Normalized defining polynomial
\( x^{30} - 2 x^{29} - 13 x^{28} + 22 x^{27} + 174 x^{26} - 256 x^{25} - 157 x^{24} + 458 x^{23} + 5658 x^{22} - 6464 x^{21} + 45434 x^{20} - 23206 x^{19} + 322087 x^{18} - 245272 x^{17} + 2223136 x^{16} - 1463962 x^{15} + 13159287 x^{14} - 5855334 x^{13} + 62967354 x^{12} - 19483620 x^{11} + 244339329 x^{10} - 69901890 x^{9} + 742997779 x^{8} - 222989416 x^{7} + 1686428990 x^{6} - 498857406 x^{5} + 2661692327 x^{4} - 653724964 x^{3} + 2600745170 x^{2} - 385672188 x + 1205881601 \)
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: | \(-27652541257338422096297668839356545284021085927702003712=-\,2^{45}\cdot 7^{20}\cdot 11^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $70.48$ | 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: | \(616=2^{3}\cdot 7\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{616}(1,·)$, $\chi_{616}(603,·)$, $\chi_{616}(515,·)$, $\chi_{616}(449,·)$, $\chi_{616}(9,·)$, $\chi_{616}(267,·)$, $\chi_{616}(499,·)$, $\chi_{616}(401,·)$, $\chi_{616}(323,·)$, $\chi_{616}(529,·)$, $\chi_{616}(345,·)$, $\chi_{616}(225,·)$, $\chi_{616}(25,·)$, $\chi_{616}(331,·)$, $\chi_{616}(155,·)$, $\chi_{616}(361,·)$, $\chi_{616}(289,·)$, $\chi_{616}(67,·)$, $\chi_{616}(291,·)$, $\chi_{616}(113,·)$, $\chi_{616}(163,·)$, $\chi_{616}(81,·)$, $\chi_{616}(379,·)$, $\chi_{616}(169,·)$, $\chi_{616}(555,·)$, $\chi_{616}(177,·)$, $\chi_{616}(179,·)$, $\chi_{616}(235,·)$, $\chi_{616}(137,·)$, $\chi_{616}(443,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{43} a^{25} - \frac{3}{43} a^{24} - \frac{14}{43} a^{23} - \frac{11}{43} a^{22} + \frac{3}{43} a^{21} - \frac{8}{43} a^{20} - \frac{21}{43} a^{19} - \frac{8}{43} a^{18} - \frac{18}{43} a^{17} - \frac{8}{43} a^{16} - \frac{4}{43} a^{15} + \frac{17}{43} a^{14} - \frac{6}{43} a^{13} - \frac{20}{43} a^{12} - \frac{3}{43} a^{11} - \frac{6}{43} a^{10} - \frac{10}{43} a^{9} + \frac{20}{43} a^{8} - \frac{13}{43} a^{7} - \frac{14}{43} a^{6} - \frac{14}{43} a^{5} - \frac{18}{43} a^{4} - \frac{20}{43} a^{3} - \frac{16}{43} a^{2} - \frac{10}{43} a - \frac{11}{43}$, $\frac{1}{43} a^{26} + \frac{20}{43} a^{24} - \frac{10}{43} a^{23} + \frac{13}{43} a^{22} + \frac{1}{43} a^{21} - \frac{2}{43} a^{20} + \frac{15}{43} a^{19} + \frac{1}{43} a^{18} - \frac{19}{43} a^{17} + \frac{15}{43} a^{16} + \frac{5}{43} a^{15} + \frac{2}{43} a^{14} + \frac{5}{43} a^{13} - \frac{20}{43} a^{12} - \frac{15}{43} a^{11} + \frac{15}{43} a^{10} - \frac{10}{43} a^{9} + \frac{4}{43} a^{8} - \frac{10}{43} a^{7} - \frac{13}{43} a^{6} - \frac{17}{43} a^{5} + \frac{12}{43} a^{4} + \frac{10}{43} a^{3} - \frac{15}{43} a^{2} + \frac{2}{43} a + \frac{10}{43}$, $\frac{1}{43} a^{27} + \frac{7}{43} a^{24} - \frac{8}{43} a^{23} + \frac{6}{43} a^{22} - \frac{19}{43} a^{21} + \frac{3}{43} a^{20} - \frac{9}{43} a^{19} + \frac{12}{43} a^{18} - \frac{12}{43} a^{17} - \frac{7}{43} a^{16} - \frac{4}{43} a^{15} + \frac{9}{43} a^{14} + \frac{14}{43} a^{13} - \frac{2}{43} a^{12} - \frac{11}{43} a^{11} - \frac{19}{43} a^{10} - \frac{11}{43} a^{9} + \frac{20}{43} a^{8} - \frac{11}{43} a^{7} + \frac{5}{43} a^{6} - \frac{9}{43} a^{5} - \frac{17}{43} a^{4} - \frac{2}{43} a^{3} + \frac{21}{43} a^{2} - \frac{5}{43} a + \frac{5}{43}$, $\frac{1}{180175181801} a^{28} - \frac{157019561}{180175181801} a^{27} + \frac{1985005653}{180175181801} a^{26} - \frac{305886139}{180175181801} a^{25} + \frac{62287195924}{180175181801} a^{24} + \frac{3072955712}{180175181801} a^{23} + \frac{74487560585}{180175181801} a^{22} - \frac{36133755933}{180175181801} a^{21} - \frac{77269191945}{180175181801} a^{20} - \frac{32517971040}{180175181801} a^{19} - \frac{15319936459}{180175181801} a^{18} + \frac{28736711641}{180175181801} a^{17} + \frac{33810517844}{180175181801} a^{16} + \frac{43431815719}{180175181801} a^{15} - \frac{15766427892}{180175181801} a^{14} + \frac{66619162112}{180175181801} a^{13} + \frac{63466494868}{180175181801} a^{12} - \frac{25807278881}{180175181801} a^{11} - \frac{88274015730}{180175181801} a^{10} + \frac{83936878599}{180175181801} a^{9} - \frac{6209520978}{180175181801} a^{8} + \frac{33736557218}{180175181801} a^{7} + \frac{74998442536}{180175181801} a^{6} - \frac{28520140898}{180175181801} a^{5} + \frac{31005092690}{180175181801} a^{4} + \frac{53048364337}{180175181801} a^{3} - \frac{30404652651}{180175181801} a^{2} - \frac{71579929061}{180175181801} a + \frac{46276336215}{180175181801}$, $\frac{1}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{29} + \frac{19067160184764329096228294121381301803745851432074574956076023558218985162570415604286258096347415646726063}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{28} + \frac{68013964606829249439116544202991455270449232564490579530637799586388715855290101935410393550470344168699441739028659}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{27} - \frac{10082182852928025874871553832337850137979166467158348112454625427869152379217604361773752681766804027727541127187777}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{26} + \frac{65074195024764482804013306318694532483382956639254362112463896945714237423387417323118344552367473652629197642199886}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{25} - \frac{122750026922498487418237707133180415930092603690088407797707944096323816794793742005800530664447480180161762096238858}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{24} + \frac{5801909320885255571465570915760557854378184165148018817427257924233304076511301317160546640485804051057957751517749295}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{23} + \frac{4889270574522129499562435778557059805066651378819954413801249960821481848214066814021850092201117565850799896176511803}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{22} + \frac{4055174889617764897487339241680416674202136873119448477396105482224343208079616023148957976075467061725129680204314837}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{21} - \frac{4473848254940155911754600349164320753749978492796000941788943594246323740358882792424422011130474087959394182271423957}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{20} + \frac{5651565905721218487227702123197924414272327215672396093631937828243834817608786208615982102178732961805427866103153041}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{19} - \frac{4404828510180104204689491000057265026583921305875838671106090761446333288488581706523866771952109079120803166007737258}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{18} - \frac{101448216848280648737092509916403878859252394343052580828372450294616425017783750122813663758912956273173998487392846}{285086505323119334743399016388415966753736705115134096791347838448535968278928831855453868950965932301419970702333793} a^{17} + \frac{2176005570605928922412593584957541491700788418471414255982031172121891421781663232580118994754846105694755108044950577}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{16} + \frac{5409288234160934936917426028949940670346048444313760515867430292736303607998482599228923237328093160766639114996614137}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{15} + \frac{2836429304064594248161341785459182392400847924273720722243215607535665689724722443071179124227677923040442419828276651}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{14} - \frac{3394142183572594047234603661384367853694705673602814189430864428576215840050550750614996940587981877094983626474384527}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{13} + \frac{5381927414480717103051207045420252011106865010835377150994268240519087216873550710334731713384495343921958782807549434}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{12} - \frac{3375029777198668535934593250143684622407471354236454435685818670456348872964012205725584475618814093046376608328243765}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{11} - \frac{3481491256184220371313740206536394319809747088015550500648847311199742598353884602495479758171494846278085373619818738}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{10} + \frac{2516934132414463968277585451188566267697462205650188383197493881958597443937495146169756155819002167939740432423693465}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{9} - \frac{3304360775241279162495506561137827265483679561969878142324352657337609078207494704734553419098270398647753902133292775}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{8} - \frac{1463207115746123736151876333965369215459340091837446981772052054543063138768202411741772659712680817291421211938929498}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{7} + \frac{2809048224359946689458718312680828734894418732295247428298029802966433554908668301959930882298308170065821313532946504}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{6} + \frac{3213746645261622667477362974223586273668223236002958435087738987316061447330376017630074648049377847211948299476157403}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{5} - \frac{4361469968970028474833062294717070516016934069954843257167168210584737935318263212250840906282997360514238181567494854}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{4} + \frac{533856475848448764351436810922459362721728077095648925437796882739945850122220545830408286194491403902567757166653958}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{3} + \frac{2599569195916280440118148647715743917569558120519614087538441876301354131701391006158026355808627037485510188969316184}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a^{2} - \frac{3856041508698190878982298915044056731635941630741512623464227184398414815918402837859464114927563537448957040427867142}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099} a - \frac{4561980823728010806321275288202747379406785732327631386786336808128710410488642817562322595956420910808065971629890013}{12258719728894131393966157704701886570410678319950766162027957053287046635993939769784516364891535088961058740200353099}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{63010}$, which has order $504080$ (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
| \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.0.1229312.1, 10.0.7024111812608.1, 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 | ${\href{/LocalNumberField/13.10.0.1}{10} }^{3}$ | $15^{2}$ | $15^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ | $30$ | $30$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{6}$ | ${\href{/LocalNumberField/43.1.0.1}{1} }^{30}$ | $30$ | $30$ | $15^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 7 | Data not computed | ||||||
| $11$ | 11.15.12.1 | $x^{15} + 165 x^{10} + 5324 x^{5} + 323433$ | $5$ | $3$ | $12$ | $C_{15}$ | $[\ ]_{5}^{3}$ |
| 11.15.12.1 | $x^{15} + 165 x^{10} + 5324 x^{5} + 323433$ | $5$ | $3$ | $12$ | $C_{15}$ | $[\ ]_{5}^{3}$ | |