Normalized defining polynomial
\( x^{27} - 513 x^{25} + 116964 x^{23} - 15617943 x^{21} + 1354686795 x^{19} - 2334150 x^{18} - 80025043581 x^{17} + 798279300 x^{16} + 3282297025608 x^{15} - 113754800250 x^{14} - 93545465229828 x^{13} + 8741424428100 x^{12} + 1824136571981646 x^{11} - 391490936887050 x^{10} - 23530968709997016 x^{9} + 10299223108874700 x^{8} + 188924064347604654 x^{7} - 152199630386703900 x^{6} - 832868015607672111 x^{5} + 1126672588576899000 x^{4} + 1268424949947516279 x^{3} - 3211016877444162150 x^{2} + 1966965268494776013 x - 342892963174292929 \)
Invariants
| Degree: | $27$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[27, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(3463254175600113063839837232871102966170246194081956872095753304432278349849=3^{94}\cdot 19^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $627.71$ | 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: | $3, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1539=3^{4}\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1539}(1,·)$, $\chi_{1539}(514,·)$, $\chi_{1539}(1027,·)$, $\chi_{1539}(1156,·)$, $\chi_{1539}(328,·)$, $\chi_{1539}(841,·)$, $\chi_{1539}(1354,·)$, $\chi_{1539}(130,·)$, $\chi_{1539}(334,·)$, $\chi_{1539}(847,·)$, $\chi_{1539}(1360,·)$, $\chi_{1539}(643,·)$, $\chi_{1539}(283,·)$, $\chi_{1539}(796,·)$, $\chi_{1539}(1309,·)$, $\chi_{1539}(484,·)$, $\chi_{1539}(997,·)$, $\chi_{1539}(1510,·)$, $\chi_{1539}(235,·)$, $\chi_{1539}(748,·)$, $\chi_{1539}(1261,·)$, $\chi_{1539}(367,·)$, $\chi_{1539}(880,·)$, $\chi_{1539}(1393,·)$, $\chi_{1539}(61,·)$, $\chi_{1539}(574,·)$, $\chi_{1539}(1087,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9}$, $\frac{1}{19} a^{10}$, $\frac{1}{19} a^{11}$, $\frac{1}{19} a^{12}$, $\frac{1}{19} a^{13}$, $\frac{1}{361} a^{14} + \frac{4}{19} a^{5}$, $\frac{1}{361} a^{15} + \frac{4}{19} a^{6}$, $\frac{1}{361} a^{16} + \frac{4}{19} a^{7}$, $\frac{1}{361} a^{17} + \frac{4}{19} a^{8}$, $\frac{1}{41111661869099} a^{18} - \frac{18}{2163771677321} a^{16} + \frac{135}{113882719859} a^{14} - \frac{546}{5993827361} a^{12} + \frac{24453}{5993827361} a^{10} + \frac{40303992444}{2163771677321} a^{9} - \frac{643302}{5993827361} a^{8} - \frac{21087772419}{113882719859} a^{7} + \frac{9506574}{5993827361} a^{6} - \frac{2668783714}{5993827361} a^{5} - \frac{70373340}{5993827361} a^{4} + \frac{1064581633}{5993827361} a^{3} + \frac{200564019}{5993827361} a^{2} + \frac{525094789}{5993827361} a + \frac{2728138315}{5993827361}$, $\frac{1}{41111661869099} a^{19} - \frac{18}{2163771677321} a^{17} + \frac{135}{113882719859} a^{15} - \frac{546}{5993827361} a^{13} + \frac{24453}{5993827361} a^{11} + \frac{40303992444}{2163771677321} a^{10} - \frac{643302}{5993827361} a^{9} - \frac{21087772419}{113882719859} a^{8} + \frac{9506574}{5993827361} a^{7} - \frac{2668783714}{5993827361} a^{6} - \frac{70373340}{5993827361} a^{5} + \frac{1064581633}{5993827361} a^{4} + \frac{200564019}{5993827361} a^{3} + \frac{525094789}{5993827361} a^{2} + \frac{2728138315}{5993827361} a$, $\frac{1}{41111661869099} a^{20} - \frac{189}{113882719859} a^{16} + \frac{1884}{5993827361} a^{14} - \frac{162279}{5993827361} a^{12} + \frac{40303992444}{2163771677321} a^{11} + \frac{7719624}{5993827361} a^{10} - \frac{2887537025}{113882719859} a^{9} - \frac{210502710}{5993827361} a^{8} + \frac{1356263848}{5993827361} a^{7} - \frac{2812952393}{5993827361} a^{6} - \frac{597689683}{5993827361} a^{5} + \frac{108191183}{5993827361} a^{4} - \frac{1011455746}{5993827361} a^{3} - \frac{604895519}{5993827361} a^{2} - \frac{232402992}{5993827361} a - \frac{2013764586}{5993827361}$, $\frac{1}{781121575512881} a^{21} - \frac{189}{2163771677321} a^{17} + \frac{1884}{113882719859} a^{15} - \frac{8541}{5993827361} a^{13} - \frac{529109606851}{41111661869099} a^{12} + \frac{406296}{5993827361} a^{11} + \frac{21087772419}{2163771677321} a^{10} - \frac{11079090}{5993827361} a^{9} + \frac{25331573292}{113882719859} a^{8} + \frac{167414472}{5993827361} a^{7} + \frac{1545865638}{5993827361} a^{6} - \frac{1256164119}{5993827361} a^{5} + \frac{262230085}{5993827361} a^{4} + \frac{41351896008}{113882719859} a^{3} - \frac{2535948520}{5993827361} a^{2} + \frac{1155870782}{5993827361} a$, $\frac{1}{781121575512881} a^{22} - \frac{1518}{113882719859} a^{16} + \frac{16974}{5993827361} a^{14} - \frac{529109606851}{41111661869099} a^{13} - \frac{1554390}{5993827361} a^{12} + \frac{21087772419}{2163771677321} a^{11} + \frac{76731633}{5993827361} a^{10} + \frac{656259933}{113882719859} a^{9} - \frac{2142683010}{5993827361} a^{8} + \frac{1852073512}{5993827361} a^{7} + \frac{2912806310}{5993827361} a^{6} + \frac{789863350}{5993827361} a^{5} + \frac{22923515226}{113882719859} a^{4} + \frac{2308666626}{5993827361} a^{3} + \frac{2121979691}{5993827361} a^{2} - \frac{2440231416}{5993827361} a + \frac{2830781291}{5993827361}$, $\frac{1}{3147775481662640898171931} a^{23} + \frac{87697790}{165672393771717942009049} a^{22} - \frac{23}{165672393771717942009049} a^{21} - \frac{20491619}{8719599672195681158371} a^{20} + \frac{230}{8719599672195681158371} a^{19} - \frac{12934528}{8719599672195681158371} a^{18} - \frac{69}{24154015712453410411} a^{17} - \frac{2455561433847}{458926298536614797809} a^{16} + \frac{4692}{24154015712453410411} a^{15} + \frac{65759309008833034666}{165672393771717942009049} a^{14} + \frac{110388924004036585740}{8719599672195681158371} a^{13} - \frac{1044562332346073919}{8719599672195681158371} a^{12} - \frac{5496462771946338940}{458926298536614797809} a^{11} + \frac{3100351770803555565}{458926298536614797809} a^{10} - \frac{5873519979831131041}{458926298536614797809} a^{9} - \frac{5474508460716428676}{24154015712453410411} a^{8} - \frac{367554110477043287}{24154015712453410411} a^{7} + \frac{514219119938289680}{1271263984865968969} a^{6} + \frac{75454859751575949742}{458926298536614797809} a^{5} + \frac{6361892360028554437}{24154015712453410411} a^{4} - \frac{947188450893467144}{24154015712453410411} a^{3} - \frac{349641849818773016}{1271263984865968969} a^{2} + \frac{335429214490050677}{1271263984865968969} a + \frac{44696266518357941}{1271263984865968969}$, $\frac{1}{3147775481662640898171931} a^{24} - \frac{24}{165672393771717942009049} a^{22} - \frac{30506222}{165672393771717942009049} a^{21} + \frac{252}{8719599672195681158371} a^{20} + \frac{4344075}{8719599672195681158371} a^{19} - \frac{80}{24154015712453410411} a^{18} + \frac{5687482608}{458926298536614797809} a^{17} + \frac{306}{1271263984865968969} a^{16} + \frac{65569968463588640785}{165672393771717942009049} a^{15} - \frac{14688}{1271263984865968969} a^{14} - \frac{65701011858782905444}{8719599672195681158371} a^{13} - \frac{217320277962597042471}{8719599672195681158371} a^{12} + \frac{7748008879058686439}{458926298536614797809} a^{11} - \frac{2281235440886581625}{458926298536614797809} a^{10} - \frac{233899796920280984}{24154015712453410411} a^{9} - \frac{7179522872090244409}{24154015712453410411} a^{8} + \frac{619239399741432676}{1271263984865968969} a^{7} - \frac{112587106746722227464}{458926298536614797809} a^{6} - \frac{518543754472603135}{1271263984865968969} a^{5} - \frac{63876157019819061}{148184145475174297} a^{4} - \frac{11778625072954894396}{24154015712453410411} a^{3} + \frac{349874221963638881}{1271263984865968969} a^{2} + \frac{102141559690931254}{1271263984865968969} a + \frac{433900617209836251}{1271263984865968969}$, $\frac{1}{59807734151590177065266689} a^{25} + \frac{71489460}{165672393771717942009049} a^{22} - \frac{300}{165672393771717942009049} a^{21} - \frac{401627}{8719599672195681158371} a^{20} + \frac{4000}{8719599672195681158371} a^{19} + \frac{74665711}{8719599672195681158371} a^{18} - \frac{1350}{24154015712453410411} a^{17} - \frac{1311222678847770063884}{3147775481662640898171931} a^{16} + \frac{97920}{24154015712453410411} a^{15} + \frac{131365039465896476096}{165672393771717942009049} a^{14} + \frac{210530531448689819118}{8719599672195681158371} a^{13} - \frac{68649801775825634524}{8719599672195681158371} a^{12} - \frac{6344243139892923164}{458926298536614797809} a^{11} + \frac{6506876055613932711}{458926298536614797809} a^{10} - \frac{970247918935739642}{458926298536614797809} a^{9} - \frac{11820153948976467608}{24154015712453410411} a^{8} - \frac{1441460099083551410590}{8719599672195681158371} a^{7} - \frac{589299170720869420}{1271263984865968969} a^{6} - \frac{152954613897463483341}{458926298536614797809} a^{5} + \frac{10619953483239593311}{24154015712453410411} a^{4} + \frac{1907612625518671815}{24154015712453410411} a^{3} - \frac{475274540087932258}{1271263984865968969} a^{2} + \frac{631407789962680434}{1271263984865968969} a + \frac{541568696423560382}{1271263984865968969}$, $\frac{1}{59807734151590177065266689} a^{26} - \frac{325}{165672393771717942009049} a^{22} + \frac{55220344}{165672393771717942009049} a^{21} + \frac{4550}{8719599672195681158371} a^{20} + \frac{82439728}{8719599672195681158371} a^{19} - \frac{1625}{24154015712453410411} a^{18} - \frac{1311208618719578124065}{3147775481662640898171931} a^{17} + \frac{6630}{1271263984865968969} a^{16} + \frac{131207719257124742153}{165672393771717942009049} a^{15} - \frac{331500}{1271263984865968969} a^{14} + \frac{128554904163962071241}{8719599672195681158371} a^{13} - \frac{133172473916696103037}{8719599672195681158371} a^{12} + \frac{11060600635688150932}{458926298536614797809} a^{11} + \frac{6952262402908372138}{458926298536614797809} a^{10} + \frac{405118111112215013}{24154015712453410411} a^{9} - \frac{853245807159065883291}{8719599672195681158371} a^{8} + \frac{362457086238706347}{1271263984865968969} a^{7} - \frac{92137698161881771212}{458926298536614797809} a^{6} + \frac{630735285980060255}{1271263984865968969} a^{5} - \frac{1404056482717669248}{24154015712453410411} a^{4} - \frac{11265507024476546487}{24154015712453410411} a^{3} + \frac{386735344372249926}{1271263984865968969} a^{2} + \frac{341690607619877758}{1271263984865968969} a - \frac{593860433547814446}{1271263984865968969}$
Class group and class number
Not computed
Unit group
| Rank: | $26$ | 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: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 27 |
| The 27 conjugacy class representatives for $C_{27}$ |
| Character table for $C_{27}$ is not computed |
Intermediate fields
| \(\Q(\zeta_{9})^+\), 9.9.1476349596018920529.2 |
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 | $27$ | R | $27$ | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ | R | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ | $27$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| $19$ | 19.9.8.1 | $x^{9} + 76$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.1 | $x^{9} + 76$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |
| 19.9.8.1 | $x^{9} + 76$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |