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Properties

Label 16.0.29865750236756037.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 16573^{3}$
Root discriminant $10.71$
Ramified primes $3, 16573$
Class number $1$
Class group Trivial
Galois group 16T1872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 28, -50, 48, -25, 19, -30, 19, 11, -16, -2, 8, 0, -2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 2*x^14 + 8*x^12 - 2*x^11 - 16*x^10 + 11*x^9 + 19*x^8 - 30*x^7 + 19*x^6 - 25*x^5 + 48*x^4 - 50*x^3 + 28*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^16 - x^15 - 2*x^14 + 8*x^12 - 2*x^11 - 16*x^10 + 11*x^9 + 19*x^8 - 30*x^7 + 19*x^6 - 25*x^5 + 48*x^4 - 50*x^3 + 28*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 2 x^{14} + 8 x^{12} - 2 x^{11} - 16 x^{10} + 11 x^{9} + 19 x^{8} - 30 x^{7} + 19 x^{6} - 25 x^{5} + 48 x^{4} - 50 x^{3} + 28 x^{2} - 8 x + 1 \)

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:  \(29865750236756037=3^{8}\cdot 16573^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.71$
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:  $3, 16573$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{3} + \frac{1}{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{12} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{3415} a^{15} - \frac{8}{3415} a^{14} + \frac{54}{3415} a^{13} + \frac{1671}{3415} a^{12} + \frac{121}{683} a^{11} - \frac{301}{683} a^{10} - \frac{409}{3415} a^{9} + \frac{142}{3415} a^{8} - \frac{1658}{3415} a^{7} + \frac{648}{3415} a^{6} + \frac{264}{3415} a^{5} + \frac{859}{3415} a^{4} + \frac{173}{683} a^{3} - \frac{1324}{3415} a^{2} - \frac{949}{3415} a - \frac{39}{683}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{21218}{3415} a^{15} - \frac{9922}{3415} a^{14} - \frac{10442}{683} a^{13} - \frac{6134}{683} a^{12} + \frac{162459}{3415} a^{11} + \frac{61362}{3415} a^{10} - \frac{324389}{3415} a^{9} + \frac{4693}{683} a^{8} + \frac{447202}{3415} a^{7} - \frac{358101}{3415} a^{6} + \frac{31062}{683} a^{5} - \frac{89255}{683} a^{4} + \frac{761539}{3415} a^{3} - \frac{581392}{3415} a^{2} + \frac{42684}{683} a - \frac{31314}{3415} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 84.9443869961 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 77 conjugacy class representatives for t16n1872 are not computed
Character table for t16n1872 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 8.0.1342413.1

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 $16$ R ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
3Data not computed
16573Data not computed