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Properties

Label 22.0.19123206450...2064.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{33}\cdot 67^{21}$
Root discriminant $156.54$
Ramified primes $2, 67$
Class number $1397102$ (GRH)
Class group $[1397102]$ (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![137216, 0, 8027136, 0, 54371840, 0, 131212800, 0, 145088768, 0, 77539904, 0, 19630464, 0, 2564224, 0, 181704, 0, 6968, 0, 134, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 134*x^20 + 6968*x^18 + 181704*x^16 + 2564224*x^14 + 19630464*x^12 + 77539904*x^10 + 145088768*x^8 + 131212800*x^6 + 54371840*x^4 + 8027136*x^2 + 137216)
 
gp: K = bnfinit(x^22 + 134*x^20 + 6968*x^18 + 181704*x^16 + 2564224*x^14 + 19630464*x^12 + 77539904*x^10 + 145088768*x^8 + 131212800*x^6 + 54371840*x^4 + 8027136*x^2 + 137216, 1)
 

Normalized defining polynomial

\( x^{22} + 134 x^{20} + 6968 x^{18} + 181704 x^{16} + 2564224 x^{14} + 19630464 x^{12} + 77539904 x^{10} + 145088768 x^{8} + 131212800 x^{6} + 54371840 x^{4} + 8027136 x^{2} + 137216 \)

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:  \(-1912320645087103459758821027834145013889251672064=-\,2^{33}\cdot 67^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $156.54$
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:  $2, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(536=2^{3}\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{536}(1,·)$, $\chi_{536}(5,·)$, $\chi_{536}(129,·)$, $\chi_{536}(9,·)$, $\chi_{536}(109,·)$, $\chi_{536}(45,·)$, $\chi_{536}(81,·)$, $\chi_{536}(405,·)$, $\chi_{536}(89,·)$, $\chi_{536}(25,·)$, $\chi_{536}(477,·)$, $\chi_{536}(133,·)$, $\chi_{536}(417,·)$, $\chi_{536}(193,·)$, $\chi_{536}(225,·)$, $\chi_{536}(125,·)$, $\chi_{536}(429,·)$, $\chi_{536}(253,·)$, $\chi_{536}(241,·)$, $\chi_{536}(53,·)$, $\chi_{536}(265,·)$, $\chi_{536}(445,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{7424} a^{16} + \frac{1}{3712} a^{14} + \frac{13}{1856} a^{12} + \frac{1}{232} a^{10} - \frac{11}{464} a^{8} + \frac{11}{232} a^{6} + \frac{5}{116} a^{4} - \frac{7}{58} a^{2} - \frac{5}{29}$, $\frac{1}{7424} a^{17} + \frac{1}{3712} a^{15} + \frac{13}{1856} a^{13} + \frac{1}{232} a^{11} - \frac{11}{464} a^{9} + \frac{11}{232} a^{7} + \frac{5}{116} a^{5} - \frac{7}{58} a^{3} - \frac{5}{29} a$, $\frac{1}{53289472} a^{18} - \frac{861}{13322368} a^{16} + \frac{13399}{13322368} a^{14} - \frac{10623}{1665296} a^{12} - \frac{18677}{3330592} a^{10} - \frac{44343}{1665296} a^{8} - \frac{10895}{416324} a^{6} + \frac{16753}{416324} a^{4} + \frac{25023}{104081} a^{2} + \frac{7629}{104081}$, $\frac{1}{53289472} a^{19} - \frac{861}{13322368} a^{17} + \frac{13399}{13322368} a^{15} - \frac{10623}{1665296} a^{13} - \frac{18677}{3330592} a^{11} - \frac{44343}{1665296} a^{9} - \frac{10895}{416324} a^{7} + \frac{16753}{416324} a^{5} + \frac{25023}{104081} a^{3} + \frac{7629}{104081} a$, $\frac{1}{12052411743922328507392} a^{20} + \frac{450965691453}{188318933498786382928} a^{18} - \frac{178006461792639605}{3013102935980582126848} a^{16} - \frac{2346707019078143843}{1506551467990291063424} a^{14} - \frac{179814978765154111}{23539866687348297866} a^{12} + \frac{2913900756043386233}{376637866997572765856} a^{10} + \frac{1374789881909054205}{94159466749393191464} a^{8} + \frac{1845313819336860951}{47079733374696595732} a^{6} + \frac{3036278358663837739}{47079733374696595732} a^{4} - \frac{2278692966717626914}{11769933343674148933} a^{2} - \frac{1757122307640656399}{11769933343674148933}$, $\frac{1}{12052411743922328507392} a^{21} + \frac{450965691453}{188318933498786382928} a^{19} - \frac{178006461792639605}{3013102935980582126848} a^{17} - \frac{2346707019078143843}{1506551467990291063424} a^{15} - \frac{179814978765154111}{23539866687348297866} a^{13} + \frac{2913900756043386233}{376637866997572765856} a^{11} + \frac{1374789881909054205}{94159466749393191464} a^{9} + \frac{1845313819336860951}{47079733374696595732} a^{7} + \frac{3036278358663837739}{47079733374696595732} a^{5} - \frac{2278692966717626914}{11769933343674148933} a^{3} - \frac{1757122307640656399}{11769933343674148933} a$

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

Class group and class number

$C_{1397102}$, which has order $1397102$ (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:  \( 338444542.042557 \) (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

\(\Q(\sqrt{-134}) \), 11.11.1822837804551761449.1

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 ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

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
67Data not computed