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Properties

Label 20.0.61917364224...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 3^{10}\cdot 5^{35}$
Root discriminant $194.80$
Ramified primes $2, 3, 5$
Class number $13351696$ (GRH)
Class group $[2, 2, 3337924]$ (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![1519222119840, 0, 1327547491200, 0, 481236228000, 0, 95094259200, 0, 11359132200, 0, 858839760, 0, 41666400, 0, 1279800, 0, 23760, 0, 240, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 240*x^18 + 23760*x^16 + 1279800*x^14 + 41666400*x^12 + 858839760*x^10 + 11359132200*x^8 + 95094259200*x^6 + 481236228000*x^4 + 1327547491200*x^2 + 1519222119840)
 
gp: K = bnfinit(x^20 + 240*x^18 + 23760*x^16 + 1279800*x^14 + 41666400*x^12 + 858839760*x^10 + 11359132200*x^8 + 95094259200*x^6 + 481236228000*x^4 + 1327547491200*x^2 + 1519222119840, 1)
 

Normalized defining polynomial

\( x^{20} + 240 x^{18} + 23760 x^{16} + 1279800 x^{14} + 41666400 x^{12} + 858839760 x^{10} + 11359132200 x^{8} + 95094259200 x^{6} + 481236228000 x^{4} + 1327547491200 x^{2} + 1519222119840 \)

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:  \(6191736422400000000000000000000000000000000000=2^{55}\cdot 3^{10}\cdot 5^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $194.80$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1200=2^{4}\cdot 3\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{1200}(1,·)$, $\chi_{1200}(83,·)$, $\chi_{1200}(323,·)$, $\chi_{1200}(961,·)$, $\chi_{1200}(649,·)$, $\chi_{1200}(587,·)$, $\chi_{1200}(721,·)$, $\chi_{1200}(1043,·)$, $\chi_{1200}(409,·)$, $\chi_{1200}(347,·)$, $\chi_{1200}(481,·)$, $\chi_{1200}(803,·)$, $\chi_{1200}(241,·)$, $\chi_{1200}(1129,·)$, $\chi_{1200}(107,·)$, $\chi_{1200}(1067,·)$, $\chi_{1200}(563,·)$, $\chi_{1200}(169,·)$, $\chi_{1200}(889,·)$, $\chi_{1200}(827,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{18} a^{4}$, $\frac{1}{18} a^{5}$, $\frac{1}{54} a^{6}$, $\frac{1}{54} a^{7}$, $\frac{1}{324} a^{8}$, $\frac{1}{324} a^{9}$, $\frac{1}{972} a^{10}$, $\frac{1}{972} a^{11}$, $\frac{1}{5832} a^{12}$, $\frac{1}{5832} a^{13}$, $\frac{1}{17496} a^{14}$, $\frac{1}{17496} a^{15}$, $\frac{1}{20890224} a^{16} + \frac{17}{870426} a^{14} - \frac{5}{145071} a^{12} - \frac{1}{32238} a^{10} + \frac{67}{64476} a^{8} + \frac{59}{10746} a^{6} + \frac{37}{3582} a^{4} + \frac{53}{597} a^{2} + \frac{92}{199}$, $\frac{1}{20890224} a^{17} + \frac{17}{870426} a^{15} - \frac{5}{145071} a^{13} - \frac{1}{32238} a^{11} + \frac{67}{64476} a^{9} + \frac{59}{10746} a^{7} + \frac{37}{3582} a^{5} + \frac{53}{597} a^{3} + \frac{92}{199} a$, $\frac{1}{180012265713450606096} a^{18} - \frac{51001040537}{15001022142787550508} a^{16} - \frac{97805107086991}{5000340714262516836} a^{14} + \frac{84237471919313}{1111186825391670408} a^{12} + \frac{69536343209071}{277796706347917602} a^{10} + \frac{89338426271681}{92598902115972534} a^{8} - \frac{15231844320487}{1714794483629121} a^{6} - \frac{4920828031068}{571598161209707} a^{4} + \frac{39995792588778}{571598161209707} a^{2} - \frac{275959384749558}{571598161209707}$, $\frac{1}{72184918551093693044496} a^{19} + \frac{4283028333071}{3007704939628903876854} a^{17} - \frac{69081581398061}{2005136626419269251236} a^{15} + \frac{11579157640773745}{334189437736544875206} a^{13} + \frac{7789262637018611}{74264319497009972268} a^{11} + \frac{4673613126224909}{37132159748504986134} a^{9} - \frac{45683194959398203}{12377386582834995378} a^{7} + \frac{38890451619065003}{2062897763805832563} a^{5} - \frac{19202328353176877}{687632587935277521} a^{3} - \frac{29114379176435278}{229210862645092507} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{3337924}$, which has order $13351696$ (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:  \( 19344397.966990974 \) (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

\(\Q(\sqrt{10}) \), 4.0.2304000.1, 5.5.390625.1, 10.10.25000000000000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
2Data not computed
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5Data not computed