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Properties

Label 21.21.1068474701...8961.1
Degree $21$
Signature $[21, 0]$
Discriminant $3^{28}\cdot 43^{20}$
Root discriminant $155.54$
Ramified primes $3, 43$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_{21}$ (as 21T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-722701, -44247, 53399808, 245278579, 303277839, -118626723, -399918748, -66235308, 187574256, 60860609, -42117210, -17262522, 4822837, 2351541, -282897, -169635, 7998, 6579, -86, -129, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 129*x^19 - 86*x^18 + 6579*x^17 + 7998*x^16 - 169635*x^15 - 282897*x^14 + 2351541*x^13 + 4822837*x^12 - 17262522*x^11 - 42117210*x^10 + 60860609*x^9 + 187574256*x^8 - 66235308*x^7 - 399918748*x^6 - 118626723*x^5 + 303277839*x^4 + 245278579*x^3 + 53399808*x^2 - 44247*x - 722701)
 
gp: K = bnfinit(x^21 - 129*x^19 - 86*x^18 + 6579*x^17 + 7998*x^16 - 169635*x^15 - 282897*x^14 + 2351541*x^13 + 4822837*x^12 - 17262522*x^11 - 42117210*x^10 + 60860609*x^9 + 187574256*x^8 - 66235308*x^7 - 399918748*x^6 - 118626723*x^5 + 303277839*x^4 + 245278579*x^3 + 53399808*x^2 - 44247*x - 722701, 1)
 

Normalized defining polynomial

\( x^{21} - 129 x^{19} - 86 x^{18} + 6579 x^{17} + 7998 x^{16} - 169635 x^{15} - 282897 x^{14} + 2351541 x^{13} + 4822837 x^{12} - 17262522 x^{11} - 42117210 x^{10} + 60860609 x^{9} + 187574256 x^{8} - 66235308 x^{7} - 399918748 x^{6} - 118626723 x^{5} + 303277839 x^{4} + 245278579 x^{3} + 53399808 x^{2} - 44247 x - 722701 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10684747015052975538074582285998174444374188961=3^{28}\cdot 43^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $155.54$
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:  $3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(387=3^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{387}(64,·)$, $\chi_{387}(1,·)$, $\chi_{387}(196,·)$, $\chi_{387}(262,·)$, $\chi_{387}(145,·)$, $\chi_{387}(268,·)$, $\chi_{387}(13,·)$, $\chi_{387}(337,·)$, $\chi_{387}(283,·)$, $\chi_{387}(160,·)$, $\chi_{387}(226,·)$, $\chi_{387}(229,·)$, $\chi_{387}(103,·)$, $\chi_{387}(169,·)$, $\chi_{387}(367,·)$, $\chi_{387}(178,·)$, $\chi_{387}(310,·)$, $\chi_{387}(58,·)$, $\chi_{387}(379,·)$, $\chi_{387}(124,·)$, $\chi_{387}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{5}$, $\frac{1}{49} a^{12} + \frac{3}{49} a^{11} - \frac{2}{49} a^{10} + \frac{3}{49} a^{9} + \frac{3}{49} a^{8} - \frac{3}{49} a^{7} - \frac{2}{49} a^{6} - \frac{22}{49} a^{5} - \frac{16}{49} a^{4} - \frac{2}{49} a^{3} + \frac{16}{49} a^{2} + \frac{3}{7} a$, $\frac{1}{49} a^{13} + \frac{3}{49} a^{11} + \frac{2}{49} a^{10} + \frac{1}{49} a^{9} + \frac{2}{49} a^{8} - \frac{2}{49} a^{6} + \frac{8}{49} a^{5} + \frac{11}{49} a^{4} + \frac{1}{49} a^{3} - \frac{13}{49} a^{2} - \frac{2}{7} a$, $\frac{1}{49} a^{14} - \frac{2}{49} a^{8} + \frac{1}{49} a^{2}$, $\frac{1}{343} a^{15} + \frac{1}{343} a^{14} + \frac{3}{343} a^{13} - \frac{2}{343} a^{12} + \frac{24}{343} a^{11} - \frac{11}{343} a^{10} - \frac{5}{343} a^{9} - \frac{9}{343} a^{8} - \frac{1}{343} a^{7} - \frac{9}{343} a^{6} - \frac{135}{343} a^{5} - \frac{89}{343} a^{4} + \frac{162}{343} a^{3} + \frac{3}{49} a^{2} - \frac{3}{7} a$, $\frac{1}{27097} a^{16} + \frac{10}{27097} a^{15} - \frac{142}{27097} a^{14} + \frac{193}{27097} a^{13} + \frac{188}{27097} a^{12} - \frac{1734}{27097} a^{11} + \frac{358}{27097} a^{10} - \frac{1202}{27097} a^{9} - \frac{117}{27097} a^{8} + \frac{1788}{27097} a^{7} - \frac{1553}{27097} a^{6} + \frac{3043}{27097} a^{5} - \frac{968}{27097} a^{4} - \frac{5136}{27097} a^{3} + \frac{375}{3871} a^{2} + \frac{31}{79} a + \frac{3}{79}$, $\frac{1}{189679} a^{17} - \frac{1}{189679} a^{16} + \frac{64}{189679} a^{15} - \frac{141}{189679} a^{14} + \frac{254}{27097} a^{13} - \frac{10}{189679} a^{12} + \frac{2131}{189679} a^{11} - \frac{8063}{189679} a^{10} + \frac{465}{189679} a^{9} - \frac{1626}{27097} a^{8} - \frac{7712}{189679} a^{7} - \frac{8709}{189679} a^{6} + \frac{52854}{189679} a^{5} - \frac{59110}{189679} a^{4} + \frac{4936}{27097} a^{3} + \frac{1219}{3871} a^{2} - \frac{22}{553} a - \frac{16}{79}$, $\frac{1}{189679} a^{18} - \frac{22}{27097} a^{15} - \frac{477}{189679} a^{14} - \frac{990}{189679} a^{13} + \frac{16}{3871} a^{12} + \frac{12065}{189679} a^{11} + \frac{10217}{189679} a^{10} - \frac{3763}{189679} a^{9} + \frac{2655}{189679} a^{8} - \frac{1875}{189679} a^{7} - \frac{10091}{189679} a^{6} - \frac{86812}{189679} a^{5} - \frac{32146}{189679} a^{4} - \frac{7141}{27097} a^{3} - \frac{19}{3871} a^{2} - \frac{33}{553} a + \frac{36}{79}$, $\frac{1}{1327753} a^{19} - \frac{2}{1327753} a^{18} - \frac{1}{189679} a^{16} + \frac{1301}{1327753} a^{15} + \frac{10058}{1327753} a^{14} + \frac{167}{1327753} a^{13} - \frac{8319}{1327753} a^{12} - \frac{67519}{1327753} a^{11} + \frac{59397}{1327753} a^{10} - \frac{65867}{1327753} a^{9} - \frac{43739}{1327753} a^{8} - \frac{10604}{1327753} a^{7} - \frac{58790}{1327753} a^{6} - \frac{181530}{1327753} a^{5} + \frac{30720}{1327753} a^{4} - \frac{80988}{189679} a^{3} - \frac{4168}{27097} a^{2} + \frac{1873}{3871} a + \frac{10}{79}$, $\frac{1}{2369528466750490297706640339087490303086563} a^{20} + \frac{187635700341608847860211986415408536}{2369528466750490297706640339087490303086563} a^{19} - \frac{4114514331518511726591231236148741238}{2369528466750490297706640339087490303086563} a^{18} - \frac{709483406605195393435380955666511692}{338504066678641471100948619869641471869509} a^{17} + \frac{7062687142415008828303132716191063202}{2369528466750490297706640339087490303086563} a^{16} + \frac{423003020190851734363929465989428383592}{2369528466750490297706640339087490303086563} a^{15} + \frac{15762645222090024707091597581241960887519}{2369528466750490297706640339087490303086563} a^{14} + \frac{5469384112249244619619175817434251490841}{2369528466750490297706640339087490303086563} a^{13} - \frac{189180626439648812056506769467631670369}{48357723811234495871564088552805924552787} a^{12} - \frac{138342795819473175192697124546078861566405}{2369528466750490297706640339087490303086563} a^{11} - \frac{42186893604365967878165381934483873655355}{2369528466750490297706640339087490303086563} a^{10} - \frac{24171982929891848003024067614310544271505}{338504066678641471100948619869641471869509} a^{9} - \frac{144066708357679178683880720092579307264352}{2369528466750490297706640339087490303086563} a^{8} - \frac{94296397342804394791480072863360034641347}{2369528466750490297706640339087490303086563} a^{7} + \frac{58534048552012014088226564643026138810966}{2369528466750490297706640339087490303086563} a^{6} - \frac{480159969261715664240040063948969838515392}{2369528466750490297706640339087490303086563} a^{5} - \frac{27770811068783842302685790064709167084283}{2369528466750490297706640339087490303086563} a^{4} + \frac{100661173986988531418269801567132154324113}{338504066678641471100948619869641471869509} a^{3} - \frac{19187571304531922853157326405467227015737}{48357723811234495871564088552805924552787} a^{2} + \frac{3414544716639378561862488458107734653634}{6908246258747785124509155507543703507541} a - \frac{39427297443977722862373618949928905876}{140984617525465002541003173623340887909}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $20$
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:  \( 104145016649537420 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{21}$ (as 21T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 21
The 21 conjugacy class representatives for $C_{21}$
Character table for $C_{21}$ is not computed

Intermediate fields

3.3.149769.1, 7.7.6321363049.1

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 $21$ R ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{21}$ $21$ $21$ $21$ $21$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ $21$ R $21$ $21$ $21$

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
43Data not computed