Normalized defining polynomial
\( x^{34} - x^{33} + 49 x^{32} - 162 x^{31} + 1646 x^{30} - 6093 x^{29} + 37764 x^{28} - 140505 x^{27} + \cdots + 8048569 \)
Invariants
| Degree: | $34$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 17]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-3325466068076643664357827857598159738994734276327509143073421552355865283\)
\(\medspace = -\,3^{17}\cdot 103^{32}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(135.83\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $3^{1/2}103^{16/17}\approx 135.82998801634537$ | ||
| Ramified primes: |
\(3\), \(103\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $34$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(309=3\cdot 103\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{309}(1,·)$, $\chi_{309}(133,·)$, $\chi_{309}(8,·)$, $\chi_{309}(137,·)$, $\chi_{309}(13,·)$, $\chi_{309}(14,·)$, $\chi_{309}(272,·)$, $\chi_{309}(278,·)$, $\chi_{309}(23,·)$, $\chi_{309}(287,·)$, $\chi_{309}(34,·)$, $\chi_{309}(164,·)$, $\chi_{309}(167,·)$, $\chi_{309}(169,·)$, $\chi_{309}(299,·)$, $\chi_{309}(175,·)$, $\chi_{309}(179,·)$, $\chi_{309}(182,·)$, $\chi_{309}(184,·)$, $\chi_{309}(61,·)$, $\chi_{309}(64,·)$, $\chi_{309}(196,·)$, $\chi_{309}(203,·)$, $\chi_{309}(76,·)$, $\chi_{309}(79,·)$, $\chi_{309}(214,·)$, $\chi_{309}(215,·)$, $\chi_{309}(220,·)$, $\chi_{309}(100,·)$, $\chi_{309}(229,·)$, $\chi_{309}(104,·)$, $\chi_{309}(236,·)$, $\chi_{309}(112,·)$, $\chi_{309}(116,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{65536}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{7003}a^{30}+\frac{111}{7003}a^{29}-\frac{137}{7003}a^{28}-\frac{2783}{7003}a^{27}+\frac{2962}{7003}a^{26}+\frac{1757}{7003}a^{25}+\frac{2457}{7003}a^{24}+\frac{933}{7003}a^{23}+\frac{319}{7003}a^{22}+\frac{116}{7003}a^{21}+\frac{243}{7003}a^{20}-\frac{193}{7003}a^{19}+\frac{2672}{7003}a^{18}-\frac{869}{7003}a^{17}+\frac{2126}{7003}a^{16}+\frac{1386}{7003}a^{15}-\frac{2142}{7003}a^{14}+\frac{2030}{7003}a^{13}+\frac{1723}{7003}a^{12}-\frac{2245}{7003}a^{11}+\frac{1361}{7003}a^{10}+\frac{337}{7003}a^{9}-\frac{3438}{7003}a^{8}-\frac{314}{7003}a^{7}+\frac{696}{7003}a^{6}+\frac{767}{7003}a^{5}-\frac{2172}{7003}a^{4}-\frac{1539}{7003}a^{3}+\frac{1712}{7003}a^{2}+\frac{2792}{7003}a+\frac{2975}{7003}$, $\frac{1}{7003}a^{31}+\frac{1548}{7003}a^{29}-\frac{1582}{7003}a^{28}-\frac{3260}{7003}a^{27}+\frac{2116}{7003}a^{26}-\frac{3489}{7003}a^{25}+\frac{1323}{7003}a^{24}+\frac{1801}{7003}a^{23}-\frac{278}{7003}a^{22}+\frac{1373}{7003}a^{21}+\frac{18}{149}a^{20}+\frac{3086}{7003}a^{19}-\frac{3335}{7003}a^{18}+\frac{543}{7003}a^{17}-\frac{3501}{7003}a^{16}-\frac{1922}{7003}a^{15}+\frac{1690}{7003}a^{14}+\frac{489}{7003}a^{13}+\frac{2586}{7003}a^{12}-\frac{1552}{7003}a^{11}+\frac{3332}{7003}a^{10}+\frac{1173}{7003}a^{9}+\frac{3142}{7003}a^{8}+\frac{535}{7003}a^{7}+\frac{544}{7003}a^{6}-\frac{3273}{7003}a^{5}+\frac{1451}{7003}a^{4}-\frac{2534}{7003}a^{3}+\frac{1841}{7003}a^{2}+\frac{1195}{7003}a-\frac{1084}{7003}$, $\frac{1}{61\!\cdots\!31}a^{32}+\frac{2740278978268}{61\!\cdots\!31}a^{31}+\frac{61711665784}{13\!\cdots\!73}a^{30}+\frac{10\!\cdots\!56}{61\!\cdots\!31}a^{29}-\frac{14\!\cdots\!80}{61\!\cdots\!31}a^{28}+\frac{13\!\cdots\!01}{61\!\cdots\!31}a^{27}+\frac{25\!\cdots\!74}{61\!\cdots\!31}a^{26}-\frac{24\!\cdots\!11}{61\!\cdots\!31}a^{25}-\frac{22\!\cdots\!37}{61\!\cdots\!31}a^{24}+\frac{23\!\cdots\!27}{61\!\cdots\!31}a^{23}-\frac{13\!\cdots\!71}{61\!\cdots\!31}a^{22}+\frac{631575503794464}{61\!\cdots\!31}a^{21}+\frac{53\!\cdots\!61}{61\!\cdots\!31}a^{20}-\frac{98\!\cdots\!06}{61\!\cdots\!31}a^{19}-\frac{17\!\cdots\!63}{61\!\cdots\!31}a^{18}+\frac{13\!\cdots\!82}{61\!\cdots\!31}a^{17}+\frac{97\!\cdots\!53}{61\!\cdots\!31}a^{16}-\frac{16\!\cdots\!31}{61\!\cdots\!31}a^{15}-\frac{25\!\cdots\!40}{61\!\cdots\!31}a^{14}+\frac{15\!\cdots\!58}{61\!\cdots\!31}a^{13}-\frac{92\!\cdots\!83}{61\!\cdots\!31}a^{12}-\frac{22\!\cdots\!72}{61\!\cdots\!31}a^{11}+\frac{938157148985112}{61\!\cdots\!31}a^{10}+\frac{62\!\cdots\!42}{61\!\cdots\!31}a^{9}-\frac{24\!\cdots\!88}{61\!\cdots\!31}a^{8}+\frac{28\!\cdots\!44}{61\!\cdots\!31}a^{7}-\frac{20\!\cdots\!19}{61\!\cdots\!31}a^{6}+\frac{29\!\cdots\!63}{61\!\cdots\!31}a^{5}-\frac{57\!\cdots\!55}{61\!\cdots\!31}a^{4}+\frac{24\!\cdots\!81}{61\!\cdots\!31}a^{3}+\frac{13\!\cdots\!98}{61\!\cdots\!31}a^{2}-\frac{10\!\cdots\!15}{61\!\cdots\!31}a-\frac{11\!\cdots\!62}{61\!\cdots\!31}$, $\frac{1}{34\!\cdots\!31}a^{33}-\frac{10\!\cdots\!37}{34\!\cdots\!31}a^{32}+\frac{37\!\cdots\!74}{34\!\cdots\!31}a^{31}+\frac{13\!\cdots\!75}{34\!\cdots\!31}a^{30}+\frac{22\!\cdots\!22}{34\!\cdots\!31}a^{29}-\frac{17\!\cdots\!85}{34\!\cdots\!31}a^{28}-\frac{43\!\cdots\!34}{34\!\cdots\!31}a^{27}-\frac{49\!\cdots\!49}{34\!\cdots\!31}a^{26}-\frac{10\!\cdots\!93}{34\!\cdots\!31}a^{25}+\frac{10\!\cdots\!11}{34\!\cdots\!31}a^{24}+\frac{15\!\cdots\!39}{34\!\cdots\!31}a^{23}+\frac{39\!\cdots\!83}{34\!\cdots\!31}a^{22}+\frac{51\!\cdots\!10}{34\!\cdots\!31}a^{21}-\frac{44\!\cdots\!72}{34\!\cdots\!31}a^{20}+\frac{29\!\cdots\!52}{34\!\cdots\!31}a^{19}-\frac{74\!\cdots\!60}{34\!\cdots\!31}a^{18}-\frac{18\!\cdots\!82}{34\!\cdots\!31}a^{17}-\frac{14\!\cdots\!36}{34\!\cdots\!31}a^{16}+\frac{47\!\cdots\!87}{34\!\cdots\!31}a^{15}-\frac{12\!\cdots\!85}{34\!\cdots\!31}a^{14}+\frac{15\!\cdots\!50}{34\!\cdots\!31}a^{13}+\frac{13\!\cdots\!92}{34\!\cdots\!31}a^{12}+\frac{38\!\cdots\!57}{34\!\cdots\!31}a^{11}-\frac{76\!\cdots\!74}{34\!\cdots\!31}a^{10}+\frac{57\!\cdots\!28}{34\!\cdots\!31}a^{9}+\frac{11\!\cdots\!14}{34\!\cdots\!31}a^{8}-\frac{77\!\cdots\!03}{34\!\cdots\!31}a^{7}+\frac{95\!\cdots\!62}{34\!\cdots\!31}a^{6}+\frac{47\!\cdots\!14}{34\!\cdots\!31}a^{5}-\frac{11\!\cdots\!34}{34\!\cdots\!31}a^{4}+\frac{10\!\cdots\!72}{34\!\cdots\!31}a^{3}-\frac{69\!\cdots\!28}{34\!\cdots\!31}a^{2}+\frac{80\!\cdots\!98}{34\!\cdots\!31}a+\frac{38\!\cdots\!22}{12\!\cdots\!63}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $16$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{10093384700223231136421203329878504243716579091494248146892569416960280312212480048387996973873248687541575249011}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{33} + \frac{8030281439862351211852410023327459654814238686568262595648858718242558463557101072130192344730408021764565599937}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{32} - \frac{493109689425189012108674684531467096955070636114660005941265398632616598791355568496290397277445671799060807928020}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{31} + \frac{1534458069507709062151911142124541323528350398805983723265310909951152939952753192086757296472271739600656121940725}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{30} - \frac{16308786358331071385799345821323064694742256744072080153735452576789167433578747845616710208389788433920933091538970}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{29} + \frac{58191445580344532442656821966429961544248911670781659075284371605048961619796267723159141213918590482578460239860618}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{28} - \frac{369561166134978958369330792568040924950301861503909080271125559464169103726643959238770106416485237612884021303910548}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{27} + \frac{1343643603969952461696881617522732752585651973376609292495611292060885007410398282063860677389651092070881032216075751}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{26} - \frac{6212207052372720443373315727111166113915576094926493582540289827852285315006848924603936062525854772788496955972712905}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{25} + \frac{20544280478499146230416865940138462015004476411103309129306745066704992676680154383590085526936923543753093094805305900}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{24} - \frac{75319613368093325551596480766390721627398727356076343794014179515496179514108448279885108505578168319488642773330136548}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{23} + \frac{221682762955598182769506256346408351061411778411876273047263294142335400672480186816126624154358672984355346762008340203}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{22} - \frac{672598838310174578751907655413208543711839521912101378480910277183028870653195012909606301322006465546245729605326051734}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{21} + \frac{1710787611734145245644058130636302997966442917720812805481235226815208746706077020900868848865688237720269407817118696526}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{20} - \frac{4300511173463311530300800684990629607865004152934200919006620796855123604747639279925755436575041894232798051432588720744}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{19} + \frac{9313554760967791279538940100762127843169197642880439249338242904743594635845103655736922912070033859161793942361087552243}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{18} - \frac{19734879055785322120541447925310944504803935532658600163044503295502822560967033551144315871659031446632104820137943626061}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{17} + \frac{36699378520259020162154655951981349829767038392316208401652763343739272385804533170675329913745075729788597252644369180412}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{16} - \frac{66329023205363481086097889877031751156424887418592657607666741674408052790572517844669086543685498535467787608685969754741}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{15} + \frac{105458750918821462940399327680187104344084196802835589611040276642704245150530109134010548404320962005786886602055270405080}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{14} - \frac{162014622792353986645097935716230632581278928154798617066382396694592461702947725928873766803274718875798385684731680963273}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{13} + \frac{218606334671211095942999797825238732798196596788437887661398688491577308893602236016482059087956291789944312189355064785239}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{12} - \frac{284360560162349369161996480034214882454101294579557098201710480241643137164667429752664252813876860951719284311311788655579}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{11} + \frac{319867682569932408077575715547072269820335127384624374739768265741651641975404167312671082367522596768143942928818740045167}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{10} - \frac{342182781966127956571827342015732095648723662545526408996079084632327885395136280963733141715789437892624900340952308175960}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{9} + \frac{307100848279672359393174136958426143572169746231502233183984071980199923969104889358217391139434741820564888991251107505643}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{8} - \frac{263893570495407101356984777289516140100257502513929163010768547137383564654706645361234141646147629358038476695861758752383}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{7} + \frac{188545473524912548263788432734020340055836074497256964942154837744881158223358801446023907020030918120664655700855615690336}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{6} - \frac{132461366563522853131626191900471861980851278123685315918415612434616314453250837411585563301525028306880663192224947736782}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{5} + \frac{72237730726434749249550036395431966540451056122386236484844719107731527328330220825598968293606770205157977715350771587268}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{4} - \frac{35668312453220422243199895636340795011213316529207115456177227901576160575949476690984014860049778372658293867254009924501}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{3} + \frac{11527164268737741401027239894727524098056912359995772696904942607013288619020789869862732167011622663395485958159883331711}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a^{2} - \frac{3604766282884167725584515044431604285573192075423670009767571282051609028257366856068335815021325523320088942728294151808}{344719541402703224315553393479283900424482766627232821683361733855383948401185085834874773985197412778557702945667110019} a + \frac{208214485819547193638735692088959558212081977832803567775274026401501043256816630189080405988556401025747011364903179}{121508474234297928909253927909511420664251944528457110216200822649060256750505846258327378916178150433048185740453687} \)
(order $6$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $
Galois group
| A cyclic group of order 34 |
| The 34 conjugacy class representatives for $C_{34}$ |
| Character table for $C_{34}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 17.17.160470643909878751793805444097921.1 |
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 | $34$ | R | $34$ | $17^{2}$ | $34$ | $17^{2}$ | $34$ | $17^{2}$ | $34$ | $34$ | $17^{2}$ | $17^{2}$ | $34$ | $17^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{17}$ | $34$ | $34$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $34$ | $2$ | $17$ | $17$ | |||
|
\(103\)
| 103.17.16.1 | $x^{17} + 103$ | $17$ | $1$ | $16$ | $C_{17}$ | $[\ ]_{17}$ |
| 103.17.16.1 | $x^{17} + 103$ | $17$ | $1$ | $16$ | $C_{17}$ | $[\ ]_{17}$ |