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Properties

Label 21.21.2377526973...2224.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 3^{28}\cdot 7^{15}\cdot 71^{3}\cdot 1326319793^{3}$
Root discriminant $1162.87$
Ramified primes $2, 3, 7, 71, 1326319793$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T70

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![310897, -4317927, 23606859, -62120671, 69115944, 17089548, -110714509, 68540517, 36692820, -51667790, 5372634, 12890748, -4547872, -962760, 707025, -40871, -34101, 4959, 590, -126, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 126*x^19 + 590*x^18 + 4959*x^17 - 34101*x^16 - 40871*x^15 + 707025*x^14 - 962760*x^13 - 4547872*x^12 + 12890748*x^11 + 5372634*x^10 - 51667790*x^9 + 36692820*x^8 + 68540517*x^7 - 110714509*x^6 + 17089548*x^5 + 69115944*x^4 - 62120671*x^3 + 23606859*x^2 - 4317927*x + 310897)
 
gp: K = bnfinit(x^21 - 3*x^20 - 126*x^19 + 590*x^18 + 4959*x^17 - 34101*x^16 - 40871*x^15 + 707025*x^14 - 962760*x^13 - 4547872*x^12 + 12890748*x^11 + 5372634*x^10 - 51667790*x^9 + 36692820*x^8 + 68540517*x^7 - 110714509*x^6 + 17089548*x^5 + 69115944*x^4 - 62120671*x^3 + 23606859*x^2 - 4317927*x + 310897, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 126 x^{19} + 590 x^{18} + 4959 x^{17} - 34101 x^{16} - 40871 x^{15} + 707025 x^{14} - 962760 x^{13} - 4547872 x^{12} + 12890748 x^{11} + 5372634 x^{10} - 51667790 x^{9} + 36692820 x^{8} + 68540517 x^{7} - 110714509 x^{6} + 17089548 x^{5} + 69115944 x^{4} - 62120671 x^{3} + 23606859 x^{2} - 4317927 x + 310897 \)

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:  \(23775269732638142813384381192164997335278189777532971313088692224=2^{18}\cdot 3^{28}\cdot 7^{15}\cdot 71^{3}\cdot 1326319793^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1162.87$
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:  $2, 3, 7, 71, 1326319793$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{8} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{9} + \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{14} a^{13} - \frac{1}{14} a^{12} - \frac{1}{14} a^{11} - \frac{1}{14} a^{10} - \frac{1}{14} a^{9} - \frac{1}{14} a^{8} - \frac{1}{14} a^{7} + \frac{5}{14} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{98} a^{14} - \frac{1}{49} a^{13} + \frac{3}{49} a^{12} - \frac{3}{49} a^{10} + \frac{1}{49} a^{9} - \frac{2}{49} a^{8} + \frac{1}{49} a^{7} + \frac{1}{49} a^{6} - \frac{3}{49} a^{5} + \frac{3}{49} a^{3} - \frac{1}{49} a^{2} + \frac{2}{49} a - \frac{3}{98}$, $\frac{1}{98} a^{15} + \frac{1}{49} a^{13} - \frac{1}{49} a^{12} - \frac{3}{49} a^{11} + \frac{2}{49} a^{10} - \frac{3}{49} a^{8} + \frac{3}{49} a^{7} + \frac{20}{49} a^{6} + \frac{22}{49} a^{5} + \frac{24}{49} a^{4} + \frac{19}{49} a^{3} + \frac{3}{7} a^{2} + \frac{47}{98} a + \frac{18}{49}$, $\frac{1}{98} a^{16} + \frac{1}{49} a^{13} - \frac{2}{49} a^{12} + \frac{2}{49} a^{11} - \frac{1}{49} a^{10} + \frac{2}{49} a^{9} - \frac{3}{49} a^{7} + \frac{6}{49} a^{6} + \frac{9}{49} a^{5} + \frac{5}{49} a^{4} + \frac{8}{49} a^{3} + \frac{9}{98} a^{2} + \frac{1}{7} a + \frac{10}{49}$, $\frac{1}{98} a^{17} + \frac{3}{49} a^{12} - \frac{1}{49} a^{11} + \frac{1}{49} a^{10} - \frac{2}{49} a^{9} + \frac{1}{49} a^{8} - \frac{3}{49} a^{7} - \frac{1}{7} a^{6} - \frac{10}{49} a^{5} - \frac{6}{49} a^{4} - \frac{17}{98} a^{3} - \frac{5}{49} a^{2} - \frac{8}{49} a - \frac{4}{49}$, $\frac{1}{98} a^{18} - \frac{1}{98} a^{13} + \frac{5}{98} a^{12} - \frac{5}{98} a^{11} + \frac{3}{98} a^{10} - \frac{5}{98} a^{9} + \frac{1}{98} a^{8} - \frac{1}{14} a^{7} + \frac{29}{98} a^{6} + \frac{23}{98} a^{5} + \frac{16}{49} a^{4} + \frac{25}{98} a^{3} + \frac{33}{98} a^{2} + \frac{27}{98} a + \frac{5}{14}$, $\frac{1}{20755224} a^{19} + \frac{13925}{2965032} a^{18} - \frac{21607}{6918408} a^{17} - \frac{10711}{2306136} a^{16} - \frac{15433}{3459204} a^{15} - \frac{13775}{3459204} a^{14} - \frac{669161}{20755224} a^{13} - \frac{4751}{391608} a^{12} - \frac{55451}{6918408} a^{11} + \frac{611}{130536} a^{10} + \frac{15161}{6918408} a^{9} + \frac{107101}{6918408} a^{8} - \frac{13895}{423576} a^{7} + \frac{860051}{20755224} a^{6} + \frac{1386541}{3459204} a^{5} + \frac{807983}{1729602} a^{4} - \frac{653791}{3459204} a^{3} + \frac{777127}{1729602} a^{2} + \frac{2984603}{20755224} a - \frac{4398071}{20755224}$, $\frac{1}{1416133176618541931985068114596416} a^{20} - \frac{2360645596998271156447325}{354033294154635482996267028649104} a^{19} + \frac{2652373603028260137508510134539}{708066588309270965992534057298208} a^{18} - \frac{160914842795094204387798819619}{59005549025772580499377838108184} a^{17} + \frac{1632456243610862073289022395501}{472044392206180643995022704865472} a^{16} + \frac{557664885090876976978229157319}{118011098051545160998755676216368} a^{15} + \frac{81626281122178546468291741139}{19945537698852703267395325557696} a^{14} - \frac{1859139787895290315880225880521}{354033294154635482996267028649104} a^{13} + \frac{6321176137552870225849280207575}{354033294154635482996267028649104} a^{12} - \frac{89985899346548403901180693949}{13112344227949462333195075135152} a^{11} + \frac{3382826172358414971982351528205}{59005549025772580499377838108184} a^{10} + \frac{11540162159466990215658869514523}{236022196103090321997511352432736} a^{9} + \frac{885470724004611278179764028795}{88508323538658870749066757162276} a^{8} - \frac{20639946656628581296149490877}{18633331271296604368224580455216} a^{7} - \frac{177503359019189299200085213183871}{1416133176618541931985068114596416} a^{6} + \frac{36248601130472274190419382111}{11239152195385253428452921544416} a^{5} - \frac{86131653715136737644537887681513}{236022196103090321997511352432736} a^{4} + \frac{13030373701845726124254766353047}{78674065367696773999170450810912} a^{3} - \frac{51927266828682910732750739335963}{202304739516934561712152587799488} a^{2} - \frac{17670885847627819295223822965357}{177016647077317741498133514324552} a - \frac{213534265704413014623502510003}{10647617869312345353271188831552}$

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:  \( 91059497445400000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T70:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24696
The 45 conjugacy class representatives for t21n70
Character table for t21n70 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 28 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 $21$ R $21$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ $21$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ $21$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ $21$

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$$[\ ]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 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$$[2]^{3}$
7Data not computed
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
1326319793Data not computed