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Properties

Label 20.4.78125000000...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 5^{27}$
Root discriminant $17.56$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $C_4\times D_5$ (as 20T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 22, 0, 0, 0, 0, -6, 0, 0, 0, 0, -22, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 22*x^15 - 6*x^10 + 22*x^5 + 1)
 
gp: K = bnfinit(x^20 - 22*x^15 - 6*x^10 + 22*x^5 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 22 x^{15} - 6 x^{10} + 22 x^{5} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7812500000000000000000000=2^{20}\cdot 5^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.56$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{10} a^{10} + \frac{2}{5} a^{5} - \frac{1}{10}$, $\frac{1}{10} a^{11} + \frac{2}{5} a^{6} - \frac{1}{10} a$, $\frac{1}{10} a^{12} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{10} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{10} a^{3} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{50} a^{14} - \frac{1}{25} a^{13} - \frac{1}{50} a^{12} + \frac{1}{25} a^{11} + \frac{1}{50} a^{10} + \frac{2}{25} a^{9} + \frac{1}{25} a^{8} - \frac{2}{25} a^{7} - \frac{6}{25} a^{6} - \frac{3}{25} a^{5} - \frac{21}{50} a^{4} + \frac{6}{25} a^{3} + \frac{21}{50} a^{2} + \frac{9}{25} a + \frac{9}{50}$, $\frac{1}{50} a^{15} + \frac{1}{50} a^{10} + \frac{17}{50} a^{5} + \frac{13}{50}$, $\frac{1}{50} a^{16} + \frac{1}{50} a^{11} + \frac{17}{50} a^{6} + \frac{13}{50} a$, $\frac{1}{50} a^{17} + \frac{1}{50} a^{12} - \frac{3}{50} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{23}{50} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{50} a^{18} + \frac{1}{50} a^{13} - \frac{3}{50} a^{8} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{23}{50} a^{3} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{50} a^{19} + \frac{1}{25} a^{13} + \frac{1}{50} a^{12} - \frac{1}{25} a^{11} - \frac{1}{50} a^{10} + \frac{3}{50} a^{9} - \frac{1}{25} a^{8} + \frac{2}{25} a^{7} - \frac{4}{25} a^{6} - \frac{12}{25} a^{5} + \frac{7}{25} a^{4} - \frac{6}{25} a^{3} - \frac{21}{50} a^{2} - \frac{4}{25} a - \frac{19}{50}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 44054.0373872 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_5$ (as 20T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 16 conjugacy class representatives for $C_4\times D_5$
Character table for $C_4\times D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 5.1.250000.1, 10.2.312500000000.1

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

Sibling fields

Galois closure: data not computed
Degree 20 sibling: 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 $20$ R $20$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$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