Normalized defining polynomial
\( x^{30} - 7 x^{29} + 35 x^{28} - 115 x^{27} + 399 x^{26} - 1141 x^{25} + 3440 x^{24} - 7757 x^{23} + 19611 x^{22} - 35947 x^{21} + 91246 x^{20} - 127051 x^{19} + 348021 x^{18} - 249983 x^{17} + 1494992 x^{16} - 1491783 x^{15} + 11747132 x^{14} - 21905478 x^{13} + 91336355 x^{12} - 176387858 x^{11} + 463852893 x^{10} - 718935363 x^{9} + 1340477375 x^{8} - 1572476398 x^{7} + 2198467858 x^{6} - 1798304614 x^{5} + 2089535021 x^{4} - 1245722827 x^{3} + 1010338243 x^{2} - 391873738 x + 718434949 \)
Invariants
| Degree: | $30$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 15]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-31923710561777007667868565434970593170029064056396484375=-\,5^{15}\cdot 7^{20}\cdot 11^{27}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $70.82$ | 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: | $5, 7, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(385=5\cdot 7\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{385}(256,·)$, $\chi_{385}(1,·)$, $\chi_{385}(134,·)$, $\chi_{385}(71,·)$, $\chi_{385}(74,·)$, $\chi_{385}(331,·)$, $\chi_{385}(204,·)$, $\chi_{385}(141,·)$, $\chi_{385}(79,·)$, $\chi_{385}(16,·)$, $\chi_{385}(81,·)$, $\chi_{385}(274,·)$, $\chi_{385}(149,·)$, $\chi_{385}(86,·)$, $\chi_{385}(219,·)$, $\chi_{385}(29,·)$, $\chi_{385}(354,·)$, $\chi_{385}(291,·)$, $\chi_{385}(36,·)$, $\chi_{385}(39,·)$, $\chi_{385}(361,·)$, $\chi_{385}(359,·)$, $\chi_{385}(109,·)$, $\chi_{385}(366,·)$, $\chi_{385}(221,·)$, $\chi_{385}(246,·)$, $\chi_{385}(184,·)$, $\chi_{385}(249,·)$, $\chi_{385}(239,·)$, $\chi_{385}(191,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{89} a^{18} - \frac{19}{89} a^{17} - \frac{27}{89} a^{16} + \frac{41}{89} a^{15} + \frac{41}{89} a^{14} - \frac{39}{89} a^{13} + \frac{23}{89} a^{12} + \frac{15}{89} a^{11} - \frac{42}{89} a^{10} - \frac{26}{89} a^{9} - \frac{22}{89} a^{8} + \frac{14}{89} a^{7} - \frac{19}{89} a^{6} - \frac{21}{89} a^{5} + \frac{4}{89} a^{4} + \frac{40}{89} a^{3} + \frac{9}{89} a^{2} - \frac{10}{89} a - \frac{24}{89}$, $\frac{1}{89} a^{19} - \frac{32}{89} a^{17} - \frac{27}{89} a^{16} + \frac{19}{89} a^{15} + \frac{28}{89} a^{14} - \frac{6}{89} a^{13} + \frac{7}{89} a^{12} - \frac{24}{89} a^{11} - \frac{23}{89} a^{10} + \frac{18}{89} a^{9} + \frac{41}{89} a^{8} - \frac{20}{89} a^{7} - \frac{26}{89} a^{6} - \frac{39}{89} a^{5} + \frac{27}{89} a^{4} - \frac{32}{89} a^{3} - \frac{17}{89} a^{2} - \frac{36}{89} a - \frac{11}{89}$, $\frac{1}{89} a^{20} - \frac{12}{89} a^{17} - \frac{44}{89} a^{16} + \frac{5}{89} a^{15} - \frac{29}{89} a^{14} + \frac{5}{89} a^{13} + \frac{12}{89} a^{11} + \frac{9}{89} a^{10} + \frac{10}{89} a^{9} - \frac{12}{89} a^{8} - \frac{23}{89} a^{7} - \frac{24}{89} a^{6} - \frac{22}{89} a^{5} + \frac{7}{89} a^{4} + \frac{17}{89} a^{3} - \frac{15}{89} a^{2} + \frac{25}{89} a + \frac{33}{89}$, $\frac{1}{89} a^{21} - \frac{5}{89} a^{17} + \frac{37}{89} a^{16} + \frac{18}{89} a^{15} - \frac{37}{89} a^{14} - \frac{23}{89} a^{13} + \frac{21}{89} a^{12} + \frac{11}{89} a^{11} + \frac{40}{89} a^{10} + \frac{32}{89} a^{9} - \frac{20}{89} a^{8} - \frac{34}{89} a^{7} + \frac{17}{89} a^{6} + \frac{22}{89} a^{5} - \frac{24}{89} a^{4} + \frac{20}{89} a^{3} + \frac{44}{89} a^{2} + \frac{2}{89} a - \frac{21}{89}$, $\frac{1}{89} a^{22} + \frac{31}{89} a^{17} - \frac{28}{89} a^{16} - \frac{10}{89} a^{15} + \frac{4}{89} a^{14} + \frac{4}{89} a^{13} + \frac{37}{89} a^{12} + \frac{26}{89} a^{11} + \frac{28}{89} a^{9} + \frac{34}{89} a^{8} - \frac{2}{89} a^{7} + \frac{16}{89} a^{6} - \frac{40}{89} a^{5} + \frac{40}{89} a^{4} - \frac{23}{89} a^{3} - \frac{42}{89} a^{2} + \frac{18}{89} a - \frac{31}{89}$, $\frac{1}{89} a^{23} + \frac{27}{89} a^{17} + \frac{26}{89} a^{16} - \frac{21}{89} a^{15} - \frac{21}{89} a^{14} + \frac{25}{89} a^{12} - \frac{20}{89} a^{11} - \frac{5}{89} a^{10} + \frac{39}{89} a^{9} - \frac{32}{89} a^{8} + \frac{27}{89} a^{7} + \frac{15}{89} a^{6} - \frac{21}{89} a^{5} + \frac{31}{89} a^{4} - \frac{36}{89} a^{3} + \frac{6}{89} a^{2} + \frac{12}{89} a + \frac{32}{89}$, $\frac{1}{89} a^{24} + \frac{5}{89} a^{17} - \frac{4}{89} a^{16} + \frac{29}{89} a^{15} - \frac{39}{89} a^{14} + \frac{10}{89} a^{13} - \frac{18}{89} a^{12} + \frac{35}{89} a^{11} + \frac{16}{89} a^{10} - \frac{42}{89} a^{9} - \frac{2}{89} a^{8} - \frac{7}{89} a^{7} - \frac{42}{89} a^{6} - \frac{25}{89} a^{5} + \frac{34}{89} a^{4} - \frac{6}{89} a^{3} + \frac{36}{89} a^{2} + \frac{35}{89} a + \frac{25}{89}$, $\frac{1}{3827} a^{25} - \frac{1}{3827} a^{23} - \frac{18}{3827} a^{22} + \frac{11}{3827} a^{21} - \frac{20}{3827} a^{20} - \frac{3}{3827} a^{19} - \frac{18}{3827} a^{18} - \frac{1473}{3827} a^{17} + \frac{1873}{3827} a^{16} + \frac{1841}{3827} a^{15} - \frac{895}{3827} a^{14} - \frac{151}{3827} a^{13} + \frac{627}{3827} a^{12} + \frac{155}{3827} a^{11} - \frac{8}{89} a^{10} - \frac{116}{3827} a^{9} + \frac{439}{3827} a^{8} - \frac{1188}{3827} a^{7} + \frac{1833}{3827} a^{6} - \frac{1236}{3827} a^{5} - \frac{1779}{3827} a^{4} - \frac{13}{3827} a^{3} - \frac{634}{3827} a^{2} + \frac{1685}{3827} a - \frac{47}{3827}$, $\frac{1}{3827} a^{26} - \frac{1}{3827} a^{24} - \frac{18}{3827} a^{23} + \frac{11}{3827} a^{22} - \frac{20}{3827} a^{21} - \frac{3}{3827} a^{20} - \frac{18}{3827} a^{19} - \frac{11}{3827} a^{18} + \frac{884}{3827} a^{17} + \frac{637}{3827} a^{16} + \frac{1642}{3827} a^{15} - \frac{1441}{3827} a^{14} + \frac{1014}{3827} a^{13} - \frac{662}{3827} a^{12} - \frac{32}{89} a^{11} - \frac{288}{3827} a^{10} + \frac{697}{3827} a^{9} + \frac{1091}{3827} a^{8} - \frac{661}{3827} a^{7} + \frac{18}{43} a^{6} - \frac{1865}{3827} a^{5} - \frac{1819}{3827} a^{4} + \frac{441}{3827} a^{3} - \frac{465}{3827} a^{2} + \frac{641}{3827} a - \frac{15}{89}$, $\frac{1}{3827} a^{27} - \frac{18}{3827} a^{24} + \frac{10}{3827} a^{23} + \frac{5}{3827} a^{22} + \frac{8}{3827} a^{21} + \frac{5}{3827} a^{20} - \frac{14}{3827} a^{19} + \frac{6}{3827} a^{18} + \frac{1013}{3827} a^{17} + \frac{677}{3827} a^{16} - \frac{632}{3827} a^{15} - \frac{1773}{3827} a^{14} - \frac{1329}{3827} a^{13} + \frac{197}{3827} a^{12} + \frac{82}{3827} a^{11} - \frac{1410}{3827} a^{10} - \frac{1820}{3827} a^{9} + \frac{509}{3827} a^{8} - \frac{1220}{3827} a^{7} + \frac{656}{3827} a^{6} + \frac{858}{3827} a^{5} + \frac{1070}{3827} a^{4} - \frac{693}{3827} a^{3} + \frac{1297}{3827} a^{2} + \frac{8}{3827} a + \frac{1544}{3827}$, $\frac{1}{1174889} a^{28} + \frac{114}{1174889} a^{27} + \frac{145}{1174889} a^{26} + \frac{3}{27323} a^{25} - \frac{2144}{1174889} a^{24} + \frac{5311}{1174889} a^{23} - \frac{10}{27323} a^{22} + \frac{1999}{1174889} a^{21} - \frac{5442}{1174889} a^{20} + \frac{3}{1174889} a^{19} + \frac{4680}{1174889} a^{18} - \frac{570408}{1174889} a^{17} - \frac{194996}{1174889} a^{16} + \frac{375083}{1174889} a^{15} - \frac{520999}{1174889} a^{14} + \frac{576728}{1174889} a^{13} - \frac{140768}{1174889} a^{12} - \frac{4938}{13201} a^{11} + \frac{531539}{1174889} a^{10} - \frac{394374}{1174889} a^{9} + \frac{484515}{1174889} a^{8} + \frac{584782}{1174889} a^{7} - \frac{358770}{1174889} a^{6} + \frac{350890}{1174889} a^{5} + \frac{275677}{1174889} a^{4} + \frac{163940}{1174889} a^{3} + \frac{291941}{1174889} a^{2} - \frac{480311}{1174889} a - \frac{385292}{1174889}$, $\frac{1}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{29} - \frac{1634428451996657270042845711273267373185661943625527420654947983098630992361172062513670755162341360892023631}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{28} + \frac{121697554744525788258992907227182131259644727915930273622389778634159107734171853317446487778057207932311157124}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{27} - \frac{1918858407267677586639241144422822512214719166959328943750314980341117816383735349533735991055485028685753131146}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{26} + \frac{3616780277206231647871794250205920445152314495380258514747709039828828365044402764145888871460091382362088862}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{25} + \frac{29628014838236042906509030298426729012445262800102022144613994028955688575574739910051483280160237992742826066111}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{24} + \frac{81430030348732163686281616329719796208564531608844201042751949885822214573339280628558030885232852048573532206750}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{23} + \frac{64715833210966363539729691578142826125390334108575154995204304455720119384175438351289995906750919744530245266270}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{22} + \frac{11481823499316793021077505936162074259566090764135505933258372519840310890839523914911635848867428941636410506251}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{21} + \frac{100784096520809057974361106834904206074761206360412912971292882195223474898494643180925147937345603161246745755000}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{20} - \frac{93066522212637015824724156003184806147054725313443412723112480883676356876139238060296663122706366343630849234433}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{19} + \frac{78684828531084596266753266666458008948110642018851953944641896088750944716688392578685915095292710663325397666916}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{18} + \frac{6386257744568528941461969980841587036605232774517514238066410860329194056791306782595864440196952229609332375444209}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{17} + \frac{2134830838805825059748700091724436591255514938026831374999348518373307000796218669796832926350870260138724838513586}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{16} + \frac{6771445892843714509510786906854363540970983908798776528847897587765195722810167777281296311788286506965256838867524}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{15} + \frac{8057496552989000719516730390400761487014222738514414129744823015541928661486294762661334061710394141311714323754310}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{14} + \frac{7721691318384608418550586010373699476243631507505467997595888432314737275889611697055940617389107141096004761082982}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{13} - \frac{6103242787743294212451871611196239064969075657171241647874199481246908912269473087714777685452929991145277203201588}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{12} - \frac{1821912056882372612209090448531212225838149239191799760122243648408912729817998867639595719813119358196425972173990}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{11} + \frac{6350320188349175129032499434825035257459987860311886903468947637912287609978326246321270428007306968655286205228969}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{10} - \frac{3641047535696363375806126977478002495228799333045924872405560463230381742279993585956383612106773428043994058990218}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{9} - \frac{3223035650668900772621452100997828841495417659893696151949590457634858392253659482478059430426880572562298723685968}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{8} - \frac{1940262443121682054331696853924795340206580134997006219090889613678557468697396025572225408154094536929333066479952}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{7} - \frac{6787209114220751010635308552346616779859228636497794373191928583822549638230287568325000276567191948270973514068529}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{6} + \frac{280110505180997937743598305208136449796850175823535047846891819148993941965854169139781906957207647026469230590220}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{5} + \frac{8385998384648175109934963386517334163655562570452032863044724754919346480836367733910235590811206480165240923665635}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{4} - \frac{8790331856864900499505630743334778045206632284260072215631606455659793778224967267085597500006160253410673723365351}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{3} - \frac{1661470284871950976822976615352695638162198086188892408097253322247441555210498529313848026803143950821638735714079}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a^{2} + \frac{1782149393578473031910029057619404661856298291061772296355467863099329451946011983591039476329484609272000644677790}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343} a + \frac{6919971880660321560203126012514764664127489020976995445922703624857884773880394887917267488340122920083792473100196}{18346704821883445934953350755830099600434093006612200212473233615006642626243792810705136386834828115286736804527343}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{12252}$, which has order $588096$ (assuming GRH)
Unit group
| Rank: | $14$ | 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: | \( 4697581952.048968 \) (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
| \(\Q(\sqrt{-55}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.0.399466375.2, 10.0.7368586534375.1, 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 | $15^{2}$ | $30$ | R | R | R | ${\href{/LocalNumberField/13.5.0.1}{5} }^{6}$ | $15^{2}$ | $30$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ | $15^{2}$ | $30$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{3}$ | ${\href{/LocalNumberField/43.1.0.1}{1} }^{30}$ | $30$ | $30$ | $15^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| $7$ | 7.15.10.1 | $x^{15} + 4116 x^{6} - 2401 x^{3} + 1075648$ | $3$ | $5$ | $10$ | $C_{15}$ | $[\ ]_{3}^{5}$ |
| 7.15.10.1 | $x^{15} + 4116 x^{6} - 2401 x^{3} + 1075648$ | $3$ | $5$ | $10$ | $C_{15}$ | $[\ ]_{3}^{5}$ | |
| 11 | Data not computed | ||||||