↑

Properties

Label 21.21.4443788844...5209.1
Degree $21$
Signature $[21, 0]$
Discriminant $3^{32}\cdot 547^{10}$
Root discriminant $107.36$
Ramified primes $3, 547$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $A_7$ (as 21T33)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-173, 2316, -5658, -32917, 163020, -98895, -558462, 1002705, -112956, -903062, 500406, 274332, -253860, -24204, 55011, -2835, -5892, 681, 304, -45, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 - 45*x^19 + 304*x^18 + 681*x^17 - 5892*x^16 - 2835*x^15 + 55011*x^14 - 24204*x^13 - 253860*x^12 + 274332*x^11 + 500406*x^10 - 903062*x^9 - 112956*x^8 + 1002705*x^7 - 558462*x^6 - 98895*x^5 + 163020*x^4 - 32917*x^3 - 5658*x^2 + 2316*x - 173)
 
gp: K = bnfinit(x^21 - 6*x^20 - 45*x^19 + 304*x^18 + 681*x^17 - 5892*x^16 - 2835*x^15 + 55011*x^14 - 24204*x^13 - 253860*x^12 + 274332*x^11 + 500406*x^10 - 903062*x^9 - 112956*x^8 + 1002705*x^7 - 558462*x^6 - 98895*x^5 + 163020*x^4 - 32917*x^3 - 5658*x^2 + 2316*x - 173, 1)
 

Normalized defining polynomial

\( x^{21} - 6 x^{20} - 45 x^{19} + 304 x^{18} + 681 x^{17} - 5892 x^{16} - 2835 x^{15} + 55011 x^{14} - 24204 x^{13} - 253860 x^{12} + 274332 x^{11} + 500406 x^{10} - 903062 x^{9} - 112956 x^{8} + 1002705 x^{7} - 558462 x^{6} - 98895 x^{5} + 163020 x^{4} - 32917 x^{3} - 5658 x^{2} + 2316 x - 173 \)

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:  \(4443788844232727236822935152611411648255209=3^{32}\cdot 547^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $107.36$
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:  $3, 547$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{63} a^{16} - \frac{1}{9} a^{15} + \frac{4}{21} a^{14} - \frac{1}{21} a^{12} - \frac{4}{21} a^{11} - \frac{13}{63} a^{10} + \frac{19}{63} a^{9} - \frac{1}{21} a^{8} - \frac{11}{63} a^{7} - \frac{16}{63} a^{6} + \frac{1}{21} a^{5} - \frac{26}{63} a^{4} + \frac{23}{63} a^{3} + \frac{4}{21} a^{2} - \frac{5}{63} a + \frac{26}{63}$, $\frac{1}{63} a^{17} + \frac{5}{63} a^{15} + \frac{1}{3} a^{14} - \frac{1}{21} a^{13} + \frac{10}{21} a^{12} + \frac{29}{63} a^{11} - \frac{1}{7} a^{10} + \frac{25}{63} a^{9} + \frac{31}{63} a^{8} - \frac{10}{21} a^{7} - \frac{4}{63} a^{6} - \frac{5}{63} a^{5} + \frac{10}{21} a^{4} + \frac{26}{63} a^{3} + \frac{16}{63} a^{2} - \frac{1}{7} a - \frac{4}{9}$, $\frac{1}{189} a^{18} - \frac{4}{27} a^{15} + \frac{10}{63} a^{13} - \frac{82}{189} a^{12} - \frac{25}{63} a^{11} + \frac{10}{21} a^{10} + \frac{83}{189} a^{9} - \frac{26}{63} a^{8} + \frac{17}{63} a^{7} + \frac{2}{7} a^{6} + \frac{5}{63} a^{5} + \frac{31}{63} a^{4} - \frac{19}{63} a^{3} + \frac{19}{63} a^{2} - \frac{1}{63} a - \frac{25}{189}$, $\frac{1}{567} a^{19} - \frac{1}{567} a^{18} - \frac{1}{567} a^{16} - \frac{5}{81} a^{15} - \frac{71}{189} a^{14} + \frac{11}{81} a^{13} - \frac{263}{567} a^{12} + \frac{10}{189} a^{11} + \frac{209}{567} a^{10} - \frac{152}{567} a^{9} + \frac{16}{189} a^{8} - \frac{5}{27} a^{7} + \frac{11}{189} a^{6} - \frac{73}{189} a^{5} - \frac{32}{189} a^{4} + \frac{5}{27} a^{3} - \frac{38}{189} a^{2} + \frac{221}{567} a + \frac{97}{567}$, $\frac{1}{63196929585779841750642861672651159} a^{20} - \frac{55339943572716551885623288252936}{63196929585779841750642861672651159} a^{19} + \frac{1123815558607178393913333159616}{3009377599322849607173469603459579} a^{18} - \frac{439363362661291936255363157895103}{63196929585779841750642861672651159} a^{17} + \frac{248642437099826764361044332072535}{63196929585779841750642861672651159} a^{16} + \frac{35010981384766914097564450751369}{1003125866440949869057823201153193} a^{15} + \frac{21039692888786697684853744677808226}{63196929585779841750642861672651159} a^{14} - \frac{14897097515253201116366905501160708}{63196929585779841750642861672651159} a^{13} + \frac{1072200071084336689799912320041949}{2340627021695549694468254136024117} a^{12} + \frac{7575678765613085160149113636421984}{63196929585779841750642861672651159} a^{11} - \frac{13767221246072411196847423344569765}{63196929585779841750642861672651159} a^{10} - \frac{2058051329505299364587463627068303}{7021881065086649083404762408072351} a^{9} + \frac{6578188016089499219386880620569469}{21065643195259947250214287224217053} a^{8} + \frac{9672573553023186442078965808025369}{21065643195259947250214287224217053} a^{7} + \frac{1303273655199764134089733147046264}{3009377599322849607173469603459579} a^{6} - \frac{4418690282144995968691447097166065}{21065643195259947250214287224217053} a^{5} - \frac{1693723486633115995396230284674528}{21065643195259947250214287224217053} a^{4} - \frac{652312062931250150234138840216975}{21065643195259947250214287224217053} a^{3} + \frac{15845967055953022542067197090190274}{63196929585779841750642861672651159} a^{2} - \frac{29325731449248580692629600273487095}{63196929585779841750642861672651159} a - \frac{7001879303915907236476000286204584}{21065643195259947250214287224217053}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 406277532862000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_7$ (as 21T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2520
The 9 conjugacy class representatives for $A_7$
Character table for $A_7$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 7 sibling: data not computed
Degree 15 siblings: data not computed
Degree 35 sibling: 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$

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$3.6.8.1$x^{6} + 6 x^{5} + 18 x^{2} + 9$$3$$2$$8$$C_3^2:C_4$$[2, 2]^{4}$
3.6.8.1$x^{6} + 6 x^{5} + 18 x^{2} + 9$$3$$2$$8$$C_3^2:C_4$$[2, 2]^{4}$
3.9.16.4$x^{9} + 3 x^{8} + 3$$9$$1$$16$$C_3^2:C_4$$[2, 2]^{4}$
547Data not computed