Normalized defining polynomial
\( x^{30} - 63 x^{28} - 10 x^{27} + 1692 x^{26} + 468 x^{25} - 25382 x^{24} - 8784 x^{23} + 234885 x^{22} + 84622 x^{21} - 1403721 x^{20} - 442344 x^{19} + 5535342 x^{18} + 1180242 x^{17} - 14466885 x^{16} - 964952 x^{15} + 24638247 x^{14} - 2607870 x^{13} - 26043061 x^{12} + 7466562 x^{11} + 15397623 x^{10} - 7303108 x^{9} - 3940323 x^{8} + 2952402 x^{7} - 9685 x^{6} - 354570 x^{5} + 76932 x^{4} - 510 x^{3} - 948 x^{2} + 30 x + 1 \)
Invariants
| Degree: | $30$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[30, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(171140406014793353454699497635681610393336745825511931904=2^{30}\cdot 3^{40}\cdot 11^{27}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $74.89$ | 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: | $2, 3, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(396=2^{2}\cdot 3^{2}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{396}(1,·)$, $\chi_{396}(343,·)$, $\chi_{396}(259,·)$, $\chi_{396}(133,·)$, $\chi_{396}(211,·)$, $\chi_{396}(7,·)$, $\chi_{396}(265,·)$, $\chi_{396}(79,·)$, $\chi_{396}(139,·)$, $\chi_{396}(271,·)$, $\chi_{396}(19,·)$, $\chi_{396}(151,·)$, $\chi_{396}(25,·)$, $\chi_{396}(283,·)$, $\chi_{396}(157,·)$, $\chi_{396}(229,·)$, $\chi_{396}(289,·)$, $\chi_{396}(391,·)$, $\chi_{396}(37,·)$, $\chi_{396}(97,·)$, $\chi_{396}(169,·)$, $\chi_{396}(43,·)$, $\chi_{396}(301,·)$, $\chi_{396}(175,·)$, $\chi_{396}(49,·)$, $\chi_{396}(307,·)$, $\chi_{396}(181,·)$, $\chi_{396}(361,·)$, $\chi_{396}(313,·)$, $\chi_{396}(127,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{136040983291270722031800194693053} a^{28} + \frac{13352133249755354312976977208782}{136040983291270722031800194693053} a^{27} - \frac{50108357925279327923559295562259}{136040983291270722031800194693053} a^{26} + \frac{19198752679032616934700230166828}{136040983291270722031800194693053} a^{25} - \frac{25739901166300404357243242472498}{136040983291270722031800194693053} a^{24} - \frac{24571288619020782417138590447426}{136040983291270722031800194693053} a^{23} + \frac{47618199222085293992230259236683}{136040983291270722031800194693053} a^{22} - \frac{20957581947116126810757886584484}{136040983291270722031800194693053} a^{21} + \frac{49786741899146855546100896241168}{136040983291270722031800194693053} a^{20} - \frac{25094462659899835258060850715836}{136040983291270722031800194693053} a^{19} + \frac{10298656317885229839600073006319}{136040983291270722031800194693053} a^{18} - \frac{55212996305862984863372489595278}{136040983291270722031800194693053} a^{17} - \frac{17717223606670091402518305884295}{136040983291270722031800194693053} a^{16} + \frac{1708393726805872278930550514252}{136040983291270722031800194693053} a^{15} + \frac{2942141064213211080375236209450}{136040983291270722031800194693053} a^{14} - \frac{34956600131814288081453662433402}{136040983291270722031800194693053} a^{13} - \frac{51839858665836212886099844103013}{136040983291270722031800194693053} a^{12} + \frac{24741325740430457878751700031507}{136040983291270722031800194693053} a^{11} + \frac{50571231008827931123931137641677}{136040983291270722031800194693053} a^{10} - \frac{62396953174085277891893472423082}{136040983291270722031800194693053} a^{9} - \frac{47261834192324042920451520055222}{136040983291270722031800194693053} a^{8} + \frac{26690303965397336891346505556984}{136040983291270722031800194693053} a^{7} - \frac{15387800587357908241303431427464}{136040983291270722031800194693053} a^{6} + \frac{37459232411885402270777722202118}{136040983291270722031800194693053} a^{5} - \frac{65769981879383633224319449216769}{136040983291270722031800194693053} a^{4} - \frac{50426591751535071340975027481617}{136040983291270722031800194693053} a^{3} + \frac{42766542904884752777977290452281}{136040983291270722031800194693053} a^{2} - \frac{5142601089594776228952693950384}{136040983291270722031800194693053} a + \frac{61723574152456106378880316404489}{136040983291270722031800194693053}$, $\frac{1}{34943952092317670515686824733723249128207} a^{29} - \frac{51734382}{34943952092317670515686824733723249128207} a^{28} + \frac{3621223398112099929169272445072046111100}{34943952092317670515686824733723249128207} a^{27} + \frac{3915597821489966785035689176010198123023}{34943952092317670515686824733723249128207} a^{26} - \frac{17464155428623819346733176128132798719673}{34943952092317670515686824733723249128207} a^{25} - \frac{12674103775163226689548296987039935066802}{34943952092317670515686824733723249128207} a^{24} + \frac{15778085207068770157589568884181215077904}{34943952092317670515686824733723249128207} a^{23} - \frac{12336900658869162295174171846677944588352}{34943952092317670515686824733723249128207} a^{22} + \frac{13822765750017910442484775245159089668922}{34943952092317670515686824733723249128207} a^{21} + \frac{14656293433654688263989571509544149646750}{34943952092317670515686824733723249128207} a^{20} + \frac{1914605828853969890803361337616781627187}{34943952092317670515686824733723249128207} a^{19} + \frac{8429331444972431438205615674011840193673}{34943952092317670515686824733723249128207} a^{18} - \frac{14129267707182033716578446622130053170628}{34943952092317670515686824733723249128207} a^{17} + \frac{2162905039682439380392363912101581052133}{34943952092317670515686824733723249128207} a^{16} - \frac{5199090109014118503628329511294579063127}{34943952092317670515686824733723249128207} a^{15} + \frac{14725797908622500653898504664143565002696}{34943952092317670515686824733723249128207} a^{14} + \frac{1565052774005741233541921180662750376842}{34943952092317670515686824733723249128207} a^{13} + \frac{2975624315275058994813607211888107795728}{34943952092317670515686824733723249128207} a^{12} + \frac{1808370796272047101064289279234921332344}{34943952092317670515686824733723249128207} a^{11} + \frac{2269772376137347987412540356966516912705}{34943952092317670515686824733723249128207} a^{10} + \frac{6313879497574864880040807127583217189913}{34943952092317670515686824733723249128207} a^{9} - \frac{12468298349396690037150634234086549207099}{34943952092317670515686824733723249128207} a^{8} + \frac{14786513898480175951994500734208202459600}{34943952092317670515686824733723249128207} a^{7} + \frac{10116774160763847781845446585438854197849}{34943952092317670515686824733723249128207} a^{6} - \frac{258695316491024015363123731679581477319}{34943952092317670515686824733723249128207} a^{5} - \frac{5970561109449001736931566017123194335503}{34943952092317670515686824733723249128207} a^{4} - \frac{6588261910277552528950319729900680218496}{34943952092317670515686824733723249128207} a^{3} - \frac{15465177452357091412041871442741223272188}{34943952092317670515686824733723249128207} a^{2} + \frac{4619644062875623077106140695766859750287}{34943952092317670515686824733723249128207} a - \frac{6658876407634580482335986139840246889551}{34943952092317670515686824733723249128207}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $29$ | 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: | \( 3435292739576321000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 30 |
| The 30 conjugacy class representatives for $C_{30}$ |
| Character table for $C_{30}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{11}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\), 6.6.558892224.1, \(\Q(\zeta_{44})^+\), 15.15.10943023107606534329121.1 |
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 | R | R | $15^{2}$ | $15^{2}$ | R | $30$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{3}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{6}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ | $30$ | $30$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{6}$ | $30$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{10}$ | $30$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{6}$ | $30$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| 11 | Data not computed | ||||||