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Properties

Label 22.0.24309170471...9375.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,5^{11}\cdot 11^{41}$
Root discriminant $195.10$
Ramified primes $5, 11$
Class number $88708$ (GRH)
Class group $[88708]$ (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![6490917711, -2522739483, 2027899907, -2592915589, 714471362, 1217292384, 1566926449, 1566072057, 720148935, 397249116, 227012456, 60773360, 24762320, 6515520, 1882144, 288222, 120120, 4026, 4895, -22, 110, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 110*x^20 - 22*x^19 + 4895*x^18 + 4026*x^17 + 120120*x^16 + 288222*x^15 + 1882144*x^14 + 6515520*x^13 + 24762320*x^12 + 60773360*x^11 + 227012456*x^10 + 397249116*x^9 + 720148935*x^8 + 1566072057*x^7 + 1566926449*x^6 + 1217292384*x^5 + 714471362*x^4 - 2592915589*x^3 + 2027899907*x^2 - 2522739483*x + 6490917711)
 
gp: K = bnfinit(x^22 + 110*x^20 - 22*x^19 + 4895*x^18 + 4026*x^17 + 120120*x^16 + 288222*x^15 + 1882144*x^14 + 6515520*x^13 + 24762320*x^12 + 60773360*x^11 + 227012456*x^10 + 397249116*x^9 + 720148935*x^8 + 1566072057*x^7 + 1566926449*x^6 + 1217292384*x^5 + 714471362*x^4 - 2592915589*x^3 + 2027899907*x^2 - 2522739483*x + 6490917711, 1)
 

Normalized defining polynomial

\( x^{22} + 110 x^{20} - 22 x^{19} + 4895 x^{18} + 4026 x^{17} + 120120 x^{16} + 288222 x^{15} + 1882144 x^{14} + 6515520 x^{13} + 24762320 x^{12} + 60773360 x^{11} + 227012456 x^{10} + 397249116 x^{9} + 720148935 x^{8} + 1566072057 x^{7} + 1566926449 x^{6} + 1217292384 x^{5} + 714471362 x^{4} - 2592915589 x^{3} + 2027899907 x^{2} - 2522739483 x + 6490917711 \)

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:  \(-243091704711882553644913533390559631408320849609375=-\,5^{11}\cdot 11^{41}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $195.10$
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:  $5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(605=5\cdot 11^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{605}(384,·)$, $\chi_{605}(1,·)$, $\chi_{605}(386,·)$, $\chi_{605}(329,·)$, $\chi_{605}(331,·)$, $\chi_{605}(274,·)$, $\chi_{605}(276,·)$, $\chi_{605}(219,·)$, $\chi_{605}(604,·)$, $\chi_{605}(221,·)$, $\chi_{605}(164,·)$, $\chi_{605}(549,·)$, $\chi_{605}(166,·)$, $\chi_{605}(551,·)$, $\chi_{605}(109,·)$, $\chi_{605}(494,·)$, $\chi_{605}(111,·)$, $\chi_{605}(496,·)$, $\chi_{605}(54,·)$, $\chi_{605}(439,·)$, $\chi_{605}(56,·)$, $\chi_{605}(441,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{8}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{16} + \frac{1}{9} a^{15} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{2}{9} a^{8} + \frac{2}{9} a^{7} + \frac{2}{9} a^{6} + \frac{2}{9} a^{5} + \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{2421} a^{19} + \frac{115}{2421} a^{18} - \frac{25}{807} a^{17} + \frac{20}{269} a^{16} + \frac{7}{807} a^{15} + \frac{16}{807} a^{14} - \frac{40}{269} a^{13} + \frac{61}{807} a^{12} - \frac{59}{2421} a^{11} - \frac{188}{2421} a^{10} - \frac{116}{807} a^{9} - \frac{17}{269} a^{8} + \frac{281}{807} a^{7} + \frac{248}{807} a^{6} + \frac{24}{269} a^{5} - \frac{280}{807} a^{4} + \frac{1192}{2421} a^{3} + \frac{1171}{2421} a^{2} + \frac{80}{269} a - \frac{110}{269}$, $\frac{1}{7263} a^{20} + \frac{1}{7263} a^{19} - \frac{4}{7263} a^{18} - \frac{49}{2421} a^{17} - \frac{12}{269} a^{16} - \frac{244}{2421} a^{15} + \frac{208}{2421} a^{14} + \frac{22}{2421} a^{13} - \frac{746}{7263} a^{12} - \frac{725}{7263} a^{11} - \frac{436}{7263} a^{10} - \frac{8}{2421} a^{9} + \frac{328}{807} a^{8} + \frac{763}{2421} a^{7} + \frac{314}{2421} a^{6} - \frac{418}{2421} a^{5} - \frac{3116}{7263} a^{4} + \frac{3280}{7263} a^{3} - \frac{964}{7263} a^{2} - \frac{353}{807} a - \frac{103}{807}$, $\frac{1}{2349819937516954832281147928639196048005159007723174306064314575106634501346212685169481135083953943811221} a^{21} - \frac{23853275434339143959185311943502647424334213143971309249202568752546518990281571607020879495782197002}{2349819937516954832281147928639196048005159007723174306064314575106634501346212685169481135083953943811221} a^{20} - \frac{72511905930013775971959469895509721479774669135627636864246693732603030071267065363631860870651513938}{783273312505651610760382642879732016001719669241058102021438191702211500448737561723160378361317981270407} a^{19} + \frac{108521781359597836995235475691643488800177429438990861819299225943579516206624697846726059901367653517101}{2349819937516954832281147928639196048005159007723174306064314575106634501346212685169481135083953943811221} a^{18} + \frac{10762121056715846124495921759267289572889923099550883421387859448035266799477274853150813388072630919960}{783273312505651610760382642879732016001719669241058102021438191702211500448737561723160378361317981270407} a^{17} + \frac{54457786921048836861935465198978460178607548333751984396632477903748976444045324687336494197644856025360}{783273312505651610760382642879732016001719669241058102021438191702211500448737561723160378361317981270407} a^{16} - \frac{25784129540961535969183537991688202251649670415167166618711915746487197510257622438501807549968777761170}{261091104168550536920127547626577338667239889747019367340479397234070500149579187241053459453772660423469} a^{15} - \frac{102131259580977548645208252590340462957660810024111843422778223083433595134496747003213133517320629305021}{783273312505651610760382642879732016001719669241058102021438191702211500448737561723160378361317981270407} a^{14} + \frac{118517275222573912870835814128854237625445045275922275467228945415445151908158351410719448026288351279376}{2349819937516954832281147928639196048005159007723174306064314575106634501346212685169481135083953943811221} a^{13} - \frac{263858554046874414360789804463882146070340670813738069681189877338232164307647773816026155792620218531567}{2349819937516954832281147928639196048005159007723174306064314575106634501346212685169481135083953943811221} a^{12} - \frac{2194279604168322190161447533748299606801940856021613768428663522708077610379565351534697227755154297955}{261091104168550536920127547626577338667239889747019367340479397234070500149579187241053459453772660423469} a^{11} + \frac{336061598974676060011822062686637857337100233588118986405826108895112989020244066251977216843037639500332}{2349819937516954832281147928639196048005159007723174306064314575106634501346212685169481135083953943811221} a^{10} - \frac{73437364613226374163942260218232297212230574179621010625284445641581585620603974759139826032352081740675}{783273312505651610760382642879732016001719669241058102021438191702211500448737561723160378361317981270407} a^{9} - \frac{77447767525771981160452064528278240491951586622957672976612734870558123294136984388891028779993577248040}{783273312505651610760382642879732016001719669241058102021438191702211500448737561723160378361317981270407} a^{8} + \frac{36200865459671094810967580202220888567077345123444642714988320145515361149733502109807981560748169172797}{261091104168550536920127547626577338667239889747019367340479397234070500149579187241053459453772660423469} a^{7} + \frac{288420271458316157508807470103624868038598334802686631757619360427092985080706488689290417474072041035819}{783273312505651610760382642879732016001719669241058102021438191702211500448737561723160378361317981270407} a^{6} + \frac{398279534639378831534006562688977455884639840597045796652042384079595329204152052451904100094632045302556}{2349819937516954832281147928639196048005159007723174306064314575106634501346212685169481135083953943811221} a^{5} + \frac{1167926181376373787461238295025615194586107192481788592903320605133525068354949692064285186706190329968235}{2349819937516954832281147928639196048005159007723174306064314575106634501346212685169481135083953943811221} a^{4} - \frac{325844655076500040934325834486118804686480291651169663936431310635182352823450189367067212205614513953721}{783273312505651610760382642879732016001719669241058102021438191702211500448737561723160378361317981270407} a^{3} + \frac{603444991053604073974366938636630012488870175830518252606932526968780487687101710680116461421173609497523}{2349819937516954832281147928639196048005159007723174306064314575106634501346212685169481135083953943811221} a^{2} - \frac{45196509709327167731473524378244327885602751108507044627989126892687636769353200032254863179330467689875}{261091104168550536920127547626577338667239889747019367340479397234070500149579187241053459453772660423469} a - \frac{115621545496787037347890992696789175963002233440110044180264324219678680431156975178288909114481288523994}{261091104168550536920127547626577338667239889747019367340479397234070500149579187241053459453772660423469}$

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

Class group and class number

$C_{88708}$, which has order $88708$ (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

\(\Q(\sqrt{-55}) \), 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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{11}$ R ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ $22$ $22$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
5Data not computed
11Data not computed