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Properties

Label 22.0.17110731628...4527.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,7^{11}\cdot 89^{21}$
Root discriminant $192.01$
Ramified primes $7, 89$
Class number $6596678$ (GRH)
Class group $[6596678]$ (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![7438698677, -6880449063, 19500553636, -19326356701, 17433518072, -13226651538, 8079316286, -3042652387, 2085803018, -732575214, 293038974, -114648803, 31913346, -9627703, 2849200, -447777, 176727, -11219, 6933, -141, 136, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 136*x^20 - 141*x^19 + 6933*x^18 - 11219*x^17 + 176727*x^16 - 447777*x^15 + 2849200*x^14 - 9627703*x^13 + 31913346*x^12 - 114648803*x^11 + 293038974*x^10 - 732575214*x^9 + 2085803018*x^8 - 3042652387*x^7 + 8079316286*x^6 - 13226651538*x^5 + 17433518072*x^4 - 19326356701*x^3 + 19500553636*x^2 - 6880449063*x + 7438698677)
 
gp: K = bnfinit(x^22 - x^21 + 136*x^20 - 141*x^19 + 6933*x^18 - 11219*x^17 + 176727*x^16 - 447777*x^15 + 2849200*x^14 - 9627703*x^13 + 31913346*x^12 - 114648803*x^11 + 293038974*x^10 - 732575214*x^9 + 2085803018*x^8 - 3042652387*x^7 + 8079316286*x^6 - 13226651538*x^5 + 17433518072*x^4 - 19326356701*x^3 + 19500553636*x^2 - 6880449063*x + 7438698677, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 136 x^{20} - 141 x^{19} + 6933 x^{18} - 11219 x^{17} + 176727 x^{16} - 447777 x^{15} + 2849200 x^{14} - 9627703 x^{13} + 31913346 x^{12} - 114648803 x^{11} + 293038974 x^{10} - 732575214 x^{9} + 2085803018 x^{8} - 3042652387 x^{7} + 8079316286 x^{6} - 13226651538 x^{5} + 17433518072 x^{4} - 19326356701 x^{3} + 19500553636 x^{2} - 6880449063 x + 7438698677 \)

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:  \(-171107316284804096938603185096461792835715406634527=-\,7^{11}\cdot 89^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $192.01$
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:  $7, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(623=7\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{623}(512,·)$, $\chi_{623}(1,·)$, $\chi_{623}(134,·)$, $\chi_{623}(449,·)$, $\chi_{623}(8,·)$, $\chi_{623}(265,·)$, $\chi_{623}(139,·)$, $\chi_{623}(78,·)$, $\chi_{623}(146,·)$, $\chi_{623}(622,·)$, $\chi_{623}(111,·)$, $\chi_{623}(477,·)$, $\chi_{623}(545,·)$, $\chi_{623}(484,·)$, $\chi_{623}(358,·)$, $\chi_{623}(615,·)$, $\chi_{623}(64,·)$, $\chi_{623}(174,·)$, $\chi_{623}(559,·)$, $\chi_{623}(372,·)$, $\chi_{623}(489,·)$, $\chi_{623}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{21} - \frac{95201848919148703934495330212036107898179690722613729460427026457134297473455648567343521030991246527719105877081503398}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{20} - \frac{146448701341727562513876624361110035732285346749063457343708276024313728235255741189795818346247320221324222032808103319}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{19} + \frac{145469735587492842274016608974900575735593939922528688724969724031896682037119661462688331734096877230800700758461085907}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{18} + \frac{66206741431971167612540105642386766595773403310318689445548395566219424079437703362065634304713521519503853756961517380}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{17} - \frac{35937435977976907946968497548297944472890176691370027042747413663975110732966521098262198396059130478276306155104334533}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{16} + \frac{74541827796156727519689249703177598195058264463692558491342509156665040756065864927297150368256966003748833886160226311}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{15} + \frac{125586120022799513611682053000235374723499853381433576704928218153700444448812499704585149272265174738968950439244657435}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{14} - \frac{89831361141364384771079670641521355815296504998328391935240015766897157415464320400141042229690394512554055551880909144}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{13} - \frac{151599869488999563064511102053261901976406792280597705085692849930873279969167144392366745157949499219003797837652015909}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{12} + \frac{112249654870078296688461291170091191214623754945879068525951897693274816428251734813522779630812033963110579212823731428}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{11} + \frac{134150764800660476278212574972666241035733797134559683934337335492439521045856331372346986418301388368133858863030703927}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{10} + \frac{47004549420464121550972080070904663209277150894450839093912231124509573330760660360890239162771618932756061726404197033}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{9} + \frac{46159223060467833705281213604568094083093375481238892893170783045062345101336308674330758633798552713937453040822077115}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{8} + \frac{97922935572817381040274233185838135054480048236661048490327053282470785608104212673146706462314516020305517916588776033}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{7} - \frac{4623305251476663895956182021806031554983200100276585732927240972245149631757813351946528745166436527159128993589756239}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{6} + \frac{60001254860515816084337644658567902565649576656286461009897290598073938821438719927801381995007109316272358146150365797}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{5} + \frac{57319144160453752064746343378757919894352430234090942743048481179898445153835594126203352828153755538423731552238489999}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{4} + \frac{141971828998974141064414918269017010632417597602985568877933716848341381873631587950962289541563546139014801919950250459}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{3} + \frac{48952883523519256941931515931880755769313265747915927969114890654093947346108732936861310352629015769394466148344454497}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a^{2} - \frac{24368352988697977330527102368527570046995067716869389368030027500987032442278016041440435732629751555796599190750723798}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017} a + \frac{105325166440600218823144924638333762006368580632803355941532793894999960840052811094229495511600026818620896471278367670}{323284824061388685705481282020718816217594122872411376771932719287792515664529421869234537909647110901149506234591060017}$

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

Class group and class number

$C_{6596678}$, which has order $6596678$ (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:  \( 866679281.3791491 \) (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

\(\Q(\sqrt{-623}) \), 11.11.31181719929966183601.1

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ R ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
7Data not computed
89Data not computed