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Properties

Label 21.21.2041566636...1216.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{30}\cdot 7^{14}\cdot 809^{6}$
Root discriminant $66.72$
Ramified primes $2, 7, 809$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times \PSL(2,7)$ (as 21T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 17, 114, -1455, -5733, 9259, 30977, -31596, -58979, 51770, 52044, -41833, -24428, 18052, 6417, -4319, -933, 568, 69, -38, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 38*x^19 + 69*x^18 + 568*x^17 - 933*x^16 - 4319*x^15 + 6417*x^14 + 18052*x^13 - 24428*x^12 - 41833*x^11 + 52044*x^10 + 51770*x^9 - 58979*x^8 - 31596*x^7 + 30977*x^6 + 9259*x^5 - 5733*x^4 - 1455*x^3 + 114*x^2 + 17*x - 1)
 
gp: K = bnfinit(x^21 - 2*x^20 - 38*x^19 + 69*x^18 + 568*x^17 - 933*x^16 - 4319*x^15 + 6417*x^14 + 18052*x^13 - 24428*x^12 - 41833*x^11 + 52044*x^10 + 51770*x^9 - 58979*x^8 - 31596*x^7 + 30977*x^6 + 9259*x^5 - 5733*x^4 - 1455*x^3 + 114*x^2 + 17*x - 1, 1)
 

Normalized defining polynomial

\( x^{21} - 2 x^{20} - 38 x^{19} + 69 x^{18} + 568 x^{17} - 933 x^{16} - 4319 x^{15} + 6417 x^{14} + 18052 x^{13} - 24428 x^{12} - 41833 x^{11} + 52044 x^{10} + 51770 x^{9} - 58979 x^{8} - 31596 x^{7} + 30977 x^{6} + 9259 x^{5} - 5733 x^{4} - 1455 x^{3} + 114 x^{2} + 17 x - 1 \)

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:  \(204156663640790261439595720457863561216=2^{30}\cdot 7^{14}\cdot 809^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.72$
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:  $2, 7, 809$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18} + \frac{4}{19} a^{17} + \frac{8}{19} a^{16} + \frac{4}{19} a^{15} + \frac{2}{19} a^{14} - \frac{9}{19} a^{13} - \frac{3}{19} a^{12} + \frac{4}{19} a^{11} + \frac{2}{19} a^{10} + \frac{6}{19} a^{9} + \frac{3}{19} a^{8} - \frac{2}{19} a^{7} + \frac{2}{19} a^{6} + \frac{8}{19} a^{5} + \frac{1}{19} a^{4} - \frac{4}{19} a^{3} - \frac{8}{19} a^{2} - \frac{9}{19} a - \frac{7}{19}$, $\frac{1}{19} a^{19} - \frac{8}{19} a^{17} - \frac{9}{19} a^{16} + \frac{5}{19} a^{15} + \frac{2}{19} a^{14} - \frac{5}{19} a^{13} - \frac{3}{19} a^{12} + \frac{5}{19} a^{11} - \frac{2}{19} a^{10} - \frac{2}{19} a^{9} + \frac{5}{19} a^{8} - \frac{9}{19} a^{7} + \frac{7}{19} a^{5} - \frac{8}{19} a^{4} + \frac{8}{19} a^{3} + \frac{4}{19} a^{2} - \frac{9}{19} a + \frac{9}{19}$, $\frac{1}{138866464374367295844465157} a^{20} - \frac{3271415163770384962677383}{138866464374367295844465157} a^{19} - \frac{863003881126673056180102}{138866464374367295844465157} a^{18} + \frac{49027299018998041066411396}{138866464374367295844465157} a^{17} - \frac{25753300433329862262933964}{138866464374367295844465157} a^{16} + \frac{56879650827033070419704170}{138866464374367295844465157} a^{15} - \frac{42711744151587582934888255}{138866464374367295844465157} a^{14} - \frac{1181439735526319339276657}{138866464374367295844465157} a^{13} + \frac{11661493119476747054993477}{138866464374367295844465157} a^{12} - \frac{49875191810354430038044815}{138866464374367295844465157} a^{11} + \frac{25323141940750671485466090}{138866464374367295844465157} a^{10} - \frac{41862939849948278696356959}{138866464374367295844465157} a^{9} + \frac{45404620420168955419278927}{138866464374367295844465157} a^{8} + \frac{44640077394931748039058944}{138866464374367295844465157} a^{7} - \frac{30064984685309821872463041}{138866464374367295844465157} a^{6} + \frac{26564992345845874642644603}{138866464374367295844465157} a^{5} - \frac{5180479627568573306048346}{138866464374367295844465157} a^{4} + \frac{61724938544331117276921486}{138866464374367295844465157} a^{3} + \frac{23074655802457431627320097}{138866464374367295844465157} a^{2} + \frac{2319358617215477573710752}{138866464374367295844465157} a - \frac{39761475573374992557469143}{138866464374367295844465157}$

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:  \( 2811089509780 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times \PSL(2,7)$ (as 21T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 504
The 18 conjugacy class representatives for $C_3\times \PSL(2,7)$
Character table for $C_3\times \PSL(2,7)$

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.7.670188544.1

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

Sibling fields

Degree 24 siblings: data not computed
Degree 42 siblings: data not computed
Arithmetically equvalently 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 $21$ $21$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
2Data not computed
7Data not computed
809Data not computed