Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 27*x^16 + 93*x^15 + 240*x^14 - 1056*x^13 - 675*x^12 + 5580*x^11 - 1083*x^10 - 15058*x^9 + 9870*x^8 + 20628*x^7 - 20619*x^6 - 12441*x^5 + 18096*x^4 + 1416*x^3 - 6102*x^2 + 654*x + 487)
gp: K = bnfinit(y^18 - 3*y^17 - 27*y^16 + 93*y^15 + 240*y^14 - 1056*y^13 - 675*y^12 + 5580*y^11 - 1083*y^10 - 15058*y^9 + 9870*y^8 + 20628*y^7 - 20619*y^6 - 12441*y^5 + 18096*y^4 + 1416*y^3 - 6102*y^2 + 654*y + 487, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 27*x^16 + 93*x^15 + 240*x^14 - 1056*x^13 - 675*x^12 + 5580*x^11 - 1083*x^10 - 15058*x^9 + 9870*x^8 + 20628*x^7 - 20619*x^6 - 12441*x^5 + 18096*x^4 + 1416*x^3 - 6102*x^2 + 654*x + 487);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 27*x^16 + 93*x^15 + 240*x^14 - 1056*x^13 - 675*x^12 + 5580*x^11 - 1083*x^10 - 15058*x^9 + 9870*x^8 + 20628*x^7 - 20619*x^6 - 12441*x^5 + 18096*x^4 + 1416*x^3 - 6102*x^2 + 654*x + 487)
\( x^{18} - 3 x^{17} - 27 x^{16} + 93 x^{15} + 240 x^{14} - 1056 x^{13} - 675 x^{12} + 5580 x^{11} + \cdots + 487 \)
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: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(102094231502838405721411584\)
\(\medspace = 2^{12}\cdot 3^{31}\cdot 7^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.86\) | 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: | $2^{2/3}3^{31/18}7^{1/2}\approx 27.857652161214947$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{21}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{445}a^{16}-\frac{37}{89}a^{15}+\frac{2}{445}a^{14}-\frac{11}{445}a^{13}-\frac{14}{89}a^{12}-\frac{15}{89}a^{11}+\frac{33}{89}a^{9}+\frac{37}{445}a^{8}+\frac{2}{5}a^{7}+\frac{117}{445}a^{6}+\frac{46}{445}a^{5}-\frac{38}{445}a^{4}+\frac{39}{445}a^{3}+\frac{191}{445}a^{2}-\frac{22}{89}a+\frac{47}{445}$, $\frac{1}{713403238525}a^{17}-\frac{593223989}{713403238525}a^{16}-\frac{8233897298}{713403238525}a^{15}-\frac{158042164929}{713403238525}a^{14}+\frac{223171034209}{713403238525}a^{13}+\frac{6626973894}{142680647705}a^{12}+\frac{775282589}{1603153345}a^{11}+\frac{10861242508}{28536129541}a^{10}+\frac{32226621467}{713403238525}a^{9}-\frac{91136511}{1603153345}a^{8}+\frac{41814917763}{142680647705}a^{7}-\frac{98973818062}{713403238525}a^{6}-\frac{84676301362}{713403238525}a^{5}+\frac{49926545241}{713403238525}a^{4}-\frac{18638122556}{142680647705}a^{3}-\frac{239664829004}{713403238525}a^{2}+\frac{143097891767}{713403238525}a-\frac{29880597}{8015766725}$
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
Trivial group, which has order $1$ (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: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{6274356038}{2632484275}a^{17}-\frac{9770030607}{2632484275}a^{16}-\frac{183527939924}{2632484275}a^{15}+\frac{318639949773}{2632484275}a^{14}+\frac{1966301340317}{2632484275}a^{13}-\frac{757367299893}{526496855}a^{12}-\frac{1941481138377}{526496855}a^{11}+\frac{839522513200}{105299371}a^{10}+\frac{23538622783796}{2632484275}a^{9}-\frac{12089667344772}{526496855}a^{8}-\frac{5092278066386}{526496855}a^{7}+\frac{92556364170169}{2632484275}a^{6}+\frac{4492282039094}{2632484275}a^{5}-\frac{71468523320592}{2632484275}a^{4}+\frac{2022233037252}{526496855}a^{3}+\frac{23470591660123}{2632484275}a^{2}-\frac{4288924676829}{2632484275}a-\frac{2107844287754}{2632484275}$, $\frac{1279189638}{526496855}a^{17}-\frac{1828060188}{526496855}a^{16}-\frac{37384689714}{526496855}a^{15}+\frac{60244085726}{526496855}a^{14}+\frac{400888007688}{526496855}a^{13}-\frac{144223867759}{105299371}a^{12}-\frac{397518994734}{105299371}a^{11}+\frac{802863169634}{105299371}a^{10}+\frac{4871870906086}{526496855}a^{9}-\frac{11593465893472}{526496855}a^{8}-\frac{5439500570138}{526496855}a^{7}+\frac{17778716397852}{526496855}a^{6}+\frac{1313640780228}{526496855}a^{5}-\frac{13732498499149}{526496855}a^{4}+\frac{1775900155366}{526496855}a^{3}+\frac{4502223541592}{526496855}a^{2}-\frac{805774056274}{526496855}a-\frac{401526587131}{526496855}$, $\frac{5624645173}{2632484275}a^{17}-\frac{10318140592}{2632484275}a^{16}-\frac{164380905629}{2632484275}a^{15}+\frac{331397616543}{2632484275}a^{14}+\frac{19667326868}{29578475}a^{13}-\frac{779862712223}{526496855}a^{12}-\frac{1699393932882}{526496855}a^{11}+\frac{860220130903}{105299371}a^{10}+\frac{19798265803141}{2632484275}a^{9}-\frac{2473322906717}{105299371}a^{8}-\frac{3777476855408}{526496855}a^{7}+\frac{94763113557434}{2632484275}a^{6}-\frac{1926539166221}{2632484275}a^{5}-\frac{73477532287547}{2632484275}a^{4}+\frac{2654346483431}{526496855}a^{3}+\frac{24390253922813}{2632484275}a^{2}-\frac{4880151710634}{2632484275}a-\frac{2257383217099}{2632484275}$, $\frac{123508291012}{142680647705}a^{17}-\frac{204107035406}{142680647705}a^{16}-\frac{3606509922501}{142680647705}a^{15}+\frac{6625684124176}{142680647705}a^{14}+\frac{38477325567761}{142680647705}a^{13}-\frac{15707782189495}{28536129541}a^{12}-\frac{37642663728734}{28536129541}a^{11}+\frac{87023193307999}{28536129541}a^{10}+\frac{447472968214529}{142680647705}a^{9}-\frac{12\!\cdots\!31}{142680647705}a^{8}-\frac{457164670377289}{142680647705}a^{7}+\frac{385488638858122}{28536129541}a^{6}+\frac{25457927962418}{142680647705}a^{5}-\frac{14\!\cdots\!54}{142680647705}a^{4}+\frac{242037775916388}{142680647705}a^{3}+\frac{496194588441669}{142680647705}a^{2}-\frac{94864907357251}{142680647705}a-\frac{45502392834212}{142680647705}$, $\frac{4174867958623}{713403238525}a^{17}-\frac{6616627550967}{713403238525}a^{16}-\frac{122107475223329}{713403238525}a^{15}+\frac{215498900194843}{713403238525}a^{14}+\frac{13\!\cdots\!02}{713403238525}a^{13}-\frac{511877776185338}{142680647705}a^{12}-\frac{12\!\cdots\!72}{142680647705}a^{11}+\frac{567436033842068}{28536129541}a^{10}+\frac{15\!\cdots\!16}{713403238525}a^{9}-\frac{16\!\cdots\!78}{28536129541}a^{8}-\frac{33\!\cdots\!38}{142680647705}a^{7}+\frac{62\!\cdots\!59}{713403238525}a^{6}+\frac{25\!\cdots\!79}{713403238525}a^{5}-\frac{48\!\cdots\!97}{713403238525}a^{4}+\frac{14\!\cdots\!61}{142680647705}a^{3}+\frac{15\!\cdots\!13}{713403238525}a^{2}-\frac{29\!\cdots\!09}{713403238525}a-\frac{14\!\cdots\!99}{713403238525}$, $\frac{370401531602}{142680647705}a^{17}-\frac{615986353264}{142680647705}a^{16}-\frac{10832540745626}{142680647705}a^{15}+\frac{3993079275608}{28536129541}a^{14}+\frac{115825010301914}{142680647705}a^{13}-\frac{47271376129583}{28536129541}a^{12}-\frac{113712831335863}{28536129541}a^{11}+\frac{261553331513526}{28536129541}a^{10}+\frac{13\!\cdots\!24}{142680647705}a^{9}-\frac{37\!\cdots\!62}{142680647705}a^{8}-\frac{14\!\cdots\!48}{142680647705}a^{7}+\frac{57\!\cdots\!79}{142680647705}a^{6}+\frac{25643430957938}{28536129541}a^{5}-\frac{892505990887868}{28536129541}a^{4}+\frac{695908366183411}{142680647705}a^{3}+\frac{14\!\cdots\!56}{142680647705}a^{2}-\frac{278257174400771}{142680647705}a-\frac{133865778416328}{142680647705}$, $\frac{1900571170118}{713403238525}a^{17}-\frac{2740225968722}{713403238525}a^{16}-\frac{55647150195839}{713403238525}a^{15}+\frac{90042770072263}{713403238525}a^{14}+\frac{598309695405582}{713403238525}a^{13}-\frac{215003497937633}{142680647705}a^{12}-\frac{595891919654372}{142680647705}a^{11}+\frac{238757248737440}{28536129541}a^{10}+\frac{73\!\cdots\!31}{713403238525}a^{9}-\frac{687582831092447}{28536129541}a^{8}-\frac{16\!\cdots\!38}{142680647705}a^{7}+\frac{26\!\cdots\!44}{713403238525}a^{6}+\frac{23\!\cdots\!89}{713403238525}a^{5}-\frac{20\!\cdots\!77}{713403238525}a^{4}+\frac{476233268568756}{142680647705}a^{3}+\frac{65\!\cdots\!83}{713403238525}a^{2}-\frac{11\!\cdots\!69}{713403238525}a-\frac{581555156414934}{713403238525}$, $\frac{2193956835048}{713403238525}a^{17}-\frac{3535984051452}{713403238525}a^{16}-\frac{64183872200029}{713403238525}a^{15}+\frac{114953602865298}{713403238525}a^{14}+\frac{687242085171037}{713403238525}a^{13}-\frac{272711927847743}{142680647705}a^{12}-\frac{677148259769417}{142680647705}a^{11}+\frac{302098556192601}{28536129541}a^{10}+\frac{81\!\cdots\!91}{713403238525}a^{9}-\frac{43\!\cdots\!04}{142680647705}a^{8}-\frac{17\!\cdots\!89}{142680647705}a^{7}+\frac{33\!\cdots\!64}{713403238525}a^{6}+\frac{12\!\cdots\!19}{713403238525}a^{5}-\frac{25\!\cdots\!92}{713403238525}a^{4}+\frac{762103904642893}{142680647705}a^{3}+\frac{84\!\cdots\!78}{713403238525}a^{2}-\frac{15\!\cdots\!34}{713403238525}a-\frac{764564735395894}{713403238525}$, $\frac{3467387416369}{713403238525}a^{17}-\frac{4933142543886}{713403238525}a^{16}-\frac{101355531369987}{713403238525}a^{15}+\frac{162627173732009}{713403238525}a^{14}+\frac{10\!\cdots\!91}{713403238525}a^{13}-\frac{389378959778104}{142680647705}a^{12}-\frac{10\!\cdots\!56}{142680647705}a^{11}+\frac{433476228895143}{28536129541}a^{10}+\frac{13\!\cdots\!48}{713403238525}a^{9}-\frac{62\!\cdots\!94}{142680647705}a^{8}-\frac{597094677347183}{28536129541}a^{7}+\frac{47\!\cdots\!82}{713403238525}a^{6}+\frac{38\!\cdots\!52}{713403238525}a^{5}-\frac{36\!\cdots\!11}{713403238525}a^{4}+\frac{183579510915048}{28536129541}a^{3}+\frac{12\!\cdots\!54}{713403238525}a^{2}-\frac{21\!\cdots\!27}{713403238525}a-\frac{10\!\cdots\!67}{713403238525}$, $\frac{1398485346058}{713403238525}a^{17}-\frac{1829031541292}{713403238525}a^{16}-\frac{40927949804909}{713403238525}a^{15}+\frac{60797381318983}{713403238525}a^{14}+\frac{440758355195402}{713403238525}a^{13}-\frac{146247575465598}{142680647705}a^{12}-\frac{441482762141247}{142680647705}a^{11}+\frac{163032616060307}{28536129541}a^{10}+\frac{55\!\cdots\!61}{713403238525}a^{9}-\frac{23\!\cdots\!84}{142680647705}a^{8}-\frac{13\!\cdots\!54}{142680647705}a^{7}+\frac{17\!\cdots\!94}{713403238525}a^{6}+\frac{23\!\cdots\!74}{713403238525}a^{5}-\frac{13\!\cdots\!82}{713403238525}a^{4}+\frac{256242130682423}{142680647705}a^{3}+\frac{45\!\cdots\!63}{713403238525}a^{2}-\frac{715151512876314}{713403238525}a-\frac{398251099014724}{713403238525}$, $\frac{14838048571783}{713403238525}a^{17}-\frac{23088407794237}{713403238525}a^{16}-\frac{434014800996709}{713403238525}a^{15}+\frac{753255766383593}{713403238525}a^{14}+\frac{46\!\cdots\!47}{713403238525}a^{13}-\frac{17\!\cdots\!23}{142680647705}a^{12}-\frac{45\!\cdots\!02}{142680647705}a^{11}+\frac{19\!\cdots\!39}{28536129541}a^{10}+\frac{55\!\cdots\!36}{713403238525}a^{9}-\frac{28\!\cdots\!57}{142680647705}a^{8}-\frac{12\!\cdots\!76}{142680647705}a^{7}+\frac{21\!\cdots\!29}{713403238525}a^{6}+\frac{10\!\cdots\!04}{713403238525}a^{5}-\frac{16\!\cdots\!97}{713403238525}a^{4}+\frac{47\!\cdots\!37}{142680647705}a^{3}+\frac{55\!\cdots\!68}{713403238525}a^{2}-\frac{10\!\cdots\!39}{713403238525}a-\frac{49\!\cdots\!14}{713403238525}$, $\frac{4373634111589}{713403238525}a^{17}-\frac{7115147429071}{713403238525}a^{16}-\frac{127891403865147}{713403238525}a^{15}+\frac{231095320745394}{713403238525}a^{14}+\frac{13\!\cdots\!76}{713403238525}a^{13}-\frac{547861862635894}{142680647705}a^{12}-\frac{13\!\cdots\!26}{142680647705}a^{11}+\frac{606598661360138}{28536129541}a^{10}+\frac{16\!\cdots\!13}{713403238525}a^{9}-\frac{87\!\cdots\!51}{142680647705}a^{8}-\frac{33\!\cdots\!88}{142680647705}a^{7}+\frac{66\!\cdots\!82}{713403238525}a^{6}+\frac{19\!\cdots\!82}{713403238525}a^{5}-\frac{51\!\cdots\!26}{713403238525}a^{4}+\frac{15\!\cdots\!86}{142680647705}a^{3}+\frac{17\!\cdots\!19}{713403238525}a^{2}-\frac{32\!\cdots\!87}{713403238525}a-\frac{15\!\cdots\!37}{713403238525}$, $\frac{221467189732}{142680647705}a^{17}-\frac{53821509577}{28536129541}a^{16}-\frac{6455075964506}{142680647705}a^{15}+\frac{9068386777773}{142680647705}a^{14}+\frac{13840694967776}{28536129541}a^{13}-\frac{22030442722747}{28536129541}a^{12}-\frac{68954782500942}{28536129541}a^{11}+\frac{123668625376791}{28536129541}a^{10}+\frac{858061974275174}{142680647705}a^{9}-\frac{17\!\cdots\!49}{142680647705}a^{8}-\frac{10\!\cdots\!71}{142680647705}a^{7}+\frac{27\!\cdots\!17}{142680647705}a^{6}+\frac{341363305373154}{142680647705}a^{5}-\frac{21\!\cdots\!52}{142680647705}a^{4}+\frac{217005466690202}{142680647705}a^{3}+\frac{138478301513593}{28536129541}a^{2}-\frac{115625734464786}{142680647705}a-\frac{12052861526783}{28536129541}$, $\frac{4574445853777}{713403238525}a^{17}-\frac{7210686605058}{713403238525}a^{16}-\frac{133746314542571}{713403238525}a^{15}+\frac{235012841193532}{713403238525}a^{14}+\frac{14\!\cdots\!98}{713403238525}a^{13}-\frac{558490478928272}{142680647705}a^{12}-\frac{14\!\cdots\!28}{142680647705}a^{11}+\frac{619272828725406}{28536129541}a^{10}+\frac{191375130918456}{8015766725}a^{9}-\frac{17\!\cdots\!81}{28536129541}a^{8}-\frac{36\!\cdots\!92}{142680647705}a^{7}+\frac{68\!\cdots\!66}{713403238525}a^{6}+\frac{27\!\cdots\!96}{713403238525}a^{5}-\frac{52\!\cdots\!78}{713403238525}a^{4}+\frac{15\!\cdots\!09}{142680647705}a^{3}+\frac{17\!\cdots\!37}{713403238525}a^{2}-\frac{32\!\cdots\!66}{713403238525}a-\frac{15\!\cdots\!26}{713403238525}$, $\frac{3911242495909}{713403238525}a^{17}-\frac{5698517474881}{713403238525}a^{16}-\frac{114298720057132}{713403238525}a^{15}+\frac{187400006425554}{713403238525}a^{14}+\frac{12\!\cdots\!11}{713403238525}a^{13}-\frac{448020043649354}{142680647705}a^{12}-\frac{12\!\cdots\!41}{142680647705}a^{11}+\frac{498450272042511}{28536129541}a^{10}+\frac{14\!\cdots\!53}{713403238525}a^{9}-\frac{71\!\cdots\!33}{142680647705}a^{8}-\frac{32\!\cdots\!46}{142680647705}a^{7}+\frac{55\!\cdots\!32}{713403238525}a^{6}+\frac{37\!\cdots\!12}{713403238525}a^{5}-\frac{42\!\cdots\!91}{713403238525}a^{4}+\frac{11\!\cdots\!27}{142680647705}a^{3}+\frac{13\!\cdots\!59}{713403238525}a^{2}-\frac{24\!\cdots\!47}{713403238525}a-\frac{12\!\cdots\!82}{713403238525}$, $\frac{1488932462253}{713403238525}a^{17}-\frac{2549623066567}{713403238525}a^{16}-\frac{43523808910344}{713403238525}a^{15}+\frac{82416005900013}{713403238525}a^{14}+\frac{464590823572577}{713403238525}a^{13}-\frac{194796582262728}{142680647705}a^{12}-\frac{454201676453657}{142680647705}a^{11}+\frac{215406014645862}{28536129541}a^{10}+\frac{53\!\cdots\!51}{713403238525}a^{9}-\frac{31\!\cdots\!72}{142680647705}a^{8}-\frac{10\!\cdots\!91}{142680647705}a^{7}+\frac{23\!\cdots\!39}{713403238525}a^{6}+\frac{108326089945189}{713403238525}a^{5}-\frac{18\!\cdots\!02}{713403238525}a^{4}+\frac{618580839439092}{142680647705}a^{3}+\frac{61\!\cdots\!13}{713403238525}a^{2}-\frac{11\!\cdots\!74}{713403238525}a-\frac{563042618636549}{713403238525}$, $\frac{676518837776}{713403238525}a^{17}-\frac{1246909871939}{713403238525}a^{16}-\frac{19701018896523}{713403238525}a^{15}+\frac{40108120441746}{713403238525}a^{14}+\frac{208497550692684}{713403238525}a^{13}-\frac{94544369445621}{142680647705}a^{12}-\frac{199968883595934}{142680647705}a^{11}+\frac{104525495416045}{28536129541}a^{10}+\frac{22\!\cdots\!92}{713403238525}a^{9}-\frac{15\!\cdots\!09}{142680647705}a^{8}-\frac{389137883953222}{142680647705}a^{7}+\frac{11\!\cdots\!88}{713403238525}a^{6}-\frac{716994071974637}{713403238525}a^{5}-\frac{90\!\cdots\!59}{713403238525}a^{4}+\frac{388605551457319}{142680647705}a^{3}+\frac{30\!\cdots\!46}{713403238525}a^{2}-\frac{669806181986308}{713403238525}a-\frac{293659911646483}{713403238525}$
| 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: | \( 15149384.4329 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 15149384.4329 \cdot 1}{2\cdot\sqrt{102094231502838405721411584}}\cr\approx \mathstrut & 0.196518893662 \end{aligned}\] (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 - 27*x^16 + 93*x^15 + 240*x^14 - 1056*x^13 - 675*x^12 + 5580*x^11 - 1083*x^10 - 15058*x^9 + 9870*x^8 + 20628*x^7 - 20619*x^6 - 12441*x^5 + 18096*x^4 + 1416*x^3 - 6102*x^2 + 654*x + 487)
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 - 27*x^16 + 93*x^15 + 240*x^14 - 1056*x^13 - 675*x^12 + 5580*x^11 - 1083*x^10 - 15058*x^9 + 9870*x^8 + 20628*x^7 - 20619*x^6 - 12441*x^5 + 18096*x^4 + 1416*x^3 - 6102*x^2 + 654*x + 487, 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 - 27*x^16 + 93*x^15 + 240*x^14 - 1056*x^13 - 675*x^12 + 5580*x^11 - 1083*x^10 - 15058*x^9 + 9870*x^8 + 20628*x^7 - 20619*x^6 - 12441*x^5 + 18096*x^4 + 1416*x^3 - 6102*x^2 + 654*x + 487);
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 - 27*x^16 + 93*x^15 + 240*x^14 - 1056*x^13 - 675*x^12 + 5580*x^11 - 1083*x^10 - 15058*x^9 + 9870*x^8 + 20628*x^7 - 20619*x^6 - 12441*x^5 + 18096*x^4 + 1416*x^3 - 6102*x^2 + 654*x + 487);
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
$C_3\times S_3$ (as 18T3):
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 9 conjugacy class representatives for $S_3 \times C_3$ |
| Character table for $S_3 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{21}) \), 3.3.756.1 x3, \(\Q(\zeta_{9})^+\), 6.6.12002256.1, 6.6.108020304.1 x2, 6.6.6751269.1, 9.9.314987206464.1 x3 |
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 6 sibling: | 6.6.108020304.1 |
| Degree 9 sibling: | 9.9.314987206464.1 |
| Minimal sibling: | 6.6.108020304.1 |
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 | ${\href{/padicField/5.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{9}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(3\)
| Deg $18$ | $18$ | $1$ | $31$ | |||
|
\(7\)
| 7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |