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Properties

Label 22.0.18983107666...7904.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{22}\cdot 11^{40}$
Root discriminant $156.48$
Ramified primes $2, 11$
Class number $14411$ (GRH)
Class group $[14411]$ (GRH)
Galois group $C_{22}$ (as 22T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59049, 0, 121207185, 0, 281918538, 0, 274663048, 0, 147421505, 0, 48063862, 0, 9871136, 0, 1279201, 0, 101783, 0, 4675, 0, 110, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 110*x^20 + 4675*x^18 + 101783*x^16 + 1279201*x^14 + 9871136*x^12 + 48063862*x^10 + 147421505*x^8 + 274663048*x^6 + 281918538*x^4 + 121207185*x^2 + 59049)
 
gp: K = bnfinit(x^22 + 110*x^20 + 4675*x^18 + 101783*x^16 + 1279201*x^14 + 9871136*x^12 + 48063862*x^10 + 147421505*x^8 + 274663048*x^6 + 281918538*x^4 + 121207185*x^2 + 59049, 1)
 

Normalized defining polynomial

\( x^{22} + 110 x^{20} + 4675 x^{18} + 101783 x^{16} + 1279201 x^{14} + 9871136 x^{12} + 48063862 x^{10} + 147421505 x^{8} + 274663048 x^{6} + 281918538 x^{4} + 121207185 x^{2} + 59049 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1898310766666226673632489495745922930975549947904=-\,2^{22}\cdot 11^{40}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $156.48$
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:  $2, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(484=2^{2}\cdot 11^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{484}(1,·)$, $\chi_{484}(67,·)$, $\chi_{484}(133,·)$, $\chi_{484}(199,·)$, $\chi_{484}(265,·)$, $\chi_{484}(331,·)$, $\chi_{484}(397,·)$, $\chi_{484}(463,·)$, $\chi_{484}(23,·)$, $\chi_{484}(89,·)$, $\chi_{484}(155,·)$, $\chi_{484}(221,·)$, $\chi_{484}(287,·)$, $\chi_{484}(353,·)$, $\chi_{484}(419,·)$, $\chi_{484}(45,·)$, $\chi_{484}(111,·)$, $\chi_{484}(177,·)$, $\chi_{484}(243,·)$, $\chi_{484}(309,·)$, $\chi_{484}(375,·)$, $\chi_{484}(441,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} - \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} - \frac{1}{9} a^{5} + \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{2}$, $\frac{1}{81} a^{9} + \frac{1}{27} a^{7} + \frac{1}{27} a^{5} + \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{10} + \frac{1}{27} a^{8} + \frac{1}{27} a^{6} + \frac{10}{81} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{81} a^{11} + \frac{1}{27} a^{7} - \frac{8}{81} a^{5} - \frac{4}{27} a^{3}$, $\frac{1}{243} a^{12} + \frac{1}{243} a^{10} - \frac{1}{81} a^{8} + \frac{4}{243} a^{6} - \frac{11}{243} a^{4} - \frac{2}{27} a^{2}$, $\frac{1}{243} a^{13} + \frac{1}{243} a^{11} + \frac{13}{243} a^{7} - \frac{2}{243} a^{5} + \frac{4}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{729} a^{14} - \frac{1}{729} a^{12} + \frac{4}{729} a^{10} - \frac{17}{729} a^{8} + \frac{8}{729} a^{6} + \frac{13}{729} a^{4} - \frac{2}{81} a^{2}$, $\frac{1}{729} a^{15} - \frac{1}{729} a^{13} + \frac{4}{729} a^{11} + \frac{1}{729} a^{9} - \frac{19}{729} a^{7} - \frac{95}{729} a^{5} + \frac{1}{9} a^{3} + \frac{2}{9} a$, $\frac{1}{6561} a^{16} - \frac{1}{2187} a^{14} - \frac{4}{2187} a^{12} + \frac{38}{6561} a^{10} - \frac{22}{2187} a^{8} + \frac{56}{2187} a^{6} + \frac{721}{6561} a^{4} + \frac{130}{729} a^{2} + \frac{1}{9}$, $\frac{1}{6561} a^{17} - \frac{1}{2187} a^{15} - \frac{4}{2187} a^{13} + \frac{38}{6561} a^{11} + \frac{5}{2187} a^{9} - \frac{106}{2187} a^{7} - \frac{494}{6561} a^{5} - \frac{104}{729} a^{3} - \frac{1}{9} a$, $\frac{1}{19683} a^{18} + \frac{1}{19683} a^{16} + \frac{1}{6561} a^{14} - \frac{37}{19683} a^{12} - \frac{49}{19683} a^{10} + \frac{301}{6561} a^{8} + \frac{880}{19683} a^{6} + \frac{1975}{19683} a^{4} - \frac{182}{2187} a^{2} - \frac{5}{27}$, $\frac{1}{59049} a^{19} - \frac{2}{59049} a^{17} - \frac{5}{19683} a^{15} + \frac{26}{59049} a^{13} - \frac{28}{59049} a^{11} + \frac{34}{19683} a^{9} + \frac{889}{59049} a^{7} + \frac{8452}{59049} a^{5} + \frac{211}{6561} a^{3} - \frac{8}{81} a$, $\frac{1}{26402245970311503} a^{20} + \frac{138754934398}{26402245970311503} a^{18} - \frac{442186375951}{8800748656770501} a^{16} - \frac{11464582516660}{26402245970311503} a^{14} + \frac{9487050996848}{26402245970311503} a^{12} - \frac{5481345992918}{977860961863389} a^{10} + \frac{623585795482060}{26402245970311503} a^{8} - \frac{1007779356535613}{26402245970311503} a^{6} + \frac{161857180309537}{8800748656770501} a^{4} + \frac{389337864680260}{977860961863389} a^{2} - \frac{25077976301}{12072357553869}$, $\frac{1}{79206737910934509} a^{21} + \frac{585879288245}{79206737910934509} a^{19} + \frac{5827430533699}{79206737910934509} a^{17} - \frac{6099090270496}{79206737910934509} a^{15} - \frac{111683648895689}{79206737910934509} a^{13} - \frac{35768128993189}{79206737910934509} a^{11} - \frac{477681488043101}{79206737910934509} a^{9} + \frac{53693859497165}{79206737910934509} a^{7} + \frac{4345148963335915}{79206737910934509} a^{5} - \frac{262784338269620}{8800748656770501} a^{3} - \frac{19748705498171}{108651217984821} a$

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

Class group and class number

$C_{14411}$, which has order $14411$ (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:  $10$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{4825964}{6419218568031} a^{21} + \frac{505973402}{6419218568031} a^{19} + \frac{19919812957}{6419218568031} a^{17} + \frac{385002011911}{6419218568031} a^{15} + \frac{4046996472904}{6419218568031} a^{13} + \frac{23912359716185}{6419218568031} a^{11} + \frac{76235839889033}{6419218568031} a^{9} + \frac{100272578425481}{6419218568031} a^{7} - \frac{77790813933239}{6419218568031} a^{5} - \frac{41046756604106}{713246507559} a^{3} - \frac{132988481978}{2935170813} a \) (order $4$)
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:  \( 285114946276.13544 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{22}$ (as 22T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 11.11.672749994932560009201.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ R ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

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
2Data not computed
11Data not computed