Normalized defining polynomial
\( x^{22} - x^{21} + 203 x^{20} + 241 x^{19} + 15645 x^{18} + 49907 x^{17} + 645544 x^{16} + 2979464 x^{15} + 17383054 x^{14} + 82563250 x^{13} + 322791094 x^{12} + 1212821214 x^{11} + 3698021698 x^{10} + 9684850830 x^{9} + 21693864639 x^{8} + 38502084388 x^{7} + 50771931117 x^{6} + 47768698105 x^{5} + 38829007897 x^{4} + 30616690773 x^{3} + 52753103834 x^{2} + 67713149325 x + 81132029807 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 11]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-398976882084318595545873356311326226309728822028879=-\,13^{11}\cdot 67^{21}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $199.55$ | 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: | $13, 67$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(871=13\cdot 67\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{871}(1,·)$, $\chi_{871}(259,·)$, $\chi_{871}(196,·)$, $\chi_{871}(454,·)$, $\chi_{871}(779,·)$, $\chi_{871}(14,·)$, $\chi_{871}(131,·)$, $\chi_{871}(142,·)$, $\chi_{871}(857,·)$, $\chi_{871}(729,·)$, $\chi_{871}(92,·)$, $\chi_{871}(417,·)$, $\chi_{871}(675,·)$, $\chi_{871}(612,·)$, $\chi_{871}(870,·)$, $\chi_{871}(40,·)$, $\chi_{871}(740,·)$, $\chi_{871}(560,·)$, $\chi_{871}(625,·)$, $\chi_{871}(246,·)$, $\chi_{871}(311,·)$, $\chi_{871}(831,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{29} a^{16} + \frac{1}{29} a^{15} - \frac{4}{29} a^{14} + \frac{2}{29} a^{13} + \frac{13}{29} a^{12} - \frac{13}{29} a^{11} + \frac{14}{29} a^{10} + \frac{6}{29} a^{9} + \frac{1}{29} a^{8} - \frac{8}{29} a^{7} - \frac{10}{29} a^{5} - \frac{8}{29} a^{4} + \frac{2}{29} a^{3} - \frac{10}{29} a^{2} - \frac{11}{29} a - \frac{9}{29}$, $\frac{1}{29} a^{17} - \frac{5}{29} a^{15} + \frac{6}{29} a^{14} + \frac{11}{29} a^{13} + \frac{3}{29} a^{12} - \frac{2}{29} a^{11} - \frac{8}{29} a^{10} - \frac{5}{29} a^{9} - \frac{9}{29} a^{8} + \frac{8}{29} a^{7} - \frac{10}{29} a^{6} + \frac{2}{29} a^{5} + \frac{10}{29} a^{4} - \frac{12}{29} a^{3} - \frac{1}{29} a^{2} + \frac{2}{29} a + \frac{9}{29}$, $\frac{1}{29} a^{18} + \frac{11}{29} a^{15} - \frac{9}{29} a^{14} + \frac{13}{29} a^{13} + \frac{5}{29} a^{12} + \frac{14}{29} a^{11} + \frac{7}{29} a^{10} - \frac{8}{29} a^{9} + \frac{13}{29} a^{8} + \frac{8}{29} a^{7} + \frac{2}{29} a^{6} - \frac{11}{29} a^{5} + \frac{6}{29} a^{4} + \frac{9}{29} a^{3} + \frac{10}{29} a^{2} + \frac{12}{29} a + \frac{13}{29}$, $\frac{1}{2813} a^{19} - \frac{1}{97} a^{18} - \frac{42}{2813} a^{17} - \frac{23}{2813} a^{16} + \frac{428}{2813} a^{15} + \frac{245}{2813} a^{14} - \frac{177}{2813} a^{13} + \frac{606}{2813} a^{12} - \frac{395}{2813} a^{11} - \frac{148}{2813} a^{10} - \frac{126}{2813} a^{9} - \frac{1301}{2813} a^{8} + \frac{895}{2813} a^{7} + \frac{32}{2813} a^{6} - \frac{260}{2813} a^{5} + \frac{151}{2813} a^{4} + \frac{243}{2813} a^{3} - \frac{766}{2813} a^{2} - \frac{248}{2813} a + \frac{392}{2813}$, $\frac{1}{745588463} a^{20} - \frac{916}{7686479} a^{19} + \frac{4851833}{745588463} a^{18} - \frac{637076}{745588463} a^{17} - \frac{3411826}{745588463} a^{16} - \frac{40111}{745588463} a^{15} - \frac{252742189}{745588463} a^{14} + \frac{135107646}{745588463} a^{13} - \frac{366216506}{745588463} a^{12} + \frac{47747220}{745588463} a^{11} - \frac{184416647}{745588463} a^{10} + \frac{351520620}{745588463} a^{9} + \frac{338588226}{745588463} a^{8} + \frac{329113619}{745588463} a^{7} + \frac{71687160}{745588463} a^{6} + \frac{142698138}{745588463} a^{5} + \frac{246138630}{745588463} a^{4} - \frac{180023488}{745588463} a^{3} - \frac{340135308}{745588463} a^{2} + \frac{103611025}{745588463} a - \frac{209759280}{745588463}$, $\frac{1}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{21} - \frac{654413152081751057400331651651984018169404500627590111390176822206609377480152883244935181345958358}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{20} - \frac{322785747059787558468341805441145451795343042438297935041231237274439997061864205817200347038473697540662}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{19} - \frac{806577910248033382250418278998569814080252638912331780518454844166595840876671511512757812503864773457060}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{18} - \frac{16145793474355671819274807390310158281387760268416691657713782435659896111753562038972791613813846850273670}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{17} + \frac{51922389200468833674191556226690938267390496664421143399699059086122656540233995616622773185020462494016121}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{16} - \frac{463858582241190982175490435986071922119882273205291333261153976480213166682967363434815558432648449837979450}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{15} - \frac{135516490191616490831869863619338535477823285985675074942462428343417562692920677030027111931212734169055783}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{14} + \frac{36814084691131369937349778326468948631682613619234283148275263595016834832640229498399081541217996419987577}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{13} + \frac{102608052160296928717929604733298199536902149074034395595025072381265380680384151208320324964891361260462204}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{12} - \frac{463888577765818324357546728197147818372943539487921940172192386884803676133979027744430483918190065211962115}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{11} + \frac{618153430144378981485349091893545400270834671198164721753423126093800258090887137740728538308125470534710765}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{10} + \frac{247150401109152640565116410078398233198506579067669555835905417826413063928618254176032154751706994613208091}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{9} + \frac{1186246323559908602080835926274663520062600478224945011565425011558496325260541803990572267959795502235456325}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{8} + \frac{264714536241662761340131273372182777606516379916374790731094952927372745458474939040574986893995342939868580}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{7} + \frac{1161619030742455665915702488800951235180066946952648143284322218753112355901363211540042218446591375915075889}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{6} - \frac{41668672251412274774699013599422625128044995862871035402086090724338897607203575235179741900247331839687369}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{5} - \frac{628671274882243027516048835466463091484092970357082973297212328814612016905358837123708478682609669015537777}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{4} - \frac{147616087955359734386362432950406918946352000860499583695567072913169760206560057242569207881029126751792067}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{3} + \frac{482870528523860179315390777437135757208881709530444225313176850588650256426321358949859728641644997601416838}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a^{2} - \frac{207390395315751750852376110100330791024749634334298998440132119518388469256074522318899293969004600723530984}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803} a - \frac{1496734409382782089670486651418644972136150438473199266296112530475821250885269281166386780406265761301022121}{3024806308361963304297639765351018265334052779749819668416154526058985038209792690266106773550762807536702803}$
Class group and class number
$C_{63265334}$, which has order $63265334$ (assuming GRH)
Unit group
| Rank: | $10$ | 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: | \( 338444542.042557 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 22 |
| The 22 conjugacy class representatives for $C_{22}$ |
| Character table for $C_{22}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-871}) \), 11.11.1822837804551761449.1 |
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 | ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ | R | ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/29.1.0.1}{1} }^{22}$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ | ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
| 13 | Data not computed | ||||||
| 67 | Data not computed | ||||||