LMFDB - Number field 22.12.56501459388151144478039723653407440896.1

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^20 - 34*x^18 + 42*x^16 + 227*x^14 - 96*x^12 - 468*x^10 + 59*x^8 + 226*x^6 - 53*x^4 - 9*x^2 + 1)

gp:K = bnfinit(y^22 - 5*y^20 - 34*y^18 + 42*y^16 + 227*y^14 - 96*y^12 - 468*y^10 + 59*y^8 + 226*y^6 - 53*y^4 - 9*y^2 + 1, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 5*x^20 - 34*x^18 + 42*x^16 + 227*x^14 - 96*x^12 - 468*x^10 + 59*x^8 + 226*x^6 - 53*x^4 - 9*x^2 + 1);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 5*x^20 - 34*x^18 + 42*x^16 + 227*x^14 - 96*x^12 - 468*x^10 + 59*x^8 + 226*x^6 - 53*x^4 - 9*x^2 + 1)

$ x^{22} - 5 x^{20} - 34 x^{18} + 42 x^{16} + 227 x^{14} - 96 x^{12} - 468 x^{10} + 59 x^{8} + 226 x^{6} + \cdots + 1 $ x^22 - 5*x^20 - 34*x^18 + 42*x^16 + 227*x^14 - 96*x^12 - 468*x^10 + 59*x^8 + 226*x^6 - 53*x^4 - 9*x^2 + 1

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$22$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$[12, 5]$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$-56501459388151144478039723653407440896$ -56501459388151144478039723653407440896 $\medspace = -\,2^{22}\cdot 1297^{10}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$52.00$	sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:(1.0 * dK)^(1/degree(K))
Galois root discriminant:	not computed
Ramified primes:	$2$, $1297$ 2, 1297	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-1}) $
$\Aut(K/\Q)$:	$C_2$	comment:Autmorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5}a^{14}-\frac{1}{5}a^{12}-\frac{2}{5}a^{10}-\frac{2}{5}a^{4}+\frac{2}{5}$, $\frac{1}{5}a^{15}-\frac{1}{5}a^{13}-\frac{2}{5}a^{11}-\frac{2}{5}a^{5}+\frac{2}{5}a$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{12}-\frac{2}{5}a^{10}-\frac{2}{5}a^{6}-\frac{2}{5}a^{4}+\frac{2}{5}a^{2}+\frac{2}{5}$, $\frac{1}{5}a^{17}+\frac{2}{5}a^{13}-\frac{2}{5}a^{11}-\frac{2}{5}a^{7}-\frac{2}{5}a^{5}+\frac{2}{5}a^{3}+\frac{2}{5}a$, $\frac{1}{85}a^{18}-\frac{3}{85}a^{16}+\frac{2}{85}a^{14}+\frac{22}{85}a^{12}-\frac{19}{85}a^{10}+\frac{8}{85}a^{8}-\frac{26}{85}a^{6}+\frac{3}{85}a^{4}+\frac{41}{85}a^{2}+\frac{19}{85}$, $\frac{1}{85}a^{19}-\frac{3}{85}a^{17}+\frac{2}{85}a^{15}+\frac{22}{85}a^{13}-\frac{19}{85}a^{11}+\frac{8}{85}a^{9}-\frac{26}{85}a^{7}+\frac{3}{85}a^{5}+\frac{41}{85}a^{3}+\frac{19}{85}a$, $\frac{1}{5218235}a^{20}-\frac{4}{306955}a^{18}-\frac{364096}{5218235}a^{16}+\frac{39247}{5218235}a^{14}-\frac{15617}{1043647}a^{12}-\frac{2140706}{5218235}a^{10}+\frac{970239}{5218235}a^{8}-\frac{132086}{474385}a^{6}-\frac{2208233}{5218235}a^{4}+\frac{2339478}{5218235}a^{2}-\frac{1767671}{5218235}$, $\frac{1}{5218235}a^{21}-\frac{4}{306955}a^{19}-\frac{364096}{5218235}a^{17}+\frac{39247}{5218235}a^{15}-\frac{15617}{1043647}a^{13}-\frac{2140706}{5218235}a^{11}+\frac{970239}{5218235}a^{9}-\frac{132086}{474385}a^{7}-\frac{2208233}{5218235}a^{5}+\frac{2339478}{5218235}a^{3}-\frac{1767671}{5218235}a$ 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, 1/5*a^14 - 1/5*a^12 - 2/5*a^10 - 2/5*a^4 + 2/5, 1/5*a^15 - 1/5*a^13 - 2/5*a^11 - 2/5*a^5 + 2/5*a, 1/5*a^16 + 2/5*a^12 - 2/5*a^10 - 2/5*a^6 - 2/5*a^4 + 2/5*a^2 + 2/5, 1/5*a^17 + 2/5*a^13 - 2/5*a^11 - 2/5*a^7 - 2/5*a^5 + 2/5*a^3 + 2/5*a, 1/85*a^18 - 3/85*a^16 + 2/85*a^14 + 22/85*a^12 - 19/85*a^10 + 8/85*a^8 - 26/85*a^6 + 3/85*a^4 + 41/85*a^2 + 19/85, 1/85*a^19 - 3/85*a^17 + 2/85*a^15 + 22/85*a^13 - 19/85*a^11 + 8/85*a^9 - 26/85*a^7 + 3/85*a^5 + 41/85*a^3 + 19/85*a, 1/5218235*a^20 - 4/306955*a^18 - 364096/5218235*a^16 + 39247/5218235*a^14 - 15617/1043647*a^12 - 2140706/5218235*a^10 + 970239/5218235*a^8 - 132086/474385*a^6 - 2208233/5218235*a^4 + 2339478/5218235*a^2 - 1767671/5218235, 1/5218235*a^21 - 4/306955*a^19 - 364096/5218235*a^17 + 39247/5218235*a^15 - 15617/1043647*a^13 - 2140706/5218235*a^11 + 970239/5218235*a^9 - 132086/474385*a^7 - 2208233/5218235*a^5 + 2339478/5218235*a^3 - 1767671/5218235*a

comment:Integral basis

sage:K.integral_basis()

gp:K.zk

magma:IntegralBasis(K);

oscar:basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Ideal class group:		Trivial group, which has order $1$ (assuming GRH)	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		Trivial group, which has order $1$ (assuming GRH)	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);

Unit group

comment:Unit group

sage:UK = K.unit_group()

magma:UK, fUK := UnitGroup(K);

oscar:UK, fUK = unit_group(OK)

Rank:	$16$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	comment:Generator for roots of unity sage:UK.torsion_generator() gp:K.tu[2] magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar:torsion_units_generator(OK)
Fundamental units:	$a$, $\frac{409225}{1043647}a^{21}-\frac{10092618}{5218235}a^{19}-\frac{14059319}{1043647}a^{17}+\frac{81517234}{5218235}a^{15}+\frac{471178966}{5218235}a^{13}-\frac{33884674}{1043647}a^{11}-\frac{976937179}{5218235}a^{9}+\frac{6821166}{474385}a^{7}+\frac{482845024}{5218235}a^{5}-\frac{98556686}{5218235}a^{3}-\frac{5625885}{1043647}a$, $\frac{409225}{1043647}a^{21}-\frac{10092618}{5218235}a^{19}-\frac{14059319}{1043647}a^{17}+\frac{81517234}{5218235}a^{15}+\frac{471178966}{5218235}a^{13}-\frac{33884674}{1043647}a^{11}-\frac{976937179}{5218235}a^{9}+\frac{6821166}{474385}a^{7}+\frac{482845024}{5218235}a^{5}-\frac{98556686}{5218235}a^{3}-\frac{4582238}{1043647}a$, $\frac{102584}{5218235}a^{20}-\frac{19984}{1043647}a^{18}-\frac{5357899}{5218235}a^{16}-\frac{2105635}{1043647}a^{14}+\frac{33054398}{5218235}a^{12}+\frac{1016102}{61391}a^{10}-\frac{35320473}{5218235}a^{8}-\frac{15449448}{474385}a^{6}-\frac{57176989}{5218235}a^{4}+\frac{33641106}{5218235}a^{2}+\frac{8481382}{5218235}$, $\frac{1130174}{5218235}a^{20}-\frac{5392708}{5218235}a^{18}-\frac{39650926}{5218235}a^{16}+\frac{38451379}{5218235}a^{14}+\frac{52904184}{1043647}a^{12}-\frac{49301226}{5218235}a^{10}-\frac{529512967}{5218235}a^{8}-\frac{4034183}{474385}a^{6}+\frac{212449694}{5218235}a^{4}-\frac{28761669}{5218235}a^{2}-\frac{1731217}{5218235}$, $\frac{312217}{5218235}a^{21}-\frac{1953982}{5218235}a^{19}-\frac{8577564}{5218235}a^{17}+\frac{26150956}{5218235}a^{15}+\frac{51636524}{5218235}a^{13}-\frac{117569711}{5218235}a^{11}-\frac{17938595}{1043647}a^{9}+\frac{18695234}{474385}a^{7}+\frac{12890198}{5218235}a^{5}-\frac{24096958}{1043647}a^{3}+\frac{1180191}{306955}a$, $\frac{621051}{5218235}a^{20}-\frac{551411}{1043647}a^{18}-\frac{22815356}{5218235}a^{16}+\frac{14255624}{5218235}a^{14}+\frac{152873058}{5218235}a^{12}+\frac{15723287}{5218235}a^{10}-\frac{302523627}{5218235}a^{8}-\frac{9781552}{474385}a^{6}+\frac{118323646}{5218235}a^{4}+\frac{21734764}{5218235}a^{2}-\frac{8193149}{5218235}$, $\frac{627027}{5218235}a^{20}-\frac{3531769}{5218235}a^{18}-\frac{3891345}{1043647}a^{16}+\frac{40373251}{5218235}a^{14}+\frac{130032637}{5218235}a^{12}-\frac{153488612}{5218235}a^{10}-\frac{283908276}{5218235}a^{8}+\frac{4030430}{94877}a^{6}+\frac{168873218}{5218235}a^{4}-\frac{102925041}{5218235}a^{2}+\frac{10048843}{5218235}$, $\frac{637250}{1043647}a^{21}-\frac{3060504}{1043647}a^{19}-\frac{111059208}{5218235}a^{17}+\frac{110430363}{5218235}a^{15}+\frac{735272201}{5218235}a^{13}-\frac{29743678}{1043647}a^{11}-\frac{291342987}{1043647}a^{9}-\frac{11065954}{474385}a^{7}+\frac{114290571}{1043647}a^{5}-\frac{63345856}{5218235}a^{3}+\frac{748756}{1043647}a$, $\frac{3891364}{5218235}a^{21}-\frac{19294316}{5218235}a^{19}-\frac{132674614}{5218235}a^{17}+\frac{155818054}{5218235}a^{15}+\frac{874505604}{5218235}a^{13}-\frac{322712682}{5218235}a^{11}-\frac{1733634681}{5218235}a^{9}+\frac{12872084}{474385}a^{7}+\frac{688280342}{5218235}a^{5}-\frac{201485611}{5218235}a^{3}+\frac{29337479}{5218235}a$, $\frac{771134}{474385}a^{21}-\frac{7741}{61391}a^{20}-\frac{3714982}{474385}a^{19}+\frac{2505972}{5218235}a^{18}-\frac{26861321}{474385}a^{17}+\frac{5202995}{1043647}a^{16}+\frac{27306902}{474385}a^{15}+\frac{125464}{5218235}a^{14}+\frac{178878827}{474385}a^{13}-\frac{171509539}{5218235}a^{12}-\frac{39849643}{474385}a^{11}-\frac{23493552}{1043647}a^{10}-\frac{360786299}{474385}a^{9}+\frac{310544796}{5218235}a^{8}-\frac{4701505}{94877}a^{7}+\frac{27585546}{474385}a^{6}+\frac{156249569}{474385}a^{5}-\frac{51516981}{5218235}a^{4}-\frac{12574127}{474385}a^{3}-\frac{80345896}{5218235}a^{2}-\frac{1074933}{94877}a-\frac{2024094}{1043647}$, $\frac{8552069}{5218235}a^{21}+\frac{5442203}{5218235}a^{20}-\frac{38782861}{5218235}a^{19}-\frac{24745429}{5218235}a^{18}-\frac{309167812}{5218235}a^{17}-\frac{196331873}{5218235}a^{16}+\frac{217116319}{5218235}a^{15}+\frac{140080196}{5218235}a^{14}+\frac{2054909453}{5218235}a^{13}+\frac{260341809}{1043647}a^{12}+\frac{122293449}{5218235}a^{11}+\frac{62567482}{5218235}a^{10}-\frac{4029112946}{5218235}a^{9}-\frac{2539603513}{5218235}a^{8}-\frac{24542812}{94877}a^{7}-\frac{74197373}{474385}a^{6}+\frac{1471234533}{5218235}a^{5}+\frac{53593163}{306955}a^{4}+\frac{228475273}{5218235}a^{3}+\frac{121261344}{5218235}a^{2}-\frac{43534692}{5218235}a-\frac{18498523}{5218235}$, $\frac{812272}{5218235}a^{21}-\frac{285887}{306955}a^{20}-\frac{3850229}{5218235}a^{19}+\frac{21995597}{5218235}a^{18}-\frac{28345354}{5218235}a^{17}+\frac{175612806}{5218235}a^{16}+\frac{25536578}{5218235}a^{15}-\frac{120536489}{5218235}a^{14}+\frac{181078762}{5218235}a^{13}-\frac{231677876}{1043647}a^{12}-\frac{25161283}{5218235}a^{11}-\frac{85173643}{5218235}a^{10}-\frac{321101336}{5218235}a^{9}+\frac{2222270994}{5218235}a^{8}-\frac{1815307}{474385}a^{7}+\frac{14044691}{94877}a^{6}+\frac{60222332}{5218235}a^{5}-\frac{142622476}{1043647}a^{4}-\frac{97240394}{5218235}a^{3}-\frac{79613218}{5218235}a^{2}-\frac{17196076}{5218235}a+\frac{10638996}{5218235}$, $\frac{180871}{306955}a^{21}-\frac{3085164}{5218235}a^{20}-\frac{13740714}{5218235}a^{19}+\frac{14076644}{5218235}a^{18}-\frac{22359417}{1043647}a^{17}+\frac{1306594}{61391}a^{16}+\frac{69296767}{5218235}a^{15}-\frac{16194485}{1043647}a^{14}+\frac{734510696}{5218235}a^{13}-\frac{736452506}{5218235}a^{12}+\frac{103827731}{5218235}a^{11}-\frac{27893281}{5218235}a^{10}-\frac{1384123516}{5218235}a^{9}+\frac{286982877}{1043647}a^{8}-\frac{10794619}{94877}a^{7}+\frac{40974928}{474385}a^{6}+\frac{384480706}{5218235}a^{5}-\frac{504372679}{5218235}a^{4}+\frac{93061399}{5218235}a^{3}-\frac{52471541}{5218235}a^{2}+\frac{2502041}{1043647}a+\frac{11913504}{5218235}$, $\frac{9909021}{5218235}a^{21}+\frac{9454}{474385}a^{20}-\frac{48177747}{5218235}a^{19}-\frac{866}{5581}a^{18}-\frac{343202966}{5218235}a^{17}-\frac{215778}{474385}a^{16}+\frac{73449939}{1043647}a^{15}+\frac{1400346}{474385}a^{14}+\frac{2287412527}{5218235}a^{13}+\frac{2014764}{474385}a^{12}-\frac{628315233}{5218235}a^{11}-\frac{7137429}{474385}a^{10}-\frac{4647933163}{5218235}a^{9}-\frac{7848768}{474385}a^{8}-\frac{4698168}{474385}a^{7}+\frac{2153360}{94877}a^{6}+\frac{2106915707}{5218235}a^{5}+\frac{10252137}{474385}a^{4}-\frac{51772464}{1043647}a^{3}-\frac{2862907}{474385}a^{2}-\frac{85997674}{5218235}a-\frac{2298779}{474385}$, $\frac{1820914}{306955}a^{21}+\frac{14539062}{5218235}a^{20}-\frac{149673234}{5218235}a^{19}-\frac{68712081}{5218235}a^{18}-\frac{215334928}{1043647}a^{17}-\frac{512686497}{5218235}a^{16}+\frac{1120334514}{5218235}a^{15}+\frac{468061529}{5218235}a^{14}+\frac{7193987573}{5218235}a^{13}+\frac{682346332}{1043647}a^{12}-\frac{354530710}{1043647}a^{11}-\frac{449095662}{5218235}a^{10}-\frac{14666507428}{5218235}a^{9}-\frac{6814162867}{5218235}a^{8}-\frac{54149009}{474385}a^{7}-\frac{90677667}{474385}a^{6}+\frac{6680047989}{5218235}a^{5}+\frac{2799468314}{5218235}a^{4}-\frac{116878880}{1043647}a^{3}-\frac{85184504}{5218235}a^{2}-\frac{280022636}{5218235}a-\frac{92107132}{5218235}$ a, 409225/1043647a^21 - 10092618/5218235a^19 - 14059319/1043647a^17 + 81517234/5218235a^15 + 471178966/5218235a^13 - 33884674/1043647a^11 - 976937179/5218235a^9 + 6821166/474385a^7 + 482845024/5218235a^5 - 98556686/5218235a^3 - 5625885/1043647a, 409225/1043647a^21 - 10092618/5218235a^19 - 14059319/1043647a^17 + 81517234/5218235a^15 + 471178966/5218235a^13 - 33884674/1043647a^11 - 976937179/5218235a^9 + 6821166/474385a^7 + 482845024/5218235a^5 - 98556686/5218235a^3 - 4582238/1043647a, 102584/5218235a^20 - 19984/1043647a^18 - 5357899/5218235a^16 - 2105635/1043647a^14 + 33054398/5218235a^12 + 1016102/61391a^10 - 35320473/5218235a^8 - 15449448/474385a^6 - 57176989/5218235a^4 + 33641106/5218235a^2 + 8481382/5218235, 1130174/5218235a^20 - 5392708/5218235a^18 - 39650926/5218235a^16 + 38451379/5218235a^14 + 52904184/1043647a^12 - 49301226/5218235a^10 - 529512967/5218235a^8 - 4034183/474385a^6 + 212449694/5218235a^4 - 28761669/5218235a^2 - 1731217/5218235, 312217/5218235a^21 - 1953982/5218235a^19 - 8577564/5218235a^17 + 26150956/5218235a^15 + 51636524/5218235a^13 - 117569711/5218235a^11 - 17938595/1043647a^9 + 18695234/474385a^7 + 12890198/5218235a^5 - 24096958/1043647a^3 + 1180191/306955a, 621051/5218235a^20 - 551411/1043647a^18 - 22815356/5218235a^16 + 14255624/5218235a^14 + 152873058/5218235a^12 + 15723287/5218235a^10 - 302523627/5218235a^8 - 9781552/474385a^6 + 118323646/5218235a^4 + 21734764/5218235a^2 - 8193149/5218235, 627027/5218235a^20 - 3531769/5218235a^18 - 3891345/1043647a^16 + 40373251/5218235a^14 + 130032637/5218235a^12 - 153488612/5218235a^10 - 283908276/5218235a^8 + 4030430/94877a^6 + 168873218/5218235a^4 - 102925041/5218235a^2 + 10048843/5218235, 637250/1043647a^21 - 3060504/1043647a^19 - 111059208/5218235a^17 + 110430363/5218235a^15 + 735272201/5218235a^13 - 29743678/1043647a^11 - 291342987/1043647a^9 - 11065954/474385a^7 + 114290571/1043647a^5 - 63345856/5218235a^3 + 748756/1043647a, 3891364/5218235a^21 - 19294316/5218235a^19 - 132674614/5218235a^17 + 155818054/5218235a^15 + 874505604/5218235a^13 - 322712682/5218235a^11 - 1733634681/5218235a^9 + 12872084/474385a^7 + 688280342/5218235a^5 - 201485611/5218235a^3 + 29337479/5218235a, 771134/474385a^21 - 7741/61391a^20 - 3714982/474385a^19 + 2505972/5218235a^18 - 26861321/474385a^17 + 5202995/1043647a^16 + 27306902/474385a^15 + 125464/5218235a^14 + 178878827/474385a^13 - 171509539/5218235a^12 - 39849643/474385a^11 - 23493552/1043647a^10 - 360786299/474385a^9 + 310544796/5218235a^8 - 4701505/94877a^7 + 27585546/474385a^6 + 156249569/474385a^5 - 51516981/5218235a^4 - 12574127/474385a^3 - 80345896/5218235a^2 - 1074933/94877a - 2024094/1043647, 8552069/5218235a^21 + 5442203/5218235a^20 - 38782861/5218235a^19 - 24745429/5218235a^18 - 309167812/5218235a^17 - 196331873/5218235a^16 + 217116319/5218235a^15 + 140080196/5218235a^14 + 2054909453/5218235a^13 + 260341809/1043647a^12 + 122293449/5218235a^11 + 62567482/5218235a^10 - 4029112946/5218235a^9 - 2539603513/5218235a^8 - 24542812/94877a^7 - 74197373/474385a^6 + 1471234533/5218235a^5 + 53593163/306955a^4 + 228475273/5218235a^3 + 121261344/5218235a^2 - 43534692/5218235a - 18498523/5218235, 812272/5218235a^21 - 285887/306955a^20 - 3850229/5218235a^19 + 21995597/5218235a^18 - 28345354/5218235a^17 + 175612806/5218235a^16 + 25536578/5218235a^15 - 120536489/5218235a^14 + 181078762/5218235a^13 - 231677876/1043647a^12 - 25161283/5218235a^11 - 85173643/5218235a^10 - 321101336/5218235a^9 + 2222270994/5218235a^8 - 1815307/474385a^7 + 14044691/94877a^6 + 60222332/5218235a^5 - 142622476/1043647a^4 - 97240394/5218235a^3 - 79613218/5218235a^2 - 17196076/5218235a + 10638996/5218235, 180871/306955a^21 - 3085164/5218235a^20 - 13740714/5218235a^19 + 14076644/5218235a^18 - 22359417/1043647a^17 + 1306594/61391a^16 + 69296767/5218235a^15 - 16194485/1043647a^14 + 734510696/5218235a^13 - 736452506/5218235a^12 + 103827731/5218235a^11 - 27893281/5218235a^10 - 1384123516/5218235a^9 + 286982877/1043647a^8 - 10794619/94877a^7 + 40974928/474385a^6 + 384480706/5218235a^5 - 504372679/5218235a^4 + 93061399/5218235a^3 - 52471541/5218235a^2 + 2502041/1043647a + 11913504/5218235, 9909021/5218235a^21 + 9454/474385a^20 - 48177747/5218235a^19 - 866/5581a^18 - 343202966/5218235a^17 - 215778/474385a^16 + 73449939/1043647a^15 + 1400346/474385a^14 + 2287412527/5218235a^13 + 2014764/474385a^12 - 628315233/5218235a^11 - 7137429/474385a^10 - 4647933163/5218235a^9 - 7848768/474385a^8 - 4698168/474385a^7 + 2153360/94877a^6 + 2106915707/5218235a^5 + 10252137/474385a^4 - 51772464/1043647a^3 - 2862907/474385a^2 - 85997674/5218235a - 2298779/474385, 1820914/306955a^21 + 14539062/5218235a^20 - 149673234/5218235a^19 - 68712081/5218235a^18 - 215334928/1043647a^17 - 512686497/5218235a^16 + 1120334514/5218235a^15 + 468061529/5218235a^14 + 7193987573/5218235a^13 + 682346332/1043647a^12 - 354530710/1043647a^11 - 449095662/5218235a^10 - 14666507428/5218235a^9 - 6814162867/5218235a^8 - 54149009/474385a^7 - 90677667/474385a^6 + 6680047989/5218235a^5 + 2799468314/5218235a^4 - 116878880/1043647a^3 - 85184504/5218235a^2 - 280022636/5218235*a - 92107132/5218235 (assuming GRH)	comment:Fundamental units sage:UK.fundamental_units() gp:K.fu magma:[K\|fUK(g): g in Generators(UK)]; oscar:[K(fUK(a)) for a in gens(UK)]
Regulator:	$ 114972984549 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{5}\cdot 114972984549 \cdot 1}{2\cdot\sqrt{56501459388151144478039723653407440896}}\cr\approx \mathstrut & 0.306757554303413 \end{aligned}\] (assuming GRH)

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^20 - 34*x^18 + 42*x^16 + 227*x^14 - 96*x^12 - 468*x^10 + 59*x^8 + 226*x^6 - 53*x^4 - 9*x^2 + 1) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^22 - 5*x^20 - 34*x^18 + 42*x^16 + 227*x^14 - 96*x^12 - 468*x^10 + 59*x^8 + 226*x^6 - 53*x^4 - 9*x^2 + 1, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 5*x^20 - 34*x^18 + 42*x^16 + 227*x^14 - 96*x^12 - 468*x^10 + 59*x^8 + 226*x^6 - 53*x^4 - 9*x^2 + 1); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 5*x^20 - 34*x^18 + 42*x^16 + 227*x^14 - 96*x^12 - 468*x^10 + 59*x^8 + 226*x^6 - 53*x^4 - 9*x^2 + 1); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_2^{10}.D_{22}$ (as 22T32):

comment:Galois group

sage:K.galois_group(type='pari')

gp:polgalois(K.pol)

magma:G = GaloisGroup(K);

oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)

A solvable group of order 45056

The 200 conjugacy class representatives for $C_2^{10}.D_{22}$

Character table for $C_2^{10}.D_{22}$

Intermediate fields

11.11.3670285774226257.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

comment:Intermediate fields

sage:K.subfields()[1:-1]

gp:L = nfsubfields(K); L[2..length(b)]

magma:L := Subfields(K); L[2..#L];

oscar:subfields(K)[2:end-1]

Sibling fields

Degree 22 siblings:	data not computed
Degree 44 siblings:	data not computed
Minimal sibling:	22.12.56501459388151144478039723653407440896.13

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	$22$	${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$	$22$	${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{7}$	${\href{/padicField/13.11.0.1}{11} }^{2}$	${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$	$22$	$22$	${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{9}$	${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$	${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{9}$	${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{9}$	${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{7}$	$22$	${\href{/padicField/53.11.0.1}{11} }^{2}$	${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

comment:Frobenius cycle types

sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.11.2.22a176.2	$x^{22} + 2 x^{21} + 2 x^{20} + 2 x^{19} + 2 x^{17} + 2 x^{16} + 2 x^{15} + 4 x^{13} + 2 x^{12} + 6 x^{11} + 4 x^{10} + 2 x^{9} + 4 x^{8} + 2 x^{7} + 4 x^{6} + 2 x^{5} + 5 x^{4} + 6 x^{2} + 9$	$2$	$11$	$22$	not computed	not computed
$1297$ 1297	$\Q_{1297}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{1297}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $4$	$2$	$2$	$2$

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