LMFDB - Number field 18.0.61773685738999443000000000000.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 + 14*x^15 - 81*x^14 + 169*x^13 - 258*x^12 - 27*x^11 + 1341*x^10 - 4988*x^9 + 10815*x^8 - 11791*x^7 + 6239*x^6 + 9090*x^5 - 16910*x^4 + 16140*x^3 + 2700*x^2 - 5400*x + 8100)

gp: K = bnfinit(y^18 - 3*y^17 + 5*y^16 + 14*y^15 - 81*y^14 + 169*y^13 - 258*y^12 - 27*y^11 + 1341*y^10 - 4988*y^9 + 10815*y^8 - 11791*y^7 + 6239*y^6 + 9090*y^5 - 16910*y^4 + 16140*y^3 + 2700*y^2 - 5400*y + 8100, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 + 5*x^16 + 14*x^15 - 81*x^14 + 169*x^13 - 258*x^12 - 27*x^11 + 1341*x^10 - 4988*x^9 + 10815*x^8 - 11791*x^7 + 6239*x^6 + 9090*x^5 - 16910*x^4 + 16140*x^3 + 2700*x^2 - 5400*x + 8100);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 + 5*x^16 + 14*x^15 - 81*x^14 + 169*x^13 - 258*x^12 - 27*x^11 + 1341*x^10 - 4988*x^9 + 10815*x^8 - 11791*x^7 + 6239*x^6 + 9090*x^5 - 16910*x^4 + 16140*x^3 + 2700*x^2 - 5400*x + 8100)

$ x^{18} - 3 x^{17} + 5 x^{16} + 14 x^{15} - 81 x^{14} + 169 x^{13} - 258 x^{12} - 27 x^{11} + 1341 x^{10} + \cdots + 8100 $ x^18 - 3*x^17 + 5*x^16 + 14*x^15 - 81*x^14 + 169*x^13 - 258*x^12 - 27*x^11 + 1341*x^10 - 4988*x^9 + 10815*x^8 - 11791*x^7 + 6239*x^6 + 9090*x^5 - 16910*x^4 + 16140*x^3 + 2700*x^2 - 5400*x + 8100

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$18$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 9]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-61773685738999443000000000000$ -61773685738999443000000000000 $\medspace = -\,2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 11^{12}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$39.76$	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:	$2^{2/3}3^{1/2}5^{2/3}11^{2/3}\approx 39.76391018034719$
Ramified primes:	$2$, $3$, $5$, $11$ 2, 3, 5, 11	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-3}) $
$\card{ \Gal(K/\Q) }$:	$18$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{9}a^{7}+\frac{1}{9}a^{6}+\frac{1}{9}a^{5}+\frac{1}{9}a^{4}+\frac{1}{9}a^{3}+\frac{1}{9}a^{2}$, $\frac{1}{9}a^{8}-\frac{1}{9}a^{2}$, $\frac{1}{9}a^{9}-\frac{1}{9}a^{3}$, $\frac{1}{9}a^{10}-\frac{1}{9}a^{4}$, $\frac{1}{27}a^{11}+\frac{1}{27}a^{10}-\frac{1}{27}a^{9}-\frac{1}{27}a^{8}-\frac{1}{9}a^{6}-\frac{4}{27}a^{5}+\frac{5}{27}a^{4}-\frac{2}{27}a^{3}+\frac{7}{27}a^{2}-\frac{1}{9}a+\frac{1}{3}$, $\frac{1}{162}a^{12}+\frac{1}{81}a^{11}+\frac{1}{162}a^{9}+\frac{4}{81}a^{8}+\frac{23}{162}a^{6}+\frac{2}{81}a^{5}+\frac{10}{27}a^{4}+\frac{5}{162}a^{3}+\frac{26}{81}a^{2}+\frac{1}{27}a+\frac{2}{9}$, $\frac{1}{162}a^{13}+\frac{1}{81}a^{11}+\frac{7}{162}a^{10}-\frac{2}{81}a^{8}+\frac{5}{162}a^{7}-\frac{4}{27}a^{6}+\frac{5}{81}a^{5}-\frac{49}{162}a^{4}+\frac{2}{27}a^{3}-\frac{19}{81}a^{2}+\frac{1}{27}a-\frac{1}{9}$, $\frac{1}{486}a^{14}-\frac{1}{486}a^{12}+\frac{7}{486}a^{11}+\frac{1}{81}a^{10}-\frac{13}{486}a^{9}+\frac{11}{486}a^{8}-\frac{1}{81}a^{7}-\frac{59}{486}a^{6}+\frac{41}{486}a^{5}+\frac{7}{81}a^{4}-\frac{101}{486}a^{3}+\frac{2}{27}a^{2}-\frac{2}{9}a-\frac{4}{9}$, $\frac{1}{53460}a^{15}-\frac{13}{53460}a^{14}-\frac{5}{2673}a^{13}-\frac{91}{53460}a^{12}-\frac{901}{53460}a^{11}-\frac{293}{26730}a^{10}-\frac{871}{17820}a^{9}+\frac{2023}{53460}a^{8}-\frac{686}{13365}a^{7}-\frac{4823}{53460}a^{6}-\frac{1105}{10692}a^{5}+\frac{1697}{26730}a^{4}+\frac{2246}{13365}a^{3}-\frac{427}{1782}a^{2}+\frac{17}{297}a+\frac{35}{99}$, $\frac{1}{80991900}a^{16}-\frac{122}{20247975}a^{15}-\frac{10313}{16198380}a^{14}-\frac{63397}{26997300}a^{13}-\frac{93403}{40495950}a^{12}-\frac{1169371}{80991900}a^{11}-\frac{348913}{7362900}a^{10}+\frac{69457}{1840725}a^{9}-\frac{3076109}{80991900}a^{8}+\frac{57313}{8999100}a^{7}+\frac{899243}{8099190}a^{6}+\frac{3812029}{80991900}a^{5}+\frac{4657607}{40495950}a^{4}+\frac{4082}{24543}a^{3}-\frac{143599}{299970}a^{2}+\frac{1208}{149985}a-\frac{2399}{9999}$, $\frac{1}{11685430340100}a^{17}-\frac{41077}{11685430340100}a^{16}-\frac{3967993}{1947571723350}a^{15}+\frac{11081269019}{11685430340100}a^{14}+\frac{414728423}{285010496100}a^{13}-\frac{980813401}{973785861675}a^{12}+\frac{20489097467}{3895143446700}a^{11}+\frac{14720688989}{779028689340}a^{10}+\frac{62864764459}{1947571723350}a^{9}+\frac{232830473743}{11685430340100}a^{8}+\frac{165940398467}{11685430340100}a^{7}-\frac{32198495743}{973785861675}a^{6}-\frac{570763617221}{5842715170050}a^{5}-\frac{2740140194323}{5842715170050}a^{4}-\frac{23010226843}{194757172335}a^{3}-\frac{1978642118}{7213228605}a^{2}-\frac{505118462}{1967244165}a-\frac{306965642}{1442645721}$ 1, a, a^2, a^3, a^4, 1/3*a^5 + 1/3*a^3 + 1/3*a, 1/3*a^6 + 1/3*a^4 + 1/3*a^2, 1/9*a^7 + 1/9*a^6 + 1/9*a^5 + 1/9*a^4 + 1/9*a^3 + 1/9*a^2, 1/9*a^8 - 1/9*a^2, 1/9*a^9 - 1/9*a^3, 1/9*a^10 - 1/9*a^4, 1/27*a^11 + 1/27*a^10 - 1/27*a^9 - 1/27*a^8 - 1/9*a^6 - 4/27*a^5 + 5/27*a^4 - 2/27*a^3 + 7/27*a^2 - 1/9*a + 1/3, 1/162*a^12 + 1/81*a^11 + 1/162*a^9 + 4/81*a^8 + 23/162*a^6 + 2/81*a^5 + 10/27*a^4 + 5/162*a^3 + 26/81*a^2 + 1/27*a + 2/9, 1/162*a^13 + 1/81*a^11 + 7/162*a^10 - 2/81*a^8 + 5/162*a^7 - 4/27*a^6 + 5/81*a^5 - 49/162*a^4 + 2/27*a^3 - 19/81*a^2 + 1/27*a - 1/9, 1/486*a^14 - 1/486*a^12 + 7/486*a^11 + 1/81*a^10 - 13/486*a^9 + 11/486*a^8 - 1/81*a^7 - 59/486*a^6 + 41/486*a^5 + 7/81*a^4 - 101/486*a^3 + 2/27*a^2 - 2/9*a - 4/9, 1/53460*a^15 - 13/53460*a^14 - 5/2673*a^13 - 91/53460*a^12 - 901/53460*a^11 - 293/26730*a^10 - 871/17820*a^9 + 2023/53460*a^8 - 686/13365*a^7 - 4823/53460*a^6 - 1105/10692*a^5 + 1697/26730*a^4 + 2246/13365*a^3 - 427/1782*a^2 + 17/297*a + 35/99, 1/80991900*a^16 - 122/20247975*a^15 - 10313/16198380*a^14 - 63397/26997300*a^13 - 93403/40495950*a^12 - 1169371/80991900*a^11 - 348913/7362900*a^10 + 69457/1840725*a^9 - 3076109/80991900*a^8 + 57313/8999100*a^7 + 899243/8099190*a^6 + 3812029/80991900*a^5 + 4657607/40495950*a^4 + 4082/24543*a^3 - 143599/299970*a^2 + 1208/149985*a - 2399/9999, 1/11685430340100*a^17 - 41077/11685430340100*a^16 - 3967993/1947571723350*a^15 + 11081269019/11685430340100*a^14 + 414728423/285010496100*a^13 - 980813401/973785861675*a^12 + 20489097467/3895143446700*a^11 + 14720688989/779028689340*a^10 + 62864764459/1947571723350*a^9 + 232830473743/11685430340100*a^8 + 165940398467/11685430340100*a^7 - 32198495743/973785861675*a^6 - 570763617221/5842715170050*a^5 - 2740140194323/5842715170050*a^4 - 23010226843/194757172335*a^3 - 1978642118/7213228605*a^2 - 505118462/1967244165*a - 306965642/1442645721

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$, $3$

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

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

oscar: UK, fUK = unit_group(OK)

Rank:	$8$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ \frac{5567302}{28924332525} a^{17} - \frac{17535853}{28924332525} a^{16} + \frac{35365543}{38565776700} a^{15} + \frac{320357357}{115697330100} a^{14} - \frac{2272106}{141094305} a^{13} + \frac{247798129}{7713155340} a^{12} - \frac{1787670457}{38565776700} a^{11} - \frac{105082927}{19282888350} a^{10} + \frac{10115550743}{38565776700} a^{9} - \frac{109961557007}{115697330100} a^{8} + \frac{59367751406}{28924332525} a^{7} - \frac{83483745331}{38565776700} a^{6} + \frac{2116186211}{2103587820} a^{5} + \frac{68880808559}{57848665050} a^{4} - \frac{8609495837}{3856577670} a^{3} + \frac{38745623}{28567242} a^{2} + \frac{125997296}{214254315} a - \frac{1722530}{14283621} $ (5567302)/(28924332525)a^(17) - (17535853)/(28924332525)a^(16) + (35365543)/(38565776700)a^(15) + (320357357)/(115697330100)a^(14) - (2272106)/(141094305)a^(13) + (247798129)/(7713155340)a^(12) - (1787670457)/(38565776700)a^(11) - (105082927)/(19282888350)a^(10) + (10115550743)/(38565776700)a^(9) - (109961557007)/(115697330100)a^(8) + (59367751406)/(28924332525)a^(7) - (83483745331)/(38565776700)a^(6) + (2116186211)/(2103587820)a^(5) + (68880808559)/(57848665050)a^(4) - (8609495837)/(3856577670)a^(3) + (38745623)/(28567242)a^(2) + (125997296)/(214254315)*a - (1722530)/(14283621) (order $6$)	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:	$\frac{594575519}{5842715170050}a^{17}-\frac{1249179323}{5842715170050}a^{16}+\frac{220230883}{973785861675}a^{15}+\frac{5259927938}{2921357585025}a^{14}-\frac{980282453}{142505248050}a^{13}+\frac{9374560802}{973785861675}a^{12}-\frac{11453390686}{973785861675}a^{11}-\frac{1775196619}{77902868934}a^{10}+\frac{127441353656}{973785861675}a^{9}-\frac{1089286605929}{2921357585025}a^{8}+\frac{3795161084023}{5842715170050}a^{7}-\frac{289468699594}{973785861675}a^{6}-\frac{136212140683}{531155924550}a^{5}+\frac{3400394390683}{2921357585025}a^{4}-\frac{111628112953}{194757172335}a^{3}-\frac{1239912524}{7213228605}a^{2}+\frac{25592614559}{21639685815}a+\frac{346058707}{1442645721}$, $\frac{858675137}{3895143446700}a^{17}-\frac{1917473791}{1947571723350}a^{16}+\frac{1209721223}{649190574450}a^{15}+\frac{3092407229}{1947571723350}a^{14}-\frac{524510504}{23750874675}a^{13}+\frac{4227450481}{72132286050}a^{12}-\frac{2651554913}{25967622978}a^{11}+\frac{26023510132}{324595287225}a^{10}+\frac{188519723149}{649190574450}a^{9}-\frac{2767131695411}{1947571723350}a^{8}+\frac{3649160577862}{973785861675}a^{7}-\frac{3692021648087}{649190574450}a^{6}+\frac{1805587904549}{354103949700}a^{5}-\frac{4419750255947}{1947571723350}a^{4}-\frac{223271407934}{64919057445}a^{3}+\frac{21762024497}{4808819070}a^{2}-\frac{26551093583}{7213228605}a+\frac{565280576}{480881907}$, $\frac{24343781}{88525987425}a^{17}-\frac{1498336547}{1947571723350}a^{16}+\frac{1156352357}{1298381148900}a^{15}+\frac{18168819431}{3895143446700}a^{14}-\frac{46049648}{2159170425}a^{13}+\frac{5010107339}{144264572100}a^{12}-\frac{2213248391}{51935245956}a^{11}-\frac{19947426437}{649190574450}a^{10}+\frac{458717832601}{1298381148900}a^{9}-\frac{4611052161029}{3895143446700}a^{8}+\frac{2246660742779}{973785861675}a^{7}-\frac{2290313365493}{1298381148900}a^{6}+\frac{926734386623}{3895143446700}a^{5}+\frac{893782384933}{973785861675}a^{4}-\frac{110280131789}{129838114890}a^{3}+\frac{1494253613}{1602939690}a^{2}-\frac{23376491}{71418105}a+\frac{219299092}{480881907}$, $\frac{565607869}{2337086068020}a^{17}+\frac{253793647}{11685430340100}a^{16}-\frac{422225443}{973785861675}a^{15}+\frac{2379573739}{467417213604}a^{14}-\frac{1965707027}{285010496100}a^{13}-\frac{9712992626}{973785861675}a^{12}+\frac{39755078131}{3895143446700}a^{11}-\frac{414758453797}{3895143446700}a^{10}+\frac{34725623911}{177051974850}a^{9}-\frac{2913514572673}{11685430340100}a^{8}-\frac{2214107149931}{11685430340100}a^{7}+\frac{443298764849}{194757172335}a^{6}-\frac{7358177058721}{5842715170050}a^{5}+\frac{5135424247937}{2921357585025}a^{4}+\frac{149785112815}{77902868934}a^{3}-\frac{2709682943}{655748055}a^{2}+\frac{61427732176}{21639685815}a+\frac{6512795222}{1442645721}$, $\frac{985869743}{3895143446700}a^{17}-\frac{170324471}{973785861675}a^{16}-\frac{7138457}{51935245956}a^{15}+\frac{19140251377}{3895143446700}a^{14}-\frac{477942469}{47501749350}a^{13}+\frac{35702951}{144264572100}a^{12}-\frac{5922806773}{1298381148900}a^{11}-\frac{24188603198}{324595287225}a^{10}+\frac{287103484901}{1298381148900}a^{9}-\frac{1855231697959}{3895143446700}a^{8}+\frac{174388366831}{389514344670}a^{7}+\frac{1461303193379}{1298381148900}a^{6}-\frac{378386519237}{973785861675}a^{5}+\frac{105742114039}{194757172335}a^{4}+\frac{58665170824}{64919057445}a^{3}-\frac{126285659}{801469845}a^{2}+\frac{155566814}{1442645721}a+\frac{97777802}{480881907}$, $\frac{348457849}{259676229780}a^{17}-\frac{3838812961}{649190574450}a^{16}+\frac{1255712243}{108198429075}a^{15}+\frac{1121842423}{129838114890}a^{14}-\frac{1048007702}{7916958225}a^{13}+\frac{25900057919}{72132286050}a^{12}-\frac{140258331503}{216396858150}a^{11}+\frac{56471469443}{108198429075}a^{10}+\frac{123241058183}{72132286050}a^{9}-\frac{5590347378761}{649190574450}a^{8}+\frac{7531123885864}{324595287225}a^{7}-\frac{1552274019877}{43279371630}a^{6}+\frac{44107394470507}{1298381148900}a^{5}-\frac{9645339727589}{649190574450}a^{4}-\frac{337887214349}{14426457210}a^{3}+\frac{432216241939}{14426457210}a^{2}-\frac{70596746896}{2404409535}a+\frac{300899827}{160293969}$, $\frac{1141363673}{508062188700}a^{17}-\frac{1421571692}{127015547175}a^{16}+\frac{1131405547}{84677031450}a^{15}+\frac{8715461201}{254031094350}a^{14}-\frac{1657920037}{6195880350}a^{13}+\frac{2029634543}{3848955975}a^{12}-\frac{10152321299}{16935406290}a^{11}+\frac{5879862091}{84677031450}a^{10}+\frac{192189211573}{42338515725}a^{9}-\frac{3763711476599}{254031094350}a^{8}+\frac{8212139169611}{254031094350}a^{7}-\frac{1396211801414}{42338515725}a^{6}-\frac{2302019549429}{508062188700}a^{5}+\frac{552640160657}{23093735850}a^{4}-\frac{1228916749969}{16935406290}a^{3}+\frac{1156567871}{209079090}a^{2}+\frac{13159591138}{940855905}a-\frac{1830446401}{62723727}$, $\frac{658780081}{973785861675}a^{17}+\frac{1688221147}{1947571723350}a^{16}+\frac{4767625969}{1298381148900}a^{15}+\frac{82201550591}{3895143446700}a^{14}+\frac{171301841}{9500349870}a^{13}+\frac{23194196461}{259676229780}a^{12}+\frac{157362770249}{1298381148900}a^{11}+\frac{27584210272}{324595287225}a^{10}+\frac{1013088567899}{1298381148900}a^{9}-\frac{2427789497561}{3895143446700}a^{8}+\frac{2193435233221}{1947571723350}a^{7}+\frac{1172938663997}{1298381148900}a^{6}-\frac{299007161513}{779028689340}a^{5}+\frac{5580528868157}{1947571723350}a^{4}+\frac{100649058806}{64919057445}a^{3}+\frac{38909789}{106862646}a^{2}+\frac{10497625928}{7213228605}a+\frac{252279727}{480881907}$ 594575519/5842715170050a^17 - 1249179323/5842715170050a^16 + 220230883/973785861675a^15 + 5259927938/2921357585025a^14 - 980282453/142505248050a^13 + 9374560802/973785861675a^12 - 11453390686/973785861675a^11 - 1775196619/77902868934a^10 + 127441353656/973785861675a^9 - 1089286605929/2921357585025a^8 + 3795161084023/5842715170050a^7 - 289468699594/973785861675a^6 - 136212140683/531155924550a^5 + 3400394390683/2921357585025a^4 - 111628112953/194757172335a^3 - 1239912524/7213228605a^2 + 25592614559/21639685815a + 346058707/1442645721, 858675137/3895143446700a^17 - 1917473791/1947571723350a^16 + 1209721223/649190574450a^15 + 3092407229/1947571723350a^14 - 524510504/23750874675a^13 + 4227450481/72132286050a^12 - 2651554913/25967622978a^11 + 26023510132/324595287225a^10 + 188519723149/649190574450a^9 - 2767131695411/1947571723350a^8 + 3649160577862/973785861675a^7 - 3692021648087/649190574450a^6 + 1805587904549/354103949700a^5 - 4419750255947/1947571723350a^4 - 223271407934/64919057445a^3 + 21762024497/4808819070a^2 - 26551093583/7213228605a + 565280576/480881907, 24343781/88525987425a^17 - 1498336547/1947571723350a^16 + 1156352357/1298381148900a^15 + 18168819431/3895143446700a^14 - 46049648/2159170425a^13 + 5010107339/144264572100a^12 - 2213248391/51935245956a^11 - 19947426437/649190574450a^10 + 458717832601/1298381148900a^9 - 4611052161029/3895143446700a^8 + 2246660742779/973785861675a^7 - 2290313365493/1298381148900a^6 + 926734386623/3895143446700a^5 + 893782384933/973785861675a^4 - 110280131789/129838114890a^3 + 1494253613/1602939690a^2 - 23376491/71418105a + 219299092/480881907, 565607869/2337086068020a^17 + 253793647/11685430340100a^16 - 422225443/973785861675a^15 + 2379573739/467417213604a^14 - 1965707027/285010496100a^13 - 9712992626/973785861675a^12 + 39755078131/3895143446700a^11 - 414758453797/3895143446700a^10 + 34725623911/177051974850a^9 - 2913514572673/11685430340100a^8 - 2214107149931/11685430340100a^7 + 443298764849/194757172335a^6 - 7358177058721/5842715170050a^5 + 5135424247937/2921357585025a^4 + 149785112815/77902868934a^3 - 2709682943/655748055a^2 + 61427732176/21639685815a + 6512795222/1442645721, 985869743/3895143446700a^17 - 170324471/973785861675a^16 - 7138457/51935245956a^15 + 19140251377/3895143446700a^14 - 477942469/47501749350a^13 + 35702951/144264572100a^12 - 5922806773/1298381148900a^11 - 24188603198/324595287225a^10 + 287103484901/1298381148900a^9 - 1855231697959/3895143446700a^8 + 174388366831/389514344670a^7 + 1461303193379/1298381148900a^6 - 378386519237/973785861675a^5 + 105742114039/194757172335a^4 + 58665170824/64919057445a^3 - 126285659/801469845a^2 + 155566814/1442645721a + 97777802/480881907, 348457849/259676229780a^17 - 3838812961/649190574450a^16 + 1255712243/108198429075a^15 + 1121842423/129838114890a^14 - 1048007702/7916958225a^13 + 25900057919/72132286050a^12 - 140258331503/216396858150a^11 + 56471469443/108198429075a^10 + 123241058183/72132286050a^9 - 5590347378761/649190574450a^8 + 7531123885864/324595287225a^7 - 1552274019877/43279371630a^6 + 44107394470507/1298381148900a^5 - 9645339727589/649190574450a^4 - 337887214349/14426457210a^3 + 432216241939/14426457210a^2 - 70596746896/2404409535a + 300899827/160293969, 1141363673/508062188700a^17 - 1421571692/127015547175a^16 + 1131405547/84677031450a^15 + 8715461201/254031094350a^14 - 1657920037/6195880350a^13 + 2029634543/3848955975a^12 - 10152321299/16935406290a^11 + 5879862091/84677031450a^10 + 192189211573/42338515725a^9 - 3763711476599/254031094350a^8 + 8212139169611/254031094350a^7 - 1396211801414/42338515725a^6 - 2302019549429/508062188700a^5 + 552640160657/23093735850a^4 - 1228916749969/16935406290a^3 + 1156567871/209079090a^2 + 13159591138/940855905a - 1830446401/62723727, 658780081/973785861675a^17 + 1688221147/1947571723350a^16 + 4767625969/1298381148900a^15 + 82201550591/3895143446700a^14 + 171301841/9500349870a^13 + 23194196461/259676229780a^12 + 157362770249/1298381148900a^11 + 27584210272/324595287225a^10 + 1013088567899/1298381148900a^9 - 2427789497561/3895143446700a^8 + 2193435233221/1947571723350a^7 + 1172938663997/1298381148900a^6 - 299007161513/779028689340a^5 + 5580528868157/1947571723350a^4 + 100649058806/64919057445a^3 + 38909789/106862646a^2 + 10497625928/7213228605a + 252279727/480881907 (assuming GRH)	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:	$ 430259533.331 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 430259533.331 \cdot 3}{6\cdot\sqrt{61773685738999443000000000000}}\cr\approx \mathstrut & 13.2104485571 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 + 14*x^15 - 81*x^14 + 169*x^13 - 258*x^12 - 27*x^11 + 1341*x^10 - 4988*x^9 + 10815*x^8 - 11791*x^7 + 6239*x^6 + 9090*x^5 - 16910*x^4 + 16140*x^3 + 2700*x^2 - 5400*x + 8100)

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))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^18 - 3*x^17 + 5*x^16 + 14*x^15 - 81*x^14 + 169*x^13 - 258*x^12 - 27*x^11 + 1341*x^10 - 4988*x^9 + 10815*x^8 - 11791*x^7 + 6239*x^6 + 9090*x^5 - 16910*x^4 + 16140*x^3 + 2700*x^2 - 5400*x + 8100, 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)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^18 - 3*x^17 + 5*x^16 + 14*x^15 - 81*x^14 + 169*x^13 - 258*x^12 - 27*x^11 + 1341*x^10 - 4988*x^9 + 10815*x^8 - 11791*x^7 + 6239*x^6 + 9090*x^5 - 16910*x^4 + 16140*x^3 + 2700*x^2 - 5400*x + 8100);

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)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 + 5*x^16 + 14*x^15 - 81*x^14 + 169*x^13 - 258*x^12 - 27*x^11 + 1341*x^10 - 4988*x^9 + 10815*x^8 - 11791*x^7 + 6239*x^6 + 9090*x^5 - 16910*x^4 + 16140*x^3 + 2700*x^2 - 5400*x + 8100);

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

$C_3:S_3$ (as 18T4):

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 18

The 6 conjugacy class representatives for $C_3^2 : C_2$

Character table for $C_3^2 : C_2$

Intermediate fields

$\Q(\sqrt{-3}) $, 3.1.300.1 x3, 3.1.36300.1 x3, 3.1.1452.1 x3, 3.1.9075.1 x3, 6.0.270000.1, 6.0.3953070000.1, 6.0.6324912.1, 6.0.247066875.1, 9.1.143496441000000.1 x9

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

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 9 sibling:	9.1.143496441000000.1
Minimal sibling:	9.1.143496441000000.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	R	${\href{/padicField/7.3.0.1}{3} }^{6}$	R	${\href{/padicField/13.3.0.1}{3} }^{6}$	${\href{/padicField/17.2.0.1}{2} }^{9}$	${\href{/padicField/19.3.0.1}{3} }^{6}$	${\href{/padicField/23.2.0.1}{2} }^{9}$	${\href{/padicField/29.2.0.1}{2} }^{9}$	${\href{/padicField/31.3.0.1}{3} }^{6}$	${\href{/padicField/37.3.0.1}{3} }^{6}$	${\href{/padicField/41.2.0.1}{2} }^{9}$	${\href{/padicField/43.3.0.1}{3} }^{6}$	${\href{/padicField/47.2.0.1}{2} }^{9}$	${\href{/padicField/53.2.0.1}{2} }^{9}$	${\href{/padicField/59.2.0.1}{2} }^{9}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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	2.6.4.1	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	2.6.4.1	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	2.6.4.1	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
$3$ 3	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$5$ 5	5.6.4.1	$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	5.6.4.1	$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	5.6.4.1	$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
$11$ 11	11.6.4.1	$x^{6} + 21 x^{5} + 153 x^{4} + 449 x^{3} + 537 x^{2} + 1569 x + 3440$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	11.6.4.1	$x^{6} + 21 x^{5} + 153 x^{4} + 449 x^{3} + 537 x^{2} + 1569 x + 3440$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	11.6.4.1	$x^{6} + 21 x^{5} + 153 x^{4} + 449 x^{3} + 537 x^{2} + 1569 x + 3440$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$

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