↑

Properties

Label 16.0.13724372348...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{14}\cdot 1361^{4}$
Root discriminant $49.67$
Ramified primes $2, 5, 1361$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group 16T1263

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1121, -7768, 27777, -59694, 104386, -109908, 68420, -10002, -15146, 10204, -1615, -706, 366, -72, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 72*x^13 + 366*x^12 - 706*x^11 - 1615*x^10 + 10204*x^9 - 15146*x^8 - 10002*x^7 + 68420*x^6 - 109908*x^5 + 104386*x^4 - 59694*x^3 + 27777*x^2 - 7768*x + 1121)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 - 72*x^13 + 366*x^12 - 706*x^11 - 1615*x^10 + 10204*x^9 - 15146*x^8 - 10002*x^7 + 68420*x^6 - 109908*x^5 + 104386*x^4 - 59694*x^3 + 27777*x^2 - 7768*x + 1121, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 18 x^{14} - 72 x^{13} + 366 x^{12} - 706 x^{11} - 1615 x^{10} + 10204 x^{9} - 15146 x^{8} - 10002 x^{7} + 68420 x^{6} - 109908 x^{5} + 104386 x^{4} - 59694 x^{3} + 27777 x^{2} - 7768 x + 1121 \)

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:  \(1372437234816400000000000000=2^{16}\cdot 5^{14}\cdot 1361^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.67$
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:  $2, 5, 1361$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} + \frac{1}{8} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{475567135393103817668311527986408} a^{15} - \frac{10567957701424308383002878225115}{475567135393103817668311527986408} a^{14} + \frac{24926972699280977905273818282121}{475567135393103817668311527986408} a^{13} - \frac{14396970389525453449582501965217}{475567135393103817668311527986408} a^{12} + \frac{8115318377132169527216146337383}{475567135393103817668311527986408} a^{11} - \frac{4697959603784161176757250903097}{475567135393103817668311527986408} a^{10} + \frac{1431260526553925890037569964849}{237783567696551908834155763993204} a^{9} + \frac{114997644892729446470515269997185}{237783567696551908834155763993204} a^{8} - \frac{47156311284313695967055602310727}{118891783848275954417077881996602} a^{7} + \frac{37910421836680806428886277306491}{237783567696551908834155763993204} a^{6} + \frac{102349517430898861307640964588349}{237783567696551908834155763993204} a^{5} - \frac{32836699175364550331678378204855}{237783567696551908834155763993204} a^{4} - \frac{8388741369414893164510211767155}{59445891924137977208538940998301} a^{3} - \frac{114152618878699891617138011857853}{237783567696551908834155763993204} a^{2} + \frac{210353857109080691097560188893383}{475567135393103817668311527986408} a + \frac{203050048780612903221847411997}{424234732732474413620260060648}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  \( 838200.79427 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.0.5444000000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
1361Data not computed