Normalized defining polynomial
\( x^{30} - 6 x^{29} - 44 x^{28} + 314 x^{27} + 713 x^{26} - 6896 x^{25} - 4258 x^{24} + 83122 x^{23} - 15378 x^{22} - 603900 x^{21} + 392708 x^{20} + 2735166 x^{19} - 2606464 x^{18} - 7727244 x^{17} + 9080215 x^{16} + 13259720 x^{15} - 18225508 x^{14} - 12998698 x^{13} + 21142858 x^{12} + 6414536 x^{11} - 13700101 x^{10} - 1060038 x^{9} + 4698934 x^{8} - 218368 x^{7} - 766340 x^{6} + 83118 x^{5} + 46787 x^{4} - 4474 x^{3} - 905 x^{2} + 10 x + 1 \)
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: | \(14183235501467494512099454351289839373429638402920153088=2^{30}\cdot 7^{25}\cdot 11^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $68.93$ | 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: | $2, 7, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(308=2^{2}\cdot 7\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{308}(1,·)$, $\chi_{308}(3,·)$, $\chi_{308}(199,·)$, $\chi_{308}(9,·)$, $\chi_{308}(75,·)$, $\chi_{308}(141,·)$, $\chi_{308}(115,·)$, $\chi_{308}(81,·)$, $\chi_{308}(251,·)$, $\chi_{308}(279,·)$, $\chi_{308}(25,·)$, $\chi_{308}(27,·)$, $\chi_{308}(93,·)$, $\chi_{308}(159,·)$, $\chi_{308}(289,·)$, $\chi_{308}(59,·)$, $\chi_{308}(113,·)$, $\chi_{308}(37,·)$, $\chi_{308}(103,·)$, $\chi_{308}(169,·)$, $\chi_{308}(225,·)$, $\chi_{308}(223,·)$, $\chi_{308}(47,·)$, $\chi_{308}(177,·)$, $\chi_{308}(221,·)$, $\chi_{308}(243,·)$, $\chi_{308}(53,·)$, $\chi_{308}(137,·)$, $\chi_{308}(111,·)$, $\chi_{308}(31,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{197} a^{27} + \frac{71}{197} a^{26} + \frac{67}{197} a^{25} - \frac{73}{197} a^{24} + \frac{94}{197} a^{23} - \frac{61}{197} a^{22} - \frac{89}{197} a^{21} - \frac{42}{197} a^{20} + \frac{18}{197} a^{19} + \frac{35}{197} a^{18} + \frac{13}{197} a^{17} - \frac{22}{197} a^{16} - \frac{84}{197} a^{15} + \frac{17}{197} a^{14} - \frac{27}{197} a^{13} + \frac{25}{197} a^{12} - \frac{63}{197} a^{11} - \frac{97}{197} a^{10} - \frac{71}{197} a^{9} + \frac{10}{197} a^{8} - \frac{28}{197} a^{7} + \frac{59}{197} a^{6} - \frac{78}{197} a^{5} - \frac{30}{197} a^{4} - \frac{68}{197} a^{3} - \frac{55}{197} a^{2} + \frac{57}{197} a + \frac{93}{197}$, $\frac{1}{238368224352821958062241113} a^{28} - \frac{513777340110616765831277}{238368224352821958062241113} a^{27} - \frac{73392601090522130454332241}{238368224352821958062241113} a^{26} - \frac{110153016316348748779683221}{238368224352821958062241113} a^{25} + \frac{79719886821111903211554897}{238368224352821958062241113} a^{24} - \frac{118843790172415510941890427}{238368224352821958062241113} a^{23} + \frac{95706154367832664105590974}{238368224352821958062241113} a^{22} - \frac{1852341786170839255956623}{238368224352821958062241113} a^{21} + \frac{79354990434999904348909580}{238368224352821958062241113} a^{20} + \frac{25959303543530838172787192}{238368224352821958062241113} a^{19} + \frac{80233903930165873311754723}{238368224352821958062241113} a^{18} + \frac{1390335673990919981202556}{238368224352821958062241113} a^{17} - \frac{115346013888937592125889450}{238368224352821958062241113} a^{16} + \frac{69547373921789175983200773}{238368224352821958062241113} a^{15} - \frac{80841621145797135791816160}{238368224352821958062241113} a^{14} - \frac{58183963186670117654412281}{238368224352821958062241113} a^{13} - \frac{108763752835410882142002133}{238368224352821958062241113} a^{12} - \frac{92353527688824523843337723}{238368224352821958062241113} a^{11} - \frac{82792233151483435318584901}{238368224352821958062241113} a^{10} + \frac{85821659917761285019702004}{238368224352821958062241113} a^{9} + \frac{78934867393645843723292176}{238368224352821958062241113} a^{8} - \frac{76475616922736830736171206}{238368224352821958062241113} a^{7} - \frac{26615673126397315095816794}{238368224352821958062241113} a^{6} + \frac{22778305508235719488566030}{238368224352821958062241113} a^{5} - \frac{107602095721344325365996979}{238368224352821958062241113} a^{4} + \frac{76583763544559508751376473}{238368224352821958062241113} a^{3} - \frac{95066722741850253309300343}{238368224352821958062241113} a^{2} + \frac{89319494661156657883307119}{238368224352821958062241113} a + \frac{79725296472685974871298724}{238368224352821958062241113}$, $\frac{1}{174335163482737429810223452178474301907422929} a^{29} - \frac{234426044237519109}{174335163482737429810223452178474301907422929} a^{28} - \frac{392044954844394141477958868752024835554807}{174335163482737429810223452178474301907422929} a^{27} + \frac{44975763997796603098595548924485698760376620}{174335163482737429810223452178474301907422929} a^{26} + \frac{86366345915096784625552651067615503166626151}{174335163482737429810223452178474301907422929} a^{25} - \frac{83970156888204591543141738730988121055464059}{174335163482737429810223452178474301907422929} a^{24} - \frac{13650244857563177709685640573753999302670265}{174335163482737429810223452178474301907422929} a^{23} + \frac{61752361835187787055085440581417844615554648}{174335163482737429810223452178474301907422929} a^{22} - \frac{12041269513834826057096492189023135118265187}{174335163482737429810223452178474301907422929} a^{21} - \frac{34158773447944635675384489949504929070346155}{174335163482737429810223452178474301907422929} a^{20} + \frac{58320706126652817695410362852085555996805726}{174335163482737429810223452178474301907422929} a^{19} + \frac{83495983468904914759120265157283674093969961}{174335163482737429810223452178474301907422929} a^{18} - \frac{63453432188121291336216919685537727511670608}{174335163482737429810223452178474301907422929} a^{17} + \frac{74048437032421095281604234332007385477319155}{174335163482737429810223452178474301907422929} a^{16} + \frac{61180633214033737753252546422629046436505334}{174335163482737429810223452178474301907422929} a^{15} - \frac{83070286605249603497145453822079012764024112}{174335163482737429810223452178474301907422929} a^{14} - \frac{5373032531328921291560238775018985146009692}{174335163482737429810223452178474301907422929} a^{13} - \frac{71015190672761457111911152554242372933491248}{174335163482737429810223452178474301907422929} a^{12} - \frac{3098676597053700345448186697037031036651912}{174335163482737429810223452178474301907422929} a^{11} - \frac{36565411961440653858747060386856919221256325}{174335163482737429810223452178474301907422929} a^{10} + \frac{39667923802476031832274755505286714009630592}{174335163482737429810223452178474301907422929} a^{9} + \frac{30026554742780008655746459361856117823182053}{174335163482737429810223452178474301907422929} a^{8} + \frac{71618784060475477520880419774733872421423149}{174335163482737429810223452178474301907422929} a^{7} + \frac{66548892515945427169075986909767246966332571}{174335163482737429810223452178474301907422929} a^{6} + \frac{73526030848862090826634936277444215001238369}{174335163482737429810223452178474301907422929} a^{5} + \frac{68435254489686071733018526189444743889871110}{174335163482737429810223452178474301907422929} a^{4} + \frac{63053856347821548156000504000689060600174040}{174335163482737429810223452178474301907422929} a^{3} - \frac{71727880106168565642000891933107939740646139}{174335163482737429810223452178474301907422929} a^{2} + \frac{74470639861570623550854277553610063885311294}{174335163482737429810223452178474301907422929} a - \frac{68930580700652034639943113407400337765355351}{174335163482737429810223452178474301907422929}$
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: | \( 827483542795204100 \) (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
| \(\Q(\sqrt{7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{28})^+\), 10.10.3689195226078208.1, 15.15.886528337182930278529.1 |
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 | $15^{2}$ | $30$ | R | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{3}$ | $30$ | $15^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ | $15^{2}$ | $15^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{3}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ | $15^{2}$ | $15^{2}$ | $15^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 7 | Data not computed | ||||||
| 11 | Data not computed | ||||||