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Properties

Label 22.0.13527660380...2359.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,17^{11}\cdot 23^{21}$
Root discriminant $82.23$
Ramified primes $17, 23$
Class number $659134$ (GRH)
Class group $[23, 28658]$ (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![2486401661, -2389932669, 2389932669, -1859353213, 1859353213, -997161597, 997161597, -350517885, 350517885, -81083005, 81083005, -12499581, 12499581, -1288829, 1288829, -87677, 87677, -3773, 3773, -93, 93, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 93*x^20 - 93*x^19 + 3773*x^18 - 3773*x^17 + 87677*x^16 - 87677*x^15 + 1288829*x^14 - 1288829*x^13 + 12499581*x^12 - 12499581*x^11 + 81083005*x^10 - 81083005*x^9 + 350517885*x^8 - 350517885*x^7 + 997161597*x^6 - 997161597*x^5 + 1859353213*x^4 - 1859353213*x^3 + 2389932669*x^2 - 2389932669*x + 2486401661)
 
gp: K = bnfinit(x^22 - x^21 + 93*x^20 - 93*x^19 + 3773*x^18 - 3773*x^17 + 87677*x^16 - 87677*x^15 + 1288829*x^14 - 1288829*x^13 + 12499581*x^12 - 12499581*x^11 + 81083005*x^10 - 81083005*x^9 + 350517885*x^8 - 350517885*x^7 + 997161597*x^6 - 997161597*x^5 + 1859353213*x^4 - 1859353213*x^3 + 2389932669*x^2 - 2389932669*x + 2486401661, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 93 x^{20} - 93 x^{19} + 3773 x^{18} - 3773 x^{17} + 87677 x^{16} - 87677 x^{15} + 1288829 x^{14} - 1288829 x^{13} + 12499581 x^{12} - 12499581 x^{11} + 81083005 x^{10} - 81083005 x^{9} + 350517885 x^{8} - 350517885 x^{7} + 997161597 x^{6} - 997161597 x^{5} + 1859353213 x^{4} - 1859353213 x^{3} + 2389932669 x^{2} - 2389932669 x + 2486401661 \)

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:  \(-1352766038082488977799200071554388212492359=-\,17^{11}\cdot 23^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.23$
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:  $17, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(391=17\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{391}(256,·)$, $\chi_{391}(1,·)$, $\chi_{391}(67,·)$, $\chi_{391}(324,·)$, $\chi_{391}(390,·)$, $\chi_{391}(135,·)$, $\chi_{391}(203,·)$, $\chi_{391}(18,·)$, $\chi_{391}(339,·)$, $\chi_{391}(84,·)$, $\chi_{391}(152,·)$, $\chi_{391}(154,·)$, $\chi_{391}(33,·)$, $\chi_{391}(35,·)$, $\chi_{391}(356,·)$, $\chi_{391}(358,·)$, $\chi_{391}(237,·)$, $\chi_{391}(239,·)$, $\chi_{391}(307,·)$, $\chi_{391}(52,·)$, $\chi_{391}(373,·)$, $\chi_{391}(188,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{603054709} a^{12} - \frac{211295624}{603054709} a^{11} + \frac{48}{603054709} a^{10} - \frac{251186821}{603054709} a^{9} + \frac{864}{603054709} a^{8} + \frac{202393827}{603054709} a^{7} + \frac{7168}{603054709} a^{6} + \frac{210647371}{603054709} a^{5} + \frac{26880}{603054709} a^{4} - \frac{173506423}{603054709} a^{3} + \frac{36864}{603054709} a^{2} + \frac{223027687}{603054709} a + \frac{8192}{603054709}$, $\frac{1}{603054709} a^{13} + \frac{52}{603054709} a^{11} + \frac{242127787}{603054709} a^{10} + \frac{1040}{603054709} a^{9} + \frac{36236136}{603054709} a^{8} + \frac{9984}{603054709} a^{7} - \frac{95748805}{603054709} a^{6} + \frac{46592}{603054709} a^{5} - \frac{116382665}{603054709} a^{4} + \frac{93184}{603054709} a^{3} - \frac{232765330}{603054709} a^{2} + \frac{53248}{603054709} a + \frac{166736978}{603054709}$, $\frac{1}{603054709} a^{14} - \frac{228539236}{603054709} a^{11} - \frac{1456}{603054709} a^{10} - \frac{169252770}{603054709} a^{9} - \frac{34944}{603054709} a^{8} + \frac{234756953}{603054709} a^{7} - \frac{326144}{603054709} a^{6} - \frac{215061195}{603054709} a^{5} - \frac{1304576}{603054709} a^{4} - \frac{256251969}{603054709} a^{3} - \frac{1863680}{603054709} a^{2} + \frac{27336725}{603054709} a - \frac{425984}{603054709}$, $\frac{1}{603054709} a^{15} - \frac{1680}{603054709} a^{11} - \frac{54354204}{603054709} a^{10} - \frac{44800}{603054709} a^{9} - \frac{109287695}{603054709} a^{8} - \frac{483840}{603054709} a^{7} + \frac{57592809}{603054709} a^{6} - \frac{2408448}{603054709} a^{5} + \frac{163145837}{603054709} a^{4} - \frac{5017600}{603054709} a^{3} + \frac{223447899}{603054709} a^{2} - \frac{2949120}{603054709} a - \frac{291450133}{603054709}$, $\frac{1}{603054709} a^{16} + \frac{168221077}{603054709} a^{11} + \frac{35840}{603054709} a^{10} + \frac{35149325}{603054709} a^{9} + \frac{967680}{603054709} a^{8} - \frac{43633707}{603054709} a^{7} + \frac{9633792}{603054709} a^{6} + \frac{57614934}{603054709} a^{5} + \frac{40140800}{603054709} a^{4} + \frac{8081706}{603054709} a^{3} + \frac{58982400}{603054709} a^{2} - \frac{101910262}{603054709} a + \frac{13762560}{603054709}$, $\frac{1}{603054709} a^{17} + \frac{43520}{603054709} a^{11} - \frac{199751154}{603054709} a^{10} + \frac{1305600}{603054709} a^{9} - \frac{50459366}{603054709} a^{8} + \frac{15040512}{603054709} a^{7} - \frac{244701711}{603054709} a^{6} + \frac{77987840}{603054709} a^{5} - \frac{70259972}{603054709} a^{4} + \frac{167116800}{603054709} a^{3} - \frac{192120143}{603054709} a^{2} + \frac{100270080}{603054709} a - \frac{87052719}{603054709}$, $\frac{1}{603054709} a^{18} + \frac{7602494}{603054709} a^{11} - \frac{783360}{603054709} a^{10} + \frac{27280511}{603054709} a^{9} - \frac{22560768}{603054709} a^{8} - \frac{206973097}{603054709} a^{7} - \frac{233963520}{603054709} a^{6} + \frac{193840326}{603054709} a^{5} + \frac{203408618}{603054709} a^{4} - \frac{40602572}{603054709} a^{3} - \frac{297941782}{603054709} a^{2} - \frac{86449604}{603054709} a + \frac{246538869}{603054709}$, $\frac{1}{603054709} a^{19} - \frac{992256}{603054709} a^{11} + \frac{265415508}{603054709} a^{10} - \frac{31752192}{603054709} a^{9} - \frac{141926114}{603054709} a^{8} + \frac{222028405}{603054709} a^{7} - \frac{25912856}{603054709} a^{6} - \frac{222976161}{603054709} a^{5} + \frac{39905059}{603054709} a^{4} - \frac{223923917}{603054709} a^{3} + \frac{75651265}{603054709} a^{2} - \frac{297301548}{603054709} a - \frac{164995821}{603054709}$, $\frac{1}{603054709} a^{20} + \frac{120968122}{603054709} a^{11} + \frac{15876096}{603054709} a^{10} + \frac{135990701}{603054709} a^{9} - \frac{126771829}{603054709} a^{8} + \frac{199373221}{603054709} a^{7} + \frac{255913048}{603054709} a^{6} - \frac{192256529}{603054709} a^{5} - \frac{86489833}{603054709} a^{4} - \frac{243064867}{603054709} a^{3} + \frac{97941096}{603054709} a^{2} - \frac{178746843}{603054709} a + \frac{288849935}{603054709}$, $\frac{1}{603054709} a^{21} + \frac{20837376}{603054709} a^{11} - \frac{242986774}{603054709} a^{10} + \frac{91524491}{603054709} a^{9} + \frac{11380470}{603054709} a^{8} + \frac{130325914}{603054709} a^{7} - \frac{99083483}{603054709} a^{6} + \frac{240509647}{603054709} a^{5} - \frac{195193299}{603054709} a^{4} - \frac{1582748}{603054709} a^{3} + \frac{41976804}{603054709} a^{2} + \frac{88619918}{603054709} a - \frac{151968537}{603054709}$

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

Class group and class number

$C_{23}\times C_{28658}$, which has order $659134$ (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:  \( 1038656.82438 \) (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

\(\Q(\sqrt{-391}) \), \(\Q(\zeta_{23})^+\)

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ R $22$ R $22$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
17Data not computed
23Data not computed