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Properties

Label 20.0.26694195914...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{15}\cdot 61^{19}$
Root discriminant $332.14$
Ramified primes $2, 5, 61$
Class number $122048800$ (GRH)
Class group $[2, 2, 122, 250100]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21649090625, 0, 148205028125, 0, 163018230000, 0, 68522062500, 0, 13257358750, 0, 1315892000, 0, 71180900, 0, 2154825, 0, 35990, 0, 305, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 305*x^18 + 35990*x^16 + 2154825*x^14 + 71180900*x^12 + 1315892000*x^10 + 13257358750*x^8 + 68522062500*x^6 + 163018230000*x^4 + 148205028125*x^2 + 21649090625)
 
gp: K = bnfinit(x^20 + 305*x^18 + 35990*x^16 + 2154825*x^14 + 71180900*x^12 + 1315892000*x^10 + 13257358750*x^8 + 68522062500*x^6 + 163018230000*x^4 + 148205028125*x^2 + 21649090625, 1)
 

Normalized defining polynomial

\( x^{20} + 305 x^{18} + 35990 x^{16} + 2154825 x^{14} + 71180900 x^{12} + 1315892000 x^{10} + 13257358750 x^{8} + 68522062500 x^{6} + 163018230000 x^{4} + 148205028125 x^{2} + 21649090625 \)

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:  \(266941959144749707507722683125783712000000000000000=2^{20}\cdot 5^{15}\cdot 61^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $332.14$
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:  $2, 5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1220=2^{2}\cdot 5\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{1220}(1,·)$, $\chi_{1220}(643,·)$, $\chi_{1220}(81,·)$, $\chi_{1220}(1089,·)$, $\chi_{1220}(587,·)$, $\chi_{1220}(461,·)$, $\chi_{1220}(1167,·)$, $\chi_{1220}(529,·)$, $\chi_{1220}(149,·)$, $\chi_{1220}(663,·)$, $\chi_{1220}(987,·)$, $\chi_{1220}(1183,·)$, $\chi_{1220}(609,·)$, $\chi_{1220}(1187,·)$, $\chi_{1220}(741,·)$, $\chi_{1220}(241,·)$, $\chi_{1220}(843,·)$, $\chi_{1220}(647,·)$, $\chi_{1220}(369,·)$, $\chi_{1220}(23,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{625} a^{16}$, $\frac{1}{625} a^{17}$, $\frac{1}{159535818330383668572406423824556048270625} a^{18} + \frac{55247626603282265639203863096684338228}{159535818330383668572406423824556048270625} a^{16} + \frac{16482952491920828175864060637020991511}{6381432733215346742896256952982241930825} a^{14} - \frac{127138908515736037609300654010544940818}{31907163666076733714481284764911209654125} a^{12} - \frac{3887049003818501549911223951342748108}{1276286546643069348579251390596448386165} a^{10} - \frac{35174362709428064613803314921898242728}{6381432733215346742896256952982241930825} a^{8} + \frac{5450527617126042537804194545561922013}{255257309328613869715850278119289677233} a^{6} + \frac{15376643292111494086002730151328690151}{255257309328613869715850278119289677233} a^{4} - \frac{69506571274289902895103773528848804371}{255257309328613869715850278119289677233} a^{2} - \frac{85615502040168296093419040069195704656}{255257309328613869715850278119289677233}$, $\frac{1}{53763570777339296308900964828875388267200625} a^{19} - \frac{926926700892023195140126857830347051599}{2150542831093571852356038593155015530688025} a^{17} + \frac{41689356183023664904562915636629322346534}{10752714155467859261780192965775077653440125} a^{15} - \frac{14421548230918112741696916228690766865866}{10752714155467859261780192965775077653440125} a^{13} - \frac{5379838740919983771782411960261796962433}{2150542831093571852356038593155015530688025} a^{11} - \frac{1998041885305073796706392832314839130963}{430108566218714370471207718631003106137605} a^{9} + \frac{36784305181406027451771461021905523131617}{430108566218714370471207718631003106137605} a^{7} + \frac{6652066685836072106698109961252860298209}{86021713243742874094241543726200621227521} a^{5} + \frac{12948616204485017452613260410554924734512}{86021713243742874094241543726200621227521} a^{3} + \frac{23398056956192307717764806546905454600780}{86021713243742874094241543726200621227521} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{122}\times C_{250100}$, which has order $122048800$ (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:  \( 398872163.22998685 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

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

Intermediate fields

\(\Q(\sqrt{305}) \), 4.0.453962000.2, 5.5.13845841.1, 10.10.36544206540106690625.1

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 $20$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
5Data not computed
61Data not computed