↑

Properties

Label 16.0.20787641712...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{10}\cdot 11^{2}$
Root discriminant $24.82$
Ramified primes $2, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T799

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![199, -348, 840, -396, 4220, -1900, 4100, -3596, 2700, -868, 72, 124, -36, -4, 12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 12*x^14 - 4*x^13 - 36*x^12 + 124*x^11 + 72*x^10 - 868*x^9 + 2700*x^8 - 3596*x^7 + 4100*x^6 - 1900*x^5 + 4220*x^4 - 396*x^3 + 840*x^2 - 348*x + 199)
 
gp: K = bnfinit(x^16 - 4*x^15 + 12*x^14 - 4*x^13 - 36*x^12 + 124*x^11 + 72*x^10 - 868*x^9 + 2700*x^8 - 3596*x^7 + 4100*x^6 - 1900*x^5 + 4220*x^4 - 396*x^3 + 840*x^2 - 348*x + 199, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 12 x^{14} - 4 x^{13} - 36 x^{12} + 124 x^{11} + 72 x^{10} - 868 x^{9} + 2700 x^{8} - 3596 x^{7} + 4100 x^{6} - 1900 x^{5} + 4220 x^{4} - 396 x^{3} + 840 x^{2} - 348 x + 199 \)

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:  \(20787641712640000000000=2^{44}\cdot 5^{10}\cdot 11^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.82$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{190} a^{12} - \frac{14}{95} a^{11} - \frac{1}{95} a^{10} - \frac{27}{190} a^{9} - \frac{9}{38} a^{8} + \frac{1}{95} a^{7} + \frac{32}{95} a^{6} - \frac{32}{95} a^{5} + \frac{9}{190} a^{4} + \frac{42}{95} a^{3} - \frac{39}{95} a^{2} + \frac{41}{190} a + \frac{29}{190}$, $\frac{1}{190} a^{13} - \frac{13}{95} a^{11} + \frac{6}{95} a^{10} - \frac{41}{190} a^{9} - \frac{23}{190} a^{8} - \frac{7}{19} a^{7} + \frac{9}{95} a^{6} - \frac{73}{190} a^{5} - \frac{22}{95} a^{4} - \frac{3}{95} a^{3} + \frac{21}{95} a^{2} + \frac{37}{190} a - \frac{43}{190}$, $\frac{1}{2470} a^{14} + \frac{2}{1235} a^{13} - \frac{1}{494} a^{12} - \frac{11}{247} a^{11} - \frac{32}{247} a^{10} - \frac{187}{1235} a^{9} + \frac{33}{2470} a^{8} - \frac{79}{247} a^{7} - \frac{557}{2470} a^{6} + \frac{3}{247} a^{5} - \frac{373}{2470} a^{4} + \frac{606}{1235} a^{3} - \frac{4}{1235} a^{2} + \frac{483}{1235} a + \frac{33}{130}$, $\frac{1}{7561471754289932050} a^{15} + \frac{196926486098159}{1512294350857986410} a^{14} + \frac{17616574406975607}{7561471754289932050} a^{13} - \frac{45383124711196}{290825836703458925} a^{12} - \frac{14140868430564237}{756147175428993205} a^{11} - \frac{384566844007770198}{3780735877144966025} a^{10} + \frac{1288083460254113113}{7561471754289932050} a^{9} + \frac{188639801609465289}{7561471754289932050} a^{8} - \frac{1015486154429818109}{7561471754289932050} a^{7} + \frac{3467534864827089003}{7561471754289932050} a^{6} - \frac{3533068850449133313}{7561471754289932050} a^{5} + \frac{341492657841079709}{3780735877144966025} a^{4} - \frac{678061837733120904}{3780735877144966025} a^{3} + \frac{1889659129339575996}{3780735877144966025} a^{2} - \frac{456224134034972297}{7561471754289932050} a - \frac{1103424889854141531}{7561471754289932050}$

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:  \( -1 \) (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:  \( 27627.5194283 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T799:

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 t16n799 are not computed
Character table for t16n799 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.2.400.1, 4.2.1600.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.40960000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.2.0.1$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$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$