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Properties

Label 20.0.18218737052...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{15}\cdot 11^{18}$
Root discriminant $81.85$
Ramified primes $2, 5, 11$
Class number $250256$ (GRH)
Class group $[2, 2, 62564]$ (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![387200000, 0, 968000000, 0, 890560000, 0, 416240000, 0, 112288000, 0, 18612000, 0, 1944800, 0, 127600, 0, 5060, 0, 110, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 110*x^18 + 5060*x^16 + 127600*x^14 + 1944800*x^12 + 18612000*x^10 + 112288000*x^8 + 416240000*x^6 + 890560000*x^4 + 968000000*x^2 + 387200000)
 
gp: K = bnfinit(x^20 + 110*x^18 + 5060*x^16 + 127600*x^14 + 1944800*x^12 + 18612000*x^10 + 112288000*x^8 + 416240000*x^6 + 890560000*x^4 + 968000000*x^2 + 387200000, 1)
 

Normalized defining polynomial

\( x^{20} + 110 x^{18} + 5060 x^{16} + 127600 x^{14} + 1944800 x^{12} + 18612000 x^{10} + 112288000 x^{8} + 416240000 x^{6} + 890560000 x^{4} + 968000000 x^{2} + 387200000 \)

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:  \(182187370528513441169408000000000000000=2^{30}\cdot 5^{15}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.85$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(440=2^{3}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(43,·)$, $\chi_{440}(9,·)$, $\chi_{440}(81,·)$, $\chi_{440}(83,·)$, $\chi_{440}(89,·)$, $\chi_{440}(347,·)$, $\chi_{440}(361,·)$, $\chi_{440}(289,·)$, $\chi_{440}(227,·)$, $\chi_{440}(401,·)$, $\chi_{440}(169,·)$, $\chi_{440}(107,·)$, $\chi_{440}(387,·)$, $\chi_{440}(307,·)$, $\chi_{440}(49,·)$, $\chi_{440}(403,·)$, $\chi_{440}(201,·)$, $\chi_{440}(283,·)$, $\chi_{440}(123,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{20} a^{4}$, $\frac{1}{20} a^{5}$, $\frac{1}{40} a^{6}$, $\frac{1}{40} a^{7}$, $\frac{1}{400} a^{8}$, $\frac{1}{400} a^{9}$, $\frac{1}{8800} a^{10}$, $\frac{1}{8800} a^{11}$, $\frac{1}{88000} a^{12}$, $\frac{1}{88000} a^{13}$, $\frac{1}{176000} a^{14}$, $\frac{1}{176000} a^{15}$, $\frac{1}{1760000} a^{16}$, $\frac{1}{1760000} a^{17}$, $\frac{1}{38720000} a^{18} - \frac{1}{1100} a^{8}$, $\frac{1}{38720000} a^{19} - \frac{1}{1100} a^{9}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{62564}$, which has order $250256$ (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:  \( 140644.599182 \) (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

\(\Q(\sqrt{5}) \), 4.0.968000.5, \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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 $20$ R $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
5Data not computed
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$