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Properties

Label 16.16.6723314169...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{16}\cdot 5^{14}\cdot 29^{6}\cdot 41^{4}$
Root discriminant $73.15$
Ramified primes $2, 5, 29, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T646)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-87719, -502312, -726420, 229470, 1140855, 401424, -507873, -278030, 101540, 66050, -11793, -7344, 915, 390, -45, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 45*x^14 + 390*x^13 + 915*x^12 - 7344*x^11 - 11793*x^10 + 66050*x^9 + 101540*x^8 - 278030*x^7 - 507873*x^6 + 401424*x^5 + 1140855*x^4 + 229470*x^3 - 726420*x^2 - 502312*x - 87719)
 
gp: K = bnfinit(x^16 - 8*x^15 - 45*x^14 + 390*x^13 + 915*x^12 - 7344*x^11 - 11793*x^10 + 66050*x^9 + 101540*x^8 - 278030*x^7 - 507873*x^6 + 401424*x^5 + 1140855*x^4 + 229470*x^3 - 726420*x^2 - 502312*x - 87719, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 45 x^{14} + 390 x^{13} + 915 x^{12} - 7344 x^{11} - 11793 x^{10} + 66050 x^{9} + 101540 x^{8} - 278030 x^{7} - 507873 x^{6} + 401424 x^{5} + 1140855 x^{4} + 229470 x^{3} - 726420 x^{2} - 502312 x - 87719 \)

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:  \(672331416948912400000000000000=2^{16}\cdot 5^{14}\cdot 29^{6}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.15$
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:  $2, 5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{10} a^{12} + \frac{2}{5} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{10} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{7} + \frac{1}{5} a^{6} + \frac{3}{10} a^{5} + \frac{1}{10} a^{3} + \frac{1}{5} a^{2} - \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{30} a^{14} + \frac{1}{30} a^{12} + \frac{1}{15} a^{11} + \frac{1}{15} a^{10} - \frac{1}{30} a^{8} - \frac{1}{5} a^{7} - \frac{4}{15} a^{6} + \frac{2}{5} a^{5} - \frac{7}{15} a^{4} + \frac{4}{15} a^{3} + \frac{7}{15} a^{2} + \frac{1}{5} a + \frac{11}{30}$, $\frac{1}{5229785950263479138068973850} a^{15} + \frac{4812082986315260661049137}{348652396684231942537931590} a^{14} + \frac{19344119477394277325885197}{522978595026347913806897385} a^{13} + \frac{4628733619023388959300971}{522978595026347913806897385} a^{12} + \frac{38381377811844694822332029}{522978595026347913806897385} a^{11} - \frac{33062533855130155343771024}{871630991710579856344828975} a^{10} + \frac{75827603577675732160977529}{1045957190052695827613794770} a^{9} - \frac{5252340245479867173507949}{69730479336846388507586318} a^{8} - \frac{233302841388010073015500429}{1045957190052695827613794770} a^{7} + \frac{21715252107449917860903227}{348652396684231942537931590} a^{6} - \frac{2238332573720474989521208073}{5229785950263479138068973850} a^{5} - \frac{280827188213304949578904571}{1045957190052695827613794770} a^{4} + \frac{269601395279829883909538059}{1045957190052695827613794770} a^{3} - \frac{171950511992793481016087897}{348652396684231942537931590} a^{2} + \frac{6222980351564405472802765}{104595719005269582761379477} a - \frac{225207190750828271586720177}{871630991710579856344828975}$

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:  \( 5631345276.64 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T646):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 4.4.58000.1, 4.4.725.1, 8.8.3364000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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}$
5Data not computed
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed