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Properties

Label 22.22.3414195660...5625.1
Degree $22$
Signature $[22, 0]$
Discriminant $3^{11}\cdot 5^{11}\cdot 23^{21}$
Root discriminant $77.25$
Ramified primes $3, 5, 23$
Class number $2$ (GRH)
Class group $[2]$ (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![7621189, -104090181, 104090181, 426489275, -426489275, -435702341, 435702341, 210941371, -210941371, -58493509, 58493509, 10089915, -10089915, -1120837, 1120837, 80315, -80315, -3589, 3589, 91, -91, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 91*x^20 + 91*x^19 + 3589*x^18 - 3589*x^17 - 80315*x^16 + 80315*x^15 + 1120837*x^14 - 1120837*x^13 - 10089915*x^12 + 10089915*x^11 + 58493509*x^10 - 58493509*x^9 - 210941371*x^8 + 210941371*x^7 + 435702341*x^6 - 435702341*x^5 - 426489275*x^4 + 426489275*x^3 + 104090181*x^2 - 104090181*x + 7621189)
 
gp: K = bnfinit(x^22 - x^21 - 91*x^20 + 91*x^19 + 3589*x^18 - 3589*x^17 - 80315*x^16 + 80315*x^15 + 1120837*x^14 - 1120837*x^13 - 10089915*x^12 + 10089915*x^11 + 58493509*x^10 - 58493509*x^9 - 210941371*x^8 + 210941371*x^7 + 435702341*x^6 - 435702341*x^5 - 426489275*x^4 + 426489275*x^3 + 104090181*x^2 - 104090181*x + 7621189, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} - 91 x^{20} + 91 x^{19} + 3589 x^{18} - 3589 x^{17} - 80315 x^{16} + 80315 x^{15} + 1120837 x^{14} - 1120837 x^{13} - 10089915 x^{12} + 10089915 x^{11} + 58493509 x^{10} - 58493509 x^{9} - 210941371 x^{8} + 210941371 x^{7} + 435702341 x^{6} - 435702341 x^{5} - 426489275 x^{4} + 426489275 x^{3} + 104090181 x^{2} - 104090181 x + 7621189 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(341419566026798986253349758444608447265625=3^{11}\cdot 5^{11}\cdot 23^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.25$
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:  $3, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(345=3\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{345}(256,·)$, $\chi_{345}(1,·)$, $\chi_{345}(194,·)$, $\chi_{345}(196,·)$, $\chi_{345}(134,·)$, $\chi_{345}(329,·)$, $\chi_{345}(74,·)$, $\chi_{345}(331,·)$, $\chi_{345}(14,·)$, $\chi_{345}(271,·)$, $\chi_{345}(16,·)$, $\chi_{345}(211,·)$, $\chi_{345}(149,·)$, $\chi_{345}(151,·)$, $\chi_{345}(344,·)$, $\chi_{345}(89,·)$, $\chi_{345}(31,·)$, $\chi_{345}(224,·)$, $\chi_{345}(44,·)$, $\chi_{345}(301,·)$, $\chi_{345}(121,·)$, $\chi_{345}(314,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3859123} a^{12} - \frac{534580}{3859123} a^{11} - \frac{48}{3859123} a^{10} + \frac{366782}{3859123} a^{9} + \frac{864}{3859123} a^{8} + \frac{1849734}{3859123} a^{7} - \frac{7168}{3859123} a^{6} - \frac{1370769}{3859123} a^{5} + \frac{26880}{3859123} a^{4} - \frac{1596550}{3859123} a^{3} - \frac{36864}{3859123} a^{2} + \frac{1277240}{3859123} a + \frac{8192}{3859123}$, $\frac{1}{3859123} a^{13} - \frac{52}{3859123} a^{11} + \frac{1720803}{3859123} a^{10} + \frac{1040}{3859123} a^{9} + \frac{632094}{3859123} a^{8} - \frac{9984}{3859123} a^{7} - \frac{1131070}{3859123} a^{6} + \frac{46592}{3859123} a^{5} + \frac{398921}{3859123} a^{4} - \frac{93184}{3859123} a^{3} - \frac{797842}{3859123} a^{2} + \frac{53248}{3859123} a - \frac{825245}{3859123}$, $\frac{1}{3859123} a^{14} + \frac{936504}{3859123} a^{11} - \frac{1456}{3859123} a^{10} + \frac{409143}{3859123} a^{9} + \frac{34944}{3859123} a^{8} - \frac{1422977}{3859123} a^{7} - \frac{326144}{3859123} a^{6} - \frac{1416853}{3859123} a^{5} + \frac{1304576}{3859123} a^{4} + \frac{1082264}{3859123} a^{3} - \frac{1863680}{3859123} a^{2} - \frac{13856}{3859123} a + \frac{425984}{3859123}$, $\frac{1}{3859123} a^{15} - \frac{1680}{3859123} a^{11} - \frac{948141}{3859123} a^{10} + \frac{44800}{3859123} a^{9} - \frac{146603}{3859123} a^{8} - \frac{483840}{3859123} a^{7} + \frac{9126}{82109} a^{6} - \frac{1450675}{3859123} a^{5} + \frac{914073}{3859123} a^{4} - \frac{1158477}{3859123} a^{3} - \frac{444758}{3859123} a^{2} - \frac{910003}{3859123} a + \frac{95756}{3859123}$, $\frac{1}{3859123} a^{16} + \frac{133118}{3859123} a^{11} - \frac{35840}{3859123} a^{10} - \frac{1412523}{3859123} a^{9} + \frac{967680}{3859123} a^{8} + \frac{1388027}{3859123} a^{7} - \frac{1915546}{3859123} a^{6} + \frac{1918584}{3859123} a^{5} + \frac{1549570}{3859123} a^{4} - \frac{558273}{3859123} a^{3} - \frac{1095555}{3859123} a^{2} + \frac{186568}{3859123} a - \frac{1673932}{3859123}$, $\frac{1}{3859123} a^{17} - \frac{43520}{3859123} a^{11} + \frac{1118018}{3859123} a^{10} + \frac{1305600}{3859123} a^{9} - \frac{1711358}{3859123} a^{8} + \frac{395980}{3859123} a^{7} - \frac{954096}{3859123} a^{6} + \frac{805380}{3859123} a^{5} - \frac{1363092}{3859123} a^{4} - \frac{1174511}{3859123} a^{3} - \frac{1355936}{3859123} a^{2} - \frac{67118}{3859123} a + \frac{1629153}{3859123}$, $\frac{1}{3859123} a^{18} - \frac{1010138}{3859123} a^{11} - \frac{783360}{3859123} a^{10} - \frac{691446}{3859123} a^{9} - \frac{593970}{3859123} a^{8} - \frac{39068}{82109} a^{7} + \frac{1442983}{3859123} a^{6} + \frac{952485}{3859123} a^{5} - \frac{671180}{3859123} a^{4} + \frac{297679}{3859123} a^{3} + \frac{1006770}{3859123} a^{2} + \frac{306261}{3859123} a + \frac{1476524}{3859123}$, $\frac{1}{3859123} a^{19} - \frac{992256}{3859123} a^{11} + \frac{990529}{3859123} a^{10} + \frac{879208}{3859123} a^{9} - \frac{1238762}{3859123} a^{8} + \frac{1026873}{3859123} a^{7} - \frac{1951}{3859123} a^{6} - \frac{1617533}{3859123} a^{5} + \frac{17691}{3859123} a^{4} + \frac{402816}{3859123} a^{3} - \frac{743144}{3859123} a^{2} + \frac{416038}{3859123} a + \frac{1090784}{3859123}$, $\frac{1}{3859123} a^{20} + \frac{1093522}{3859123} a^{11} - \frac{439604}{3859123} a^{10} - \frac{1911331}{3859123} a^{9} + \frac{1610751}{3859123} a^{8} - \frac{818216}{3859123} a^{7} - \frac{1744852}{3859123} a^{6} + \frac{1872423}{3859123} a^{5} + \frac{1845043}{3859123} a^{4} - \frac{1631952}{3859123} a^{3} - \frac{1341352}{3859123} a^{2} - \frac{1285468}{3859123} a + \frac{1248114}{3859123}$, $\frac{1}{3859123} a^{21} - \frac{1541761}{3859123} a^{11} + \frac{409126}{3859123} a^{10} - \frac{62940}{3859123} a^{9} - \frac{136089}{3859123} a^{8} + \frac{1879466}{3859123} a^{7} - \frac{31911}{82109} a^{6} - \frac{370445}{3859123} a^{5} - \frac{563421}{3859123} a^{4} + \frac{1680794}{3859123} a^{3} + \frac{1769805}{3859123} a^{2} + \frac{1145871}{3859123} a - \frac{1107741}{3859123}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $21$
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:  \( 25638523812500 \) (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

\(\Q(\sqrt{345}) \), \(\Q(\zeta_{23})^+\)

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ R R ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ $22$ $22$ R $22$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ $22$ $22$

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
3Data not computed
5Data not computed
23Data not computed