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Properties

Label 20.0.26334194176...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{35}\cdot 29^{15}$
Root discriminant $417.86$
Ramified primes $2, 5, 29$
Class number $3757921600$ (GRH)
Class group $[2, 2, 2, 469740200]$ (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![16279165773981745, 0, 5699366824174475, 0, 812534226323875, 0, 62830388169700, 0, 2930498707675, 0, 86282306695, 0, 1627250900, 0, 19490175, 0, 142680, 0, 580, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 580*x^18 + 142680*x^16 + 19490175*x^14 + 1627250900*x^12 + 86282306695*x^10 + 2930498707675*x^8 + 62830388169700*x^6 + 812534226323875*x^4 + 5699366824174475*x^2 + 16279165773981745)
 
gp: K = bnfinit(x^20 + 580*x^18 + 142680*x^16 + 19490175*x^14 + 1627250900*x^12 + 86282306695*x^10 + 2930498707675*x^8 + 62830388169700*x^6 + 812534226323875*x^4 + 5699366824174475*x^2 + 16279165773981745, 1)
 

Normalized defining polynomial

\( x^{20} + 580 x^{18} + 142680 x^{16} + 19490175 x^{14} + 1627250900 x^{12} + 86282306695 x^{10} + 2930498707675 x^{8} + 62830388169700 x^{6} + 812534226323875 x^{4} + 5699366824174475 x^{2} + 16279165773981745 \)

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:  \(26334194176019849978482055664062500000000000000000000=2^{20}\cdot 5^{35}\cdot 29^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $417.86$
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:  $2, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2900=2^{2}\cdot 5^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{2900}(1,·)$, $\chi_{2900}(2883,·)$, $\chi_{2900}(581,·)$, $\chi_{2900}(1161,·)$, $\chi_{2900}(1143,·)$, $\chi_{2900}(1741,·)$, $\chi_{2900}(2321,·)$, $\chi_{2900}(2627,·)$, $\chi_{2900}(563,·)$, $\chi_{2900}(1467,·)$, $\chi_{2900}(289,·)$, $\chi_{2900}(1723,·)$, $\chi_{2900}(869,·)$, $\chi_{2900}(1449,·)$, $\chi_{2900}(2029,·)$, $\chi_{2900}(2609,·)$, $\chi_{2900}(307,·)$, $\chi_{2900}(887,·)$, $\chi_{2900}(2047,·)$, $\chi_{2900}(2303,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{29} a^{4}$, $\frac{1}{29} a^{5}$, $\frac{1}{29} a^{6}$, $\frac{1}{29} a^{7}$, $\frac{1}{841} a^{8}$, $\frac{1}{841} a^{9}$, $\frac{1}{841} a^{10}$, $\frac{1}{841} a^{11}$, $\frac{1}{24389} a^{12}$, $\frac{1}{24389} a^{13}$, $\frac{1}{24389} a^{14}$, $\frac{1}{24389} a^{15}$, $\frac{1}{1378490669} a^{16} + \frac{338}{47534161} a^{14} - \frac{23}{1639109} a^{12} - \frac{584}{1639109} a^{10} - \frac{582}{1639109} a^{8} - \frac{803}{56521} a^{6} - \frac{193}{56521} a^{4} - \frac{917}{1949} a^{2} - \frac{293}{1949}$, $\frac{1}{59275098767} a^{17} - \frac{17203}{2043968923} a^{15} - \frac{33800}{2043968923} a^{13} + \frac{15008}{70481687} a^{11} - \frac{8378}{70481687} a^{9} + \frac{6993}{2430403} a^{7} + \frac{9552}{2430403} a^{5} - \frac{30152}{83807} a^{3} - \frac{41222}{83807} a$, $\frac{1}{3748841971558366171433600255595319} a^{18} - \frac{21611251770818404852409}{3748841971558366171433600255595319} a^{16} + \frac{769576964382975747336572593}{129270412812357454187365526055011} a^{14} - \frac{98308143322893001161292609}{129270412812357454187365526055011} a^{12} - \frac{2590192962061554651956413239}{4457600441805429454736742277759} a^{10} + \frac{2158262763314548716717814004}{4457600441805429454736742277759} a^{8} - \frac{2623730776780336022703735695}{153710360062256188094370423371} a^{6} + \frac{2076583666921117120088701307}{153710360062256188094370423371} a^{4} - \frac{1956928363047905569226818848}{5300357243526075451530014599} a^{2} + \frac{777260839979512106453518}{2866607487034113278274751}$, $\frac{1}{1098410697666601288230044874889428467} a^{19} + \frac{4848238614606785014525580}{1098410697666601288230044874889428467} a^{17} + \frac{696146064511422155498689173879}{37876230954020734076898099134118223} a^{15} + \frac{36714341627104978582003233953}{37876230954020734076898099134118223} a^{13} + \frac{102298657293689970175248533267}{1306076929448990830237865487383387} a^{11} + \frac{411889026282519400649220263077}{1306076929448990830237865487383387} a^{9} + \frac{17350445655607550340523373817}{1553004672353140107298294277507} a^{7} + \frac{250006964844550852140127806997}{45037135498241063111650534047703} a^{5} - \frac{678828365886672949023313403076}{1553004672353140107298294277507} a^{3} + \frac{6705234197807057936558010124}{36116387729142793192983587849} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{469740200}$, which has order $3757921600$ (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:  \( 257696579.12792215 \) (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

\(\Q(\sqrt{145}) \), 4.0.48778000.3, 5.5.390625.1, 10.10.15648764801025390625.1

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ $20$ R $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
5Data not computed
29Data not computed