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Properties

Label 16.0.18240099233...0813.4
Degree $16$
Signature $[0, 8]$
Discriminant $13^{5}\cdot 53^{12}$
Root discriminant $43.78$
Ramified primes $13, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1281

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1053, -3456, 1980, 5025, -6521, -1880, 6695, -1387, -3178, 1632, 814, -605, -102, 104, 2, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 2*x^14 + 104*x^13 - 102*x^12 - 605*x^11 + 814*x^10 + 1632*x^9 - 3178*x^8 - 1387*x^7 + 6695*x^6 - 1880*x^5 - 6521*x^4 + 5025*x^3 + 1980*x^2 - 3456*x + 1053)
 
gp: K = bnfinit(x^16 - 8*x^15 + 2*x^14 + 104*x^13 - 102*x^12 - 605*x^11 + 814*x^10 + 1632*x^9 - 3178*x^8 - 1387*x^7 + 6695*x^6 - 1880*x^5 - 6521*x^4 + 5025*x^3 + 1980*x^2 - 3456*x + 1053, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 2 x^{14} + 104 x^{13} - 102 x^{12} - 605 x^{11} + 814 x^{10} + 1632 x^{9} - 3178 x^{8} - 1387 x^{7} + 6695 x^{6} - 1880 x^{5} - 6521 x^{4} + 5025 x^{3} + 1980 x^{2} - 3456 x + 1053 \)

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:  \(182400992338192624135120813=13^{5}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.78$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{53} a^{12} + \frac{16}{53} a^{11} + \frac{11}{53} a^{10} + \frac{2}{53} a^{9} - \frac{25}{53} a^{8} + \frac{10}{53} a^{7} + \frac{21}{53} a^{6} + \frac{5}{53} a^{5} + \frac{3}{53} a^{4} - \frac{20}{53} a^{3} + \frac{2}{53} a^{2} - \frac{22}{53} a + \frac{24}{53}$, $\frac{1}{159} a^{13} + \frac{1}{159} a^{12} - \frac{70}{159} a^{11} - \frac{4}{159} a^{10} + \frac{17}{53} a^{9} + \frac{67}{159} a^{8} - \frac{23}{159} a^{7} - \frac{15}{53} a^{6} - \frac{19}{159} a^{5} + \frac{41}{159} a^{4} - \frac{16}{159} a^{3} + \frac{1}{159} a^{2} - \frac{17}{159} a - \frac{14}{53}$, $\frac{1}{954} a^{14} + \frac{1}{954} a^{13} + \frac{1}{477} a^{12} - \frac{283}{954} a^{11} - \frac{37}{318} a^{10} - \frac{425}{954} a^{9} - \frac{196}{477} a^{8} + \frac{13}{318} a^{7} - \frac{415}{954} a^{6} - \frac{235}{954} a^{5} + \frac{359}{954} a^{4} - \frac{4}{477} a^{3} - \frac{16}{477} a^{2} - \frac{65}{318} a + \frac{33}{106}$, $\frac{1}{425261510914675266} a^{15} - \frac{208452382989983}{425261510914675266} a^{14} - \frac{555417197766566}{212630755457337633} a^{13} + \frac{1562048295885563}{425261510914675266} a^{12} - \frac{55022912106750859}{141753836971558422} a^{11} + \frac{167135206925300227}{425261510914675266} a^{10} + \frac{2958702570852269}{16356211958256741} a^{9} - \frac{58704430026727097}{141753836971558422} a^{8} + \frac{141394547380528907}{425261510914675266} a^{7} + \frac{13816740842113715}{425261510914675266} a^{6} - \frac{51747267243682081}{425261510914675266} a^{5} - \frac{6948349252581460}{212630755457337633} a^{4} - \frac{2461919671744346}{12507691497490449} a^{3} + \frac{402156904221355}{10904141305504494} a^{2} + \frac{22343889689294263}{47251278990519474} a + \frac{155007206630958}{605785628083583}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1312913.5276 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1281:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1281
Character table for t16n1281 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 4.0.148877.1, 8.0.3745777030801.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{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}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$