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Properties

Label 21.21.4723624648...4224.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 3^{28}\cdot 31^{12}$
Root discriminant $55.77$
Ramified primes $2, 3, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times C_7:C_3$ (as 21T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, -1188, 7767, 18503, -79299, -71394, 270894, 104895, -428406, -59437, 354021, -1233, -158528, 15282, 37887, -6390, -4470, 1017, 208, -57, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 57*x^19 + 208*x^18 + 1017*x^17 - 4470*x^16 - 6390*x^15 + 37887*x^14 + 15282*x^13 - 158528*x^12 - 1233*x^11 + 354021*x^10 - 59437*x^9 - 428406*x^8 + 104895*x^7 + 270894*x^6 - 71394*x^5 - 79299*x^4 + 18503*x^3 + 7767*x^2 - 1188*x - 19)
 
gp: K = bnfinit(x^21 - 3*x^20 - 57*x^19 + 208*x^18 + 1017*x^17 - 4470*x^16 - 6390*x^15 + 37887*x^14 + 15282*x^13 - 158528*x^12 - 1233*x^11 + 354021*x^10 - 59437*x^9 - 428406*x^8 + 104895*x^7 + 270894*x^6 - 71394*x^5 - 79299*x^4 + 18503*x^3 + 7767*x^2 - 1188*x - 19, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 57 x^{19} + 208 x^{18} + 1017 x^{17} - 4470 x^{16} - 6390 x^{15} + 37887 x^{14} + 15282 x^{13} - 158528 x^{12} - 1233 x^{11} + 354021 x^{10} - 59437 x^{9} - 428406 x^{8} + 104895 x^{7} + 270894 x^{6} - 71394 x^{5} - 79299 x^{4} + 18503 x^{3} + 7767 x^{2} - 1188 x - 19 \)

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:  \(4723624648173806587548558901725364224=2^{18}\cdot 3^{28}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.77$
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:  $2, 3, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{168495672868149678045651526556631321007739} a^{20} + \frac{65321183589421942731705843296382011930700}{168495672868149678045651526556631321007739} a^{19} - \frac{27040149276198091257416711841256329204962}{168495672868149678045651526556631321007739} a^{18} - \frac{52370979985405125584225466803518755533657}{168495672868149678045651526556631321007739} a^{17} - \frac{2115977479180287156618423630909016194261}{168495672868149678045651526556631321007739} a^{16} - \frac{43048925389534183956762990864999780563177}{168495672868149678045651526556631321007739} a^{15} - \frac{89980922758252415452993136456218353065}{168495672868149678045651526556631321007739} a^{14} - \frac{13086260898018416672375820334489651932745}{168495672868149678045651526556631321007739} a^{13} - \frac{71946067805288133040279484220193200032086}{168495672868149678045651526556631321007739} a^{12} - \frac{76581439344568012630303903566860265233205}{168495672868149678045651526556631321007739} a^{11} + \frac{41299310970962515611934771526327486087395}{168495672868149678045651526556631321007739} a^{10} + \frac{84097016722782157258795746000400400376373}{168495672868149678045651526556631321007739} a^{9} - \frac{17997863467833199617642707658966059156756}{168495672868149678045651526556631321007739} a^{8} - \frac{59561719129223119371824719885276571356470}{168495672868149678045651526556631321007739} a^{7} - \frac{45872934620064756841408162664481634208046}{168495672868149678045651526556631321007739} a^{6} - \frac{66506493717726258206650225615070637672757}{168495672868149678045651526556631321007739} a^{5} + \frac{39360077057539309875002851618481552077645}{168495672868149678045651526556631321007739} a^{4} - \frac{54130716207356361042963774812356556766283}{168495672868149678045651526556631321007739} a^{3} - \frac{55851702766593620093265421351144928500699}{168495672868149678045651526556631321007739} a^{2} - \frac{16810526701107901768979525428353312980657}{168495672868149678045651526556631321007739} a + \frac{81933900780038762181038080759939357036284}{168495672868149678045651526556631321007739}$

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:  $20$
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:  \( 741765605490 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_7:C_3$ (as 21T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 63
The 15 conjugacy class representatives for $C_3\times C_7:C_3$
Character table for $C_3\times C_7:C_3$

Intermediate fields

\(\Q(\zeta_{9})^+\), 7.7.387790161984.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R $21$ $21$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ $21$ $21$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
2Data not computed
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$31$31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.9.6.1$x^{9} + 837 x^{6} + 232562 x^{3} + 21717639$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
31.9.6.1$x^{9} + 837 x^{6} + 232562 x^{3} + 21717639$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$