↑

Properties

Label 16.0.40144896000...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{4}\cdot 5^{12}\cdot 11^{2}$
Root discriminant $16.80$
Ramified primes $2, 3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T781

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121, -776, 1758, -1226, -1258, 2530, -1188, -428, 855, -438, 52, 40, 2, -26, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 26*x^13 + 2*x^12 + 40*x^11 + 52*x^10 - 438*x^9 + 855*x^8 - 428*x^7 - 1188*x^6 + 2530*x^5 - 1258*x^4 - 1226*x^3 + 1758*x^2 - 776*x + 121)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 - 26*x^13 + 2*x^12 + 40*x^11 + 52*x^10 - 438*x^9 + 855*x^8 - 428*x^7 - 1188*x^6 + 2530*x^5 - 1258*x^4 - 1226*x^3 + 1758*x^2 - 776*x + 121, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 18 x^{14} - 26 x^{13} + 2 x^{12} + 40 x^{11} + 52 x^{10} - 438 x^{9} + 855 x^{8} - 428 x^{7} - 1188 x^{6} + 2530 x^{5} - 1258 x^{4} - 1226 x^{3} + 1758 x^{2} - 776 x + 121 \)

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:  \(40144896000000000000=2^{24}\cdot 3^{4}\cdot 5^{12}\cdot 11^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.80$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{22} a^{12} - \frac{5}{22} a^{11} - \frac{1}{11} a^{10} - \frac{5}{22} a^{9} - \frac{3}{22} a^{8} + \frac{5}{22} a^{7} - \frac{2}{11} a^{6} - \frac{1}{2} a^{5} + \frac{7}{22} a^{4} + \frac{9}{22} a^{3} + \frac{4}{11} a^{2} - \frac{7}{22} a$, $\frac{1}{22} a^{13} - \frac{5}{22} a^{11} - \frac{2}{11} a^{10} + \frac{5}{22} a^{9} + \frac{1}{22} a^{8} - \frac{1}{22} a^{7} + \frac{1}{11} a^{6} + \frac{7}{22} a^{5} - \frac{1}{2} a^{4} + \frac{9}{22} a^{3} - \frac{1}{11} a - \frac{1}{2}$, $\frac{1}{242} a^{14} - \frac{1}{121} a^{13} - \frac{5}{242} a^{12} - \frac{30}{121} a^{11} + \frac{12}{121} a^{10} - \frac{21}{121} a^{9} + \frac{26}{121} a^{8} + \frac{46}{121} a^{7} + \frac{40}{121} a^{6} + \frac{48}{121} a^{5} + \frac{32}{121} a^{4} + \frac{35}{121} a^{3} + \frac{97}{242} a^{2} + \frac{2}{121} a - \frac{1}{2}$, $\frac{1}{2088407362822} a^{15} + \frac{305480767}{1044203681411} a^{14} + \frac{20790809891}{1044203681411} a^{13} + \frac{5621664441}{1044203681411} a^{12} - \frac{89752497334}{1044203681411} a^{11} + \frac{477721326575}{2088407362822} a^{10} + \frac{186840773781}{1044203681411} a^{9} - \frac{335184435763}{2088407362822} a^{8} - \frac{194074904702}{1044203681411} a^{7} + \frac{521912849889}{2088407362822} a^{6} - \frac{462516246124}{1044203681411} a^{5} + \frac{580611229019}{2088407362822} a^{4} + \frac{138247449913}{2088407362822} a^{3} - \frac{267081978859}{2088407362822} a^{2} + \frac{312033089353}{1044203681411} a + \frac{6305168391}{17259564982}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( \frac{7035381750811}{2088407362822} a^{15} - \frac{38505867612955}{2088407362822} a^{14} + \frac{106243905583479}{2088407362822} a^{13} - \frac{63215424857782}{1044203681411} a^{12} - \frac{26908585459021}{1044203681411} a^{11} + \frac{127070977825081}{1044203681411} a^{10} + \frac{501305295559019}{2088407362822} a^{9} - \frac{1410177711055551}{1044203681411} a^{8} + \frac{2259972466804898}{1044203681411} a^{7} - \frac{296545658573135}{1044203681411} a^{6} - \frac{8715202380650441}{2088407362822} a^{5} + \frac{6593370356079656}{1044203681411} a^{4} - \frac{1781635967897361}{2088407362822} a^{3} - \frac{9688798805620505}{2088407362822} a^{2} + \frac{3604890603878855}{1044203681411} a - \frac{6382576304202}{8629782491} \) (order $10$)
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:  \( 8357.0551607 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n781 are not computed
Character table for t16n781 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.2.400.1, 4.2.2000.1, 8.0.4000000.2

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$