Normalized defining polynomial
\( x^{30} - 15 x^{29} + 81 x^{28} - 111 x^{27} - 597 x^{26} + 1935 x^{25} + 4777 x^{24} - 37983 x^{23} + 84639 x^{22} - 65485 x^{21} + 42693 x^{20} - 760743 x^{19} + 3959403 x^{18} - 11598279 x^{17} + 25260000 x^{16} - 49060283 x^{15} + 102637572 x^{14} - 241637424 x^{13} + 592252406 x^{12} - 1355130954 x^{11} + 2839183851 x^{10} - 5299560927 x^{9} + 8966269281 x^{8} - 13378531284 x^{7} + 17917235515 x^{6} - 20619001905 x^{5} + 20898921987 x^{4} - 17214632474 x^{3} + 12116563935 x^{2} - 5846344410 x + 2284044751 \)
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: | \(-568519326398404118019654805600981921999434432632975849463=-\,3^{40}\cdot 7^{15}\cdot 11^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $77.95$ | 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: | $3, 7, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(693=3^{2}\cdot 7\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{693}(64,·)$, $\chi_{693}(1,·)$, $\chi_{693}(643,·)$, $\chi_{693}(454,·)$, $\chi_{693}(328,·)$, $\chi_{693}(265,·)$, $\chi_{693}(202,·)$, $\chi_{693}(652,·)$, $\chi_{693}(610,·)$, $\chi_{693}(526,·)$, $\chi_{693}(463,·)$, $\chi_{693}(400,·)$, $\chi_{693}(148,·)$, $\chi_{693}(664,·)$, $\chi_{693}(412,·)$, $\chi_{693}(223,·)$, $\chi_{693}(97,·)$, $\chi_{693}(34,·)$, $\chi_{693}(421,·)$, $\chi_{693}(295,·)$, $\chi_{693}(232,·)$, $\chi_{693}(169,·)$, $\chi_{693}(685,·)$, $\chi_{693}(559,·)$, $\chi_{693}(496,·)$, $\chi_{693}(433,·)$, $\chi_{693}(181,·)$, $\chi_{693}(631,·)$, $\chi_{693}(379,·)$, $\chi_{693}(190,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{22103532267192199} a^{28} + \frac{9093939390438912}{22103532267192199} a^{27} + \frac{9298967655442911}{22103532267192199} a^{26} + \frac{10677951330824312}{22103532267192199} a^{25} - \frac{4388793086700469}{22103532267192199} a^{24} + \frac{8780201266215898}{22103532267192199} a^{23} - \frac{6201231632443689}{22103532267192199} a^{22} - \frac{6092189795940518}{22103532267192199} a^{21} + \frac{6225337655457757}{22103532267192199} a^{20} + \frac{5996907687927856}{22103532267192199} a^{19} + \frac{9632086152172915}{22103532267192199} a^{18} + \frac{8918796064949118}{22103532267192199} a^{17} - \frac{6800071157782612}{22103532267192199} a^{16} - \frac{5156627396402168}{22103532267192199} a^{15} + \frac{7853620376203680}{22103532267192199} a^{14} + \frac{3396795868543661}{22103532267192199} a^{13} + \frac{9934434816917593}{22103532267192199} a^{12} - \frac{313317778341577}{22103532267192199} a^{11} + \frac{1988877646618633}{22103532267192199} a^{10} - \frac{3568430615166155}{22103532267192199} a^{9} - \frac{10146891539597848}{22103532267192199} a^{8} + \frac{2087672598042201}{22103532267192199} a^{7} - \frac{8670033691160472}{22103532267192199} a^{6} + \frac{3534751893327399}{22103532267192199} a^{5} - \frac{6392795080404970}{22103532267192199} a^{4} - \frac{8024724885704933}{22103532267192199} a^{3} - \frac{7223775518741128}{22103532267192199} a^{2} - \frac{3083688008610202}{22103532267192199} a + \frac{59739729983747}{202784699699011}$, $\frac{1}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{29} - \frac{1010961171541369027109754962871341997782278283885739551207912333021883418481323255576558132601292345911321}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{28} - \frac{9153731771783476716287126717883564980297764912374685007912836730496338043606497499377452985913219258100242084532794261628}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{27} - \frac{30449601405351548587699320021135002244281622214845276821256956933677966434459765041930387925138932862570386414327683265286}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{26} - \frac{16570421740412435726805116985517772306206651899479828457747129477812432997688432994327014021420335024557069971375808711548}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{25} - \frac{6166843128202802171225848652976169762505776438964235146490052860635125249213924715154081194777050088816305021393794837330}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{24} - \frac{24339452895481819154369761411659206261538743563398505350391792275066930667035185035098968850589655360917044448172970679281}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{23} + \frac{26823844503202667355173166111111063811465082804420291373595882736299246231136039034090457272147774934810762761507001340262}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{22} - \frac{26441303331890452687277234765214136926501146992867070063491166117438698099552634435717392651646075526705164540241196866063}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{21} + \frac{14260048626916844799332933330676990921411432483577967222091420111198783665789633340136515742937969692575896159775818956981}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{20} - \frac{33881434955069114011076068837019013944786257730370943460319263066541504063150426432420370478515880426699200645808809057609}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{19} - \frac{3497733653303487550933566643140379441187881302107397884445777827408497318940886407660087842094271251631614364182570100763}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{18} - \frac{16585509736503057712184914672263760525270098877523904845404224782048559624754435881872323683998754331779603803718117177136}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{17} + \frac{25539310479460015327180425559563449686970000436806412789380640989903631389897048547661768584942655428548463903345767606166}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{16} + \frac{12142091974646984725955664293636435990156267533019827502863716618278525021909447753328508623788693497796904516060218184583}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{15} - \frac{21232025321508192216121714534315782423120474701375389634136510039127935319266046548029348747930869093965433395153329552557}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{14} + \frac{31866701836142156810717598208767153012916933717152482927479236984983484767039947701999825564252439839944715223493345046437}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{13} - \frac{36089864652676886731995263577424960195350961215183027378147189073049756200930902097507268143348541975988013673323523009186}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{12} + \frac{23391366869207163433322621840406415360365240563456969290544315106615988441865097599529326351637553118075998964320218169338}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{11} + \frac{24361548445930140856532989833972071052190955391567457916074486445284019097579286786118369413880206415195039361617289639778}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{10} + \frac{25245952385724903411941319719940747306144561183934307499793186837151493865920681430832006769361279339490824109635953059632}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{9} + \frac{26903676027816817954367220945235134810023770329580504522594034224173234958145261177815010145905777887885079999788987974398}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{8} - \frac{24299051009377792884192902021043333470285034923050705003087378034386953203658126141709473293876883487211737971646724859169}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{7} - \frac{568901015562585175664899793861059355347842070478287532986269338444076652479601474137768323658872240166173179056901891174}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{6} + \frac{9014024819914376435743431936626444985201657471958198425139467009001503370764067992203522207692278151956086105229640890647}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{5} + \frac{19639666862773242585282013767541933550907260771831224509422314442179922902420421895412344508016200307346902651488957276864}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{4} - \frac{163459168388742629014670211926759766492791578615206442152598857170591245832510170650215108449255630485775610499406939416}{827472531940194185331505990486329638374891409340783932637901892553458468601139817714714321409960503512691127910219359401} a^{3} + \frac{16619456884957902032014369597747132199857498756884621120020778503897958401418223483777381189219817567696565344411342117271}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a^{2} + \frac{32015123903000343636272955310671519785193184563036974623931969243581562417724646835841986715908273733211852693210690718458}{73645055342677282494504033153283337815365335431329770004773268437257803705501443776609574605486484812629510384009522986689} a + \frac{253914340696558603901544150618387421219519212752756278423454311863441813235220437966628356921699899687069832777396188644}{675642709565846628389945258287003099223535187443392385364892370984016547756893979601922702802628301033298260403757091621}$
Class group and class number
$C_{423155}$, which has order $423155$ (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: | \( 15853905121.091976 \) (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
| \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\), 6.0.2250423.1, 10.0.3602729712967.1, 15.15.10943023107606534329121.1 |
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 | $15^{2}$ | R | $30$ | R | R | $30$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{3}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{3}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{10}$ | $15^{2}$ | $30$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{6}$ | $30$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{10}$ | $30$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{6}$ | $30$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | 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 | $x^{15} + 165 x^{10} + 5324 x^{5} + 323433$ | $5$ | $3$ | $12$ | $C_{15}$ | $[\ ]_{5}^{3}$ | |