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Properties

Label 21.3.35485930488...7447.2
Degree $21$
Signature $[3, 9]$
Discriminant $-\,3^{18}\cdot 7^{17}\cdot 13^{14}$
Root discriminant $68.50$
Ramified primes $3, 7, 13$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times F_7$ (as 21T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2527, -20482, 57281, -6943, -296659, 783536, -1124295, 1106479, -823628, 486885, -229903, 85460, -27129, 11287, -6722, 3114, -685, -100, 92, -13, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 13*x^19 + 92*x^18 - 100*x^17 - 685*x^16 + 3114*x^15 - 6722*x^14 + 11287*x^13 - 27129*x^12 + 85460*x^11 - 229903*x^10 + 486885*x^9 - 823628*x^8 + 1106479*x^7 - 1124295*x^6 + 783536*x^5 - 296659*x^4 - 6943*x^3 + 57281*x^2 - 20482*x + 2527)
 
gp: K = bnfinit(x^21 - 4*x^20 - 13*x^19 + 92*x^18 - 100*x^17 - 685*x^16 + 3114*x^15 - 6722*x^14 + 11287*x^13 - 27129*x^12 + 85460*x^11 - 229903*x^10 + 486885*x^9 - 823628*x^8 + 1106479*x^7 - 1124295*x^6 + 783536*x^5 - 296659*x^4 - 6943*x^3 + 57281*x^2 - 20482*x + 2527, 1)
 

Normalized defining polynomial

\( x^{21} - 4 x^{20} - 13 x^{19} + 92 x^{18} - 100 x^{17} - 685 x^{16} + 3114 x^{15} - 6722 x^{14} + 11287 x^{13} - 27129 x^{12} + 85460 x^{11} - 229903 x^{10} + 486885 x^{9} - 823628 x^{8} + 1106479 x^{7} - 1124295 x^{6} + 783536 x^{5} - 296659 x^{4} - 6943 x^{3} + 57281 x^{2} - 20482 x + 2527 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-354859304882011896712926901003999617447=-\,3^{18}\cdot 7^{17}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.50$
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:  $3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{18} - \frac{1}{3}$, $\frac{1}{3} a^{19} - \frac{1}{3} a$, $\frac{1}{600987921080090310797672647231715551238360181} a^{20} + \frac{28454343398081313821592493738184784714981436}{200329307026696770265890882410571850412786727} a^{19} + \frac{74541224405004767578846114852740643447807160}{600987921080090310797672647231715551238360181} a^{18} + \frac{5102860637205499320950894782478082872242375}{600987921080090310797672647231715551238360181} a^{17} - \frac{3964479774555503646802424361767625361544582}{200329307026696770265890882410571850412786727} a^{16} - \frac{19224803538577649137358154131646215963023531}{600987921080090310797672647231715551238360181} a^{15} + \frac{51870338229855807628841136532675253890882619}{600987921080090310797672647231715551238360181} a^{14} - \frac{32319414356773681552084055142828334405022845}{200329307026696770265890882410571850412786727} a^{13} + \frac{25314474905150434648276561554442381026533116}{600987921080090310797672647231715551238360181} a^{12} - \frac{79783345345282616518195142059592832659336252}{600987921080090310797672647231715551238360181} a^{11} + \frac{56367023941700192832668283585684324036217391}{200329307026696770265890882410571850412786727} a^{10} + \frac{119520542773890730910383289987817943141292819}{600987921080090310797672647231715551238360181} a^{9} + \frac{254931112300035824650658626707210994946032955}{600987921080090310797672647231715551238360181} a^{8} - \frac{98603150530279370821965849898746939683832914}{200329307026696770265890882410571850412786727} a^{7} + \frac{41377020213928885327042244176232483285973370}{600987921080090310797672647231715551238360181} a^{6} + \frac{100430852462235811492760544758229390379617043}{600987921080090310797672647231715551238360181} a^{5} - \frac{19539218360620845119127387394140994923352508}{66776435675565590088630294136857283470928909} a^{4} - \frac{194858511715803218448147401394294714211582540}{600987921080090310797672647231715551238360181} a^{3} + \frac{75866954870610880295699210448552767736532099}{600987921080090310797672647231715551238360181} a^{2} + \frac{66120939067925626033151812735075617636059864}{200329307026696770265890882410571850412786727} a + \frac{14456738584707571301573961342366135341415851}{31630943214741595305140665643774502696755799}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (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:  $11$
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:  \( 94137132744.11961 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times F_7$ (as 21T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 126
The 21 conjugacy class representatives for $C_3\times F_7$
Character table for $C_3\times F_7$ is not computed

Intermediate fields

3.3.8281.2, 7.1.349938025983.1

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
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 $21$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ $21$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
7Data not computed
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$