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Properties

Label 16.16.1144754599...1184.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{52}\cdot 3^{26}$
Root discriminant $56.71$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\GL(2,Z/4)$ (as 16T193)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, 904, 6288, -10288, -16594, 22992, 15796, -20816, -6669, 8920, 1228, -1920, -46, 200, -12, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 12*x^14 + 200*x^13 - 46*x^12 - 1920*x^11 + 1228*x^10 + 8920*x^9 - 6669*x^8 - 20816*x^7 + 15796*x^6 + 22992*x^5 - 16594*x^4 - 10288*x^3 + 6288*x^2 + 904*x + 19)
 
gp: K = bnfinit(x^16 - 8*x^15 - 12*x^14 + 200*x^13 - 46*x^12 - 1920*x^11 + 1228*x^10 + 8920*x^9 - 6669*x^8 - 20816*x^7 + 15796*x^6 + 22992*x^5 - 16594*x^4 - 10288*x^3 + 6288*x^2 + 904*x + 19, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 12 x^{14} + 200 x^{13} - 46 x^{12} - 1920 x^{11} + 1228 x^{10} + 8920 x^{9} - 6669 x^{8} - 20816 x^{7} + 15796 x^{6} + 22992 x^{5} - 16594 x^{4} - 10288 x^{3} + 6288 x^{2} + 904 x + 19 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11447545997288281555215581184=2^{52}\cdot 3^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.71$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{2} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{9} + \frac{1}{9} a^{3} + \frac{1}{9}$, $\frac{1}{99} a^{13} - \frac{2}{99} a^{12} + \frac{2}{99} a^{11} - \frac{7}{99} a^{10} + \frac{4}{33} a^{9} + \frac{4}{33} a^{8} + \frac{1}{11} a^{7} + \frac{14}{33} a^{5} + \frac{10}{99} a^{4} - \frac{47}{99} a^{3} - \frac{49}{99} a^{2} + \frac{20}{99} a + \frac{16}{33}$, $\frac{1}{136719} a^{14} + \frac{202}{45573} a^{13} - \frac{1831}{45573} a^{12} + \frac{5114}{136719} a^{11} + \frac{18350}{136719} a^{10} - \frac{1382}{136719} a^{9} - \frac{2492}{15191} a^{8} - \frac{4028}{45573} a^{7} + \frac{2423}{45573} a^{6} + \frac{46765}{136719} a^{5} + \frac{4590}{15191} a^{4} + \frac{436}{15191} a^{3} + \frac{47756}{136719} a^{2} + \frac{28532}{136719} a - \frac{40787}{136719}$, $\frac{1}{14968406277} a^{15} + \frac{7016}{4989468759} a^{14} - \frac{2153335}{4989468759} a^{13} - \frac{193627904}{4989468759} a^{12} + \frac{213935929}{4989468759} a^{11} - \frac{335653192}{4989468759} a^{10} - \frac{572146978}{14968406277} a^{9} - \frac{347052260}{4989468759} a^{8} - \frac{74179408}{453588069} a^{7} + \frac{1880276944}{14968406277} a^{6} - \frac{1897469936}{4989468759} a^{5} - \frac{1791742745}{4989468759} a^{4} - \frac{102816697}{1663156253} a^{3} + \frac{1174686020}{4989468759} a^{2} + \frac{361700346}{1663156253} a - \frac{635228389}{14968406277}$

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:  $15$
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:  \( 2550688765.33 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\GL(2,Z/4)$ (as 16T193):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 14 conjugacy class representatives for $\GL(2,Z/4)$
Character table for $\GL(2,Z/4)$

Intermediate fields

\(\Q(\sqrt{6}) \), 4.4.497664.1, 4.4.27648.1, 8.8.2229025112064.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.12.23.50$x^{12} - 9 x^{11} + 6 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 9 x^{4} + 3 x^{3} - 9 x^{2} + 9 x + 3$$12$$1$$23$$(C_6\times C_2):C_2$$[5/2]_{4}^{2}$