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Properties

Label 16.0.13665147289...0000.2
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![1271, 4772, 11688, 20344, 25512, 22092, 11808, 2192, -1715, -1268, -124, 184, 62, -16, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 - 16*x^13 + 62*x^12 + 184*x^11 - 124*x^10 - 1268*x^9 - 1715*x^8 + 2192*x^7 + 11808*x^6 + 22092*x^5 + 25512*x^4 + 20344*x^3 + 11688*x^2 + 4772*x + 1271)
 
gp: K = bnfinit(x^16 - 12*x^14 - 16*x^13 + 62*x^12 + 184*x^11 - 124*x^10 - 1268*x^9 - 1715*x^8 + 2192*x^7 + 11808*x^6 + 22092*x^5 + 25512*x^4 + 20344*x^3 + 11688*x^2 + 4772*x + 1271, 1)
 

Normalized defining polynomial

\( x^{16} - 12 x^{14} - 16 x^{13} + 62 x^{12} + 184 x^{11} - 124 x^{10} - 1268 x^{9} - 1715 x^{8} + 2192 x^{7} + 11808 x^{6} + 22092 x^{5} + 25512 x^{4} + 20344 x^{3} + 11688 x^{2} + 4772 x + 1271 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{38} a^{12} + \frac{4}{19} a^{10} + \frac{3}{38} a^{9} + \frac{4}{19} a^{8} + \frac{8}{19} a^{7} - \frac{2}{19} a^{6} - \frac{13}{38} a^{5} + \frac{4}{19} a^{4} - \frac{5}{19} a^{3} + \frac{9}{19} a^{2} - \frac{7}{38} a + \frac{11}{38}$, $\frac{1}{38} a^{13} + \frac{4}{19} a^{11} + \frac{3}{38} a^{10} + \frac{4}{19} a^{9} - \frac{3}{38} a^{8} - \frac{2}{19} a^{7} - \frac{13}{38} a^{6} + \frac{4}{19} a^{5} + \frac{9}{38} a^{4} + \frac{9}{19} a^{3} - \frac{7}{38} a^{2} + \frac{11}{38} a - \frac{1}{2}$, $\frac{1}{38} a^{14} + \frac{3}{38} a^{11} + \frac{1}{38} a^{10} - \frac{4}{19} a^{9} + \frac{4}{19} a^{8} + \frac{11}{38} a^{7} - \frac{17}{38} a^{6} + \frac{9}{19} a^{5} - \frac{4}{19} a^{4} - \frac{3}{38} a^{3} + \frac{9}{19} a - \frac{6}{19}$, $\frac{1}{4016855779118063398} a^{15} + \frac{546665339375795}{211413462058845442} a^{14} - \frac{46001184495275821}{4016855779118063398} a^{13} - \frac{32160299411684263}{4016855779118063398} a^{12} + \frac{226744350034867147}{4016855779118063398} a^{11} - \frac{368997199644722587}{2008427889559031699} a^{10} - \frac{852047447127689009}{4016855779118063398} a^{9} + \frac{50791118317610760}{2008427889559031699} a^{8} - \frac{115656562325310395}{365168707192551218} a^{7} - \frac{328623564089398912}{2008427889559031699} a^{6} - \frac{712568532213642315}{4016855779118063398} a^{5} + \frac{401366787854933524}{2008427889559031699} a^{4} + \frac{280687100150871945}{2008427889559031699} a^{3} - \frac{1370784352124810137}{4016855779118063398} a^{2} + \frac{49991942933281097}{105706731029422721} a - \frac{770089270104203983}{4016855779118063398}$

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:  \( -\frac{63467130865228}{16598577599661419} a^{15} + \frac{104811197031577}{16598577599661419} a^{14} + \frac{666985542003363}{16598577599661419} a^{13} - \frac{397318456410717}{33197155199322838} a^{12} - \frac{4507203073922996}{16598577599661419} a^{11} - \frac{4030088176998308}{16598577599661419} a^{10} + \frac{20524095426219854}{16598577599661419} a^{9} + \frac{2740820929876587}{873609347350601} a^{8} - \frac{4270690606184008}{16598577599661419} a^{7} - \frac{201233002102114842}{16598577599661419} a^{6} - \frac{409138077272766760}{16598577599661419} a^{5} - \frac{457119225824278139}{16598577599661419} a^{4} - \frac{342184947292584278}{16598577599661419} a^{3} - \frac{174589464088802650}{16598577599661419} a^{2} - \frac{54779931927067619}{16598577599661419} a + \frac{10033525485001195}{33197155199322838} \) (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:  \( 407604.224227 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 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$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$