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Properties

Label 20.20.3206128490...5625.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{10}\cdot 5^{10}\cdot 11^{18}$
Root discriminant $33.52$
Ramified primes $3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (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![1, 12, -108, -316, 1391, 1922, -5641, -4995, 10781, 6622, -11153, -4730, 6524, 1794, -2132, -338, 368, 30, -31, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 31*x^18 + 30*x^17 + 368*x^16 - 338*x^15 - 2132*x^14 + 1794*x^13 + 6524*x^12 - 4730*x^11 - 11153*x^10 + 6622*x^9 + 10781*x^8 - 4995*x^7 - 5641*x^6 + 1922*x^5 + 1391*x^4 - 316*x^3 - 108*x^2 + 12*x + 1)
 
gp: K = bnfinit(x^20 - x^19 - 31*x^18 + 30*x^17 + 368*x^16 - 338*x^15 - 2132*x^14 + 1794*x^13 + 6524*x^12 - 4730*x^11 - 11153*x^10 + 6622*x^9 + 10781*x^8 - 4995*x^7 - 5641*x^6 + 1922*x^5 + 1391*x^4 - 316*x^3 - 108*x^2 + 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 31 x^{18} + 30 x^{17} + 368 x^{16} - 338 x^{15} - 2132 x^{14} + 1794 x^{13} + 6524 x^{12} - 4730 x^{11} - 11153 x^{10} + 6622 x^{9} + 10781 x^{8} - 4995 x^{7} - 5641 x^{6} + 1922 x^{5} + 1391 x^{4} - 316 x^{3} - 108 x^{2} + 12 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3206128490667995866421572265625=3^{10}\cdot 5^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.52$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(165=3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{165}(64,·)$, $\chi_{165}(1,·)$, $\chi_{165}(131,·)$, $\chi_{165}(4,·)$, $\chi_{165}(134,·)$, $\chi_{165}(136,·)$, $\chi_{165}(74,·)$, $\chi_{165}(16,·)$, $\chi_{165}(149,·)$, $\chi_{165}(91,·)$, $\chi_{165}(29,·)$, $\chi_{165}(31,·)$, $\chi_{165}(161,·)$, $\chi_{165}(34,·)$, $\chi_{165}(164,·)$, $\chi_{165}(101,·)$, $\chi_{165}(41,·)$, $\chi_{165}(49,·)$, $\chi_{165}(116,·)$, $\chi_{165}(124,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{26069} a^{18} - \frac{4705}{26069} a^{17} - \frac{3752}{26069} a^{16} + \frac{99}{199} a^{15} - \frac{4790}{26069} a^{14} + \frac{215}{26069} a^{13} - \frac{3285}{26069} a^{12} + \frac{7618}{26069} a^{11} - \frac{9726}{26069} a^{10} - \frac{7242}{26069} a^{9} + \frac{5982}{26069} a^{8} + \frac{756}{26069} a^{7} + \frac{1007}{26069} a^{6} - \frac{6225}{26069} a^{5} + \frac{10298}{26069} a^{4} + \frac{1838}{26069} a^{3} - \frac{10479}{26069} a^{2} - \frac{2223}{26069} a - \frac{1424}{26069}$, $\frac{1}{3622628295995963} a^{19} - \frac{52228984885}{3622628295995963} a^{18} - \frac{1654130183225409}{3622628295995963} a^{17} - \frac{1775671016444522}{3622628295995963} a^{16} - \frac{1744134740482159}{3622628295995963} a^{15} - \frac{105651025952296}{3622628295995963} a^{14} - \frac{449763897425804}{3622628295995963} a^{13} - \frac{819545829119549}{3622628295995963} a^{12} - \frac{1053295058849499}{3622628295995963} a^{11} - \frac{211058445567393}{3622628295995963} a^{10} + \frac{122815159815711}{3622628295995963} a^{9} - \frac{876393995990442}{3622628295995963} a^{8} + \frac{1129504663697236}{3622628295995963} a^{7} - \frac{760355620215612}{3622628295995963} a^{6} + \frac{924223330454500}{3622628295995963} a^{5} - \frac{1045968996168922}{3622628295995963} a^{4} + \frac{1148589467768929}{3622628295995963} a^{3} + \frac{1411465484119934}{3622628295995963} a^{2} + \frac{760622789543558}{3622628295995963} a - \frac{759684186778156}{3622628295995963}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $19$
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:  \( 527561758.587 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{5}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.10.1790566527853125.1, \(\Q(\zeta_{33})^+\)

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
5Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$