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Properties

Label 21.3.24011181716...4512.2
Degree $21$
Signature $[3, 9]$
Discriminant $-\,2^{18}\cdot 7^{17}\cdot 13^{14}$
Root discriminant $48.39$
Ramified primes $2, 7, 13$
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![5888, -53248, 149248, -215232, 172576, -56240, -47992, 95820, -90396, 61024, -35784, 20752, -11638, 6088, -2881, 1181, -439, 159, -59, 23, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 23*x^19 - 59*x^18 + 159*x^17 - 439*x^16 + 1181*x^15 - 2881*x^14 + 6088*x^13 - 11638*x^12 + 20752*x^11 - 35784*x^10 + 61024*x^9 - 90396*x^8 + 95820*x^7 - 47992*x^6 - 56240*x^5 + 172576*x^4 - 215232*x^3 + 149248*x^2 - 53248*x + 5888)
 
gp: K = bnfinit(x^21 - 7*x^20 + 23*x^19 - 59*x^18 + 159*x^17 - 439*x^16 + 1181*x^15 - 2881*x^14 + 6088*x^13 - 11638*x^12 + 20752*x^11 - 35784*x^10 + 61024*x^9 - 90396*x^8 + 95820*x^7 - 47992*x^6 - 56240*x^5 + 172576*x^4 - 215232*x^3 + 149248*x^2 - 53248*x + 5888, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 23 x^{19} - 59 x^{18} + 159 x^{17} - 439 x^{16} + 1181 x^{15} - 2881 x^{14} + 6088 x^{13} - 11638 x^{12} + 20752 x^{11} - 35784 x^{10} + 61024 x^{9} - 90396 x^{8} + 95820 x^{7} - 47992 x^{6} - 56240 x^{5} + 172576 x^{4} - 215232 x^{3} + 149248 x^{2} - 53248 x + 5888 \)

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:  \(-240111817160475801918451219385024512=-\,2^{18}\cdot 7^{17}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.39$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{12} a^{15} + \frac{1}{12} a^{14} - \frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{4} a^{11} - \frac{1}{12} a^{10} - \frac{1}{4} a^{9} - \frac{1}{12} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{24} a^{16} - \frac{1}{24} a^{15} - \frac{1}{8} a^{14} - \frac{5}{24} a^{13} + \frac{5}{24} a^{12} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} + \frac{5}{24} a^{9} + \frac{1}{12} a^{8} - \frac{5}{12} a^{7} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{48} a^{17} - \frac{1}{48} a^{16} + \frac{1}{48} a^{15} - \frac{1}{48} a^{14} - \frac{11}{48} a^{13} - \frac{5}{48} a^{12} + \frac{11}{48} a^{11} - \frac{11}{48} a^{10} + \frac{1}{24} a^{9} + \frac{5}{24} a^{8} - \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{5}{12} a^{4} + \frac{1}{12} a^{3} + \frac{1}{3}$, $\frac{1}{96} a^{18} - \frac{1}{96} a^{17} + \frac{1}{96} a^{16} + \frac{1}{32} a^{15} - \frac{7}{96} a^{14} - \frac{3}{32} a^{13} - \frac{17}{96} a^{12} + \frac{1}{96} a^{11} + \frac{11}{48} a^{10} + \frac{11}{48} a^{9} - \frac{1}{24} a^{8} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{7}{24} a^{5} + \frac{3}{8} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{192} a^{19} - \frac{1}{192} a^{18} + \frac{1}{192} a^{17} + \frac{1}{64} a^{16} - \frac{7}{192} a^{15} - \frac{3}{64} a^{14} - \frac{17}{192} a^{13} - \frac{47}{192} a^{12} + \frac{11}{96} a^{11} + \frac{11}{96} a^{10} - \frac{1}{48} a^{9} - \frac{1}{12} a^{8} - \frac{5}{12} a^{7} + \frac{17}{48} a^{6} + \frac{3}{16} a^{5} + \frac{1}{6} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{25474224818440218834187149304795008} a^{20} - \frac{7657964077075995812137281475609}{25474224818440218834187149304795008} a^{19} - \frac{1265417830430558361948601558873}{499494604283141545768375476564608} a^{18} - \frac{75002931684064734666633732767297}{25474224818440218834187149304795008} a^{17} - \frac{344186466298958211172069173015587}{25474224818440218834187149304795008} a^{16} + \frac{516224437399090061356563408446659}{25474224818440218834187149304795008} a^{15} + \frac{551210090114284205407746072233649}{8491408272813406278062383101598336} a^{14} - \frac{1085494454887263683701059893064961}{8491408272813406278062383101598336} a^{13} + \frac{1631754294274461823328485298158297}{12737112409220109417093574652397504} a^{12} - \frac{32915215670323306194366157291613}{4245704136406703139031191550799168} a^{11} - \frac{431562368110253033998208353815297}{2122852068203351569515595775399584} a^{10} - \frac{531406220125017051429617759681905}{3184278102305027354273393663099376} a^{9} + \frac{6357301050848539812423602580373}{46827619151544519915785200927932} a^{8} + \frac{65373000443897700442841433980747}{2122852068203351569515595775399584} a^{7} + \frac{778586044997643227665467794661745}{6368556204610054708546787326198752} a^{6} - \frac{4681237890937883959272519760189}{93655238303089039831570401855864} a^{5} + \frac{298336856009268365100884306197033}{796069525576256838568348415774844} a^{4} + \frac{103112661151602647289643906772041}{265356508525418946189449471924948} a^{3} - \frac{184156938459282797102981316380699}{398034762788128419284174207887422} a^{2} + \frac{68472492952189331878644078464782}{199017381394064209642087103943711} a + \frac{3144481111103523075001196853544}{66339127131354736547362367981237}$

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:  \( 5199297416.013432 \) (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

3.3.8281.1, 7.1.30721582528.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R $21$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
7Data not computed
$13$13.3.2.2$x^{3} - 13$$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}$