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Properties

Label 20.0.51651070346...9984.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 61^{19}$
Root discriminant $172.05$
Ramified primes $2, 3, 61$
Class number $10120400$ (GRH)
Class group $[2, 2, 2530100]$ (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![3601989, 0, 81645084, 0, 511482438, 0, 601532163, 0, 284912883, 0, 64153944, 0, 7258329, 0, 421632, 0, 12627, 0, 183, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 183*x^18 + 12627*x^16 + 421632*x^14 + 7258329*x^12 + 64153944*x^10 + 284912883*x^8 + 601532163*x^6 + 511482438*x^4 + 81645084*x^2 + 3601989)
 
gp: K = bnfinit(x^20 + 183*x^18 + 12627*x^16 + 421632*x^14 + 7258329*x^12 + 64153944*x^10 + 284912883*x^8 + 601532163*x^6 + 511482438*x^4 + 81645084*x^2 + 3601989, 1)
 

Normalized defining polynomial

\( x^{20} + 183 x^{18} + 12627 x^{16} + 421632 x^{14} + 7258329 x^{12} + 64153944 x^{10} + 284912883 x^{8} + 601532163 x^{6} + 511482438 x^{4} + 81645084 x^{2} + 3601989 \)

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:  \(516510703469799849283535395746427778167209984=2^{20}\cdot 3^{10}\cdot 61^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $172.05$
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:  $2, 3, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(732=2^{2}\cdot 3\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{732}(1,·)$, $\chi_{732}(11,·)$, $\chi_{732}(325,·)$, $\chi_{732}(455,·)$, $\chi_{732}(587,·)$, $\chi_{732}(529,·)$, $\chi_{732}(599,·)$, $\chi_{732}(601,·)$, $\chi_{732}(155,·)$, $\chi_{732}(419,·)$, $\chi_{732}(613,·)$, $\chi_{732}(647,·)$, $\chi_{732}(253,·)$, $\chi_{732}(241,·)$, $\chi_{732}(695,·)$, $\chi_{732}(23,·)$, $\chi_{732}(121,·)$, $\chi_{732}(217,·)$, $\chi_{732}(637,·)$, $\chi_{732}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{3159} a^{10} - \frac{1}{1053} a^{8} + \frac{1}{351} a^{6} - \frac{1}{117} a^{4} + \frac{1}{39} a^{2} - \frac{1}{13}$, $\frac{1}{3159} a^{11} - \frac{1}{1053} a^{9} + \frac{1}{351} a^{7} - \frac{1}{117} a^{5} + \frac{1}{39} a^{3} - \frac{1}{13} a$, $\frac{1}{9477} a^{12} - \frac{1}{13}$, $\frac{1}{9477} a^{13} - \frac{1}{13} a$, $\frac{1}{369603} a^{14} - \frac{1}{123201} a^{12} + \frac{5}{41067} a^{10} - \frac{70}{13689} a^{8} + \frac{19}{1521} a^{6} - \frac{2}{169} a^{4} + \frac{43}{507} a^{2} - \frac{30}{169}$, $\frac{1}{369603} a^{15} - \frac{1}{123201} a^{13} + \frac{5}{41067} a^{11} - \frac{70}{13689} a^{9} + \frac{19}{1521} a^{7} - \frac{2}{169} a^{5} + \frac{43}{507} a^{3} - \frac{30}{169} a$, $\frac{1}{52114023} a^{16} - \frac{17}{17371341} a^{14} + \frac{190}{5790447} a^{12} - \frac{11}{643383} a^{10} + \frac{3257}{643383} a^{8} + \frac{877}{214461} a^{6} + \frac{1397}{71487} a^{4} - \frac{313}{7943} a^{2} + \frac{1377}{7943}$, $\frac{1}{52114023} a^{17} - \frac{17}{17371341} a^{15} + \frac{190}{5790447} a^{13} - \frac{11}{643383} a^{11} + \frac{3257}{643383} a^{9} + \frac{877}{214461} a^{7} + \frac{1397}{71487} a^{5} - \frac{313}{7943} a^{3} + \frac{1377}{7943} a$, $\frac{1}{85450475854946241} a^{18} + \frac{94544462}{28483491951648747} a^{16} + \frac{4736255416}{9494497317216249} a^{14} + \frac{131691240659}{3164832439072083} a^{12} - \frac{23798126569}{351648048785787} a^{10} + \frac{96905103688}{39072005420643} a^{8} - \frac{1858616811560}{117216016261929} a^{6} + \frac{1849417613825}{39072005420643} a^{4} - \frac{1451954339066}{13024001806881} a^{2} - \frac{1026470992862}{4341333935627}$, $\frac{1}{85450475854946241} a^{19} + \frac{94544462}{28483491951648747} a^{17} + \frac{4736255416}{9494497317216249} a^{15} + \frac{131691240659}{3164832439072083} a^{13} - \frac{23798126569}{351648048785787} a^{11} + \frac{96905103688}{39072005420643} a^{9} - \frac{1858616811560}{117216016261929} a^{7} + \frac{1849417613825}{39072005420643} a^{5} - \frac{1451954339066}{13024001806881} a^{3} - \frac{1026470992862}{4341333935627} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2530100}$, which has order $10120400$ (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:  \( 36549838.47150319 \) (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

\(\Q(\sqrt{61}) \), 4.0.32685264.1, 5.5.13845841.1, 10.10.11694146092834141.1

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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{20}$ $20$ $20$

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}$
61Data not computed