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Properties

Label 22.0.40781203438...3104.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{22}\cdot 89^{20}$
Root discriminant $118.36$
Ramified primes $2, 89$
Class number $117139$ (GRH)
Class group $[117139]$ (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![1, 0, 1687, 0, 250429, 0, 9845506, 0, 13598526, 0, 7761381, 0, 2360004, 0, 416416, 0, 43459, 0, 2602, 0, 81, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 81*x^20 + 2602*x^18 + 43459*x^16 + 416416*x^14 + 2360004*x^12 + 7761381*x^10 + 13598526*x^8 + 9845506*x^6 + 250429*x^4 + 1687*x^2 + 1)
 
gp: K = bnfinit(x^22 + 81*x^20 + 2602*x^18 + 43459*x^16 + 416416*x^14 + 2360004*x^12 + 7761381*x^10 + 13598526*x^8 + 9845506*x^6 + 250429*x^4 + 1687*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{22} + 81 x^{20} + 2602 x^{18} + 43459 x^{16} + 416416 x^{14} + 2360004 x^{12} + 7761381 x^{10} + 13598526 x^{8} + 9845506 x^{6} + 250429 x^{4} + 1687 x^{2} + 1 \)

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:  \(-4078120343870794568028632139836020814044463104=-\,2^{22}\cdot 89^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $118.36$
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:  $2, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(356=2^{2}\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{356}(1,·)$, $\chi_{356}(67,·)$, $\chi_{356}(331,·)$, $\chi_{356}(269,·)$, $\chi_{356}(271,·)$, $\chi_{356}(275,·)$, $\chi_{356}(153,·)$, $\chi_{356}(217,·)$, $\chi_{356}(283,·)$, $\chi_{356}(93,·)$, $\chi_{356}(223,·)$, $\chi_{356}(97,·)$, $\chi_{356}(91,·)$, $\chi_{356}(167,·)$, $\chi_{356}(105,·)$, $\chi_{356}(299,·)$, $\chi_{356}(45,·)$, $\chi_{356}(179,·)$, $\chi_{356}(245,·)$, $\chi_{356}(39,·)$, $\chi_{356}(121,·)$, $\chi_{356}(345,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{6623} a^{18} - \frac{2613}{6623} a^{16} - \frac{2954}{6623} a^{14} - \frac{690}{6623} a^{12} - \frac{1477}{6623} a^{10} - \frac{1298}{6623} a^{8} + \frac{3184}{6623} a^{6} - \frac{683}{6623} a^{4} - \frac{521}{6623} a^{2} + \frac{2759}{6623}$, $\frac{1}{6623} a^{19} - \frac{2613}{6623} a^{17} - \frac{2954}{6623} a^{15} - \frac{690}{6623} a^{13} - \frac{1477}{6623} a^{11} - \frac{1298}{6623} a^{9} + \frac{3184}{6623} a^{7} - \frac{683}{6623} a^{5} - \frac{521}{6623} a^{3} + \frac{2759}{6623} a$, $\frac{1}{59999649265306446430139} a^{20} - \frac{1957641981089856857}{59999649265306446430139} a^{18} + \frac{13203172452822583116888}{59999649265306446430139} a^{16} - \frac{8361853895425950293933}{59999649265306446430139} a^{14} + \frac{10416280292646196281415}{59999649265306446430139} a^{12} + \frac{29355330266147790008535}{59999649265306446430139} a^{10} - \frac{24854331917112092083603}{59999649265306446430139} a^{8} + \frac{16205052955264457126882}{59999649265306446430139} a^{6} - \frac{13217949487366278099118}{59999649265306446430139} a^{4} - \frac{4197991993675682776}{16055565765401778547} a^{2} - \frac{24720034784300627693482}{59999649265306446430139}$, $\frac{1}{59999649265306446430139} a^{21} - \frac{1957641981089856857}{59999649265306446430139} a^{19} + \frac{13203172452822583116888}{59999649265306446430139} a^{17} - \frac{8361853895425950293933}{59999649265306446430139} a^{15} + \frac{10416280292646196281415}{59999649265306446430139} a^{13} + \frac{29355330266147790008535}{59999649265306446430139} a^{11} - \frac{24854331917112092083603}{59999649265306446430139} a^{9} + \frac{16205052955264457126882}{59999649265306446430139} a^{7} - \frac{13217949487366278099118}{59999649265306446430139} a^{5} - \frac{4197991993675682776}{16055565765401778547} a^{3} - \frac{24720034784300627693482}{59999649265306446430139} a$

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

Class group and class number

$C_{117139}$, which has order $117139$ (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:  \( -\frac{299199715661299}{15849521979666119} a^{21} - \frac{135394620934031}{88544815528861} a^{19} - \frac{778554563412431617}{15849521979666119} a^{17} - \frac{13004089685279283474}{15849521979666119} a^{15} - \frac{124610727144956560274}{15849521979666119} a^{13} - \frac{706292230713837293384}{15849521979666119} a^{11} - \frac{2323195132748317167612}{15849521979666119} a^{9} - \frac{4071844310951381513727}{15849521979666119} a^{7} - \frac{2951161919911943756766}{15849521979666119} a^{5} - \frac{78746404027271354940}{15849521979666119} a^{3} - \frac{669695610289093554}{15849521979666119} a \) (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:  \( 866679281.3791491 \) (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

\(\Q(\sqrt{-1}) \), 11.11.31181719929966183601.1

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 $22$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

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
$89$89.11.10.1$x^{11} - 89$$11$$1$$10$$C_{11}$$[\ ]_{11}$
89.11.10.1$x^{11} - 89$$11$$1$$10$$C_{11}$$[\ ]_{11}$