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Properties

Label 20.4.17533216858...3125.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{10}\cdot 5^{11}\cdot 239^{10}$
Root discriminant $64.89$
Ramified primes $3, 5, 239$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![786559, -1370685, 1568454, -1205067, 999706, 56136, -742381, 970940, -688895, 516232, -225519, 84298, -28502, -329, 1571, -1520, 669, -163, 49, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 49*x^18 - 163*x^17 + 669*x^16 - 1520*x^15 + 1571*x^14 - 329*x^13 - 28502*x^12 + 84298*x^11 - 225519*x^10 + 516232*x^9 - 688895*x^8 + 970940*x^7 - 742381*x^6 + 56136*x^5 + 999706*x^4 - 1205067*x^3 + 1568454*x^2 - 1370685*x + 786559)
 
gp: K = bnfinit(x^20 - 5*x^19 + 49*x^18 - 163*x^17 + 669*x^16 - 1520*x^15 + 1571*x^14 - 329*x^13 - 28502*x^12 + 84298*x^11 - 225519*x^10 + 516232*x^9 - 688895*x^8 + 970940*x^7 - 742381*x^6 + 56136*x^5 + 999706*x^4 - 1205067*x^3 + 1568454*x^2 - 1370685*x + 786559, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 49 x^{18} - 163 x^{17} + 669 x^{16} - 1520 x^{15} + 1571 x^{14} - 329 x^{13} - 28502 x^{12} + 84298 x^{11} - 225519 x^{10} + 516232 x^{9} - 688895 x^{8} + 970940 x^{7} - 742381 x^{6} + 56136 x^{5} + 999706 x^{4} - 1205067 x^{3} + 1568454 x^{2} - 1370685 x + 786559 \)

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:  \(1753321685810638349237472141064453125=3^{10}\cdot 5^{11}\cdot 239^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.89$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{25} a^{14} - \frac{1}{25} a^{13} - \frac{8}{25} a^{11} + \frac{7}{25} a^{10} + \frac{12}{25} a^{9} + \frac{4}{25} a^{8} - \frac{2}{25} a^{7} - \frac{2}{25} a^{6} - \frac{9}{25} a^{5} + \frac{1}{25} a^{4} + \frac{1}{5} a^{3} - \frac{2}{25} a^{2} - \frac{9}{25} a + \frac{4}{25}$, $\frac{1}{25} a^{15} - \frac{1}{25} a^{13} + \frac{2}{25} a^{12} - \frac{6}{25} a^{11} - \frac{6}{25} a^{10} - \frac{9}{25} a^{9} - \frac{3}{25} a^{8} + \frac{1}{25} a^{7} - \frac{1}{25} a^{6} - \frac{3}{25} a^{5} - \frac{9}{25} a^{4} - \frac{12}{25} a^{3} - \frac{6}{25} a^{2} - \frac{6}{25}$, $\frac{1}{89625} a^{16} - \frac{4}{89625} a^{15} + \frac{499}{89625} a^{14} + \frac{6496}{89625} a^{13} + \frac{1631}{89625} a^{12} + \frac{22273}{89625} a^{11} + \frac{706}{5975} a^{10} + \frac{33638}{89625} a^{9} + \frac{921}{29875} a^{8} + \frac{7426}{17925} a^{7} - \frac{4588}{29875} a^{6} - \frac{31862}{89625} a^{5} + \frac{12868}{29875} a^{4} - \frac{26348}{89625} a^{3} + \frac{13514}{89625} a^{2} + \frac{5624}{89625} a + \frac{42109}{89625}$, $\frac{1}{448125} a^{17} + \frac{2}{448125} a^{16} + \frac{812}{89625} a^{15} - \frac{1687}{89625} a^{14} + \frac{1172}{448125} a^{13} + \frac{21304}{448125} a^{12} + \frac{58831}{149375} a^{11} + \frac{219068}{448125} a^{10} + \frac{69392}{149375} a^{9} - \frac{10822}{448125} a^{8} - \frac{24733}{149375} a^{7} + \frac{114994}{448125} a^{6} + \frac{71034}{149375} a^{5} + \frac{101311}{448125} a^{4} + \frac{99206}{448125} a^{3} + \frac{118973}{448125} a^{2} + \frac{111703}{448125} a - \frac{6602}{149375}$, $\frac{1}{1344375} a^{18} - \frac{4}{1344375} a^{16} + \frac{1174}{89625} a^{15} - \frac{18223}{1344375} a^{14} + \frac{1949}{53775} a^{13} + \frac{7}{10755} a^{12} + \frac{156127}{1344375} a^{11} - \frac{55412}{268875} a^{10} + \frac{150482}{448125} a^{9} - \frac{99482}{268875} a^{8} - \frac{649333}{1344375} a^{7} - \frac{204446}{1344375} a^{6} + \frac{3178}{149375} a^{5} - \frac{44909}{149375} a^{4} + \frac{8197}{448125} a^{3} + \frac{520642}{1344375} a^{2} + \frac{423023}{1344375} a - \frac{671278}{1344375}$, $\frac{1}{896726630591584490134963685339184475490625} a^{19} - \frac{97519194100123921141596055529094926}{896726630591584490134963685339184475490625} a^{18} - \frac{75013077878346504862254286649337632}{179345326118316898026992737067836895098125} a^{17} - \frac{2249984362736033216958823299936360568}{896726630591584490134963685339184475490625} a^{16} - \frac{17320145700513772996464085746510228961963}{896726630591584490134963685339184475490625} a^{15} - \frac{5445417055689517097412629660947044130247}{896726630591584490134963685339184475490625} a^{14} + \frac{7923489634553956925366054772649634567173}{896726630591584490134963685339184475490625} a^{13} + \frac{28771706029166712772117685547880013356436}{298908876863861496711654561779728158496875} a^{12} + \frac{1753842812150388325152564115369256453896}{19927258457590766447443637451981877233125} a^{11} - \frac{430540835893239023112932798577225396723652}{896726630591584490134963685339184475490625} a^{10} + \frac{104203053801357721899335521062973762923878}{896726630591584490134963685339184475490625} a^{9} - \frac{407765476570084052624644816119268268651426}{896726630591584490134963685339184475490625} a^{8} + \frac{85846165257928314898852620318448093137077}{298908876863861496711654561779728158496875} a^{7} + \frac{171442083598663803043512193162709071474489}{896726630591584490134963685339184475490625} a^{6} + \frac{27945316123943392946105560289630453636453}{59781775372772299342330912355945631699375} a^{5} + \frac{98176920227158384864183089458608593125692}{298908876863861496711654561779728158496875} a^{4} + \frac{11919445784511202486688998243508778939238}{35869065223663379605398547413567379019625} a^{3} + \frac{18846439101720941134185578043766828973082}{99636292287953832237218187259909386165625} a^{2} + \frac{10137297225771019338235723264160866254301}{896726630591584490134963685339184475490625} a - \frac{196850298473823284968897031731340370917641}{896726630591584490134963685339184475490625}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4646462497.73 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 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

Degree 20 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 $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
239Data not computed