↑

Properties

Label 16.0.11463919801...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 5^{12}\cdot 41^{2}\cdot 101^{4}$
Root discriminant $56.72$
Ramified primes $2, 5, 41, 101$
Class number $1152$ (GRH)
Class group $[2, 2, 2, 144]$ (GRH)
Galois group 16T1161

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![581341, -46464, 3152, 165288, 67782, 86476, 40612, 1164, 15990, -1724, 3724, -252, 482, -8, 32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 32*x^14 - 8*x^13 + 482*x^12 - 252*x^11 + 3724*x^10 - 1724*x^9 + 15990*x^8 + 1164*x^7 + 40612*x^6 + 86476*x^5 + 67782*x^4 + 165288*x^3 + 3152*x^2 - 46464*x + 581341)
 
gp: K = bnfinit(x^16 + 32*x^14 - 8*x^13 + 482*x^12 - 252*x^11 + 3724*x^10 - 1724*x^9 + 15990*x^8 + 1164*x^7 + 40612*x^6 + 86476*x^5 + 67782*x^4 + 165288*x^3 + 3152*x^2 - 46464*x + 581341, 1)
 

Normalized defining polynomial

\( x^{16} + 32 x^{14} - 8 x^{13} + 482 x^{12} - 252 x^{11} + 3724 x^{10} - 1724 x^{9} + 15990 x^{8} + 1164 x^{7} + 40612 x^{6} + 86476 x^{5} + 67782 x^{4} + 165288 x^{3} + 3152 x^{2} - 46464 x + 581341 \)

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:  \(11463919801532416000000000000=2^{28}\cdot 5^{12}\cdot 41^{2}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.72$
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:  $2, 5, 41, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{75324243123240936004367450121671079646538} a^{15} + \frac{1823410987450546644700184854942351810271}{37662121561620468002183725060835539823269} a^{14} + \frac{7714701197203880846990859185495609632790}{37662121561620468002183725060835539823269} a^{13} - \frac{434088412038429665125567628511457328104}{37662121561620468002183725060835539823269} a^{12} - \frac{4470387030643958819689946281863090022265}{75324243123240936004367450121671079646538} a^{11} + \frac{9084289313763209841609316767297126859741}{37662121561620468002183725060835539823269} a^{10} - \frac{800552830614323200759852831506634472864}{37662121561620468002183725060835539823269} a^{9} - \frac{12388098682872271692120402806804318311839}{75324243123240936004367450121671079646538} a^{8} + \frac{19736809394635340610304792764642317700489}{75324243123240936004367450121671079646538} a^{7} + \frac{18490303380607763580274458583948816759311}{37662121561620468002183725060835539823269} a^{6} - \frac{16925533447242704301850726662710166207454}{37662121561620468002183725060835539823269} a^{5} + \frac{13118649551071830216189058801659668742719}{37662121561620468002183725060835539823269} a^{4} - \frac{8116595191443032158558577372473150798415}{75324243123240936004367450121671079646538} a^{3} - \frac{15233342923979708081248185057848049658543}{37662121561620468002183725060835539823269} a^{2} + \frac{10877657723355602288426894096354874053454}{37662121561620468002183725060835539823269} a - \frac{1329129591641186422555118360256342898309}{75324243123240936004367450121671079646538}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{144}$, which has order $1152$ (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:  \( 20635.0035043 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

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 t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.6464000000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
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}$
41Data not computed
101Data not computed