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Properties

Label 20.0.21822182114...9209.3
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 883^{8}$
Root discriminant $26.12$
Ramified primes $3, 883$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T230

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 11, -4, 63, -56, 99, -89, 96, -153, 77, -153, 96, -89, 99, -56, 63, -4, 11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 11*x^18 - 4*x^17 + 63*x^16 - 56*x^15 + 99*x^14 - 89*x^13 + 96*x^12 - 153*x^11 + 77*x^10 - 153*x^9 + 96*x^8 - 89*x^7 + 99*x^6 - 56*x^5 + 63*x^4 - 4*x^3 + 11*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 11*x^18 - 4*x^17 + 63*x^16 - 56*x^15 + 99*x^14 - 89*x^13 + 96*x^12 - 153*x^11 + 77*x^10 - 153*x^9 + 96*x^8 - 89*x^7 + 99*x^6 - 56*x^5 + 63*x^4 - 4*x^3 + 11*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 11 x^{18} - 4 x^{17} + 63 x^{16} - 56 x^{15} + 99 x^{14} - 89 x^{13} + 96 x^{12} - 153 x^{11} + 77 x^{10} - 153 x^{9} + 96 x^{8} - 89 x^{7} + 99 x^{6} - 56 x^{5} + 63 x^{4} - 4 x^{3} + 11 x^{2} - 2 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(21822182114965484993955979209=3^{10}\cdot 883^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.12$
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:  $3, 883$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{3} a^{9} + \frac{4}{9} a^{8} + \frac{1}{9} a^{7} - \frac{4}{9} a^{6} - \frac{2}{9} a^{5} + \frac{2}{9} a^{4} + \frac{2}{9} a^{3} - \frac{1}{9} a + \frac{2}{9}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{4}{9} a^{9} + \frac{4}{9} a^{7} - \frac{1}{9} a^{6} + \frac{4}{9} a^{5} - \frac{1}{3} a^{4} - \frac{2}{9} a^{3} + \frac{2}{9} a^{2} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{10} - \frac{4}{9} a^{8} + \frac{1}{3} a^{6} + \frac{4}{9} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} + \frac{1}{3} a^{2} - \frac{4}{9}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{4}{9} a^{8} + \frac{4}{9} a^{7} + \frac{1}{9} a^{5} - \frac{1}{3} a^{4} - \frac{4}{9} a^{3} + \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{10} - \frac{2}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} - \frac{1}{3} a^{4} - \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9} a^{7} - \frac{1}{3} a^{6} - \frac{1}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a^{8} - \frac{1}{3} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{9} a^{2} + \frac{1}{9} a$, $\frac{1}{1899621} a^{18} - \frac{55387}{1899621} a^{17} - \frac{67841}{1899621} a^{16} - \frac{21170}{1899621} a^{15} + \frac{80059}{1899621} a^{14} + \frac{30317}{633207} a^{13} - \frac{3149}{211069} a^{12} + \frac{66161}{1899621} a^{11} + \frac{70361}{1899621} a^{10} + \frac{126262}{633207} a^{9} - \frac{117259}{633207} a^{8} + \frac{66161}{1899621} a^{7} - \frac{3149}{211069} a^{6} + \frac{302020}{1899621} a^{5} + \frac{80059}{1899621} a^{4} + \frac{189899}{1899621} a^{3} - \frac{304039}{633207} a^{2} + \frac{262963}{633207} a + \frac{844277}{1899621}$, $\frac{1}{66486735} a^{19} - \frac{1}{22162245} a^{18} + \frac{98372}{9498105} a^{17} + \frac{533569}{22162245} a^{16} + \frac{197514}{7387415} a^{15} + \frac{3529228}{66486735} a^{14} - \frac{197683}{22162245} a^{13} - \frac{392464}{13297347} a^{12} - \frac{3203759}{66486735} a^{11} + \frac{402613}{9498105} a^{10} - \frac{4164922}{9498105} a^{9} + \frac{22557221}{66486735} a^{8} + \frac{563708}{13297347} a^{7} - \frac{25366189}{66486735} a^{6} - \frac{8041459}{22162245} a^{5} - \frac{5122678}{22162245} a^{4} - \frac{3858068}{66486735} a^{3} - \frac{440827}{1055345} a^{2} + \frac{177858}{7387415} a + \frac{10079417}{22162245}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{5488102}{22162245} a^{19} + \frac{8971576}{22162245} a^{18} - \frac{2642753}{1055345} a^{17} - \frac{1659859}{22162245} a^{16} - \frac{328633552}{22162245} a^{15} + \frac{59857178}{7387415} a^{14} - \frac{378133517}{22162245} a^{13} + \frac{16878762}{1477483} a^{12} - \frac{92001314}{7387415} a^{11} + \frac{85305434}{3166035} a^{10} - \frac{2480397}{1055345} a^{9} + \frac{198778266}{7387415} a^{8} - \frac{37343095}{4432449} a^{7} + \frac{67214586}{7387415} a^{6} - \frac{101010417}{7387415} a^{5} + \frac{60868708}{22162245} a^{4} - \frac{184511129}{22162245} a^{3} - \frac{6813843}{1055345} a^{2} - \frac{9634208}{7387415} a + \frac{3943313}{22162245} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2114484.68113 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T230:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 24 conjugacy class representatives for t20n230
Character table for t20n230 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.7017201.1, 10.0.147723329623203.1, 10.4.147723329623203.1, 10.6.49241109874401.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 sibling: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
883Data not computed