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Properties

Label 21.3.29588990284...5887.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,3^{28}\cdot 7^{17}\cdot 11^{18}$
Root discriminant $163.27$
Ramified primes $3, 7, 11$
Class number $21$ (GRH)
Class group $[21]$ (GRH)
Galois group $C_3\times F_7$ (as 21T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-57365, -85701, 557865, -8525, -1504272, 658350, 698269, -95907, 12474, -29151, -30030, 3213, 2519, 3759, 693, -978, 0, 189, -11, -21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^19 - 11*x^18 + 189*x^17 - 978*x^15 + 693*x^14 + 3759*x^13 + 2519*x^12 + 3213*x^11 - 30030*x^10 - 29151*x^9 + 12474*x^8 - 95907*x^7 + 698269*x^6 + 658350*x^5 - 1504272*x^4 - 8525*x^3 + 557865*x^2 - 85701*x - 57365)
 
gp: K = bnfinit(x^21 - 21*x^19 - 11*x^18 + 189*x^17 - 978*x^15 + 693*x^14 + 3759*x^13 + 2519*x^12 + 3213*x^11 - 30030*x^10 - 29151*x^9 + 12474*x^8 - 95907*x^7 + 698269*x^6 + 658350*x^5 - 1504272*x^4 - 8525*x^3 + 557865*x^2 - 85701*x - 57365, 1)
 

Normalized defining polynomial

\( x^{21} - 21 x^{19} - 11 x^{18} + 189 x^{17} - 978 x^{15} + 693 x^{14} + 3759 x^{13} + 2519 x^{12} + 3213 x^{11} - 30030 x^{10} - 29151 x^{9} + 12474 x^{8} - 95907 x^{7} + 698269 x^{6} + 658350 x^{5} - 1504272 x^{4} - 8525 x^{3} + 557865 x^{2} - 85701 x - 57365 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-29588990284336432332939370460310604619437105887=-\,3^{28}\cdot 7^{17}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $163.27$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} - \frac{4}{11} a^{17} - \frac{3}{11} a^{16} + \frac{1}{11} a^{14} - \frac{3}{11} a^{13} + \frac{4}{11} a^{12} - \frac{4}{11} a^{11} - \frac{2}{11} a^{9} + \frac{3}{11} a^{8} - \frac{5}{11} a^{7}$, $\frac{1}{11} a^{19} + \frac{3}{11} a^{17} - \frac{1}{11} a^{16} + \frac{1}{11} a^{15} + \frac{1}{11} a^{14} + \frac{3}{11} a^{13} + \frac{1}{11} a^{12} - \frac{5}{11} a^{11} - \frac{2}{11} a^{10} - \frac{5}{11} a^{9} - \frac{4}{11} a^{8} + \frac{2}{11} a^{7}$, $\frac{1}{651259630870552923321144887811584423048671914315679007626919} a^{20} - \frac{14629453198616827607837693054565770665311613701101801986788}{651259630870552923321144887811584423048671914315679007626919} a^{19} + \frac{19916198824384363269958468531882288508639981305519057403627}{651259630870552923321144887811584423048671914315679007626919} a^{18} - \frac{149593578816991583655142253287626925465405169221985676352249}{651259630870552923321144887811584423048671914315679007626919} a^{17} + \frac{145346765488057271245489764404784391166571812931111254649392}{651259630870552923321144887811584423048671914315679007626919} a^{16} + \frac{2446800661903588442113666004497875340407011066385684256538}{651259630870552923321144887811584423048671914315679007626919} a^{15} - \frac{35830071821877989342363615619214010753656632781833338816992}{651259630870552923321144887811584423048671914315679007626919} a^{14} - \frac{285772873301654447540304666697309897425330323507000480597748}{651259630870552923321144887811584423048671914315679007626919} a^{13} + \frac{96668418467061224626482003862765820680681046668507896647661}{651259630870552923321144887811584423048671914315679007626919} a^{12} - \frac{48348331364552917407470373113572666756471334471693798924437}{651259630870552923321144887811584423048671914315679007626919} a^{11} + \frac{93904399381497556179790992065835364170341857861126287655388}{651259630870552923321144887811584423048671914315679007626919} a^{10} + \frac{173869665646000778889585465343383608550209724665884560918056}{651259630870552923321144887811584423048671914315679007626919} a^{9} - \frac{308863295318798428374037705162229219156436107198464712617825}{651259630870552923321144887811584423048671914315679007626919} a^{8} - \frac{214932226200400992442318683230287633465701975493515624925173}{651259630870552923321144887811584423048671914315679007626919} a^{7} + \frac{9917739609224327280488548144838627273510953026938812494654}{59205420988232083938285898891962220277151992210516273420629} a^{6} + \frac{16791473526838832723948211121847337005761031118615011814180}{59205420988232083938285898891962220277151992210516273420629} a^{5} + \frac{6238080181879763134373571440540660514490281481051185104327}{59205420988232083938285898891962220277151992210516273420629} a^{4} - \frac{20861416044370500100751815411227939636606862109110671373447}{59205420988232083938285898891962220277151992210516273420629} a^{3} - \frac{20996920271410282324789337123403459080478396017064753359911}{59205420988232083938285898891962220277151992210516273420629} a^{2} + \frac{22760519422194052378607337619669339810862200245730224145870}{59205420988232083938285898891962220277151992210516273420629} a + \frac{19243718220906218987843684344714176029289458444634629254749}{59205420988232083938285898891962220277151992210516273420629}$

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

Class group and class number

$C_{21}$, which has order $21$ (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:  $11$
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:  \( 81427960974978.42 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times F_7$ (as 21T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 126
The 21 conjugacy class representatives for $C_3\times F_7$
Character table for $C_3\times F_7$ is not computed

Intermediate fields

3.3.3969.2, Deg 7

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 21 siblings: data not computed
Degree 42 siblings: data not computed

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ R R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
7Data not computed
$11$11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$