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Properties

Label 20.0.15013659868...3125.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{13}\cdot 97^{2}\cdot 1039^{2}\cdot 1049^{4}$
Root discriminant $36.21$
Ramified primes $5, 97, 1039, 1049$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1039

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4326689, -4233714, 3568744, 359049, -1135731, 1208576, -69140, -209120, 243515, -82331, 21460, 1165, -3887, 2216, -816, 58, 54, -27, 22, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 22*x^18 - 27*x^17 + 54*x^16 + 58*x^15 - 816*x^14 + 2216*x^13 - 3887*x^12 + 1165*x^11 + 21460*x^10 - 82331*x^9 + 243515*x^8 - 209120*x^7 - 69140*x^6 + 1208576*x^5 - 1135731*x^4 + 359049*x^3 + 3568744*x^2 - 4233714*x + 4326689)
 
gp: K = bnfinit(x^20 - 2*x^19 + 22*x^18 - 27*x^17 + 54*x^16 + 58*x^15 - 816*x^14 + 2216*x^13 - 3887*x^12 + 1165*x^11 + 21460*x^10 - 82331*x^9 + 243515*x^8 - 209120*x^7 - 69140*x^6 + 1208576*x^5 - 1135731*x^4 + 359049*x^3 + 3568744*x^2 - 4233714*x + 4326689, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 22 x^{18} - 27 x^{17} + 54 x^{16} + 58 x^{15} - 816 x^{14} + 2216 x^{13} - 3887 x^{12} + 1165 x^{11} + 21460 x^{10} - 82331 x^{9} + 243515 x^{8} - 209120 x^{7} - 69140 x^{6} + 1208576 x^{5} - 1135731 x^{4} + 359049 x^{3} + 3568744 x^{2} - 4233714 x + 4326689 \)

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:  \(15013659868612228667344970703125=5^{13}\cdot 97^{2}\cdot 1039^{2}\cdot 1049^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.21$
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:  $5, 97, 1039, 1049$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{300980726658663510396044049272317920275179814106501638415945063451} a^{19} - \frac{30824326463036933973680347254181638217232539400408143046367814410}{300980726658663510396044049272317920275179814106501638415945063451} a^{18} + \frac{110175514171742317113027380090361294194390096654704048671316592880}{300980726658663510396044049272317920275179814106501638415945063451} a^{17} - \frac{68623917590244924414820381504037236912933240954025687094164761683}{300980726658663510396044049272317920275179814106501638415945063451} a^{16} + \frac{111745328625039044359693607348922036137148921252280394079147373483}{300980726658663510396044049272317920275179814106501638415945063451} a^{15} - \frac{40003569117192732128639576336055365846737664048481987978933290474}{300980726658663510396044049272317920275179814106501638415945063451} a^{14} - \frac{87702606562607840577368699945236475252102525765075118019197857379}{300980726658663510396044049272317920275179814106501638415945063451} a^{13} + \frac{141645614929865244440205468457734052134298823733988396403315692062}{300980726658663510396044049272317920275179814106501638415945063451} a^{12} - \frac{58938095308397292177351242691904561787477979340360495475921141302}{300980726658663510396044049272317920275179814106501638415945063451} a^{11} - \frac{45729877846469912713378782125051260343534848171874359768866113741}{300980726658663510396044049272317920275179814106501638415945063451} a^{10} - \frac{2656474093262700289887312741367875150529598547468081204021850487}{7340993333138134399903513396885802933540971075768332644291343011} a^{9} - \frac{24066384613884366791849588762449609231319822370354852364435201017}{300980726658663510396044049272317920275179814106501638415945063451} a^{8} - \frac{144807866249544384686413776422974829215479467451924387727364531753}{300980726658663510396044049272317920275179814106501638415945063451} a^{7} + \frac{136525254493258821472051929834794196163542488793160188717715168310}{300980726658663510396044049272317920275179814106501638415945063451} a^{6} + \frac{74799818647701322523978585808161082100896764769141018588044815369}{300980726658663510396044049272317920275179814106501638415945063451} a^{5} - \frac{88156777395043839872180059930308422587969382401406114113870965007}{300980726658663510396044049272317920275179814106501638415945063451} a^{4} - \frac{109476203376594818624611749645913584305116390441270070902772376559}{300980726658663510396044049272317920275179814106501638415945063451} a^{3} + \frac{84354616768623049676457967598228801933668614791505359671271482637}{300980726658663510396044049272317920275179814106501638415945063451} a^{2} - \frac{65421755833390424051698729378790943882913497805729301245726734236}{300980726658663510396044049272317920275179814106501638415945063451} a + \frac{1587835093692527324182353770712132985445770498460026493460437868}{7340993333138134399903513396885802933540971075768332644291343011}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -1 \) (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:  \( 6261261.99419 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1039:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1039 are not computed
Character table for t20n1039 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.4.3405971875.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
1039Data not computed
1049Data not computed