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Properties

Label 22.0.13826007828...2683.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,3^{11}\cdot 7^{11}\cdot 23^{21}$
Root discriminant $91.40$
Ramified primes $3, 7, 23$
Class number $2866868$ (GRH)
Class group $[2, 1433434]$ (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![17926283491, -16803236616, 16803236616, -11861830366, 11861830366, -5438002241, 5438002241, -1583705366, 1583705366, -298939741, 298939741, -37314741, 37314741, -3102241, 3102241, -169741, 169741, -5866, 5866, -116, 116, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 116*x^20 - 116*x^19 + 5866*x^18 - 5866*x^17 + 169741*x^16 - 169741*x^15 + 3102241*x^14 - 3102241*x^13 + 37314741*x^12 - 37314741*x^11 + 298939741*x^10 - 298939741*x^9 + 1583705366*x^8 - 1583705366*x^7 + 5438002241*x^6 - 5438002241*x^5 + 11861830366*x^4 - 11861830366*x^3 + 16803236616*x^2 - 16803236616*x + 17926283491)
 
gp: K = bnfinit(x^22 - x^21 + 116*x^20 - 116*x^19 + 5866*x^18 - 5866*x^17 + 169741*x^16 - 169741*x^15 + 3102241*x^14 - 3102241*x^13 + 37314741*x^12 - 37314741*x^11 + 298939741*x^10 - 298939741*x^9 + 1583705366*x^8 - 1583705366*x^7 + 5438002241*x^6 - 5438002241*x^5 + 11861830366*x^4 - 11861830366*x^3 + 16803236616*x^2 - 16803236616*x + 17926283491, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 116 x^{20} - 116 x^{19} + 5866 x^{18} - 5866 x^{17} + 169741 x^{16} - 169741 x^{15} + 3102241 x^{14} - 3102241 x^{13} + 37314741 x^{12} - 37314741 x^{11} + 298939741 x^{10} - 298939741 x^{9} + 1583705366 x^{8} - 1583705366 x^{7} + 5438002241 x^{6} - 5438002241 x^{5} + 11861830366 x^{4} - 11861830366 x^{3} + 16803236616 x^{2} - 16803236616 x + 17926283491 \)

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:  \(-13826007828239234871378695182440742234972683=-\,3^{11}\cdot 7^{11}\cdot 23^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.40$
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:  $3, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(483=3\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{483}(64,·)$, $\chi_{483}(1,·)$, $\chi_{483}(398,·)$, $\chi_{483}(463,·)$, $\chi_{483}(272,·)$, $\chi_{483}(83,·)$, $\chi_{483}(20,·)$, $\chi_{483}(85,·)$, $\chi_{483}(400,·)$, $\chi_{483}(482,·)$, $\chi_{483}(419,·)$, $\chi_{483}(356,·)$, $\chi_{483}(293,·)$, $\chi_{483}(358,·)$, $\chi_{483}(232,·)$, $\chi_{483}(169,·)$, $\chi_{483}(211,·)$, $\chi_{483}(314,·)$, $\chi_{483}(251,·)$, $\chi_{483}(125,·)$, $\chi_{483}(190,·)$, $\chi_{483}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3912125981} a^{12} - \frac{1695528413}{3912125981} a^{11} + \frac{60}{3912125981} a^{10} + \frac{636960829}{3912125981} a^{9} + \frac{1350}{3912125981} a^{8} + \frac{1002838637}{3912125981} a^{7} + \frac{14000}{3912125981} a^{6} + \frac{1928617607}{3912125981} a^{5} + \frac{65625}{3912125981} a^{4} - \frac{936331937}{3912125981} a^{3} + \frac{112500}{3912125981} a^{2} - \frac{936331937}{3912125981} a + \frac{31250}{3912125981}$, $\frac{1}{3912125981} a^{13} + \frac{65}{3912125981} a^{11} + \frac{653390103}{3912125981} a^{10} + \frac{1625}{3912125981} a^{9} + \frac{1372497302}{3912125981} a^{8} + \frac{19500}{3912125981} a^{7} + \frac{545946899}{3912125981} a^{6} + \frac{113750}{3912125981} a^{5} - \frac{571380414}{3912125981} a^{4} + \frac{284375}{3912125981} a^{3} - \frac{1428451035}{3912125981} a^{2} + \frac{203125}{3912125981} a - \frac{571380414}{3912125981}$, $\frac{1}{3912125981} a^{14} + \frac{1323209480}{3912125981} a^{11} - \frac{2275}{3912125981} a^{10} - \frac{908696773}{3912125981} a^{9} - \frac{68250}{3912125981} a^{8} + \frac{1867577171}{3912125981} a^{7} - \frac{796250}{3912125981} a^{6} - \frac{743493477}{3912125981} a^{5} - \frac{3981250}{3912125981} a^{4} + \frac{751235155}{3912125981} a^{3} - \frac{7109375}{3912125981} a^{2} + \frac{1608305776}{3912125981} a - \frac{2031250}{3912125981}$, $\frac{1}{3912125981} a^{15} - \frac{2625}{3912125981} a^{11} + \frac{1853380028}{3912125981} a^{10} - \frac{87500}{3912125981} a^{9} - \frac{535773493}{3912125981} a^{8} - \frac{1181250}{3912125981} a^{7} - \frac{37440286}{83236723} a^{6} - \frac{7350000}{3912125981} a^{5} - \frac{1322615569}{3912125981} a^{4} - \frac{19140625}{3912125981} a^{3} + \frac{847508807}{3912125981} a^{2} - \frac{14062500}{3912125981} a + \frac{875369170}{3912125981}$, $\frac{1}{3912125981} a^{16} - \frac{821463700}{3912125981} a^{11} + \frac{70000}{3912125981} a^{10} + \frac{1008608745}{3912125981} a^{9} + \frac{2362500}{3912125981} a^{8} + \frac{1743069451}{3912125981} a^{7} + \frac{29400000}{3912125981} a^{6} - \frac{992416608}{3912125981} a^{5} + \frac{153125000}{3912125981} a^{4} - \frac{208709750}{3912125981} a^{3} + \frac{281250000}{3912125981} a^{2} - \frac{180849387}{3912125981} a + \frac{82031250}{3912125981}$, $\frac{1}{3912125981} a^{17} + \frac{85000}{3912125981} a^{11} - \frac{561207008}{3912125981} a^{10} + \frac{3187500}{3912125981} a^{9} - \frac{324714153}{3912125981} a^{8} + \frac{45900000}{3912125981} a^{7} + \frac{1761125233}{3912125981} a^{6} + \frac{297500000}{3912125981} a^{5} - \frac{749415430}{3912125981} a^{4} + \frac{796875000}{3912125981} a^{3} - \frac{1666648550}{3912125981} a^{2} + \frac{597656250}{3912125981} a - \frac{630062322}{3912125981}$, $\frac{1}{3912125981} a^{18} + \frac{544883933}{3912125981} a^{11} - \frac{1912500}{3912125981} a^{10} + \frac{1828397887}{3912125981} a^{9} - \frac{68850000}{3912125981} a^{8} + \frac{38084686}{83236723} a^{7} - \frac{892500000}{3912125981} a^{6} + \frac{481097394}{3912125981} a^{5} - \frac{869124019}{3912125981} a^{4} - \frac{1742961014}{3912125981} a^{3} - \frac{1140591788}{3912125981} a^{2} - \frac{706374786}{3912125981} a + \frac{1255875981}{3912125981}$, $\frac{1}{3912125981} a^{19} - \frac{2422500}{3912125981} a^{11} + \frac{432369755}{3912125981} a^{10} - \frac{96900000}{3912125981} a^{9} + \frac{1676355120}{3912125981} a^{8} - \frac{1453500000}{3912125981} a^{7} + \frac{751698344}{3912125981} a^{6} - \frac{1865748038}{3912125981} a^{5} + \frac{992528182}{3912125981} a^{4} + \frac{888788117}{3912125981} a^{3} - \frac{1046840997}{3912125981} a^{2} - \frac{626870095}{3912125981} a + \frac{1861489043}{3912125981}$, $\frac{1}{3912125981} a^{20} - \frac{1750277206}{3912125981} a^{11} + \frac{48450000}{3912125981} a^{10} - \frac{1005448305}{3912125981} a^{9} + \frac{1816875000}{3912125981} a^{8} + \frac{1885393578}{3912125981} a^{7} + \frac{752244114}{3912125981} a^{6} + \frac{1132046527}{3912125981} a^{5} - \frac{531814604}{3912125981} a^{4} + \frac{1128064227}{3912125981} a^{3} - \frac{1944438765}{3912125981} a^{2} + \frac{124268286}{3912125981} a + \frac{1372731361}{3912125981}$, $\frac{1}{3912125981} a^{21} + \frac{63590625}{3912125981} a^{11} - \frac{1616217432}{3912125981} a^{10} - \frac{1262516606}{3912125981} a^{9} + \frac{1835529154}{3912125981} a^{8} + \frac{1758427690}{3912125981} a^{7} - \frac{11579031}{83236723} a^{6} + \frac{448039724}{3912125981} a^{5} - \frac{861220164}{3912125981} a^{4} - \frac{1798209863}{3912125981} a^{3} + \frac{1185067594}{3912125981} a^{2} + \frac{1518960263}{3912125981} a + \frac{729347139}{3912125981}$

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

Class group and class number

$C_{2}\times C_{1433434}$, which has order $2866868$ (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

\(\Q(\sqrt{-483}) \), \(\Q(\zeta_{23})^+\)

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 $22$ R $22$ R ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R $22$ $22$ $22$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
3Data not computed
7Data not computed
23Data not computed