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Properties

Label 22.0.92690955403...0000.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{22}\cdot 5^{11}\cdot 11^{40}$
Root discriminant $349.91$
Ramified primes $2, 5, 11$
Class number $89838230198$ (GRH)
Class group $[89838230198]$ (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![588511917549, -2644056756, 173666905632, 7426919808, 36670510998, 3469742298, 5990640084, 627037026, 582513800, 89782902, 76849531, 8316900, 2279805, 764610, 488939, 22176, -8855, 2772, 1650, -66, -55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 55*x^20 - 66*x^19 + 1650*x^18 + 2772*x^17 - 8855*x^16 + 22176*x^15 + 488939*x^14 + 764610*x^13 + 2279805*x^12 + 8316900*x^11 + 76849531*x^10 + 89782902*x^9 + 582513800*x^8 + 627037026*x^7 + 5990640084*x^6 + 3469742298*x^5 + 36670510998*x^4 + 7426919808*x^3 + 173666905632*x^2 - 2644056756*x + 588511917549)
 
gp: K = bnfinit(x^22 - 55*x^20 - 66*x^19 + 1650*x^18 + 2772*x^17 - 8855*x^16 + 22176*x^15 + 488939*x^14 + 764610*x^13 + 2279805*x^12 + 8316900*x^11 + 76849531*x^10 + 89782902*x^9 + 582513800*x^8 + 627037026*x^7 + 5990640084*x^6 + 3469742298*x^5 + 36670510998*x^4 + 7426919808*x^3 + 173666905632*x^2 - 2644056756*x + 588511917549, 1)
 

Normalized defining polynomial

\( x^{22} - 55 x^{20} - 66 x^{19} + 1650 x^{18} + 2772 x^{17} - 8855 x^{16} + 22176 x^{15} + 488939 x^{14} + 764610 x^{13} + 2279805 x^{12} + 8316900 x^{11} + 76849531 x^{10} + 89782902 x^{9} + 582513800 x^{8} + 627037026 x^{7} + 5990640084 x^{6} + 3469742298 x^{5} + 36670510998 x^{4} + 7426919808 x^{3} + 173666905632 x^{2} - 2644056756 x + 588511917549 \)

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:  \(-92690955403624349298461401159468893114040524800000000000=-\,2^{22}\cdot 5^{11}\cdot 11^{40}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $349.91$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2420=2^{2}\cdot 5\cdot 11^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{2420}(1,·)$, $\chi_{2420}(2179,·)$, $\chi_{2420}(1541,·)$, $\chi_{2420}(199,·)$, $\chi_{2420}(1739,·)$, $\chi_{2420}(1101,·)$, $\chi_{2420}(1299,·)$, $\chi_{2420}(661,·)$, $\chi_{2420}(2201,·)$, $\chi_{2420}(859,·)$, $\chi_{2420}(221,·)$, $\chi_{2420}(2399,·)$, $\chi_{2420}(1761,·)$, $\chi_{2420}(419,·)$, $\chi_{2420}(1959,·)$, $\chi_{2420}(1321,·)$, $\chi_{2420}(1519,·)$, $\chi_{2420}(881,·)$, $\chi_{2420}(1079,·)$, $\chi_{2420}(441,·)$, $\chi_{2420}(1981,·)$, $\chi_{2420}(639,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{2}$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{10} - \frac{1}{27} a^{8} + \frac{1}{27} a^{6} - \frac{10}{81} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{81} a^{11} + \frac{1}{27} a^{7} + \frac{8}{81} a^{5} - \frac{4}{27} a^{3}$, $\frac{1}{243} a^{12} - \frac{1}{243} a^{10} - \frac{1}{81} a^{8} - \frac{4}{243} a^{6} - \frac{11}{243} a^{4} + \frac{2}{27} a^{2}$, $\frac{1}{243} a^{13} - \frac{1}{243} a^{11} - \frac{13}{243} a^{7} - \frac{2}{243} a^{5} - \frac{4}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{243} a^{14} - \frac{1}{243} a^{10} + \frac{11}{243} a^{8} - \frac{2}{81} a^{6} - \frac{23}{243} a^{4} + \frac{2}{27} a^{2}$, $\frac{1}{2187} a^{15} - \frac{1}{729} a^{14} - \frac{2}{2187} a^{13} + \frac{7}{2187} a^{11} - \frac{2}{729} a^{10} - \frac{1}{2187} a^{9} - \frac{2}{729} a^{8} + \frac{20}{2187} a^{7} - \frac{10}{243} a^{6} - \frac{142}{2187} a^{5} - \frac{55}{729} a^{4} - \frac{14}{243} a^{3} + \frac{37}{81} a^{2} - \frac{2}{9} a$, $\frac{1}{6561} a^{16} + \frac{7}{6561} a^{14} + \frac{4}{2187} a^{13} - \frac{11}{6561} a^{12} - \frac{1}{2187} a^{11} + \frac{8}{6561} a^{10} + \frac{2}{729} a^{9} + \frac{173}{6561} a^{8} + \frac{47}{2187} a^{7} + \frac{119}{6561} a^{6} - \frac{263}{2187} a^{5} + \frac{31}{243} a^{4} - \frac{4}{243} a^{3} + \frac{31}{81} a^{2} - \frac{4}{9} a$, $\frac{1}{6561} a^{17} + \frac{1}{6561} a^{15} + \frac{1}{2187} a^{14} + \frac{1}{6561} a^{13} - \frac{1}{2187} a^{12} - \frac{34}{6561} a^{11} + \frac{17}{6561} a^{9} + \frac{41}{2187} a^{8} - \frac{244}{6561} a^{7} - \frac{110}{2187} a^{6} + \frac{158}{2187} a^{5} + \frac{14}{729} a^{4} - \frac{8}{243} a^{3} - \frac{17}{81} a^{2} + \frac{2}{9} a$, $\frac{1}{19683} a^{18} + \frac{1}{6561} a^{14} - \frac{4}{2187} a^{13} - \frac{23}{19683} a^{12} + \frac{1}{2187} a^{11} - \frac{2}{729} a^{10} + \frac{4}{729} a^{9} - \frac{52}{6561} a^{8} - \frac{20}{2187} a^{7} - \frac{1076}{19683} a^{6} - \frac{88}{2187} a^{5} + \frac{352}{2187} a^{4} - \frac{34}{243} a^{3} - \frac{77}{243} a^{2} + \frac{11}{27} a$, $\frac{1}{177147} a^{19} - \frac{1}{59049} a^{18} + \frac{1}{59049} a^{17} - \frac{4}{59049} a^{16} - \frac{13}{59049} a^{15} + \frac{113}{59049} a^{14} + \frac{115}{177147} a^{13} - \frac{95}{59049} a^{12} + \frac{224}{59049} a^{11} + \frac{202}{59049} a^{10} + \frac{286}{59049} a^{9} + \frac{1291}{59049} a^{8} + \frac{9667}{177147} a^{7} - \frac{763}{19683} a^{6} - \frac{1505}{19683} a^{5} + \frac{308}{2187} a^{4} - \frac{161}{2187} a^{3} - \frac{64}{243} a^{2} + \frac{2}{9} a$, $\frac{1}{80956179} a^{20} - \frac{194}{80956179} a^{19} + \frac{83}{8995131} a^{18} + \frac{511}{8995131} a^{17} - \frac{392}{26985393} a^{16} - \frac{320}{26985393} a^{15} - \frac{148874}{80956179} a^{14} + \frac{2860}{80956179} a^{13} - \frac{11372}{8995131} a^{12} + \frac{17207}{8995131} a^{11} - \frac{94582}{26985393} a^{10} + \frac{164555}{26985393} a^{9} - \frac{4438499}{80956179} a^{8} + \frac{3161689}{80956179} a^{7} - \frac{403073}{8995131} a^{6} - \frac{1194917}{8995131} a^{5} - \frac{42281}{999459} a^{4} + \frac{36484}{999459} a^{3} + \frac{1490}{12339} a^{2} - \frac{2668}{12339} a + \frac{201}{457}$, $\frac{1}{2940771999750793437643204689103254079356106567932305488934739990177} a^{21} + \frac{1067724782588981703681506223260899005083777433145746857447}{980257333250264479214401563034418026452035522644101829644913330059} a^{20} + \frac{4584840994833932058075173756513774733125114723821469687207729}{2940771999750793437643204689103254079356106567932305488934739990177} a^{19} - \frac{702741193519552003534265934629441615303705665900258147195651}{36305827157417202933866724556830297276001315653485252949811604817} a^{18} - \frac{70237897618260242331627041226435086042743278084735532347459221}{980257333250264479214401563034418026452035522644101829644913330059} a^{17} + \frac{123905911304344344868724559475254532971222892665199580124206}{12101942385805734311288908185610099092000438551161750983270534939} a^{16} + \frac{165649988832290727150728426349674756314533784662905013378735868}{2940771999750793437643204689103254079356106567932305488934739990177} a^{15} + \frac{418937804845749241762537527998369977651931981910435535995169520}{980257333250264479214401563034418026452035522644101829644913330059} a^{14} - \frac{5451862002652190244020873075073557430083846991552645828838241767}{2940771999750793437643204689103254079356106567932305488934739990177} a^{13} + \frac{2253321157973636889645726903576949462270305289478890095727447}{1355819271438816707073861083035156329809177762993225213893379433} a^{12} + \frac{61300925843461970605952248000832370269684839316648158909372145}{980257333250264479214401563034418026452035522644101829644913330059} a^{11} - \frac{490209669866691723110730662256511213236246850999841869558126648}{108917481472251608801600173670490891828003946960455758849434814451} a^{10} - \frac{1545118528357389356831219256045939149138505868526781287611322861}{2940771999750793437643204689103254079356106567932305488934739990177} a^{9} + \frac{50935668809533911904907864206587767847087410957534779939163382110}{980257333250264479214401563034418026452035522644101829644913330059} a^{8} - \frac{70051471888028129210159880616052433577258273955965085236141458085}{2940771999750793437643204689103254079356106567932305488934739990177} a^{7} + \frac{2253243426624363962941781938530505984505627404825640765818366904}{326752444416754826404800521011472675484011840881367276548304443353} a^{6} + \frac{45508429726784718062657960238002229800954922610936953492055286399}{326752444416754826404800521011472675484011840881367276548304443353} a^{5} + \frac{2185943320968405783573738466039129745163416851088632140038159654}{36305827157417202933866724556830297276001315653485252949811604817} a^{4} - \frac{4711365630773762985141279210173872806990198602553560633413980160}{36305827157417202933866724556830297276001315653485252949811604817} a^{3} - \frac{1597553656934069009620888863724840515692388674527536568173098716}{4033980795268578103762969395203366364000146183720583661090178313} a^{2} + \frac{211596418011513172412938831400414675334658585663465070515172684}{448220088363175344862552155022596262666682909302287073454464257} a - \frac{3091769017428070517396876338510886144733722322291577125903262}{16600744013450938698613042778614676395062329974158780498313491}$

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

Class group and class number

$C_{89838230198}$, which has order $89838230198$ (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:  \( 285114946276.13544 \) (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

\(\Q(\sqrt{-5}) \), 11.11.672749994932560009201.1

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 R ${\href{/LocalNumberField/3.1.0.1}{1} }^{22}$ R ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ R $22$ $22$ $22$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ $22$ $22$

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
2Data not computed
5Data not computed
11Data not computed