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Properties

Label 20.0.48883259296...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{8}\cdot 5^{27}$
Root discriminant $13.63$
Ramified primes $3, 5$
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, 5, 10, -25, -35, 92, 65, -275, 50, 375, -261, -285, 535, -295, -5, 48, 45, -75, 40, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 40*x^18 - 75*x^17 + 45*x^16 + 48*x^15 - 5*x^14 - 295*x^13 + 535*x^12 - 285*x^11 - 261*x^10 + 375*x^9 + 50*x^8 - 275*x^7 + 65*x^6 + 92*x^5 - 35*x^4 - 25*x^3 + 10*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 40*x^18 - 75*x^17 + 45*x^16 + 48*x^15 - 5*x^14 - 295*x^13 + 535*x^12 - 285*x^11 - 261*x^10 + 375*x^9 + 50*x^8 - 275*x^7 + 65*x^6 + 92*x^5 - 35*x^4 - 25*x^3 + 10*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 40 x^{18} - 75 x^{17} + 45 x^{16} + 48 x^{15} - 5 x^{14} - 295 x^{13} + 535 x^{12} - 285 x^{11} - 261 x^{10} + 375 x^{9} + 50 x^{8} - 275 x^{7} + 65 x^{6} + 92 x^{5} - 35 x^{4} - 25 x^{3} + 10 x^{2} + 5 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(48883259296417236328125=3^{8}\cdot 5^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.63$
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:  $3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{10} - \frac{4}{9} a^{9} - \frac{1}{3} a^{8} - \frac{2}{9} a^{7} - \frac{2}{9} a^{6} - \frac{2}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{27} a^{17} - \frac{1}{27} a^{16} + \frac{1}{27} a^{15} - \frac{4}{27} a^{14} + \frac{1}{27} a^{13} + \frac{2}{27} a^{12} - \frac{1}{27} a^{11} + \frac{1}{27} a^{10} - \frac{10}{27} a^{9} - \frac{5}{27} a^{8} - \frac{10}{27} a^{7} - \frac{2}{27} a^{6} + \frac{1}{27} a^{5} + \frac{5}{27} a^{4} - \frac{2}{27} a^{3} + \frac{1}{27} a^{2} - \frac{13}{27} a - \frac{2}{27}$, $\frac{1}{27} a^{18} - \frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{27} a^{12} - \frac{2}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{8}{27} a^{6} - \frac{4}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{27} a^{3} - \frac{4}{9} a^{2} + \frac{1}{9} a + \frac{7}{27}$, $\frac{1}{7209} a^{19} + \frac{124}{7209} a^{18} + \frac{34}{2403} a^{17} - \frac{8}{2403} a^{16} + \frac{11}{2403} a^{15} - \frac{379}{2403} a^{14} - \frac{173}{7209} a^{13} + \frac{553}{7209} a^{12} + \frac{137}{2403} a^{11} + \frac{374}{2403} a^{10} - \frac{701}{2403} a^{9} - \frac{626}{2403} a^{8} - \frac{88}{7209} a^{7} - \frac{52}{7209} a^{6} + \frac{814}{2403} a^{5} + \frac{2915}{7209} a^{4} + \frac{2891}{7209} a^{3} - \frac{728}{2403} a^{2} + \frac{520}{7209} a + \frac{1600}{7209}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{5327}{7209} a^{19} - \frac{55546}{7209} a^{18} + \frac{2947}{89} a^{17} - \frac{172022}{2403} a^{16} + \frac{167444}{2403} a^{15} - \frac{214}{267} a^{14} - \frac{35932}{7209} a^{13} - \frac{1505059}{7209} a^{12} + \frac{44022}{89} a^{11} - \frac{1111933}{2403} a^{10} + \frac{111463}{2403} a^{9} + \frac{209786}{801} a^{8} - \frac{861800}{7209} a^{7} - \frac{877220}{7209} a^{6} + \frac{84553}{801} a^{5} + \frac{76381}{7209} a^{4} - \frac{208997}{7209} a^{3} - \frac{2482}{801} a^{2} + \frac{44504}{7209} a + \frac{7502}{7209} \) (order $10$)
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:  \( 6025.48578502 \)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.140625.1, 10.2.98876953125.1

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 $20$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$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