Properties

Label 16.8.826...041.1
Degree $16$
Signature $[8, 4]$
Discriminant $8.264\times 10^{29}$
Root discriminant $74.10$
Ramified primes $3, 41$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 19*x^14 + 105*x^13 - 37*x^12 + 1620*x^11 - 3133*x^10 - 16161*x^9 + 16123*x^8 + 55743*x^7 + 14233*x^6 - 83412*x^5 - 113170*x^4 + 4836*x^3 + 39049*x^2 - 12594*x - 5129)
 
gp: K = bnfinit(x^16 - 6*x^15 - 19*x^14 + 105*x^13 - 37*x^12 + 1620*x^11 - 3133*x^10 - 16161*x^9 + 16123*x^8 + 55743*x^7 + 14233*x^6 - 83412*x^5 - 113170*x^4 + 4836*x^3 + 39049*x^2 - 12594*x - 5129, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5129, -12594, 39049, 4836, -113170, -83412, 14233, 55743, 16123, -16161, -3133, 1620, -37, 105, -19, -6, 1]);
 

\( x^{16} - 6 x^{15} - 19 x^{14} + 105 x^{13} - 37 x^{12} + 1620 x^{11} - 3133 x^{10} - 16161 x^{9} + 16123 x^{8} + 55743 x^{7} + 14233 x^{6} - 83412 x^{5} - 113170 x^{4} + 4836 x^{3} + 39049 x^{2} - 12594 x - 5129 \)

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:  \(826443003617417898900549004041\)\(\medspace = 3^{12}\cdot 41^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $74.10$
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:  $3, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1229624425791640105004594150517815905517} a^{15} + \frac{23465798972543656003256449793999519683}{1229624425791640105004594150517815905517} a^{14} - \frac{291331717717700033613282857861327614877}{1229624425791640105004594150517815905517} a^{13} - \frac{258804998749309737524477384078255380506}{1229624425791640105004594150517815905517} a^{12} - \frac{464216801883741823550199638799302291468}{1229624425791640105004594150517815905517} a^{11} + \frac{8391787859998347609757877002759664771}{53461931556158265434982354370339821979} a^{10} + \frac{241297399171699038908240186161340257738}{1229624425791640105004594150517815905517} a^{9} - \frac{350847097587027272629792972252066427187}{1229624425791640105004594150517815905517} a^{8} + \frac{20848020112836954946774656373558474954}{1229624425791640105004594150517815905517} a^{7} - \frac{321613825707977565953390623310388759339}{1229624425791640105004594150517815905517} a^{6} + \frac{365873439753561650066385323136409132567}{1229624425791640105004594150517815905517} a^{5} + \frac{16482693483633764371166989157356288514}{1229624425791640105004594150517815905517} a^{4} + \frac{588763646437183611068470615706841644292}{1229624425791640105004594150517815905517} a^{3} + \frac{192575227492347971449840262956404201884}{1229624425791640105004594150517815905517} a^{2} - \frac{76655874489217882981844134090441370754}{1229624425791640105004594150517815905517} a + \frac{6843193778382876999354992851942587411}{53461931556158265434982354370339821979}$

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:  \( 847278146.588 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 847278146.588 \cdot 1}{2\sqrt{826443003617417898900549004041}}\approx 0.185929796680$ (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

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.15775096184361.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
41Data not computed