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Properties

Label 16.0.13665147289...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 19^{4}$
Root discriminant $27.92$
Ramified primes $2, 5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3$ (as 16T203)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2201, -3272, 11388, -13424, 21047, -18172, 17748, -11072, 8090, -3352, 2076, -464, 322, -24, 28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 28*x^14 - 24*x^13 + 322*x^12 - 464*x^11 + 2076*x^10 - 3352*x^9 + 8090*x^8 - 11072*x^7 + 17748*x^6 - 18172*x^5 + 21047*x^4 - 13424*x^3 + 11388*x^2 - 3272*x + 2201)
 
gp: K = bnfinit(x^16 + 28*x^14 - 24*x^13 + 322*x^12 - 464*x^11 + 2076*x^10 - 3352*x^9 + 8090*x^8 - 11072*x^7 + 17748*x^6 - 18172*x^5 + 21047*x^4 - 13424*x^3 + 11388*x^2 - 3272*x + 2201, 1)
 

Normalized defining polynomial

\( x^{16} + 28 x^{14} - 24 x^{13} + 322 x^{12} - 464 x^{11} + 2076 x^{10} - 3352 x^{9} + 8090 x^{8} - 11072 x^{7} + 17748 x^{6} - 18172 x^{5} + 21047 x^{4} - 13424 x^{3} + 11388 x^{2} - 3272 x + 2201 \)

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:  \(136651472896000000000000=2^{32}\cdot 5^{12}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.92$
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:  $2, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} + \frac{5}{11} a^{12} + \frac{4}{11} a^{11} + \frac{2}{11} a^{10} + \frac{5}{11} a^{9} + \frac{5}{11} a^{8} + \frac{4}{11} a^{7} - \frac{1}{11} a^{6} + \frac{3}{11} a^{5} - \frac{5}{11} a^{4} - \frac{3}{11} a^{2} - \frac{1}{11} a - \frac{5}{11}$, $\frac{1}{341} a^{14} - \frac{1}{341} a^{13} + \frac{84}{341} a^{12} - \frac{12}{31} a^{11} + \frac{114}{341} a^{10} + \frac{129}{341} a^{9} + \frac{95}{341} a^{8} + \frac{52}{341} a^{7} + \frac{119}{341} a^{6} - \frac{23}{341} a^{5} + \frac{74}{341} a^{4} + \frac{19}{341} a^{3} + \frac{50}{341} a^{2} - \frac{153}{341} a - \frac{4}{11}$, $\frac{1}{1386216628434009331759921} a^{15} + \frac{55193057750886206590}{44716665433355139734191} a^{14} - \frac{4537952551083224660157}{1386216628434009331759921} a^{13} - \frac{292211647446472044784002}{1386216628434009331759921} a^{12} - \frac{412679157761195626071160}{1386216628434009331759921} a^{11} + \frac{76215745666063661025895}{1386216628434009331759921} a^{10} + \frac{210691501761909244198744}{1386216628434009331759921} a^{9} - \frac{233154876189462028718319}{1386216628434009331759921} a^{8} + \frac{544076508763673309549589}{1386216628434009331759921} a^{7} - \frac{261155725148765339780483}{1386216628434009331759921} a^{6} - \frac{185050001823487474898608}{1386216628434009331759921} a^{5} + \frac{15385022034494561190312}{44716665433355139734191} a^{4} + \frac{243732964454707061457285}{1386216628434009331759921} a^{3} + \frac{195169596288074814306726}{1386216628434009331759921} a^{2} + \frac{366533073587438436995739}{1386216628434009331759921} a - \frac{16264630325301540444732}{44716665433355139734191}$

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:  \( -\frac{104946920180765418720}{126019693494000848341811} a^{15} - \frac{35916813025010465556}{126019693494000848341811} a^{14} - \frac{2824288593373935904252}{126019693494000848341811} a^{13} + \frac{1535582371084391883205}{126019693494000848341811} a^{12} - \frac{29903164134058703223136}{126019693494000848341811} a^{11} + \frac{35056465625682938996120}{126019693494000848341811} a^{10} - \frac{169341482662055023179584}{126019693494000848341811} a^{9} + \frac{236313122848646189364357}{126019693494000848341811} a^{8} - \frac{547053725427163451177920}{126019693494000848341811} a^{7} + \frac{611209865805135064641968}{126019693494000848341811} a^{6} - \frac{869784601331702981489832}{126019693494000848341811} a^{5} + \frac{625405454500323757186088}{126019693494000848341811} a^{4} - \frac{53651517354208138961200}{11456335772181895303801} a^{3} - \frac{28505017861626358917948}{126019693494000848341811} a^{2} + \frac{39744327627145959830104}{126019693494000848341811} a - \frac{4352820653989564348342}{4065151403032285430381} \) (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:  \( 402011.397978 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T203):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 41 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.8000.2, \(\Q(\zeta_{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.64000000.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
$19$19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$