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Properties

Label 20.0.21315882145...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{8}\cdot 5^{15}\cdot 239^{8}$
Root discriminant $46.39$
Ramified primes $3, 5, 239$
Class number $116$ (GRH)
Class group $[2, 58]$ (GRH)
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, -45, 732, 3157, 9379, 15269, 21892, 15388, 24764, 5719, 13426, -772, 3653, -633, 406, -151, 94, -17, 13, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 13 x^{18} - 17 x^{17} + 94 x^{16} - 151 x^{15} + 406 x^{14} - 633 x^{13} + 3653 x^{12} - 772 x^{11} + 13426 x^{10} + 5719 x^{9} + 24764 x^{8} + 15388 x^{7} + 21892 x^{6} + 15269 x^{5} + 9379 x^{4} + 3157 x^{3} + 732 x^{2} - 45 x + 1 \)

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:  \(2131588214553606414985382080078125=3^{8}\cdot 5^{15}\cdot 239^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.39$
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:  $3, 5, 239$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{19} a^{16} - \frac{3}{19} a^{15} + \frac{2}{19} a^{14} - \frac{6}{19} a^{13} - \frac{6}{19} a^{12} + \frac{5}{19} a^{11} + \frac{5}{19} a^{10} - \frac{7}{19} a^{9} + \frac{6}{19} a^{8} - \frac{2}{19} a^{7} - \frac{1}{19} a^{6} + \frac{9}{19} a^{5} + \frac{4}{19} a^{4} + \frac{2}{19} a^{3} - \frac{1}{19} a^{2} - \frac{8}{19}$, $\frac{1}{19} a^{17} - \frac{7}{19} a^{15} - \frac{5}{19} a^{13} + \frac{6}{19} a^{12} + \frac{1}{19} a^{11} + \frac{8}{19} a^{10} + \frac{4}{19} a^{9} - \frac{3}{19} a^{8} - \frac{7}{19} a^{7} + \frac{6}{19} a^{6} - \frac{7}{19} a^{5} - \frac{5}{19} a^{4} + \frac{5}{19} a^{3} - \frac{3}{19} a^{2} - \frac{8}{19} a - \frac{5}{19}$, $\frac{1}{19} a^{18} - \frac{2}{19} a^{15} + \frac{9}{19} a^{14} + \frac{2}{19} a^{13} - \frac{3}{19} a^{12} + \frac{5}{19} a^{11} + \frac{1}{19} a^{10} + \frac{5}{19} a^{9} - \frac{3}{19} a^{8} - \frac{8}{19} a^{7} + \frac{5}{19} a^{6} + \frac{1}{19} a^{5} - \frac{5}{19} a^{4} - \frac{8}{19} a^{3} + \frac{4}{19} a^{2} - \frac{5}{19} a + \frac{1}{19}$, $\frac{1}{15512369205008039820354470756707694779} a^{19} - \frac{323252816855046070455389182688129516}{15512369205008039820354470756707694779} a^{18} + \frac{228568345040343823113967698155206840}{15512369205008039820354470756707694779} a^{17} + \frac{9875554837163375767244664536011767}{816440484474107358966024776668826041} a^{16} - \frac{6235113483377371664772176503554996236}{15512369205008039820354470756707694779} a^{15} - \frac{122178574679704887050251438836464324}{15512369205008039820354470756707694779} a^{14} - \frac{1476555159434645100164456683350338014}{15512369205008039820354470756707694779} a^{13} + \frac{3371131001866041241379831279730066107}{15512369205008039820354470756707694779} a^{12} - \frac{4639066402397403657397313451609707101}{15512369205008039820354470756707694779} a^{11} - \frac{6815730492821960225415334958385178807}{15512369205008039820354470756707694779} a^{10} - \frac{1117009524367977755880157439057141537}{15512369205008039820354470756707694779} a^{9} - \frac{3065720927126047345366829822029928746}{15512369205008039820354470756707694779} a^{8} - \frac{4803278884894523203016612453513018358}{15512369205008039820354470756707694779} a^{7} + \frac{782561422108092912947815474263345800}{15512369205008039820354470756707694779} a^{6} - \frac{99096795565569851985925441562595662}{15512369205008039820354470756707694779} a^{5} - \frac{885770062296444773556619676142698479}{15512369205008039820354470756707694779} a^{4} + \frac{2259070241367831694113368303013671357}{15512369205008039820354470756707694779} a^{3} + \frac{4659113921577682132592373823052802805}{15512369205008039820354470756707694779} a^{2} - \frac{3335523516769060600562477160408302914}{15512369205008039820354470756707694779} a + \frac{27936620578177830466262836616446355}{15512369205008039820354470756707694779}$

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

Class group and class number

$C_{2}\times C_{58}$, which has order $116$ (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:  \( \frac{462818769889456179651310650579408872}{15512369205008039820354470756707694779} a^{19} - \frac{2326769500830031245644024226778428991}{15512369205008039820354470756707694779} a^{18} + \frac{6085575599032599985406446477638831728}{15512369205008039820354470756707694779} a^{17} - \frac{8053672957771882543603366298796728394}{15512369205008039820354470756707694779} a^{16} + \frac{43756386035050661857706352506913597896}{15512369205008039820354470756707694779} a^{15} - \frac{71071874341284803146102106521707118592}{15512369205008039820354470756707694779} a^{14} + \frac{190193241180812691445636467388065236429}{15512369205008039820354470756707694779} a^{13} - \frac{298259079478417551045537111755855720027}{15512369205008039820354470756707694779} a^{12} + \frac{1699680945567312614727430986045645406878}{15512369205008039820354470756707694779} a^{11} - \frac{403976620459062232554254939005395478672}{15512369205008039820354470756707694779} a^{10} + \frac{6238463018903085141089317736446009784682}{15512369205008039820354470756707694779} a^{9} + \frac{2499055278537939215613511471752905341233}{15512369205008039820354470756707694779} a^{8} + \frac{11448874579390192561509181938382218220782}{15512369205008039820354470756707694779} a^{7} + \frac{6933857183413525225748035473749834037042}{15512369205008039820354470756707694779} a^{6} + \frac{10074706929359956358303595959856066709952}{15512369205008039820354470756707694779} a^{5} + \frac{7034096685754392810557671000997154146244}{15512369205008039820354470756707694779} a^{4} + \frac{4294624621290086941447127430527707227167}{15512369205008039820354470756707694779} a^{3} + \frac{1532784301452226441723579802687424978039}{15512369205008039820354470756707694779} a^{2} + \frac{380233125383883954419361157061624425244}{15512369205008039820354470756707694779} a + \frac{1934096680512204750531796611047313906}{15512369205008039820354470756707694779} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 10123076.9997 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.12852225.1, 10.10.825898437253125.1

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 $20$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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$$[\ ]_{2}^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
239Data not computed