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Properties

Label 21.9.51760737730...3264.1
Degree $21$
Signature $[9, 6]$
Discriminant $2^{18}\cdot 3^{30}\cdot 17^{2}\cdot 57605311^{2}$
Root discriminant $62.50$
Ramified primes $2, 3, 17, 57605311$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T143

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5491, -91392, -526482, -1402417, -1853052, -1236021, -505489, -119745, 110757, 103482, 59316, 30555, 5757, 1677, -894, -292, -279, 18, -42, 15, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 + 15*x^19 - 42*x^18 + 18*x^17 - 279*x^16 - 292*x^15 - 894*x^14 + 1677*x^13 + 5757*x^12 + 30555*x^11 + 59316*x^10 + 103482*x^9 + 110757*x^8 - 119745*x^7 - 505489*x^6 - 1236021*x^5 - 1853052*x^4 - 1402417*x^3 - 526482*x^2 - 91392*x - 5491)
 
gp: K = bnfinit(x^21 - 3*x^20 + 15*x^19 - 42*x^18 + 18*x^17 - 279*x^16 - 292*x^15 - 894*x^14 + 1677*x^13 + 5757*x^12 + 30555*x^11 + 59316*x^10 + 103482*x^9 + 110757*x^8 - 119745*x^7 - 505489*x^6 - 1236021*x^5 - 1853052*x^4 - 1402417*x^3 - 526482*x^2 - 91392*x - 5491, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} + 15 x^{19} - 42 x^{18} + 18 x^{17} - 279 x^{16} - 292 x^{15} - 894 x^{14} + 1677 x^{13} + 5757 x^{12} + 30555 x^{11} + 59316 x^{10} + 103482 x^{9} + 110757 x^{8} - 119745 x^{7} - 505489 x^{6} - 1236021 x^{5} - 1853052 x^{4} - 1402417 x^{3} - 526482 x^{2} - 91392 x - 5491 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(51760737730687241530618054097890443264=2^{18}\cdot 3^{30}\cdot 17^{2}\cdot 57605311^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.50$
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:  $2, 3, 17, 57605311$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} - \frac{1}{17} a^{17} - \frac{5}{17} a^{16} + \frac{6}{17} a^{14} + \frac{5}{17} a^{13} + \frac{1}{17} a^{12} + \frac{4}{17} a^{11} + \frac{1}{17} a^{10} - \frac{8}{17} a^{9} + \frac{6}{17} a^{8} + \frac{6}{17} a^{7} - \frac{8}{17} a^{6} - \frac{3}{17} a^{5} + \frac{5}{17} a^{4} + \frac{2}{17} a^{3} - \frac{3}{17} a^{2} + \frac{8}{17} a$, $\frac{1}{17} a^{19} - \frac{6}{17} a^{17} - \frac{5}{17} a^{16} + \frac{6}{17} a^{15} - \frac{6}{17} a^{14} + \frac{6}{17} a^{13} + \frac{5}{17} a^{12} + \frac{5}{17} a^{11} - \frac{7}{17} a^{10} - \frac{2}{17} a^{9} - \frac{5}{17} a^{8} - \frac{2}{17} a^{7} + \frac{6}{17} a^{6} + \frac{2}{17} a^{5} + \frac{7}{17} a^{4} - \frac{1}{17} a^{3} + \frac{5}{17} a^{2} + \frac{8}{17} a$, $\frac{1}{128015364719278042580962850430471982990255802872301239} a^{20} - \frac{1231628894957908216088912834513204580691899806221916}{128015364719278042580962850430471982990255802872301239} a^{19} + \frac{1144967954217953440637774386902226132359539193247764}{128015364719278042580962850430471982990255802872301239} a^{18} + \frac{2479972279071142207656860403391319603155756118882530}{7530315571722237798880167672380704881779753110135367} a^{17} - \frac{60421570652922655131412995601733706386427104444978706}{128015364719278042580962850430471982990255802872301239} a^{16} + \frac{40915655928289689059799830701606491648279903422081113}{128015364719278042580962850430471982990255802872301239} a^{15} - \frac{9502260369434377912401366234208776257501850259913992}{128015364719278042580962850430471982990255802872301239} a^{14} - \frac{23549225249891677367934598533422774139857477197586766}{128015364719278042580962850430471982990255802872301239} a^{13} - \frac{62703702946920667865734781801750899962886844059347178}{128015364719278042580962850430471982990255802872301239} a^{12} + \frac{36012584311443346514819369144056379353749844745688951}{128015364719278042580962850430471982990255802872301239} a^{11} - \frac{24873102612339987956853145136511682338215287720023269}{128015364719278042580962850430471982990255802872301239} a^{10} - \frac{16627010667210104531933654152249622089126176489144318}{128015364719278042580962850430471982990255802872301239} a^{9} + \frac{25755249335315685835247892527600714975522965901774075}{128015364719278042580962850430471982990255802872301239} a^{8} - \frac{20286604808752613227886654876109648040051897276249140}{128015364719278042580962850430471982990255802872301239} a^{7} - \frac{53961791278845167726303495075884424251890099053451399}{128015364719278042580962850430471982990255802872301239} a^{6} + \frac{62941020931662448268299911590705270327882838290671020}{128015364719278042580962850430471982990255802872301239} a^{5} + \frac{56918016946502305720457379383485761261755457580816191}{128015364719278042580962850430471982990255802872301239} a^{4} - \frac{25194058101625835699832507696217721878194277946762192}{128015364719278042580962850430471982990255802872301239} a^{3} - \frac{3672838268773892753251435806292819178715377532817264}{7530315571722237798880167672380704881779753110135367} a^{2} + \frac{3742642386913372908666028011443434314290753136090015}{7530315571722237798880167672380704881779753110135367} a - \frac{3161575374707191062436067400312503179558560542729743}{7530315571722237798880167672380704881779753110135367}$

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:  $14$
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:  \( 223805105406 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T143:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14224896
The 88 conjugacy class representatives for t21n143 are not computed
Character table for t21n143 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 24 sibling: 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 R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $21$ $21$ R ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.12.12.2$x^{12} - 6 x^{10} - 13 x^{8} - 28 x^{6} + 15 x^{4} - 30 x^{2} - 3$$2$$6$$12$12T105$[2, 2, 2, 2, 2]^{6}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
3.12.18.74$x^{12} - 6 x^{11} - 3 x^{10} - 12 x^{9} + 9 x^{8} + 9 x^{7} + 12 x^{6} - 9 x^{3} - 9$$6$$2$$18$$C_{12}$$[2]_{2}^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.7.0.1$x^{7} - x + 3$$1$$7$$0$$C_7$$[\ ]^{7}$
57605311Data not computed