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Properties

Label 22.0.13936571865...2704.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{22}\cdot 67^{20}$
Root discriminant $91.43$
Ramified primes $2, 67$
Class number $36961$ (GRH)
Class group $[36961]$ (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![841, 0, 57019, 0, 969293, 0, 2944762, 0, 3334358, 0, 1953909, 0, 672568, 0, 142812, 0, 18767, 0, 1466, 0, 61, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 61*x^20 + 1466*x^18 + 18767*x^16 + 142812*x^14 + 672568*x^12 + 1953909*x^10 + 3334358*x^8 + 2944762*x^6 + 969293*x^4 + 57019*x^2 + 841)
 
gp: K = bnfinit(x^22 + 61*x^20 + 1466*x^18 + 18767*x^16 + 142812*x^14 + 672568*x^12 + 1953909*x^10 + 3334358*x^8 + 2944762*x^6 + 969293*x^4 + 57019*x^2 + 841, 1)
 

Normalized defining polynomial

\( x^{22} + 61 x^{20} + 1466 x^{18} + 18767 x^{16} + 142812 x^{14} + 672568 x^{12} + 1953909 x^{10} + 3334358 x^{8} + 2944762 x^{6} + 969293 x^{4} + 57019 x^{2} + 841 \)

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:  \(-13936571865431899047915848208912554030792704=-\,2^{22}\cdot 67^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.43$
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:  \(268=2^{2}\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{268}(1,·)$, $\chi_{268}(131,·)$, $\chi_{268}(263,·)$, $\chi_{268}(9,·)$, $\chi_{268}(143,·)$, $\chi_{268}(81,·)$, $\chi_{268}(107,·)$, $\chi_{268}(149,·)$, $\chi_{268}(215,·)$, $\chi_{268}(25,·)$, $\chi_{268}(15,·)$, $\chi_{268}(159,·)$, $\chi_{268}(225,·)$, $\chi_{268}(91,·)$, $\chi_{268}(129,·)$, $\chi_{268}(193,·)$, $\chi_{268}(135,·)$, $\chi_{268}(223,·)$, $\chi_{268}(241,·)$, $\chi_{268}(89,·)$, $\chi_{268}(265,·)$, $\chi_{268}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3589} a^{18} - \frac{1385}{3589} a^{16} - \frac{1598}{3589} a^{14} - \frac{55}{3589} a^{12} - \frac{42}{97} a^{10} + \frac{683}{3589} a^{8} - \frac{1074}{3589} a^{6} - \frac{1664}{3589} a^{4} + \frac{636}{3589} a^{2} - \frac{1168}{3589}$, $\frac{1}{104081} a^{19} - \frac{40864}{104081} a^{17} + \frac{9169}{104081} a^{15} - \frac{55}{104081} a^{13} + \frac{152}{2813} a^{11} + \frac{47340}{104081} a^{9} - \frac{11841}{104081} a^{7} + \frac{41404}{104081} a^{5} - \frac{2953}{104081} a^{3} - \frac{47825}{104081} a$, $\frac{1}{55922909710652711} a^{20} - \frac{3790645056271}{55922909710652711} a^{18} + \frac{12431823211438717}{55922909710652711} a^{16} - \frac{14784484195636425}{55922909710652711} a^{14} - \frac{10046397035901170}{55922909710652711} a^{12} - \frac{12590692015905581}{55922909710652711} a^{10} + \frac{680609036849393}{55922909710652711} a^{8} - \frac{10217098261483437}{55922909710652711} a^{6} - \frac{6571963048437087}{55922909710652711} a^{4} + \frac{6954739873882007}{55922909710652711} a^{2} + \frac{933518806678822}{1928376196919059}$, $\frac{1}{55922909710652711} a^{21} - \frac{29532384654}{55922909710652711} a^{19} + \frac{26506444130439762}{55922909710652711} a^{17} + \frac{532463726768104}{1511429992179803} a^{15} - \frac{10253258232840105}{55922909710652711} a^{13} + \frac{8561805649268427}{55922909710652711} a^{11} + \frac{10962953779240040}{55922909710652711} a^{9} + \frac{1170476304552377}{55922909710652711} a^{7} - \frac{18615583124764952}{55922909710652711} a^{5} - \frac{4151825845402994}{55922909710652711} a^{3} + \frac{14965561005560946}{55922909710652711} a$

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

Class group and class number

$C_{36961}$, which has order $36961$ (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:  \( -\frac{72933840167}{233987069919049} a^{21} - \frac{4205981313173}{233987069919049} a^{19} - \frac{93216845469217}{233987069919049} a^{17} - \frac{1075618941279740}{233987069919049} a^{15} - \frac{7206450912301406}{233987069919049} a^{13} - \frac{1005608112077194}{8068519652381} a^{11} - \frac{71087326003407904}{233987069919049} a^{9} - \frac{100037379277050157}{233987069919049} a^{7} - \frac{1997989584513736}{6323974862677} a^{5} - \frac{24100784453974422}{233987069919049} a^{3} - \frac{2053811813163438}{233987069919049} a \) (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:  \( 338444542.043 \) (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

\(\Q(\sqrt{-1}) \), 11.11.1822837804551761449.1

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 $22$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{22}$ $22$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

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
67Data not computed