↑

Properties

Label 16.0.29548117155...4256.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{52}\cdot 3^{8}$
Root discriminant $16.48$
Ramified primes $2, 3$
Class number $2$
Class group $[2]$
Galois group $(C_2^2\times C_4):C_2$ (as 16T54)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![34, 112, 168, 32, 24, 40, 232, -216, 386, -344, 308, -152, 84, -24, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 12*x^14 - 24*x^13 + 84*x^12 - 152*x^11 + 308*x^10 - 344*x^9 + 386*x^8 - 216*x^7 + 232*x^6 + 40*x^5 + 24*x^4 + 32*x^3 + 168*x^2 + 112*x + 34)
 
gp: K = bnfinit(x^16 + 12*x^14 - 24*x^13 + 84*x^12 - 152*x^11 + 308*x^10 - 344*x^9 + 386*x^8 - 216*x^7 + 232*x^6 + 40*x^5 + 24*x^4 + 32*x^3 + 168*x^2 + 112*x + 34, 1)
 

Normalized defining polynomial

\( x^{16} + 12 x^{14} - 24 x^{13} + 84 x^{12} - 152 x^{11} + 308 x^{10} - 344 x^{9} + 386 x^{8} - 216 x^{7} + 232 x^{6} + 40 x^{5} + 24 x^{4} + 32 x^{3} + 168 x^{2} + 112 x + 34 \)

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:  \(29548117155177824256=2^{52}\cdot 3^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.48$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{323170275062549395633} a^{15} - \frac{71548258336619893971}{323170275062549395633} a^{14} - \frac{71675527049003191289}{323170275062549395633} a^{13} - \frac{110177084914883370707}{323170275062549395633} a^{12} + \frac{96745760782700945790}{323170275062549395633} a^{11} - \frac{118827697108891590142}{323170275062549395633} a^{10} + \frac{120557781887279585933}{323170275062549395633} a^{9} + \frac{20952036489840373596}{323170275062549395633} a^{8} + \frac{108130872564268776745}{323170275062549395633} a^{7} - \frac{9404387513894359129}{19010016180149964449} a^{6} + \frac{41176294121792763339}{323170275062549395633} a^{5} + \frac{30137711837688941591}{323170275062549395633} a^{4} - \frac{13092710215073275636}{323170275062549395633} a^{3} + \frac{69336171337597643904}{323170275062549395633} a^{2} + \frac{145600609683554433807}{323170275062549395633} a + \frac{7181061747310054431}{19010016180149964449}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{15928045468531106}{560087131824175729} a^{15} + \frac{4531148122189282}{560087131824175729} a^{14} + \frac{187147024370474498}{560087131824175729} a^{13} - \frac{319332162314409247}{560087131824175729} a^{12} + \frac{1180664922544109016}{560087131824175729} a^{11} - \frac{1847103759096823147}{560087131824175729} a^{10} + \frac{3659473510665487888}{560087131824175729} a^{9} - \frac{2811544734275069385}{560087131824175729} a^{8} + \frac{2017616026463779182}{560087131824175729} a^{7} + \frac{124276870909111328}{32946301872010337} a^{6} - \frac{1769064121228298574}{560087131824175729} a^{5} + \frac{5364167736391768695}{560087131824175729} a^{4} - \frac{2467956549639797004}{560087131824175729} a^{3} + \frac{1885656905433907158}{560087131824175729} a^{2} + \frac{2532501746555408516}{560087131824175729} a + \frac{99165483487280513}{32946301872010337} \) (order $8$)
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:  \( 4666.08220865 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4):C_2$ (as 16T54):

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 $(C_2^2\times C_4):C_2$
Character table for $(C_2^2\times C_4):C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.0.18432.1 x2, \(\Q(\zeta_{8})\), 4.2.18432.3 x2, 8.0.1358954496.9, 8.0.150994944.1, 8.0.9437184.1

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 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$