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Properties

Label 20.2.23224632608...4375.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,3^{10}\cdot 5^{10}\cdot 3319^{5}$
Root discriminant $29.40$
Ramified primes $3, 5, 3319$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times D_5\wr C_2$ (as 20T92)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10161, 49824, -133473, 243846, -333566, 354922, -299095, 197534, -96544, 27074, 6132, -14162, 11274, -6346, 2823, -1028, 308, -80, 20, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 20*x^18 - 80*x^17 + 308*x^16 - 1028*x^15 + 2823*x^14 - 6346*x^13 + 11274*x^12 - 14162*x^11 + 6132*x^10 + 27074*x^9 - 96544*x^8 + 197534*x^7 - 299095*x^6 + 354922*x^5 - 333566*x^4 + 243846*x^3 - 133473*x^2 + 49824*x - 10161)
 
gp: K = bnfinit(x^20 - 4*x^19 + 20*x^18 - 80*x^17 + 308*x^16 - 1028*x^15 + 2823*x^14 - 6346*x^13 + 11274*x^12 - 14162*x^11 + 6132*x^10 + 27074*x^9 - 96544*x^8 + 197534*x^7 - 299095*x^6 + 354922*x^5 - 333566*x^4 + 243846*x^3 - 133473*x^2 + 49824*x - 10161, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 20 x^{18} - 80 x^{17} + 308 x^{16} - 1028 x^{15} + 2823 x^{14} - 6346 x^{13} + 11274 x^{12} - 14162 x^{11} + 6132 x^{10} + 27074 x^{9} - 96544 x^{8} + 197534 x^{7} - 299095 x^{6} + 354922 x^{5} - 333566 x^{4} + 243846 x^{3} - 133473 x^{2} + 49824 x - 10161 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-232246326086930752913583984375=-\,3^{10}\cdot 5^{10}\cdot 3319^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.40$
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:  $3, 5, 3319$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{18} - \frac{1}{6} a^{17} - \frac{1}{6} a^{16} + \frac{1}{6} a^{15} - \frac{1}{6} a^{14} + \frac{1}{6} a^{13} - \frac{1}{6} a^{11} - \frac{1}{3} a^{9} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6683009793898668433216882918597378326} a^{19} - \frac{171852049624179723268241508323307727}{2227669931299556144405627639532459442} a^{18} + \frac{466746432789045402702679012139439191}{3341504896949334216608441459298689163} a^{17} - \frac{19279107556955475353446383532095375}{2227669931299556144405627639532459442} a^{16} + \frac{76909404187345733907716157971650401}{2227669931299556144405627639532459442} a^{15} - \frac{472663471207955521269695602481851841}{2227669931299556144405627639532459442} a^{14} - \frac{1385060304845471725294054474212297563}{6683009793898668433216882918597378326} a^{13} + \frac{365832965310497148991574577278297849}{6683009793898668433216882918597378326} a^{12} + \frac{564484521683527452752262021803638619}{6683009793898668433216882918597378326} a^{11} + \frac{561505789932581073609110583318167828}{3341504896949334216608441459298689163} a^{10} + \frac{686555022318429451752553953603522491}{3341504896949334216608441459298689163} a^{9} - \frac{339077311250825922832606582099700810}{3341504896949334216608441459298689163} a^{8} + \frac{755019168554568537262395763466113975}{6683009793898668433216882918597378326} a^{7} - \frac{3286730320234873879193172126556466201}{6683009793898668433216882918597378326} a^{6} - \frac{1523833761860603262791744882247524747}{6683009793898668433216882918597378326} a^{5} - \frac{305965931345841096879203285165912391}{2227669931299556144405627639532459442} a^{4} + \frac{2363244595099188472951043279742648851}{6683009793898668433216882918597378326} a^{3} + \frac{1079476511748163924618185015003734557}{6683009793898668433216882918597378326} a^{2} - \frac{458449664345569735149475565138039119}{2227669931299556144405627639532459442} a - \frac{85624820489735844972378053005150586}{1113834965649778072202813819766229721}$

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:  $10$
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:  \( 2406364.96464 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_2\times D_5\wr C_2$
Character table for $C_2\times D_5\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.746775.1, 10.6.34424253125.1

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

Sibling fields

Degree 20 siblings: 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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$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}$
5Data not computed
3319Data not computed