Normalized defining polynomial
\( x^{30} - 2 x^{29} - 73 x^{28} + 134 x^{27} + 2230 x^{26} - 3768 x^{25} - 37749 x^{24} + 58634 x^{23} + 395706 x^{22} - 561432 x^{21} - 2721406 x^{20} + 3480410 x^{19} + 12690151 x^{18} - 14319232 x^{17} - 40831660 x^{16} + 39414022 x^{15} + 91167079 x^{14} - 72111502 x^{13} - 140553034 x^{12} + 85687148 x^{11} + 147256817 x^{10} - 62879794 x^{9} - 101697857 x^{8} + 25294552 x^{7} + 43762382 x^{6} - 3478846 x^{5} - 10467281 x^{4} - 815316 x^{3} + 1021626 x^{2} + 235124 x + 12473 \)
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: | \(27652541257338422096297668839356545284021085927702003712=2^{45}\cdot 7^{20}\cdot 11^{24}\) | 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$$\chi_{616}(1,·)$, $\chi_{616}(389,·)$, $\chi_{616}(449,·)$, $\chi_{616}(9,·)$, $\chi_{616}(141,·)$, $\chi_{616}(597,·)$, $\chi_{616}(333,·)$, $\chi_{616}(401,·)$, $\chi_{616}(529,·)$, $\chi_{616}(533,·)$, $\chi_{616}(345,·)$, $\chi_{616}(485,·)$, $\chi_{616}(25,·)$, $\chi_{616}(361,·)$, $\chi_{616}(221,·)$, $\chi_{616}(37,·)$, $\chi_{616}(289,·)$, $\chi_{616}(113,·)$, $\chi_{616}(421,·)$, $\chi_{616}(81,·)$, $\chi_{616}(169,·)$, $\chi_{616}(225,·)$, $\chi_{616}(93,·)$, $\chi_{616}(177,·)$, $\chi_{616}(53,·)$, $\chi_{616}(137,·)$, $\chi_{616}(445,·)$, $\chi_{616}(477,·)$, $\chi_{616}(317,·)$, $\chi_{616}(309,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{2927} a^{27} + \frac{637}{2927} a^{26} - \frac{282}{2927} a^{25} - \frac{394}{2927} a^{24} - \frac{533}{2927} a^{23} - \frac{908}{2927} a^{22} - \frac{624}{2927} a^{21} - \frac{1215}{2927} a^{20} + \frac{1162}{2927} a^{19} - \frac{901}{2927} a^{18} - \frac{815}{2927} a^{17} - \frac{431}{2927} a^{16} + \frac{1378}{2927} a^{15} + \frac{294}{2927} a^{14} - \frac{1354}{2927} a^{13} - \frac{314}{2927} a^{12} - \frac{172}{2927} a^{11} - \frac{220}{2927} a^{10} - \frac{1180}{2927} a^{9} - \frac{926}{2927} a^{8} - \frac{1084}{2927} a^{7} + \frac{294}{2927} a^{6} - \frac{919}{2927} a^{5} + \frac{542}{2927} a^{4} - \frac{414}{2927} a^{3} + \frac{578}{2927} a^{2} + \frac{1353}{2927} a - \frac{185}{2927}$, $\frac{1}{586889843} a^{28} - \frac{21014}{586889843} a^{27} + \frac{229110980}{586889843} a^{26} + \frac{191088661}{586889843} a^{25} + \frac{160643224}{586889843} a^{24} - \frac{1044098}{586889843} a^{23} - \frac{180360988}{586889843} a^{22} + \frac{65931579}{586889843} a^{21} + \frac{187687272}{586889843} a^{20} + \frac{243231902}{586889843} a^{19} - \frac{247602503}{586889843} a^{18} - \frac{112129265}{586889843} a^{17} + \frac{229829723}{586889843} a^{16} - \frac{201755056}{586889843} a^{15} + \frac{106764729}{586889843} a^{14} - \frac{222412714}{586889843} a^{13} - \frac{23338150}{586889843} a^{12} - \frac{275617420}{586889843} a^{11} + \frac{3785968}{13648601} a^{10} - \frac{205873074}{586889843} a^{9} - \frac{79719053}{586889843} a^{8} - \frac{148634695}{586889843} a^{7} - \frac{127722660}{586889843} a^{6} + \frac{123071634}{586889843} a^{5} - \frac{1066341}{586889843} a^{4} + \frac{233895261}{586889843} a^{3} - \frac{98688}{200509} a^{2} - \frac{278384615}{586889843} a - \frac{213713606}{586889843}$, $\frac{1}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{29} + \frac{1131830682750750306023245407744282070032502869759651636849809642869}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{28} + \frac{6349490097782357304055228106917782759786914926545725182348980425069832}{46699597704533450413809764070792920534249933524823718812876378719141953079} a^{27} + \frac{136713471688747251152225231428789824673897021058511793019612164390449127766}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{26} - \frac{190017366546881613823163184025619911328606547186363244293017405419299214963}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{25} + \frac{728480184114033486713398043813139242400529666019630506014644493011245418103}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{24} + \frac{806954079462658716708887768559093247460379114111956089073155430762975531623}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{23} - \frac{261499418420706925447114672949324483024136293487804061613344499556421601507}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{22} + \frac{813678817673770845099481970715716912396172385515935114753194812102266749994}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{21} - \frac{324514715871396554355691686418302223476648919509312109422529177848141651842}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{20} - \frac{264622523362001399080689675150957048204417860423824057920543402502012334607}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{19} - \frac{609332717375968195426326819729540865642672349175528865519045016079919903035}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{18} + \frac{948760669346271390401534026340997865524400079389101358588411706703779748622}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{17} + \frac{625154030249420694500952090894236645565125549818164022009078356235837726373}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{16} + \frac{205601918454298067761655606560044959189135331084371156291539502178745742175}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{15} + \frac{411893113553792840062830286000657200004292769708131154722613194058139523682}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{14} + \frac{12672087190905256057107079324391610698584277596735734271067257087380167913}{46699597704533450413809764070792920534249933524823718812876378719141953079} a^{13} - \frac{52033368247096915707149177134996747963417355954145431997794963632298545206}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{12} - \frac{717540395330731138211826518993605157705034460867946199727517204438976976547}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{11} + \frac{813602930719223189903762871718448232199506278282690447703031125608052360956}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{10} - \frac{625694269669914060163596097660648631826260186043651581011008659211717934268}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{9} + \frac{804599060829563286286134306280568581164760053019603343485841825151942061242}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{8} - \frac{985674712957736942669742960644387414680077647278584932236204543333998523038}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{7} + \frac{90553925267813509419924072904023748126307067734497583288147296852838895985}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{6} - \frac{7107892462548372255808278273096016011091278848713879098884472982457337140}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{5} - \frac{251914059828991604851137680045263505228925618606146543926474547889891370786}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{4} + \frac{794179710525676892543901891226396332341292544917134769190203041986682545684}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{3} - \frac{11822050970670595931535860725747670886100674114818100674127707574063232483}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a^{2} - \frac{473694932620876230232129935064375823147409845086901186429883354567477967712}{2008082701294938367793819855044095582972747141567419908953684284923103982397} a + \frac{938099230429071923742150181134812276975594328040339693011557757822837332564}{2008082701294938367793819855044095582972747141567419908953684284923103982397}$
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: | \( 1002666442794481000 \) (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
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.6.1229312.1, 10.10.7024111812608.1, 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 | $30$ | $30$ | R | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{3}$ | $15^{2}$ | $30$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ | $15^{2}$ | $30$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{6}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ | $15^{2}$ | $30$ | $30$ |
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 | ||||||
| $7$ | 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 | ||||||