↑

Properties

Label 16.0.531312517142544384.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 13^{6}$
Root discriminant $12.82$
Ramified primes $2, 3, 13$
Class number $1$
Class group Trivial
Galois group $C_2^2\wr C_2$ (as 16T46)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -72, 258, -510, 634, -642, 698, -720, 591, -426, 322, -228, 126, -54, 20, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 20*x^14 - 54*x^13 + 126*x^12 - 228*x^11 + 322*x^10 - 426*x^9 + 591*x^8 - 720*x^7 + 698*x^6 - 642*x^5 + 634*x^4 - 510*x^3 + 258*x^2 - 72*x + 9)
 
gp: K = bnfinit(x^16 - 6*x^15 + 20*x^14 - 54*x^13 + 126*x^12 - 228*x^11 + 322*x^10 - 426*x^9 + 591*x^8 - 720*x^7 + 698*x^6 - 642*x^5 + 634*x^4 - 510*x^3 + 258*x^2 - 72*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 20 x^{14} - 54 x^{13} + 126 x^{12} - 228 x^{11} + 322 x^{10} - 426 x^{9} + 591 x^{8} - 720 x^{7} + 698 x^{6} - 642 x^{5} + 634 x^{4} - 510 x^{3} + 258 x^{2} - 72 x + 9 \)

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:  \(531312517142544384=2^{24}\cdot 3^{8}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.82$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{1}{6} a^{2}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{10} - \frac{1}{6} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{21838914} a^{15} - \frac{206009}{7279638} a^{14} - \frac{72245}{949518} a^{13} + \frac{450299}{7279638} a^{12} + \frac{128069}{3639819} a^{11} - \frac{254059}{3639819} a^{10} - \frac{2144188}{10919457} a^{9} + \frac{552999}{2426546} a^{8} + \frac{388139}{3639819} a^{7} - \frac{634936}{3639819} a^{6} - \frac{4394396}{10919457} a^{5} + \frac{3574525}{7279638} a^{4} + \frac{9729673}{21838914} a^{3} - \frac{930805}{2426546} a^{2} - \frac{2963675}{7279638} a + \frac{311511}{1213273}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{17360959}{10919457} a^{15} + \frac{20457609}{2426546} a^{14} - \frac{24641123}{949518} a^{13} + \frac{164723255}{2426546} a^{12} - \frac{1115942683}{7279638} a^{11} + \frac{933187892}{3639819} a^{10} - \frac{3655498930}{10919457} a^{9} + \frac{3244813889}{7279638} a^{8} - \frac{1530480337}{2426546} a^{7} + \frac{2578306688}{3639819} a^{6} - \frac{6769219775}{10919457} a^{5} + \frac{4295053055}{7279638} a^{4} - \frac{13055213141}{21838914} a^{3} + \frac{2892637961}{7279638} a^{2} - \frac{973814915}{7279638} a + \frac{23292528}{1213273} \) (order $12$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1024.23596776 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\wr C_2$ (as 16T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 4.4.7488.1, 4.0.7488.1, 4.0.832.1, 4.0.7488.4, 4.0.117.1, \(\Q(\zeta_{12})\), 4.0.1872.1, 8.0.3504384.2, 8.0.56070144.2, 8.0.56070144.5

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 siblings: data not computed

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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

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$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$