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Properties

Label 22.22.1126170360...9653.1
Degree $22$
Signature $[22, 0]$
Discriminant $11^{11}\cdot 23^{21}$
Root discriminant $66.15$
Ramified primes $11, 23$
Class number $1$ (GRH)
Class group Trivial (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![-268133, -3806248, 3806248, 26072546, -26072546, -38664841, 38664841, 26072546, -26072546, -9892669, 9892669, 2313707, -2313707, -346657, 346657, 33395, -33395, -2002, 2002, 68, -68, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 68*x^20 + 68*x^19 + 2002*x^18 - 2002*x^17 - 33395*x^16 + 33395*x^15 + 346657*x^14 - 346657*x^13 - 2313707*x^12 + 2313707*x^11 + 9892669*x^10 - 9892669*x^9 - 26072546*x^8 + 26072546*x^7 + 38664841*x^6 - 38664841*x^5 - 26072546*x^4 + 26072546*x^3 + 3806248*x^2 - 3806248*x - 268133)
 
gp: K = bnfinit(x^22 - x^21 - 68*x^20 + 68*x^19 + 2002*x^18 - 2002*x^17 - 33395*x^16 + 33395*x^15 + 346657*x^14 - 346657*x^13 - 2313707*x^12 + 2313707*x^11 + 9892669*x^10 - 9892669*x^9 - 26072546*x^8 + 26072546*x^7 + 38664841*x^6 - 38664841*x^5 - 26072546*x^4 + 26072546*x^3 + 3806248*x^2 - 3806248*x - 268133, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} - 68 x^{20} + 68 x^{19} + 2002 x^{18} - 2002 x^{17} - 33395 x^{16} + 33395 x^{15} + 346657 x^{14} - 346657 x^{13} - 2313707 x^{12} + 2313707 x^{11} + 9892669 x^{10} - 9892669 x^{9} - 26072546 x^{8} + 26072546 x^{7} + 38664841 x^{6} - 38664841 x^{5} - 26072546 x^{4} + 26072546 x^{3} + 3806248 x^{2} - 3806248 x - 268133 \)

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:  \(11261703607138248537697553999262299669653=11^{11}\cdot 23^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.15$
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:  $11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(253=11\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{253}(1,·)$, $\chi_{253}(133,·)$, $\chi_{253}(65,·)$, $\chi_{253}(76,·)$, $\chi_{253}(10,·)$, $\chi_{253}(12,·)$, $\chi_{253}(78,·)$, $\chi_{253}(144,·)$, $\chi_{253}(210,·)$, $\chi_{253}(21,·)$, $\chi_{253}(153,·)$, $\chi_{253}(100,·)$, $\chi_{253}(241,·)$, $\chi_{253}(232,·)$, $\chi_{253}(252,·)$, $\chi_{253}(43,·)$, $\chi_{253}(109,·)$, $\chi_{253}(175,·)$, $\chi_{253}(177,·)$, $\chi_{253}(243,·)$, $\chi_{253}(120,·)$, $\chi_{253}(188,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{166427} a^{12} - \frac{44732}{166427} a^{11} - \frac{36}{166427} a^{10} - \frac{21687}{166427} a^{9} + \frac{486}{166427} a^{8} - \frac{72610}{166427} a^{7} - \frac{3024}{166427} a^{6} - \frac{34865}{166427} a^{5} + \frac{8505}{166427} a^{4} - \frac{67941}{166427} a^{3} - \frac{8748}{166427} a^{2} + \frac{74050}{166427} a + \frac{1458}{166427}$, $\frac{1}{166427} a^{13} - \frac{39}{166427} a^{11} + \frac{32231}{166427} a^{10} + \frac{585}{166427} a^{9} + \frac{31632}{166427} a^{8} - \frac{4212}{166427} a^{7} + \frac{718}{166427} a^{6} + \frac{14742}{166427} a^{5} - \frac{74403}{166427} a^{4} - \frac{22113}{166427} a^{3} + \frac{28391}{166427} a^{2} + \frac{9477}{166427} a - \frac{20128}{166427}$, $\frac{1}{166427} a^{14} - \frac{48047}{166427} a^{11} - \frac{819}{166427} a^{10} + \frac{17974}{166427} a^{9} + \frac{14742}{166427} a^{8} - \frac{1813}{166427} a^{7} + \frac{63233}{166427} a^{6} + \frac{63705}{166427} a^{5} - \frac{23272}{166427} a^{4} + \frac{41524}{166427} a^{3} + \frac{1159}{166427} a^{2} + \frac{38563}{166427} a + \frac{56862}{166427}$, $\frac{1}{166427} a^{15} - \frac{945}{166427} a^{11} - \frac{47448}{166427} a^{10} + \frac{18900}{166427} a^{9} + \frac{49249}{166427} a^{8} + \frac{13337}{166427} a^{7} + \frac{1284}{3541} a^{6} + \frac{72255}{166427} a^{5} - \frac{63453}{166427} a^{4} - \frac{60890}{166427} a^{3} - \frac{48418}{166427} a^{2} + \frac{60806}{166427} a - \frac{13241}{166427}$, $\frac{1}{166427} a^{16} - \frac{46730}{166427} a^{11} - \frac{15120}{166427} a^{10} + \frac{25555}{166427} a^{9} - \frac{26674}{166427} a^{8} + \frac{11822}{166427} a^{7} + \frac{43834}{166427} a^{6} - \frac{58332}{166427} a^{5} - \frac{12161}{166427} a^{4} - \frac{11841}{166427} a^{3} - \frac{51131}{166427} a^{2} + \frac{64669}{166427} a + \frac{46394}{166427}$, $\frac{1}{166427} a^{17} - \frac{18360}{166427} a^{11} + \frac{7545}{166427} a^{10} + \frac{80246}{166427} a^{9} - \frac{77897}{166427} a^{8} - \frac{74217}{166427} a^{7} - \frac{73329}{166427} a^{6} + \frac{66719}{166427} a^{5} - \frac{867}{166427} a^{4} - \frac{6182}{166427} a^{3} + \frac{15341}{166427} a^{2} + \frac{52710}{166427} a + \frac{63697}{166427}$, $\frac{1}{166427} a^{18} + \frac{45270}{166427} a^{11} - \frac{81433}{166427} a^{10} + \frac{8594}{166427} a^{9} + \frac{28112}{166427} a^{8} + \frac{1144}{3541} a^{7} - \frac{33730}{166427} a^{6} - \frac{44025}{166427} a^{5} + \frac{37092}{166427} a^{4} - \frac{11054}{166427} a^{3} + \frac{41485}{166427} a^{2} + \frac{79534}{166427} a - \frac{25867}{166427}$, $\frac{1}{166427} a^{19} + \frac{18898}{166427} a^{11} - \frac{25956}{166427} a^{10} + \frac{45729}{166427} a^{9} + \frac{20912}{166427} a^{8} - \frac{78707}{166427} a^{7} + \frac{49461}{166427} a^{6} - \frac{18026}{166427} a^{5} + \frac{79674}{166427} a^{4} - \frac{6832}{166427} a^{3} + \frac{5234}{166427} a^{2} + \frac{69694}{166427} a + \frac{67859}{166427}$, $\frac{1}{166427} a^{20} + \frac{36647}{166427} a^{11} + \frac{60349}{166427} a^{10} - \frac{47863}{166427} a^{9} + \frac{56777}{166427} a^{8} + \frac{42626}{166427} a^{7} + \frac{45065}{166427} a^{6} + \frac{73951}{166427} a^{5} + \frac{34160}{166427} a^{4} - \frac{30053}{166427} a^{3} - \frac{39040}{166427} a^{2} - \frac{10825}{166427} a + \frac{73598}{166427}$, $\frac{1}{166427} a^{21} + \frac{48003}{166427} a^{11} - \frac{59987}{166427} a^{10} - \frac{35086}{166427} a^{9} + \frac{39873}{166427} a^{8} - \frac{17568}{166427} a^{7} + \frac{1151}{3541} a^{6} + \frac{71736}{166427} a^{5} + \frac{4983}{166427} a^{4} + \frac{46867}{166427} a^{3} + \frac{38729}{166427} a^{2} - \frac{44517}{166427} a - \frac{8259}{166427}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 7739860067820 \) (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

\(\Q(\sqrt{253}) \), \(\Q(\zeta_{23})^+\)

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 $22$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ R $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R $22$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
11Data not computed
23Data not computed