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Properties

Label 20.0.14399360760...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{15}\cdot 41^{19}$
Root discriminant $322.05$
Ramified primes $2, 5, 41$
Class number $26613490832$ (GRH)
Class group $[2, 2, 6653372708]$ (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![95644800000, 0, 350697600000, 0, 427036320000, 0, 201697040000, 0, 41636320000, 0, 3804144000, 0, 177726800, 0, 4510000, 0, 61500, 0, 410, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 410*x^18 + 61500*x^16 + 4510000*x^14 + 177726800*x^12 + 3804144000*x^10 + 41636320000*x^8 + 201697040000*x^6 + 427036320000*x^4 + 350697600000*x^2 + 95644800000)
 
gp: K = bnfinit(x^20 + 410*x^18 + 61500*x^16 + 4510000*x^14 + 177726800*x^12 + 3804144000*x^10 + 41636320000*x^8 + 201697040000*x^6 + 427036320000*x^4 + 350697600000*x^2 + 95644800000, 1)
 

Normalized defining polynomial

\( x^{20} + 410 x^{18} + 61500 x^{16} + 4510000 x^{14} + 177726800 x^{12} + 3804144000 x^{10} + 41636320000 x^{8} + 201697040000 x^{6} + 427036320000 x^{4} + 350697600000 x^{2} + 95644800000 \)

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:  \(143993607607707341901348341362229248000000000000000=2^{30}\cdot 5^{15}\cdot 41^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $322.05$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1640=2^{3}\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{1640}(1,·)$, $\chi_{1640}(197,·)$, $\chi_{1640}(1089,·)$, $\chi_{1640}(961,·)$, $\chi_{1640}(201,·)$, $\chi_{1640}(1281,·)$, $\chi_{1640}(717,·)$, $\chi_{1640}(77,·)$, $\chi_{1640}(209,·)$, $\chi_{1640}(237,·)$, $\chi_{1640}(213,·)$, $\chi_{1640}(333,·)$, $\chi_{1640}(409,·)$, $\chi_{1640}(1437,·)$, $\chi_{1640}(613,·)$, $\chi_{1640}(1041,·)$, $\chi_{1640}(173,·)$, $\chi_{1640}(1009,·)$, $\chi_{1640}(1333,·)$, $\chi_{1640}(769,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{6} a^{3} + \frac{1}{3} a$, $\frac{1}{60} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{60} a^{5} + \frac{1}{3} a$, $\frac{1}{360} a^{6} - \frac{1}{180} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{360} a^{7} - \frac{1}{180} a^{5} - \frac{1}{18} a^{3} + \frac{1}{3} a$, $\frac{1}{10800} a^{8} + \frac{1}{180} a^{4} + \frac{1}{54} a^{2} + \frac{1}{3}$, $\frac{1}{32400} a^{9} - \frac{1}{1080} a^{7} - \frac{1}{135} a^{5} + \frac{2}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{64800} a^{10} - \frac{1}{1080} a^{6} + \frac{1}{810} a^{4} - \frac{13}{54} a^{2} - \frac{1}{3}$, $\frac{1}{64800} a^{11} - \frac{1}{1080} a^{7} + \frac{1}{810} a^{5} - \frac{2}{27} a^{3}$, $\frac{1}{5832000} a^{12} - \frac{1}{583200} a^{10} + \frac{1}{97200} a^{8} + \frac{17}{14580} a^{6} - \frac{113}{14580} a^{4} - \frac{17}{162} a^{2} + \frac{2}{9}$, $\frac{1}{17496000} a^{13} - \frac{1}{1749600} a^{11} + \frac{1}{291600} a^{9} - \frac{47}{87480} a^{7} - \frac{8}{10935} a^{5} + \frac{19}{486} a^{3} - \frac{7}{27} a$, $\frac{1}{34992000} a^{14} + \frac{1}{17496000} a^{12} - \frac{1}{583200} a^{10} + \frac{13}{437400} a^{8} - \frac{71}{87480} a^{6} - \frac{103}{14580} a^{4} - \frac{37}{162} a^{2} + \frac{4}{9}$, $\frac{1}{34992000} a^{15} - \frac{1}{874800} a^{11} - \frac{1}{218700} a^{9} + \frac{19}{29160} a^{7} + \frac{47}{43740} a^{5} + \frac{10}{243} a^{3} + \frac{7}{27} a$, $\frac{1}{9447840000} a^{16} + \frac{1}{944784000} a^{14} - \frac{1}{29524500} a^{12} - \frac{11}{1889568} a^{10} + \frac{1}{118098} a^{8} - \frac{1193}{1180980} a^{6} + \frac{17}{4374} a^{4} - \frac{89}{486} a^{2} - \frac{11}{81}$, $\frac{1}{9447840000} a^{17} + \frac{1}{944784000} a^{15} + \frac{11}{472392000} a^{13} - \frac{151}{23619600} a^{11} + \frac{281}{23619600} a^{9} + \frac{1453}{1180980} a^{7} - \frac{7}{2916} a^{5} - \frac{8}{243} a^{3} + \frac{22}{81} a$, $\frac{1}{57383893782720000} a^{18} + \frac{126329}{9563982297120000} a^{16} - \frac{3084179}{239099557428000} a^{14} - \frac{49291709}{1434597344568000} a^{12} - \frac{4991617}{11954977871400} a^{10} - \frac{51997313}{2988744467850} a^{8} + \frac{1948065721}{1434597344568} a^{6} - \frac{112945819}{33208271865} a^{4} + \frac{183429151}{1475923194} a^{2} - \frac{23511395}{245987199}$, $\frac{1}{57383893782720000} a^{19} + \frac{126329}{9563982297120000} a^{17} - \frac{3084179}{239099557428000} a^{15} + \frac{4088003}{179324668071000} a^{13} - \frac{47298379}{47819911485600} a^{11} - \frac{333982771}{23909955742800} a^{9} + \frac{2943264577}{3586493361420} a^{7} - \frac{137240851}{33208271865} a^{5} - \frac{4857347}{1475923194} a^{3} + \frac{76705612}{245987199} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6653372708}$, which has order $26613490832$ (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:  \( 3541438824.6395073 \) (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

\(\Q(\sqrt{205}) \), 4.0.551368000.3, 5.5.2825761.1, 10.10.1023068544981128125.1

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 ${\href{/LocalNumberField/3.1.0.1}{1} }^{20}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ $20$ $20$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ R $20$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
41Data not computed