↑

Properties

Label 16.0.45905397709...736.11
Degree $16$
Signature $[0, 8]$
Discriminant $2^{64}\cdot 3^{8}\cdot 41^{14}$
Root discriminant $714.28$
Ramified primes $2, 3, 41$
Class number $7678861312$ (GRH)
Class group $[2, 2, 2, 2, 8, 8, 7498888]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1353632190849, 0, 8655861520224, 0, 1163849279472, 0, 59754143904, 0, 1543733476, 0, 21800192, 0, 168264, 0, 656, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 656*x^14 + 168264*x^12 + 21800192*x^10 + 1543733476*x^8 + 59754143904*x^6 + 1163849279472*x^4 + 8655861520224*x^2 + 1353632190849)
 
gp: K = bnfinit(x^16 + 656*x^14 + 168264*x^12 + 21800192*x^10 + 1543733476*x^8 + 59754143904*x^6 + 1163849279472*x^4 + 8655861520224*x^2 + 1353632190849, 1)
 

Normalized defining polynomial

\( x^{16} + 656 x^{14} + 168264 x^{12} + 21800192 x^{10} + 1543733476 x^{8} + 59754143904 x^{6} + 1163849279472 x^{4} + 8655861520224 x^{2} + 1353632190849 \)

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:  \(4590539770923916035789355095649376383046516736=2^{64}\cdot 3^{8}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $714.28$
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:  $2, 3, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3936=2^{5}\cdot 3\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{3936}(1,·)$, $\chi_{3936}(1859,·)$, $\chi_{3936}(2053,·)$, $\chi_{3936}(647,·)$, $\chi_{3936}(73,·)$, $\chi_{3936}(3863,·)$, $\chi_{3936}(3289,·)$, $\chi_{3936}(1883,·)$, $\chi_{3936}(2077,·)$, $\chi_{3936}(3935,·)$, $\chi_{3936}(1643,·)$, $\chi_{3936}(301,·)$, $\chi_{3936}(2543,·)$, $\chi_{3936}(1393,·)$, $\chi_{3936}(3635,·)$, $\chi_{3936}(2293,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{4}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{1107} a^{8} - \frac{1}{27} a^{6} + \frac{2}{27} a^{4} - \frac{4}{9} a^{2}$, $\frac{1}{1107} a^{9} - \frac{1}{27} a^{7} + \frac{2}{27} a^{5} - \frac{4}{9} a^{3}$, $\frac{1}{96309} a^{10} + \frac{13}{32103} a^{8} + \frac{70}{783} a^{6} - \frac{293}{2349} a^{4} - \frac{152}{783} a^{2} - \frac{34}{87}$, $\frac{1}{288927} a^{11} + \frac{14}{32103} a^{9} + \frac{41}{2349} a^{7} - \frac{119}{7047} a^{5} + \frac{283}{2349} a^{3} - \frac{34}{261} a$, $\frac{1}{288927} a^{12} + \frac{43}{96309} a^{8} - \frac{740}{7047} a^{6} + \frac{61}{2349} a^{4} - \frac{9}{29} a^{2} + \frac{12}{29}$, $\frac{1}{303662277} a^{13} - \frac{323}{303662277} a^{11} - \frac{4214}{101220759} a^{9} - \frac{17501}{7406397} a^{7} - \frac{477116}{7406397} a^{5} + \frac{771511}{2468799} a^{3} - \frac{124978}{274311} a$, $\frac{1}{201643249075073153036841807} a^{14} - \frac{326646440289778748342}{201643249075073153036841807} a^{12} + \frac{52489833594963041684}{67214416358357717678947269} a^{10} + \frac{30226096830872907625583}{201643249075073153036841807} a^{8} + \frac{229285905608037907081174}{4918128026221296415532727} a^{6} + \frac{250824931202660414902373}{1639376008740432138510909} a^{4} - \frac{55089591989296665107585}{546458669580144046170303} a^{2} - \frac{24189792274992408937}{57771293961321920517}$, $\frac{1}{604929747225219459110525421} a^{15} + \frac{724225491045467921}{604929747225219459110525421} a^{13} + \frac{166040307932733713045}{201643249075073153036841807} a^{11} + \frac{150215082274953663369719}{604929747225219459110525421} a^{9} + \frac{808324264793059394473610}{14754384078663889246598181} a^{7} - \frac{53814767754188432568253}{1639376008740432138510909} a^{5} + \frac{333205219519842074874883}{1639376008740432138510909} a^{3} - \frac{30381842393766341108884}{182152889860048015390101} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{8}\times C_{8}\times C_{7498888}$, which has order $7678861312$ (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:  \( 69702662.454808 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

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

Intermediate fields

\(\Q(\sqrt{123}) \), \(\Q(\sqrt{82}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{6}, \sqrt{82})\), 4.4.141150208.1, 4.4.1270351872.2, 8.8.6455175514775617536.1, 8.0.418231618537560997888.74, 8.0.33876761101542440828928.4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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

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
2Data not computed
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
41Data not computed