Properties

Label 20.0.12453009040...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{4}\cdot 5^{17}\cdot 67^{4}$
Root discriminant $11.34$
Ramified primes $3, 5, 67$
Class number $1$
Class group Trivial
Galois group $C_4\times S_5$ (as 20T123)

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Show commands for: SageMath / Pari/GP / Magma

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

Normalized defining polynomial

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

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(1245300904083251953125=3^{4}\cdot 5^{17}\cdot 67^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $11.34$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 5, 67$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

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

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

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{8}{25} a^{19} + \frac{17}{25} a^{18} - \frac{16}{25} a^{17} + \frac{24}{25} a^{16} - \frac{32}{25} a^{15} + \frac{56}{25} a^{14} - \frac{81}{25} a^{13} + \frac{72}{25} a^{12} - \frac{72}{25} a^{11} + \frac{64}{25} a^{10} - \frac{24}{5} a^{9} + \frac{172}{25} a^{8} - \frac{16}{5} a^{7} + \frac{64}{25} a^{6} - \frac{8}{5} a^{5} + \frac{72}{25} a^{4} - \frac{121}{25} a^{3} + \frac{24}{25} a^{2} + \frac{8}{25} a + \frac{8}{25} \) (order $10$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 676.304774 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_4\times S_5$ (as 20T123):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 480
The 28 conjugacy class representatives for $C_4\times S_5$
Character table for $C_4\times S_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.5025.1, 10.2.3156328125.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 24 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 $20$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
$67$67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.8.4.1$x^{8} + 17956 x^{4} - 300763 x^{2} + 80604484$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$