↑

Properties

Label 16.0.51671137288...9856.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{66}\cdot 3^{14}\cdot 11^{4}$
Root discriminant $83.10$
Ramified primes $2, 3, 11$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T1276

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1089, 0, 199584, 0, 191916, 0, 73296, 0, 20406, 0, 3744, 0, 684, 0, 48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 48*x^14 + 684*x^12 + 3744*x^10 + 20406*x^8 + 73296*x^6 + 191916*x^4 + 199584*x^2 + 1089)
 
gp: K = bnfinit(x^16 + 48*x^14 + 684*x^12 + 3744*x^10 + 20406*x^8 + 73296*x^6 + 191916*x^4 + 199584*x^2 + 1089, 1)
 

Normalized defining polynomial

\( x^{16} + 48 x^{14} + 684 x^{12} + 3744 x^{10} + 20406 x^{8} + 73296 x^{6} + 191916 x^{4} + 199584 x^{2} + 1089 \)

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:  \(5167113728869511408443358969856=2^{66}\cdot 3^{14}\cdot 11^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.10$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{12} a^{8} + \frac{1}{4}$, $\frac{1}{12} a^{9} + \frac{1}{4} a$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{3}{8} a^{2} + \frac{1}{8}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{72} a^{12} - \frac{1}{24} a^{8} + \frac{5}{24} a^{4} + \frac{3}{8}$, $\frac{1}{144} a^{13} - \frac{1}{144} a^{12} + \frac{1}{48} a^{9} - \frac{1}{48} a^{8} - \frac{7}{48} a^{5} + \frac{7}{48} a^{4} + \frac{1}{16} a - \frac{1}{16}$, $\frac{1}{1640845853160624} a^{14} + \frac{752966381803}{182316205906736} a^{12} - \frac{11089526271475}{546948617720208} a^{10} - \frac{21301850428249}{546948617720208} a^{8} - \frac{68119700178823}{546948617720208} a^{6} - \frac{38408735814775}{182316205906736} a^{4} - \frac{78158719854627}{182316205906736} a^{2} + \frac{3394562869477}{16574200536976}$, $\frac{1}{1640845853160624} a^{15} - \frac{288629089559}{102552865822539} a^{13} - \frac{1}{144} a^{12} - \frac{11089526271475}{546948617720208} a^{11} + \frac{805152386204}{34184288607513} a^{9} - \frac{1}{48} a^{8} - \frac{68119700178823}{546948617720208} a^{7} - \frac{2216429210008}{34184288607513} a^{5} + \frac{7}{48} a^{4} - \frac{78158719854627}{182316205906736} a^{3} + \frac{406389091885}{1035887533561} a - \frac{1}{16}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 135234869.093 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1276:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1276
Character table for t16n1276 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.13824.1, 8.4.1076291960832.9

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
3Data not computed
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$