Properties

Label 16.0.225...433.1
Degree $16$
Signature $[0, 8]$
Discriminant $2.259\times 10^{25}$
Root discriminant $38.43$
Ramified primes $17, 53$
Class number $162$ (GRH)
Class group $[9, 18]$ (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 - x^15 + 35*x^14 - 52*x^13 + 426*x^12 - 834*x^11 + 2908*x^10 - 5441*x^9 + 12734*x^8 - 20894*x^7 + 36823*x^6 - 49114*x^5 + 65638*x^4 - 66318*x^3 + 63428*x^2 - 39339*x + 19211)
 
gp: K = bnfinit(x^16 - x^15 + 35*x^14 - 52*x^13 + 426*x^12 - 834*x^11 + 2908*x^10 - 5441*x^9 + 12734*x^8 - 20894*x^7 + 36823*x^6 - 49114*x^5 + 65638*x^4 - 66318*x^3 + 63428*x^2 - 39339*x + 19211, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19211, -39339, 63428, -66318, 65638, -49114, 36823, -20894, 12734, -5441, 2908, -834, 426, -52, 35, -1, 1]);
 

\( x^{16} - x^{15} + 35 x^{14} - 52 x^{13} + 426 x^{12} - 834 x^{11} + 2908 x^{10} - 5441 x^{9} + 12734 x^{8} - 20894 x^{7} + 36823 x^{6} - 49114 x^{5} + 65638 x^{4} - 66318 x^{3} + 63428 x^{2} - 39339 x + 19211 \)

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:  \(22585894701900222828166433\)\(\medspace = 17^{15}\cdot 53^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $38.43$
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:  $17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{16013} a^{14} - \frac{2829}{16013} a^{13} - \frac{970}{16013} a^{12} + \frac{4752}{16013} a^{11} - \frac{1829}{16013} a^{10} + \frac{6191}{16013} a^{9} + \frac{3715}{16013} a^{8} - \frac{3625}{16013} a^{7} + \frac{3919}{16013} a^{6} - \frac{1729}{16013} a^{5} - \frac{5464}{16013} a^{4} + \frac{1202}{16013} a^{3} + \frac{6129}{16013} a^{2} - \frac{199}{16013} a - \frac{5898}{16013}$, $\frac{1}{690566194599920398863276991} a^{15} + \frac{14428737279049097286310}{690566194599920398863276991} a^{14} - \frac{2033326798308122436183773}{10306958128357020878556373} a^{13} - \frac{36113245427720514890159247}{690566194599920398863276991} a^{12} + \frac{110332151766398763375216972}{690566194599920398863276991} a^{11} + \frac{102236046910001773818919986}{690566194599920398863276991} a^{10} - \frac{41483784668995925869731019}{690566194599920398863276991} a^{9} - \frac{250213828182844428450639574}{690566194599920398863276991} a^{8} + \frac{233398456884159139127045696}{690566194599920398863276991} a^{7} - \frac{888340964055777080246548}{2889398303765357317419569} a^{6} + \frac{118208132180122613468225145}{690566194599920398863276991} a^{5} - \frac{57431778574801815406655822}{690566194599920398863276991} a^{4} + \frac{156953381015147112874735861}{690566194599920398863276991} a^{3} + \frac{42087680167378838161571195}{690566194599920398863276991} a^{2} - \frac{329097055046218809165784404}{690566194599920398863276991} a - \frac{317678849449758609706028972}{690566194599920398863276991}$

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

Class group and class number

$C_{9}\times C_{18}$, which has order $162$ (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:  \( 3640.01221338 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.01221338 \cdot 162}{2\sqrt{22585894701900222828166433}}\approx 0.150698231956$ (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

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
$53$53.8.4.2$x^{8} - 148877 x^{2} + 142028658$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
53.8.0.1$x^{8} + x^{2} - x + 33$$1$$8$$0$$C_8$$[\ ]^{8}$