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Properties

Label 22.22.1002912833...8713.1
Degree $22$
Signature $[22, 0]$
Discriminant $23^{21}\cdot 31^{11}$
Root discriminant $111.05$
Ramified primes $23, 31$
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![39405042377, -236973537993, 236973537993, 306339824951, -306339824951, -135102282441, 135102282441, 30438507831, -30438507831, -4049156809, 4049156809, 340182327, -340182327, -18561737, 18561737, 656695, -656695, -14537, 14537, 183, -183, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 183*x^20 + 183*x^19 + 14537*x^18 - 14537*x^17 - 656695*x^16 + 656695*x^15 + 18561737*x^14 - 18561737*x^13 - 340182327*x^12 + 340182327*x^11 + 4049156809*x^10 - 4049156809*x^9 - 30438507831*x^8 + 30438507831*x^7 + 135102282441*x^6 - 135102282441*x^5 - 306339824951*x^4 + 306339824951*x^3 + 236973537993*x^2 - 236973537993*x + 39405042377)
 
gp: K = bnfinit(x^22 - x^21 - 183*x^20 + 183*x^19 + 14537*x^18 - 14537*x^17 - 656695*x^16 + 656695*x^15 + 18561737*x^14 - 18561737*x^13 - 340182327*x^12 + 340182327*x^11 + 4049156809*x^10 - 4049156809*x^9 - 30438507831*x^8 + 30438507831*x^7 + 135102282441*x^6 - 135102282441*x^5 - 306339824951*x^4 + 306339824951*x^3 + 236973537993*x^2 - 236973537993*x + 39405042377, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} - 183 x^{20} + 183 x^{19} + 14537 x^{18} - 14537 x^{17} - 656695 x^{16} + 656695 x^{15} + 18561737 x^{14} - 18561737 x^{13} - 340182327 x^{12} + 340182327 x^{11} + 4049156809 x^{10} - 4049156809 x^{9} - 30438507831 x^{8} + 30438507831 x^{7} + 135102282441 x^{6} - 135102282441 x^{5} - 306339824951 x^{4} + 306339824951 x^{3} + 236973537993 x^{2} - 236973537993 x + 39405042377 \)

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:  \(1002912833195191049488840533168697988824668713=23^{21}\cdot 31^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $111.05$
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:  $23, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(713=23\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{713}(1,·)$, $\chi_{713}(619,·)$, $\chi_{713}(712,·)$, $\chi_{713}(247,·)$, $\chi_{713}(652,·)$, $\chi_{713}(526,·)$, $\chi_{713}(402,·)$, $\chi_{713}(404,·)$, $\chi_{713}(280,·)$, $\chi_{713}(156,·)$, $\chi_{713}(30,·)$, $\chi_{713}(32,·)$, $\chi_{713}(681,·)$, $\chi_{713}(683,·)$, $\chi_{713}(557,·)$, $\chi_{713}(466,·)$, $\chi_{713}(433,·)$, $\chi_{713}(94,·)$, $\chi_{713}(309,·)$, $\chi_{713}(311,·)$, $\chi_{713}(187,·)$, $\chi_{713}(61,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5106687193} a^{12} - \frac{177687979}{5106687193} a^{11} - \frac{96}{5106687193} a^{10} + \frac{316480573}{5106687193} a^{9} + \frac{3456}{5106687193} a^{8} + \frac{85996050}{5106687193} a^{7} - \frac{57344}{5106687193} a^{6} - \frac{1203944700}{5106687193} a^{5} + \frac{430080}{5106687193} a^{4} + \frac{1772996807}{5106687193} a^{3} - \frac{1179648}{5106687193} a^{2} - \frac{794120014}{5106687193} a + \frac{524288}{5106687193}$, $\frac{1}{5106687193} a^{13} - \frac{104}{5106687193} a^{11} - \frac{1421503832}{5106687193} a^{10} + \frac{4160}{5106687193} a^{9} + \frac{1373188314}{5106687193} a^{8} - \frac{79872}{5106687193} a^{7} + \frac{2404224752}{5106687193} a^{6} + \frac{745472}{5106687193} a^{5} + \frac{245161882}{5106687193} a^{4} - \frac{2981888}{5106687193} a^{3} - \frac{980647528}{5106687193} a^{2} + \frac{3407872}{5106687193} a - \frac{1619327947}{5106687193}$, $\frac{1}{5106687193} a^{14} + \frac{525695124}{5106687193} a^{11} - \frac{5824}{5106687193} a^{10} - \frac{1459642445}{5106687193} a^{9} + \frac{279552}{5106687193} a^{8} + \frac{1134439566}{5106687193} a^{7} - \frac{5218304}{5106687193} a^{6} - \frac{2404594286}{5106687193} a^{5} + \frac{41746432}{5106687193} a^{4} - \frac{429718548}{5106687193} a^{3} - \frac{119275520}{5106687193} a^{2} - \frac{2500814315}{5106687193} a + \frac{54525952}{5106687193}$, $\frac{1}{5106687193} a^{15} - \frac{6720}{5106687193} a^{11} - \frac{2059782471}{5106687193} a^{10} + \frac{358400}{5106687193} a^{9} + \frac{2312731730}{5106687193} a^{8} - \frac{7741440}{5106687193} a^{7} - \frac{36551147}{108652919} a^{6} + \frac{77070336}{5106687193} a^{5} + \frac{2080134414}{5106687193} a^{4} - \frac{321126400}{5106687193} a^{3} + \frac{2141540082}{5106687193} a^{2} + \frac{377487360}{5106687193} a + \frac{2476008884}{5106687193}$, $\frac{1}{5106687193} a^{16} - \frac{1158198189}{5106687193} a^{11} - \frac{286720}{5106687193} a^{10} - \frac{426377191}{5106687193} a^{9} + \frac{15482880}{5106687193} a^{8} - \frac{880100718}{5106687193} a^{7} - \frac{308281344}{5106687193} a^{6} + \frac{564264126}{5106687193} a^{5} - \frac{2537675993}{5106687193} a^{4} - \frac{2327825340}{5106687193} a^{3} - \frac{2443060007}{5106687193} a^{2} + \frac{2477631489}{5106687193} a - \frac{1583471833}{5106687193}$, $\frac{1}{5106687193} a^{17} - \frac{348160}{5106687193} a^{11} + \frac{733714911}{5106687193} a^{10} + \frac{20889600}{5106687193} a^{9} - \frac{1789918846}{5106687193} a^{8} - \frac{481296384}{5106687193} a^{7} + \frac{2420946268}{5106687193} a^{6} - \frac{115465433}{5106687193} a^{5} - \frac{932879826}{5106687193} a^{4} - \frac{964201628}{5106687193} a^{3} - \frac{181261991}{5106687193} a^{2} + \frac{135704515}{5106687193} a - \frac{1655318005}{5106687193}$, $\frac{1}{5106687193} a^{18} - \frac{704397727}{5106687193} a^{11} - \frac{12533760}{5106687193} a^{10} + \frac{2203500666}{5106687193} a^{9} + \frac{721944576}{5106687193} a^{8} + \frac{48908547}{108652919} a^{7} + \frac{346396299}{5106687193} a^{6} + \frac{778544000}{5106687193} a^{5} + \frac{678522575}{5106687193} a^{4} + \frac{252547675}{5106687193} a^{3} - \frac{2035567725}{5106687193} a^{2} - \frac{1328076032}{5106687193} a - \frac{1304628868}{5106687193}$, $\frac{1}{5106687193} a^{19} - \frac{15876096}{5106687193} a^{11} + \frac{968252383}{5106687193} a^{10} + \frac{1016070144}{5106687193} a^{9} + \frac{807455160}{5106687193} a^{8} + \frac{1147752509}{5106687193} a^{7} + \frac{1690983542}{5106687193} a^{6} - \frac{327089979}{5106687193} a^{5} - \frac{1484061697}{5106687193} a^{4} + \frac{792682759}{5106687193} a^{3} + \frac{2122047253}{5106687193} a^{2} - \frac{1744479888}{5106687193} a + \frac{1871070002}{5106687193}$, $\frac{1}{5106687193} a^{20} - \frac{265425278}{5106687193} a^{11} - \frac{508035072}{5106687193} a^{10} + \frac{2130658275}{5106687193} a^{9} - \frac{158018838}{5106687193} a^{8} + \frac{201981406}{5106687193} a^{7} - \frac{1735618649}{5106687193} a^{6} - \frac{1064780565}{5106687193} a^{5} + \frac{1143273398}{5106687193} a^{4} + \frac{455681812}{5106687193} a^{3} + \frac{1379249828}{5106687193} a^{2} - \frac{1590769924}{5106687193} a - \frac{253504942}{5106687193}$, $\frac{1}{5106687193} a^{21} - \frac{666796032}{5106687193} a^{11} + \frac{2183267552}{5106687193} a^{10} - \frac{1507115937}{5106687193} a^{9} - \frac{1691952566}{5106687193} a^{8} + \frac{581990975}{5106687193} a^{7} + \frac{30268088}{108652919} a^{6} - \frac{663138180}{5106687193} a^{5} - \frac{326268270}{5106687193} a^{4} - \frac{2159406393}{5106687193} a^{3} + \frac{1429439534}{5106687193} a^{2} - \frac{719679620}{5106687193} a + \frac{2062142814}{5106687193}$

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:  \( 3138976630957292.5 \) (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

\(\Q(\sqrt{713}) \), \(\Q(\zeta_{23})^+\)

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ $22$ $22$ $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ R $22$ R ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
23Data not computed
31Data not computed