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Properties

Label 20.0.34425542535...000.14
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}$
Root discriminant $94.81$
Ramified primes $2, 3, 5, 11$
Class number $130328$ (GRH)
Class group $[2, 65164]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![762871264, -218747328, 285885936, -143467744, 96711800, -35695792, 21822312, -9383840, 3679294, -1258764, 628957, -220234, 56457, -17312, 8274, -2332, 368, -84, 41, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 41*x^18 - 84*x^17 + 368*x^16 - 2332*x^15 + 8274*x^14 - 17312*x^13 + 56457*x^12 - 220234*x^11 + 628957*x^10 - 1258764*x^9 + 3679294*x^8 - 9383840*x^7 + 21822312*x^6 - 35695792*x^5 + 96711800*x^4 - 143467744*x^3 + 285885936*x^2 - 218747328*x + 762871264)
 
gp: K = bnfinit(x^20 - 10*x^19 + 41*x^18 - 84*x^17 + 368*x^16 - 2332*x^15 + 8274*x^14 - 17312*x^13 + 56457*x^12 - 220234*x^11 + 628957*x^10 - 1258764*x^9 + 3679294*x^8 - 9383840*x^7 + 21822312*x^6 - 35695792*x^5 + 96711800*x^4 - 143467744*x^3 + 285885936*x^2 - 218747328*x + 762871264, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 41 x^{18} - 84 x^{17} + 368 x^{16} - 2332 x^{15} + 8274 x^{14} - 17312 x^{13} + 56457 x^{12} - 220234 x^{11} + 628957 x^{10} - 1258764 x^{9} + 3679294 x^{8} - 9383840 x^{7} + 21822312 x^{6} - 35695792 x^{5} + 96711800 x^{4} - 143467744 x^{3} + 285885936 x^{2} - 218747328 x + 762871264 \)

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:  \(3442554253548220860035959357440000000000=2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.81$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1320=2^{3}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1320}(1,·)$, $\chi_{1320}(899,·)$, $\chi_{1320}(211,·)$, $\chi_{1320}(961,·)$, $\chi_{1320}(841,·)$, $\chi_{1320}(1169,·)$, $\chi_{1320}(659,·)$, $\chi_{1320}(1049,·)$, $\chi_{1320}(89,·)$, $\chi_{1320}(1051,·)$, $\chi_{1320}(571,·)$, $\chi_{1320}(929,·)$, $\chi_{1320}(931,·)$, $\chi_{1320}(361,·)$, $\chi_{1320}(811,·)$, $\chi_{1320}(449,·)$, $\chi_{1320}(1139,·)$, $\chi_{1320}(1081,·)$, $\chi_{1320}(1019,·)$, $\chi_{1320}(299,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{6}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{7}$, $\frac{1}{688} a^{16} - \frac{1}{86} a^{15} + \frac{1}{86} a^{14} - \frac{1}{344} a^{13} + \frac{5}{344} a^{12} - \frac{27}{344} a^{11} + \frac{5}{172} a^{10} - \frac{9}{344} a^{9} - \frac{35}{688} a^{8} - \frac{81}{344} a^{7} + \frac{7}{172} a^{6} - \frac{17}{172} a^{5} + \frac{10}{43} a^{4} - \frac{27}{86} a^{3} - \frac{15}{43} a^{2} - \frac{7}{43} a - \frac{16}{43}$, $\frac{1}{688} a^{17} + \frac{15}{344} a^{15} - \frac{3}{86} a^{14} - \frac{3}{344} a^{13} + \frac{13}{344} a^{12} - \frac{17}{172} a^{11} - \frac{15}{344} a^{10} - \frac{7}{688} a^{9} + \frac{37}{344} a^{8} + \frac{11}{344} a^{7} + \frac{35}{344} a^{6} + \frac{33}{172} a^{5} - \frac{35}{172} a^{4} - \frac{31}{86} a^{3} - \frac{39}{86} a^{2} + \frac{14}{43} a + \frac{1}{43}$, $\frac{1}{338269495001706265316863792} a^{18} - \frac{9}{338269495001706265316863792} a^{17} + \frac{195502883023995830638609}{338269495001706265316863792} a^{16} - \frac{391005766047991661277167}{84567373750426566329215948} a^{15} + \frac{9825460851850971534096015}{169134747500853132658431896} a^{14} + \frac{1041563270192854019117543}{21141843437606641582303987} a^{13} + \frac{96176218258611690936575}{84567373750426566329215948} a^{12} - \frac{18207158803887549961860321}{169134747500853132658431896} a^{11} - \frac{25065404024718428061305561}{338269495001706265316863792} a^{10} - \frac{14405031502383831675633619}{338269495001706265316863792} a^{9} - \frac{17315160566981531065469625}{338269495001706265316863792} a^{8} - \frac{26564529790511440721970149}{169134747500853132658431896} a^{7} - \frac{27746998925499177214988091}{169134747500853132658431896} a^{6} - \frac{3170127383205331990515017}{84567373750426566329215948} a^{5} - \frac{19089167995290381561637513}{84567373750426566329215948} a^{4} - \frac{1104004225684383721063019}{42283686875213283164607974} a^{3} - \frac{4976535174509816597999462}{21141843437606641582303987} a^{2} + \frac{357195970320922969058678}{21141843437606641582303987} a + \frac{8168576523610084290415535}{21141843437606641582303987}$, $\frac{1}{9244371847403014540329482741160016} a^{19} + \frac{6832101}{4622185923701507270164741370580008} a^{18} - \frac{5316103061772311295378752609797}{9244371847403014540329482741160016} a^{17} + \frac{51888394456184313629756568076}{577773240462688408770592671322501} a^{16} + \frac{287024108053151950260147136456293}{4622185923701507270164741370580008} a^{15} - \frac{43039570242834465151470488383537}{1155546480925376817541185342645002} a^{14} + \frac{115547271243325461400351238431949}{2311092961850753635082370685290004} a^{13} + \frac{68673074134370112183204132924531}{4622185923701507270164741370580008} a^{12} - \frac{703769517377491083015165327944009}{9244371847403014540329482741160016} a^{11} + \frac{14650022348863754000089882892403}{577773240462688408770592671322501} a^{10} - \frac{5893172561271547860033720426697}{214985391800070105589057738166512} a^{9} - \frac{343990211333280178589813155054239}{4622185923701507270164741370580008} a^{8} + \frac{857295116316530813354242882591571}{4622185923701507270164741370580008} a^{7} - \frac{195000071755906999944852901047989}{4622185923701507270164741370580008} a^{6} + \frac{409847457420903964916886326019587}{2311092961850753635082370685290004} a^{5} + \frac{136513133670785708175389993759891}{1155546480925376817541185342645002} a^{4} - \frac{48431397109645422431442432263771}{577773240462688408770592671322501} a^{3} - \frac{65499880402198532376206498469327}{1155546480925376817541185342645002} a^{2} - \frac{273394480536784766674784082623179}{577773240462688408770592671322501} a - \frac{222633480435993725827555667584732}{577773240462688408770592671322501}$

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

Class group and class number

$C_{2}\times C_{65164}$, which has order $130328$ (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:  \( 1589230.00872 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-330}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-15}, \sqrt{22})\), \(\Q(\zeta_{11})^+\), 10.0.58673283984691200000.1, 10.10.77265229938688.1, 10.0.162778775259375.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5Data not computed
11Data not computed