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Properties

Label 20.20.1388819463...0973.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{10}\cdot 11^{16}\cdot 13^{15}$
Root discriminant $80.75$
Ramified primes $3, 11, 13$
Class number $1$ (GRH)
Class group Trivial (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![529, 17940, 2500, -1007476, -3020232, -1687071, 3433304, 4213355, -343368, -2291342, -537473, 517750, 205169, -53182, -30387, 2188, 2161, -8, -74, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 74*x^18 - 8*x^17 + 2161*x^16 + 2188*x^15 - 30387*x^14 - 53182*x^13 + 205169*x^12 + 517750*x^11 - 537473*x^10 - 2291342*x^9 - 343368*x^8 + 4213355*x^7 + 3433304*x^6 - 1687071*x^5 - 3020232*x^4 - 1007476*x^3 + 2500*x^2 + 17940*x + 529)
 
gp: K = bnfinit(x^20 - x^19 - 74*x^18 - 8*x^17 + 2161*x^16 + 2188*x^15 - 30387*x^14 - 53182*x^13 + 205169*x^12 + 517750*x^11 - 537473*x^10 - 2291342*x^9 - 343368*x^8 + 4213355*x^7 + 3433304*x^6 - 1687071*x^5 - 3020232*x^4 - 1007476*x^3 + 2500*x^2 + 17940*x + 529, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 74 x^{18} - 8 x^{17} + 2161 x^{16} + 2188 x^{15} - 30387 x^{14} - 53182 x^{13} + 205169 x^{12} + 517750 x^{11} - 537473 x^{10} - 2291342 x^{9} - 343368 x^{8} + 4213355 x^{7} + 3433304 x^{6} - 1687071 x^{5} - 3020232 x^{4} - 1007476 x^{3} + 2500 x^{2} + 17940 x + 529 \)

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:  \(138881946372451702408146629446494920973=3^{10}\cdot 11^{16}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.75$
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:  $3, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(429=3\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{429}(64,·)$, $\chi_{429}(1,·)$, $\chi_{429}(196,·)$, $\chi_{429}(5,·)$, $\chi_{429}(203,·)$, $\chi_{429}(320,·)$, $\chi_{429}(278,·)$, $\chi_{429}(25,·)$, $\chi_{429}(47,·)$, $\chi_{429}(86,·)$, $\chi_{429}(157,·)$, $\chi_{429}(356,·)$, $\chi_{429}(103,·)$, $\chi_{429}(298,·)$, $\chi_{429}(235,·)$, $\chi_{429}(125,·)$, $\chi_{429}(181,·)$, $\chi_{429}(313,·)$, $\chi_{429}(122,·)$, $\chi_{429}(317,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} + \frac{1}{23} a^{15} - \frac{7}{23} a^{14} - \frac{2}{23} a^{13} + \frac{5}{23} a^{12} - \frac{11}{23} a^{11} - \frac{10}{23} a^{10} + \frac{2}{23} a^{9} + \frac{8}{23} a^{8} + \frac{11}{23} a^{7} - \frac{4}{23} a^{5} + \frac{4}{23} a^{4} + \frac{5}{23} a^{3} + \frac{8}{23} a^{2} + \frac{5}{23} a$, $\frac{1}{23} a^{18} + \frac{1}{23} a^{16} - \frac{7}{23} a^{15} - \frac{2}{23} a^{14} + \frac{5}{23} a^{13} - \frac{11}{23} a^{12} - \frac{10}{23} a^{11} + \frac{2}{23} a^{10} + \frac{8}{23} a^{9} + \frac{11}{23} a^{8} - \frac{4}{23} a^{6} + \frac{4}{23} a^{5} + \frac{5}{23} a^{4} + \frac{8}{23} a^{3} + \frac{5}{23} a^{2}$, $\frac{1}{261334788618030046408395899190328738989529371233} a^{19} - \frac{1501608622150987724490928468497661184995128869}{261334788618030046408395899190328738989529371233} a^{18} + \frac{122115070631039355155581868674280319453675261}{261334788618030046408395899190328738989529371233} a^{17} + \frac{26684727905042557119422528286752854778005417267}{261334788618030046408395899190328738989529371233} a^{16} - \frac{4652221956501420327723738196974310853472534774}{261334788618030046408395899190328738989529371233} a^{15} - \frac{37401788979079343767968061101809339971749738305}{261334788618030046408395899190328738989529371233} a^{14} + \frac{26492299485223874755144485596265056141899082366}{261334788618030046408395899190328738989529371233} a^{13} - \frac{108244544749041322162469111922597667114594480935}{261334788618030046408395899190328738989529371233} a^{12} + \frac{47196915187298484283851228760496691327052087573}{261334788618030046408395899190328738989529371233} a^{11} + \frac{86646883333486757692342157522075019370239135884}{261334788618030046408395899190328738989529371233} a^{10} + \frac{11448280354625523872364223324325336610876178912}{261334788618030046408395899190328738989529371233} a^{9} + \frac{20084044481348892606784435986265696626040679166}{261334788618030046408395899190328738989529371233} a^{8} + \frac{88360803142170666786138127140160524829108722487}{261334788618030046408395899190328738989529371233} a^{7} + \frac{72183620228358129040182443688367749132295675834}{261334788618030046408395899190328738989529371233} a^{6} - \frac{111828002190882877389509613613365272385389691429}{261334788618030046408395899190328738989529371233} a^{5} - \frac{100238791203652220781941419147153374976137117245}{261334788618030046408395899190328738989529371233} a^{4} - \frac{95828430607812848596180078823053685035351407075}{261334788618030046408395899190328738989529371233} a^{3} + \frac{38398862462746697082400232276793605955466699362}{261334788618030046408395899190328738989529371233} a^{2} - \frac{72224969531283605126655764702461681837493466316}{261334788618030046408395899190328738989529371233} a - \frac{4368243884320833119336013535495074297858081068}{11362382113827393322104169530014292999544755271}$

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:  \( 5124246016990 \) (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

\(\Q(\sqrt{13}) \), 4.4.19773.1, \(\Q(\zeta_{11})^+\), 10.10.79589952003133.1

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 $20$ R $20$ $20$ R R ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
$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}$
11Data not computed
13Data not computed