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Properties

Label 21.21.8594175910...9392.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 3^{28}\cdot 7^{21}\cdot 37^{6}$
Root discriminant $153.94$
Ramified primes $2, 3, 7, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T43

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-405224, -4254852, -14891982, -14384083, 20929272, 35970438, -11030292, -30526707, 2910348, 13380332, -411600, -3420396, 29820, 533022, -870, -50722, 0, 2835, 0, -84, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 84*x^19 + 2835*x^17 - 50722*x^15 - 870*x^14 + 533022*x^13 + 29820*x^12 - 3420396*x^11 - 411600*x^10 + 13380332*x^9 + 2910348*x^8 - 30526707*x^7 - 11030292*x^6 + 35970438*x^5 + 20929272*x^4 - 14384083*x^3 - 14891982*x^2 - 4254852*x - 405224)
 
gp: K = bnfinit(x^21 - 84*x^19 + 2835*x^17 - 50722*x^15 - 870*x^14 + 533022*x^13 + 29820*x^12 - 3420396*x^11 - 411600*x^10 + 13380332*x^9 + 2910348*x^8 - 30526707*x^7 - 11030292*x^6 + 35970438*x^5 + 20929272*x^4 - 14384083*x^3 - 14891982*x^2 - 4254852*x - 405224, 1)
 

Normalized defining polynomial

\( x^{21} - 84 x^{19} + 2835 x^{17} - 50722 x^{15} - 870 x^{14} + 533022 x^{13} + 29820 x^{12} - 3420396 x^{11} - 411600 x^{10} + 13380332 x^{9} + 2910348 x^{8} - 30526707 x^{7} - 11030292 x^{6} + 35970438 x^{5} + 20929272 x^{4} - 14384083 x^{3} - 14891982 x^{2} - 4254852 x - 405224 \)

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:  \(8594175910050075590605445090065778283994939392=2^{18}\cdot 3^{28}\cdot 7^{21}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $153.94$
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:  $2, 3, 7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{15} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{16} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{444} a^{17} + \frac{9}{148} a^{15} + \frac{5}{37} a^{13} + \frac{7}{74} a^{11} + \frac{3}{74} a^{10} + \frac{1}{3} a^{9} - \frac{1}{222} a^{8} + \frac{15}{37} a^{7} - \frac{43}{222} a^{6} - \frac{50}{111} a^{5} + \frac{25}{74} a^{4} + \frac{217}{444} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{6}$, $\frac{1}{444} a^{18} + \frac{9}{148} a^{16} + \frac{5}{37} a^{14} + \frac{7}{74} a^{12} + \frac{3}{74} a^{11} - \frac{25}{74} a^{9} + \frac{15}{37} a^{8} + \frac{35}{74} a^{7} + \frac{8}{37} a^{6} + \frac{25}{74} a^{5} + \frac{23}{148} a^{4} + \frac{1}{6} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{65712} a^{19} + \frac{4}{4107} a^{17} - \frac{1}{12} a^{16} + \frac{1355}{65712} a^{15} - \frac{1}{12} a^{14} + \frac{3721}{32856} a^{13} - \frac{145}{10952} a^{12} - \frac{113}{888} a^{11} - \frac{31}{5476} a^{10} - \frac{6319}{16428} a^{9} - \frac{139}{1369} a^{8} - \frac{7783}{16428} a^{7} - \frac{6641}{16428} a^{6} - \frac{11179}{65712} a^{5} - \frac{3}{148} a^{4} + \frac{341}{888} a^{3} - \frac{1}{4} a^{2} - \frac{1}{16} a + \frac{11}{24}$, $\frac{1}{55989182536980900576607663365044129202432} a^{20} + \frac{36874433806061189408965886284058905}{27994591268490450288303831682522064601216} a^{19} - \frac{347266003237500095188754617443604937}{3499323908561306286037978960315258075152} a^{18} + \frac{257046127209507850683771842631567481}{1749661954280653143018989480157629037576} a^{17} - \frac{3278466789052420925895414968220929239853}{55989182536980900576607663365044129202432} a^{16} + \frac{231214518201071138119097886129215762107}{27994591268490450288303831682522064601216} a^{15} - \frac{677667505281435871623476036624024743321}{9331530422830150096101277227507354867072} a^{14} + \frac{1039701619013433651351695498999661761549}{9331530422830150096101277227507354867072} a^{13} - \frac{332931262566045852251855816832912871409}{9331530422830150096101277227507354867072} a^{12} - \frac{137746425399510019599927909133512383053}{1166441302853768762012659653438419358384} a^{11} + \frac{182647320028513174992703903827650004367}{4665765211415075048050638613753677433536} a^{10} - \frac{390199395404860824168051662153254671911}{2332882605707537524025319306876838716768} a^{9} - \frac{4807257220799834079491083706554295129881}{13997295634245225144151915841261032300608} a^{8} - \frac{3244199967383401061040850441201967325503}{13997295634245225144151915841261032300608} a^{7} - \frac{24593113725924887339264625859207107823531}{55989182536980900576607663365044129202432} a^{6} - \frac{3595717501216972993926368038100844842173}{27994591268490450288303831682522064601216} a^{5} - \frac{245981475932827762217343177600145527707}{756610574824066224008211667095190935168} a^{4} - \frac{106542929321441003197595344969004513069}{378305287412033112004105833547595467584} a^{3} - \frac{3343546125723876762323483138837892371}{40897868909408985081524954978118428928} a^{2} - \frac{2566999861519063911340568472497566621}{10224467227352246270381238744529607232} a - \frac{4146194300626640929507745671864077683}{10224467227352246270381238744529607232}$

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:  \( 56207658188100000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T43:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6174
The 45 conjugacy class representatives for t21n43
Character table for t21n43 is not computed

Intermediate fields

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

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $21$ $21$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ $21$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$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}$
7Data not computed
37Data not computed