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Properties

Label 20.0.66362369058...000.11
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}$
Root discriminant $309.81$
Ramified primes $2, 3, 5, 41$
Class number $135692800$ (GRH)
Class group $[2, 2, 2, 2, 2, 4240400]$ (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![559121089536, 0, 17217802752, 0, 1597110191104, 0, 96595229824, 0, 5045005888, 0, 249048352, 0, 13285840, 0, 233896, 0, 4720, 0, 106, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 106*x^18 + 4720*x^16 + 233896*x^14 + 13285840*x^12 + 249048352*x^10 + 5045005888*x^8 + 96595229824*x^6 + 1597110191104*x^4 + 17217802752*x^2 + 559121089536)
 
gp: K = bnfinit(x^20 + 106*x^18 + 4720*x^16 + 233896*x^14 + 13285840*x^12 + 249048352*x^10 + 5045005888*x^8 + 96595229824*x^6 + 1597110191104*x^4 + 17217802752*x^2 + 559121089536, 1)
 

Normalized defining polynomial

\( x^{20} + 106 x^{18} + 4720 x^{16} + 233896 x^{14} + 13285840 x^{12} + 249048352 x^{10} + 5045005888 x^{8} + 96595229824 x^{6} + 1597110191104 x^{4} + 17217802752 x^{2} + 559121089536 \)

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:  \(66362369058556182102889507973449950167040000000000=2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $309.81$
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:  $2, 3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4920=2^{3}\cdot 3\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4920}(1,·)$, $\chi_{4920}(4699,·)$, $\chi_{4920}(2051,·)$, $\chi_{4920}(961,·)$, $\chi_{4920}(4049,·)$, $\chi_{4920}(4561,·)$, $\chi_{4920}(3139,·)$, $\chi_{4920}(3481,·)$, $\chi_{4920}(4459,·)$, $\chi_{4920}(4099,·)$, $\chi_{4920}(4321,·)$, $\chi_{4920}(611,·)$, $\chi_{4920}(1691,·)$, $\chi_{4920}(209,·)$, $\chi_{4920}(2729,·)$, $\chi_{4920}(4289,·)$, $\chi_{4920}(619,·)$, $\chi_{4920}(3689,·)$, $\chi_{4920}(1451,·)$, $\chi_{4920}(3011,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{24} a^{6} - \frac{1}{12} a^{4} + \frac{1}{6} a^{2}$, $\frac{1}{24} a^{7} - \frac{1}{12} a^{5} + \frac{1}{6} a^{3}$, $\frac{1}{48} a^{8} + \frac{1}{6} a^{2}$, $\frac{1}{144} a^{9} - \frac{1}{12} a^{5} + \frac{2}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{288} a^{10} + \frac{1}{36} a^{4}$, $\frac{1}{288} a^{11} + \frac{1}{36} a^{5}$, $\frac{1}{576} a^{12} + \frac{1}{72} a^{6}$, $\frac{1}{576} a^{13} + \frac{1}{72} a^{7}$, $\frac{1}{3456} a^{14} - \frac{1}{1728} a^{12} + \frac{1}{864} a^{10} + \frac{1}{108} a^{8} - \frac{1}{216} a^{6} + \frac{5}{54} a^{4} - \frac{1}{9} a^{2} + \frac{1}{3}$, $\frac{1}{3456} a^{15} - \frac{1}{1728} a^{13} + \frac{1}{864} a^{11} + \frac{1}{432} a^{9} - \frac{1}{216} a^{7} - \frac{2}{27} a^{5} + \frac{1}{6} a^{3} - \frac{1}{3} a$, $\frac{1}{20736} a^{16} + \frac{1}{1728} a^{12} - \frac{1}{2592} a^{10} + \frac{1}{108} a^{8} + \frac{1}{54} a^{6} - \frac{1}{162} a^{4} + \frac{2}{27} a^{2} + \frac{1}{9}$, $\frac{1}{161512704} a^{17} + \frac{37}{2243232} a^{15} - \frac{731}{3364848} a^{13} + \frac{26009}{20189088} a^{11} + \frac{1141}{3364848} a^{9} + \frac{21487}{1682424} a^{7} + \frac{298573}{2523636} a^{5} - \frac{7630}{210303} a^{3} + \frac{13024}{70101} a$, $\frac{1}{1483791102355389860482096296073415624350161408} a^{18} + \frac{2878896007206903393160834220050734185035}{185473887794423732560262037009176953043770176} a^{16} + \frac{6977143680619353673382918463618195492595}{123649258529615821706841358006117968695846784} a^{14} - \frac{29266046542064135255480387824360545132163}{185473887794423732560262037009176953043770176} a^{12} - \frac{8518103588348594955283947486839443183603}{92736943897211866280131018504588476521885088} a^{10} + \frac{75215231892077428131570565824233727952015}{7728078658100988856677584875382373043490424} a^{8} - \frac{93717939750771491410841734928492274063767}{23184235974302966570032754626147119130471272} a^{6} + \frac{1375647597884828516171589030059708677096}{2898029496787870821254094328268389891308909} a^{4} + \frac{180568233723412173576540258388671569006701}{1932019664525247214169396218845593260872606} a^{2} - \frac{8374248347361999127148509813135080239}{41340772553713510809461980974142878009}$, $\frac{1}{1483791102355389860482096296073415624350161408} a^{19} - \frac{117836967916605059017153372196056367}{741895551177694930241048148036707812175080704} a^{17} + \frac{5139906032871899851802746138498471595657}{61824629264807910853420679003058984347923392} a^{15} - \frac{7611982065659723041933234143405007117553}{92736943897211866280131018504588476521885088} a^{13} + \frac{78654460981470783285268413764776502797249}{46368471948605933140065509252294238260942544} a^{11} - \frac{37297984382258196322625723955115353134839}{15456157316201977713355169750764746086980848} a^{9} - \frac{41243514936839660487135615048591729435661}{11592117987151483285016377313073559565235636} a^{7} - \frac{7122177871489615372440040392402994024754}{2898029496787870821254094328268389891308909} a^{5} + \frac{101861829886332187232186676199372131616871}{966009832262623607084698109422796630436303} a^{3} + \frac{7799157628945988226287654124037912750327}{322003277420874535694899369807598876812101} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4240400}$, which has order $135692800$ (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:  \( 9452086796.91891 \) (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

\(\Q(\sqrt{-615}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-410}) \), \(\Q(\sqrt{6}, \sqrt{-410})\), 5.5.2825761.1, 10.0.248605656430414134375.1, 10.10.63580957267604373504.1, 10.0.33523910081941606400000.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
41Data not computed