609}{7389324319736262592108518124114460172881516793479794872596} a^{3} + \frac{579686387763508900167651869954111330745746862857692992701}{1343513512679320471292457840748083667796639416996326340472} a^{2} + \frac{3685825934110806504499391922872634126248867745189578042057}{7389324319736262592108518124114460172881516793479794872596} a + \frac{4102725199396662539687852882598720333486928007971956183457}{14778648639472525184217036248228920345763033586959589745192}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 507241262351 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 507241262351 \cdot 1}{2\sqrt{92534813992455271438265413802888409}}\approx 0.332652996301041$ (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

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.8.3582738126537849.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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

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
3Data not computed
89Data not computed