Normalized defining polynomial
\( x^{30} - 60 x^{28} + 1530 x^{26} - 15 x^{25} - 21700 x^{24} + 585 x^{23} + 188175 x^{22} - 8970 x^{21} - 1033569 x^{20} + 69450 x^{19} + 3609230 x^{18} - 289800 x^{17} - 7868280 x^{16} + 644651 x^{15} + 10355625 x^{14} - 712725 x^{13} - 7965350 x^{12} + 365580 x^{11} + 3575034 x^{10} - 62875 x^{9} - 924615 x^{8} - 12450 x^{7} + 131005 x^{6} + 5823 x^{5} - 9000 x^{4} - 620 x^{3} + 225 x^{2} + 15 x - 1 \)
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: | \(5399088047333990303844331037907977588474750518798828125=3^{40}\cdot 5^{51}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $66.74$ | 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: | $3, 5$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(225=3^{2}\cdot 5^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{225}(64,·)$, $\chi_{225}(1,·)$, $\chi_{225}(4,·)$, $\chi_{225}(199,·)$, $\chi_{225}(136,·)$, $\chi_{225}(139,·)$, $\chi_{225}(76,·)$, $\chi_{225}(79,·)$, $\chi_{225}(16,·)$, $\chi_{225}(19,·)$, $\chi_{225}(214,·)$, $\chi_{225}(151,·)$, $\chi_{225}(196,·)$, $\chi_{225}(154,·)$, $\chi_{225}(91,·)$, $\chi_{225}(94,·)$, $\chi_{225}(31,·)$, $\chi_{225}(34,·)$, $\chi_{225}(166,·)$, $\chi_{225}(169,·)$, $\chi_{225}(106,·)$, $\chi_{225}(109,·)$, $\chi_{225}(46,·)$, $\chi_{225}(49,·)$, $\chi_{225}(211,·)$, $\chi_{225}(181,·)$, $\chi_{225}(184,·)$, $\chi_{225}(121,·)$, $\chi_{225}(124,·)$, $\chi_{225}(61,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{199} a^{28} - \frac{23}{199} a^{27} + \frac{49}{199} a^{26} - \frac{24}{199} a^{25} + \frac{9}{199} a^{24} - \frac{92}{199} a^{23} - \frac{81}{199} a^{22} + \frac{94}{199} a^{21} - \frac{61}{199} a^{20} - \frac{83}{199} a^{19} - \frac{95}{199} a^{18} + \frac{30}{199} a^{17} - \frac{26}{199} a^{16} + \frac{81}{199} a^{15} + \frac{83}{199} a^{14} - \frac{19}{199} a^{13} + \frac{67}{199} a^{12} - \frac{35}{199} a^{11} - \frac{49}{199} a^{10} - \frac{76}{199} a^{9} + \frac{39}{199} a^{8} - \frac{12}{199} a^{7} - \frac{46}{199} a^{6} + \frac{16}{199} a^{5} - \frac{89}{199} a^{4} - \frac{44}{199} a^{3} - \frac{60}{199} a^{2} - \frac{63}{199} a + \frac{9}{199}$, $\frac{1}{64346547406834497057088827049183303114576151256014179599368851} a^{29} - \frac{157631624333296610204326908428284912821066737090161708088403}{64346547406834497057088827049183303114576151256014179599368851} a^{28} + \frac{7300220506428136333070321861324396816136141757134810243311301}{64346547406834497057088827049183303114576151256014179599368851} a^{27} - \frac{457766238696799395370243731449128986826064662116336744216045}{64346547406834497057088827049183303114576151256014179599368851} a^{26} + \frac{12077909239378769109453966809915580650033060938847849078961037}{64346547406834497057088827049183303114576151256014179599368851} a^{25} + \frac{13745918339847808093943349137985756739037335961299262309907354}{64346547406834497057088827049183303114576151256014179599368851} a^{24} - \frac{31464163701778586288464294771612016658659597680880744961717715}{64346547406834497057088827049183303114576151256014179599368851} a^{23} - \frac{19637672558132707359818664806009344292288032365272869638044800}{64346547406834497057088827049183303114576151256014179599368851} a^{22} + \frac{7070206615498945003084013192370854985961891118304794002243613}{64346547406834497057088827049183303114576151256014179599368851} a^{21} + \frac{13869306929560028236045490466437796361814814933550769631895791}{64346547406834497057088827049183303114576151256014179599368851} a^{20} - \frac{12484431312569871440091824519776031455509992503020915329005413}{64346547406834497057088827049183303114576151256014179599368851} a^{19} - \frac{14031691488337534291797383222674304785779881746212939697493460}{64346547406834497057088827049183303114576151256014179599368851} a^{18} + \frac{11555103587361275096494687986513191486729467473262361328872265}{64346547406834497057088827049183303114576151256014179599368851} a^{17} + \frac{5723733994115114062752462191132158560729149521023160123949254}{64346547406834497057088827049183303114576151256014179599368851} a^{16} - \frac{1117145732913986924704164182566867537336705084190938931876512}{64346547406834497057088827049183303114576151256014179599368851} a^{15} + \frac{8536277371766163667347131879333846232342477304408072581767539}{64346547406834497057088827049183303114576151256014179599368851} a^{14} - \frac{31812410905145264887852890957722144459270119090517008270792139}{64346547406834497057088827049183303114576151256014179599368851} a^{13} + \frac{6345661630096498044464671163178461691582309415968788173553770}{64346547406834497057088827049183303114576151256014179599368851} a^{12} + \frac{1288750365691119816438166627341220423961354458178059009934365}{64346547406834497057088827049183303114576151256014179599368851} a^{11} + \frac{13182625897398904706647880124013294788924640785111682168372242}{64346547406834497057088827049183303114576151256014179599368851} a^{10} - \frac{28164455929657357459767245783485671497450294388681074792808414}{64346547406834497057088827049183303114576151256014179599368851} a^{9} + \frac{22112972890924720712712428230894477590689480596963511632689669}{64346547406834497057088827049183303114576151256014179599368851} a^{8} + \frac{14438208509781358752171234104920952999977104854213503209011955}{64346547406834497057088827049183303114576151256014179599368851} a^{7} - \frac{13322075617988117381846711457458730482510120158575055270881841}{64346547406834497057088827049183303114576151256014179599368851} a^{6} - \frac{20668572974802042140153818910799354098080042679764133334760716}{64346547406834497057088827049183303114576151256014179599368851} a^{5} + \frac{15995672671511252251725617150528277588250523780145208120213542}{64346547406834497057088827049183303114576151256014179599368851} a^{4} - \frac{13160281912495116527487920385869300356966536304384576313878315}{64346547406834497057088827049183303114576151256014179599368851} a^{3} - \frac{17831109831022244537634810946251748140603692274776966215528841}{64346547406834497057088827049183303114576151256014179599368851} a^{2} + \frac{15974789462817112504515144121457517373263148446083495341671582}{64346547406834497057088827049183303114576151256014179599368851} a - \frac{11872765741106150594563129042490073176173169404672618596286617}{64346547406834497057088827049183303114576151256014179599368851}$
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: | \( 495122325212282700 \) (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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 5.5.390625.1, 6.6.820125.1, \(\Q(\zeta_{25})^+\), 15.15.207828545629978179931640625.1 |
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 | $30$ | R | R | ${\href{/LocalNumberField/7.6.0.1}{6} }^{5}$ | $15^{2}$ | $30$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{3}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{6}$ | $30$ | $15^{2}$ | $15^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{3}$ | $15^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{5}$ | $30$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{3}$ | $15^{2}$ |
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 | ||||||
| 5 | Data not computed | ||||||