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Properties

Label 21.21.2041566636...1216.2
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![8, 48, -168, -924, 1376, 6120, -4888, -19278, 8240, 31979, -6976, -29068, 3152, 14642, -828, -4024, 128, 579, -8, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 40*x^19 - 8*x^18 + 579*x^17 + 128*x^16 - 4024*x^15 - 828*x^14 + 14642*x^13 + 3152*x^12 - 29068*x^11 - 6976*x^10 + 31979*x^9 + 8240*x^8 - 19278*x^7 - 4888*x^6 + 6120*x^5 + 1376*x^4 - 924*x^3 - 168*x^2 + 48*x + 8)
 
gp: K = bnfinit(x^21 - 40*x^19 - 8*x^18 + 579*x^17 + 128*x^16 - 4024*x^15 - 828*x^14 + 14642*x^13 + 3152*x^12 - 29068*x^11 - 6976*x^10 + 31979*x^9 + 8240*x^8 - 19278*x^7 - 4888*x^6 + 6120*x^5 + 1376*x^4 - 924*x^3 - 168*x^2 + 48*x + 8, 1)
 

Normalized defining polynomial

\( x^{21} - 40 x^{19} - 8 x^{18} + 579 x^{17} + 128 x^{16} - 4024 x^{15} - 828 x^{14} + 14642 x^{13} + 3152 x^{12} - 29068 x^{11} - 6976 x^{10} + 31979 x^{9} + 8240 x^{8} - 19278 x^{7} - 4888 x^{6} + 6120 x^{5} + 1376 x^{4} - 924 x^{3} - 168 x^{2} + 48 x + 8 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{14} - \frac{1}{2} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{15} - \frac{1}{2} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{76342517945518036292804} a^{20} - \frac{1793403187379517189601}{38171258972759018146402} a^{19} - \frac{5346931838987985191633}{76342517945518036292804} a^{18} - \frac{8157280095287941838993}{38171258972759018146402} a^{17} + \frac{4790920699203445918791}{76342517945518036292804} a^{16} + \frac{2083805194288127307021}{38171258972759018146402} a^{15} - \frac{11602141230647421453587}{76342517945518036292804} a^{14} + \frac{8601249126359852102363}{38171258972759018146402} a^{13} - \frac{6816834634626926629149}{38171258972759018146402} a^{12} + \frac{5049748360817596045151}{19085629486379509073201} a^{11} + \frac{14714087495260414170227}{38171258972759018146402} a^{10} - \frac{295792401266424843801}{19085629486379509073201} a^{9} + \frac{13887807622717603645419}{76342517945518036292804} a^{8} - \frac{18816226199906806144641}{38171258972759018146402} a^{7} - \frac{31672393497775671790189}{76342517945518036292804} a^{6} + \frac{12727317211641084967901}{38171258972759018146402} a^{5} - \frac{11645932808255050582375}{38171258972759018146402} a^{4} - \frac{7024165789655291598374}{19085629486379509073201} a^{3} - \frac{215551755992400816048}{19085629486379509073201} a^{2} + \frac{7703751187703341315194}{19085629486379509073201} a - \frac{1235904188266802577622}{19085629486379509073201}$

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

\(\Q(\zeta_{7})^+\), 7.7.670188544.2

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
$2$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.12.24.142$x^{12} + 8 x^{11} + 32 x^{10} - 92 x^{9} - 70 x^{8} + 96 x^{7} + 80 x^{6} + 112 x^{5} + 76 x^{4} + 96 x^{3} - 32 x^{2} - 112 x + 120$$4$$3$$24$12T45$[8/3, 8/3]_{3}^{6}$
7Data not computed
809Data not computed