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Properties

Label 22.0.42765144951...1312.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{33}\cdot 11^{41}$
Root discriminant $246.79$
Ramified primes $2, 11$
Class number $914894$ (GRH)
Class group $[914894]$ (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![1330255872, 0, 9845535744, 0, 23394438144, 0, 20992435200, 0, 9128116096, 0, 2128407424, 0, 275949696, 0, 20058896, 0, 825704, 0, 18876, 0, 220, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 220*x^20 + 18876*x^18 + 825704*x^16 + 20058896*x^14 + 275949696*x^12 + 2128407424*x^10 + 9128116096*x^8 + 20992435200*x^6 + 23394438144*x^4 + 9845535744*x^2 + 1330255872)
 
gp: K = bnfinit(x^22 + 220*x^20 + 18876*x^18 + 825704*x^16 + 20058896*x^14 + 275949696*x^12 + 2128407424*x^10 + 9128116096*x^8 + 20992435200*x^6 + 23394438144*x^4 + 9845535744*x^2 + 1330255872, 1)
 

Normalized defining polynomial

\( x^{22} + 220 x^{20} + 18876 x^{18} + 825704 x^{16} + 20058896 x^{14} + 275949696 x^{12} + 2128407424 x^{10} + 9128116096 x^{8} + 20992435200 x^{6} + 23394438144 x^{4} + 9845535744 x^{2} + 1330255872 \)

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:  \(-42765144951456754503592723360164151789017189226381312=-\,2^{33}\cdot 11^{41}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $246.79$
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:  \(968=2^{3}\cdot 11^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{968}(1,·)$, $\chi_{968}(901,·)$, $\chi_{968}(705,·)$, $\chi_{968}(265,·)$, $\chi_{968}(461,·)$, $\chi_{968}(109,·)$, $\chi_{968}(529,·)$, $\chi_{968}(21,·)$, $\chi_{968}(89,·)$, $\chi_{968}(793,·)$, $\chi_{968}(285,·)$, $\chi_{968}(197,·)$, $\chi_{968}(353,·)$, $\chi_{968}(549,·)$, $\chi_{968}(881,·)$, $\chi_{968}(617,·)$, $\chi_{968}(813,·)$, $\chi_{968}(177,·)$, $\chi_{968}(373,·)$, $\chi_{968}(441,·)$, $\chi_{968}(637,·)$, $\chi_{968}(725,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{6} a^{3} - \frac{1}{3} a$, $\frac{1}{12} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{12} a^{5} - \frac{1}{3} a$, $\frac{1}{72} a^{6} + \frac{1}{36} a^{4} - \frac{1}{9} a^{2}$, $\frac{1}{72} a^{7} + \frac{1}{36} a^{5} + \frac{1}{18} a^{3} - \frac{1}{3} a$, $\frac{1}{144} a^{8} - \frac{1}{18} a^{2}$, $\frac{1}{1296} a^{9} - \frac{1}{216} a^{7} + \frac{1}{108} a^{5} - \frac{5}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{2592} a^{10} - \frac{1}{432} a^{8} + \frac{1}{216} a^{6} - \frac{5}{162} a^{4} + \frac{1}{18} a^{2}$, $\frac{1}{2592} a^{11} + \frac{1}{216} a^{7} + \frac{2}{81} a^{5} - \frac{2}{27} a^{3}$, $\frac{1}{15552} a^{12} - \frac{1}{7776} a^{10} - \frac{1}{1296} a^{8} - \frac{1}{486} a^{6} - \frac{11}{972} a^{4} + \frac{1}{27} a^{2}$, $\frac{1}{15552} a^{13} - \frac{1}{7776} a^{11} - \frac{13}{1944} a^{7} - \frac{1}{486} a^{5} - \frac{2}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{31104} a^{14} - \frac{1}{7776} a^{10} + \frac{11}{3888} a^{8} - \frac{1}{324} a^{6} - \frac{23}{972} a^{4} + \frac{1}{27} a^{2}$, $\frac{1}{93312} a^{15} + \frac{1}{46656} a^{13} + \frac{1}{5832} a^{11} - \frac{1}{11664} a^{9} - \frac{19}{5832} a^{7} + \frac{95}{2916} a^{5} + \frac{1}{18} a^{3} - \frac{2}{9} a$, $\frac{1}{559872} a^{16} + \frac{1}{69984} a^{14} + \frac{1}{139968} a^{12} - \frac{1}{69984} a^{10} + \frac{13}{4374} a^{8} + \frac{1}{2187} a^{6} - \frac{31}{972} a^{4} + \frac{1}{27} a^{2}$, $\frac{1}{559872} a^{17} + \frac{1}{279936} a^{15} - \frac{1}{69984} a^{13} - \frac{13}{69984} a^{11} - \frac{1}{34992} a^{9} - \frac{97}{17496} a^{7} + \frac{7}{729} a^{5} - \frac{4}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{10077696} a^{18} - \frac{1}{1679616} a^{16} - \frac{1}{93312} a^{14} + \frac{19}{1259712} a^{12} + \frac{7}{52488} a^{10} + \frac{25}{8748} a^{8} + \frac{1015}{157464} a^{6} + \frac{283}{8748} a^{4} - \frac{28}{243} a^{2} - \frac{1}{3}$, $\frac{1}{30233088} a^{19} - \frac{1}{5038848} a^{17} - \frac{1}{279936} a^{15} - \frac{31}{1889568} a^{13} - \frac{107}{629856} a^{11} - \frac{35}{104976} a^{9} + \frac{517}{118098} a^{7} + \frac{679}{26244} a^{5} - \frac{11}{1458} a^{3} - \frac{2}{9} a$, $\frac{1}{51863933309841912259584} a^{20} + \frac{6850221677681}{180083101770284417568} a^{18} - \frac{291064062306739}{1440664814162275340544} a^{16} - \frac{28862362723814999}{6482991663730239032448} a^{14} - \frac{7316806874925065}{540249305310853252704} a^{12} - \frac{34728852883766767}{180083101770284417568} a^{10} + \frac{76046670093793969}{50648372372892492441} a^{8} + \frac{281186432189229419}{67531163163856656588} a^{6} + \frac{70405034275271389}{7503462573761850732} a^{4} - \frac{10389600570027751}{208429515937829187} a^{2} + \frac{18410041417379}{2573203900467027}$, $\frac{1}{51863933309841912259584} a^{21} + \frac{128697288097055}{25931966654920956129792} a^{19} - \frac{1932194178901}{540249305310853252704} a^{17} - \frac{1425881904902939}{1620747915932559758112} a^{15} + \frac{1159838253346021}{405186978983139939528} a^{13} - \frac{6204476433988169}{270124652655426626352} a^{11} + \frac{236356035422766217}{810373957966279879056} a^{9} - \frac{1962542272360356781}{405186978983139939528} a^{7} + \frac{115632837609523451}{5627596930321388049} a^{5} + \frac{16006603131812242}{625288547813487561} a^{3} + \frac{18410041417379}{2573203900467027} a$

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

Class group and class number

$C_{914894}$, which has order $914894$ (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:  \( -1 \) (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:  \( 285114946276.13544 \) (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

\(\Q(\sqrt{-22}) \), 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{11}$ $22$ $22$ R ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ $22$ $22$

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