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Properties

Label 20.0.22369715839...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{15}\cdot 31^{18}$
Root discriminant $147.06$
Ramified primes $2, 5, 31$
Class number $1347200$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 21050]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3003125, 0, 63065625, 0, 306318750, 0, 299711875, 0, 124930000, 0, 27167625, 0, 3309250, 0, 227850, 0, 8525, 0, 155, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 155*x^18 + 8525*x^16 + 227850*x^14 + 3309250*x^12 + 27167625*x^10 + 124930000*x^8 + 299711875*x^6 + 306318750*x^4 + 63065625*x^2 + 3003125)
 
gp: K = bnfinit(x^20 + 155*x^18 + 8525*x^16 + 227850*x^14 + 3309250*x^12 + 27167625*x^10 + 124930000*x^8 + 299711875*x^6 + 306318750*x^4 + 63065625*x^2 + 3003125, 1)
 

Normalized defining polynomial

\( x^{20} + 155 x^{18} + 8525 x^{16} + 227850 x^{14} + 3309250 x^{12} + 27167625 x^{10} + 124930000 x^{8} + 299711875 x^{6} + 306318750 x^{4} + 63065625 x^{2} + 3003125 \)

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:  \(22369715839969441233253447712000000000000000=2^{20}\cdot 5^{15}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $147.06$
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:  $2, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(620=2^{2}\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{620}(1,·)$, $\chi_{620}(387,·)$, $\chi_{620}(263,·)$, $\chi_{620}(587,·)$, $\chi_{620}(463,·)$, $\chi_{620}(529,·)$, $\chi_{620}(147,·)$, $\chi_{620}(469,·)$, $\chi_{620}(23,·)$, $\chi_{620}(281,·)$, $\chi_{620}(27,·)$, $\chi_{620}(349,·)$, $\chi_{620}(523,·)$, $\chi_{620}(481,·)$, $\chi_{620}(101,·)$, $\chi_{620}(109,·)$, $\chi_{620}(221,·)$, $\chi_{620}(247,·)$, $\chi_{620}(249,·)$, $\chi_{620}(123,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{25} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{125} a^{8} + \frac{2}{25} a^{4} + \frac{1}{5}$, $\frac{1}{125} a^{9} + \frac{2}{25} a^{5} + \frac{1}{5} a$, $\frac{1}{19375} a^{10} + \frac{2}{125} a^{6} + \frac{11}{25} a^{2}$, $\frac{1}{19375} a^{11} + \frac{2}{125} a^{7} + \frac{11}{25} a^{3}$, $\frac{1}{96875} a^{12} - \frac{2}{625} a^{8} - \frac{12}{125} a^{4} + \frac{11}{25}$, $\frac{1}{96875} a^{13} - \frac{2}{625} a^{9} - \frac{12}{125} a^{5} + \frac{11}{25} a$, $\frac{1}{96875} a^{14} + \frac{2}{125} a^{6} + \frac{8}{25} a^{2}$, $\frac{1}{96875} a^{15} + \frac{2}{125} a^{7} + \frac{8}{25} a^{3}$, $\frac{1}{147734375} a^{16} + \frac{27}{5909375} a^{14} + \frac{89}{29546875} a^{12} - \frac{11}{1181875} a^{10} - \frac{9}{190625} a^{8} + \frac{47}{7625} a^{6} + \frac{3209}{38125} a^{4} - \frac{33}{305} a^{2} - \frac{984}{7625}$, $\frac{1}{147734375} a^{17} + \frac{27}{5909375} a^{15} + \frac{89}{29546875} a^{13} - \frac{11}{1181875} a^{11} - \frac{9}{190625} a^{9} + \frac{47}{7625} a^{7} + \frac{3209}{38125} a^{5} - \frac{33}{305} a^{3} - \frac{984}{7625} a$, $\frac{1}{5107325078125} a^{18} - \frac{9053}{5107325078125} a^{16} + \frac{4235064}{1021465015625} a^{14} + \frac{3467068}{1021465015625} a^{12} - \frac{3400654}{204293003125} a^{10} - \frac{19987593}{6590096875} a^{8} + \frac{12371159}{1318019375} a^{6} - \frac{6258247}{1318019375} a^{4} - \frac{104660684}{263603875} a^{2} - \frac{39718088}{263603875}$, $\frac{1}{5107325078125} a^{19} - \frac{9053}{5107325078125} a^{17} + \frac{4235064}{1021465015625} a^{15} + \frac{3467068}{1021465015625} a^{13} - \frac{3400654}{204293003125} a^{11} - \frac{19987593}{6590096875} a^{9} + \frac{12371159}{1318019375} a^{7} - \frac{6258247}{1318019375} a^{5} - \frac{104660684}{263603875} a^{3} - \frac{39718088}{263603875} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{21050}$, which has order $1347200$ (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:  \( -1 \) (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:  \( 24173706.832424585 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1922000.2, 5.5.923521.1, 10.10.2665284492003125.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$31$31.10.9.1$x^{10} - 31$$10$$1$$9$$C_{10}$$[\ ]_{10}$
31.10.9.1$x^{10} - 31$$10$$1$$9$$C_{10}$$[\ ]_{10}$