Normalized defining polynomial
\( x^{38} - x^{37} + 109 x^{36} - 318 x^{35} + 7607 x^{34} - 26082 x^{33} + 346271 x^{32} + \cdots + 63175314409 \)
Invariants
Degree: | $38$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 19]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-104\!\cdots\!707\) \(\medspace = -\,3^{19}\cdot 229^{36}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(297.98\) | 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: | $3^{1/2}229^{18/19}\approx 297.98488962998783$ | ||
Ramified primes: | \(3\), \(229\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Gal(K/\Q) }$: | $38$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(687=3\cdot 229\) | ||
Dirichlet character group: | $\lbrace$$\chi_{687}(256,·)$, $\chi_{687}(1,·)$, $\chi_{687}(515,·)$, $\chi_{687}(214,·)$, $\chi_{687}(518,·)$, $\chi_{687}(394,·)$, $\chi_{687}(271,·)$, $\chi_{687}(272,·)$, $\chi_{687}(17,·)$, $\chi_{687}(661,·)$, $\chi_{687}(286,·)$, $\chi_{687}(161,·)$, $\chi_{687}(290,·)$, $\chi_{687}(676,·)$, $\chi_{687}(683,·)$, $\chi_{687}(44,·)$, $\chi_{687}(562,·)$, $\chi_{687}(53,·)$, $\chi_{687}(443,·)$, $\chi_{687}(61,·)$, $\chi_{687}(454,·)$, $\chi_{687}(289,·)$, $\chi_{687}(203,·)$, $\chi_{687}(43,·)$, $\chi_{687}(218,·)$, $\chi_{687}(475,·)$, $\chi_{687}(350,·)$, $\chi_{687}(16,·)$, $\chi_{687}(485,·)$, $\chi_{687}(230,·)$, $\chi_{687}(104,·)$, $\chi_{687}(619,·)$, $\chi_{687}(623,·)$, $\chi_{687}(500,·)$, $\chi_{687}(245,·)$, $\chi_{687}(502,·)$, $\chi_{687}(121,·)$, $\chi_{687}(511,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{262144}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{18779}a^{34}-\frac{1805}{18779}a^{33}+\frac{3105}{18779}a^{32}+\frac{7534}{18779}a^{31}+\frac{784}{18779}a^{30}-\frac{4677}{18779}a^{29}-\frac{5655}{18779}a^{28}-\frac{4996}{18779}a^{27}+\frac{2872}{18779}a^{26}-\frac{7351}{18779}a^{25}-\frac{6429}{18779}a^{24}-\frac{8866}{18779}a^{23}-\frac{7249}{18779}a^{22}+\frac{8899}{18779}a^{21}+\frac{6020}{18779}a^{20}+\frac{5750}{18779}a^{19}+\frac{6965}{18779}a^{18}+\frac{4291}{18779}a^{17}+\frac{8703}{18779}a^{16}+\frac{1645}{18779}a^{15}-\frac{6735}{18779}a^{14}+\frac{6263}{18779}a^{13}+\frac{2548}{18779}a^{12}-\frac{3228}{18779}a^{11}-\frac{4611}{18779}a^{10}+\frac{5579}{18779}a^{9}+\frac{8903}{18779}a^{8}+\frac{6950}{18779}a^{7}+\frac{4683}{18779}a^{6}-\frac{4035}{18779}a^{5}-\frac{8562}{18779}a^{4}+\frac{3332}{18779}a^{3}+\frac{5424}{18779}a^{2}+\frac{7714}{18779}a-\frac{4015}{18779}$, $\frac{1}{8582003}a^{35}-\frac{20}{8582003}a^{34}+\frac{574538}{8582003}a^{33}-\frac{3370066}{8582003}a^{32}-\frac{90685}{8582003}a^{31}+\frac{1432321}{8582003}a^{30}-\frac{4091267}{8582003}a^{29}-\frac{653334}{8582003}a^{28}-\frac{821239}{8582003}a^{27}+\frac{1569938}{8582003}a^{26}+\frac{2871744}{8582003}a^{25}+\frac{2562061}{8582003}a^{24}-\frac{3870836}{8582003}a^{23}-\frac{2902580}{8582003}a^{22}+\frac{78817}{8582003}a^{21}+\frac{460558}{8582003}a^{20}-\frac{3325281}{8582003}a^{19}+\frac{2690515}{8582003}a^{18}-\frac{2265953}{8582003}a^{17}-\frac{143965}{8582003}a^{16}+\frac{1934303}{8582003}a^{15}-\frac{2289}{96427}a^{14}-\frac{1756728}{8582003}a^{13}-\frac{3736587}{8582003}a^{12}+\frac{768501}{8582003}a^{11}-\frac{1858975}{8582003}a^{10}-\frac{3478346}{8582003}a^{9}-\frac{795726}{8582003}a^{8}-\frac{3701949}{8582003}a^{7}+\frac{2702641}{8582003}a^{6}-\frac{1013967}{8582003}a^{5}+\frac{738649}{8582003}a^{4}+\frac{1896780}{8582003}a^{3}-\frac{3568420}{8582003}a^{2}+\frac{113142}{8582003}a-\frac{2134003}{8582003}$, $\frac{1}{48\!\cdots\!41}a^{36}-\frac{23\!\cdots\!33}{48\!\cdots\!41}a^{35}+\frac{56\!\cdots\!70}{48\!\cdots\!41}a^{34}-\frac{20\!\cdots\!93}{48\!\cdots\!41}a^{33}-\frac{19\!\cdots\!00}{48\!\cdots\!41}a^{32}+\frac{99\!\cdots\!12}{48\!\cdots\!41}a^{31}+\frac{61\!\cdots\!21}{48\!\cdots\!41}a^{30}-\frac{91\!\cdots\!64}{48\!\cdots\!41}a^{29}+\frac{51\!\cdots\!14}{48\!\cdots\!41}a^{28}-\frac{13\!\cdots\!80}{48\!\cdots\!41}a^{27}-\frac{21\!\cdots\!67}{48\!\cdots\!41}a^{26}-\frac{69\!\cdots\!38}{48\!\cdots\!41}a^{25}-\frac{50\!\cdots\!95}{48\!\cdots\!41}a^{24}+\frac{13\!\cdots\!27}{48\!\cdots\!41}a^{23}+\frac{35\!\cdots\!37}{48\!\cdots\!41}a^{22}+\frac{30\!\cdots\!56}{48\!\cdots\!41}a^{21}+\frac{18\!\cdots\!26}{48\!\cdots\!41}a^{20}+\frac{15\!\cdots\!66}{48\!\cdots\!41}a^{19}-\frac{23\!\cdots\!75}{48\!\cdots\!41}a^{18}-\frac{18\!\cdots\!92}{48\!\cdots\!41}a^{17}+\frac{23\!\cdots\!34}{48\!\cdots\!41}a^{16}-\frac{16\!\cdots\!11}{48\!\cdots\!41}a^{15}+\frac{24\!\cdots\!17}{48\!\cdots\!41}a^{14}-\frac{91\!\cdots\!59}{48\!\cdots\!41}a^{13}-\frac{11\!\cdots\!56}{48\!\cdots\!41}a^{12}+\frac{16\!\cdots\!67}{48\!\cdots\!41}a^{11}+\frac{15\!\cdots\!95}{48\!\cdots\!41}a^{10}+\frac{45\!\cdots\!72}{48\!\cdots\!41}a^{9}+\frac{10\!\cdots\!41}{48\!\cdots\!41}a^{8}+\frac{20\!\cdots\!24}{48\!\cdots\!41}a^{7}-\frac{27\!\cdots\!10}{48\!\cdots\!41}a^{6}-\frac{22\!\cdots\!09}{48\!\cdots\!41}a^{5}-\frac{75\!\cdots\!49}{48\!\cdots\!41}a^{4}+\frac{16\!\cdots\!09}{48\!\cdots\!41}a^{3}+\frac{16\!\cdots\!01}{48\!\cdots\!41}a^{2}+\frac{99\!\cdots\!71}{48\!\cdots\!41}a+\frac{71\!\cdots\!00}{48\!\cdots\!41}$, $\frac{1}{18\!\cdots\!63}a^{37}+\frac{44\!\cdots\!67}{18\!\cdots\!63}a^{36}+\frac{58\!\cdots\!01}{18\!\cdots\!63}a^{35}+\frac{24\!\cdots\!83}{18\!\cdots\!63}a^{34}+\frac{57\!\cdots\!21}{18\!\cdots\!63}a^{33}+\frac{16\!\cdots\!75}{18\!\cdots\!63}a^{32}+\frac{89\!\cdots\!69}{18\!\cdots\!63}a^{31}+\frac{63\!\cdots\!21}{18\!\cdots\!63}a^{30}-\frac{31\!\cdots\!35}{18\!\cdots\!63}a^{29}+\frac{26\!\cdots\!53}{18\!\cdots\!63}a^{28}+\frac{82\!\cdots\!98}{18\!\cdots\!63}a^{27}-\frac{51\!\cdots\!02}{18\!\cdots\!63}a^{26}-\frac{48\!\cdots\!17}{18\!\cdots\!63}a^{25}-\frac{85\!\cdots\!01}{18\!\cdots\!63}a^{24}+\frac{33\!\cdots\!42}{18\!\cdots\!63}a^{23}+\frac{39\!\cdots\!10}{18\!\cdots\!63}a^{22}-\frac{60\!\cdots\!15}{18\!\cdots\!63}a^{21}+\frac{41\!\cdots\!18}{18\!\cdots\!63}a^{20}-\frac{32\!\cdots\!95}{18\!\cdots\!63}a^{19}-\frac{59\!\cdots\!99}{18\!\cdots\!63}a^{18}-\frac{75\!\cdots\!14}{18\!\cdots\!63}a^{17}-\frac{79\!\cdots\!81}{18\!\cdots\!63}a^{16}-\frac{61\!\cdots\!05}{18\!\cdots\!63}a^{15}+\frac{91\!\cdots\!87}{18\!\cdots\!63}a^{14}+\frac{10\!\cdots\!75}{18\!\cdots\!63}a^{13}+\frac{82\!\cdots\!84}{18\!\cdots\!63}a^{12}-\frac{21\!\cdots\!13}{18\!\cdots\!63}a^{11}-\frac{65\!\cdots\!79}{18\!\cdots\!63}a^{10}-\frac{31\!\cdots\!57}{18\!\cdots\!63}a^{9}-\frac{48\!\cdots\!61}{18\!\cdots\!63}a^{8}+\frac{89\!\cdots\!06}{18\!\cdots\!63}a^{7}-\frac{52\!\cdots\!57}{18\!\cdots\!63}a^{6}-\frac{42\!\cdots\!02}{18\!\cdots\!63}a^{5}+\frac{38\!\cdots\!59}{18\!\cdots\!63}a^{4}-\frac{11\!\cdots\!41}{18\!\cdots\!63}a^{3}-\frac{86\!\cdots\!19}{18\!\cdots\!63}a^{2}+\frac{28\!\cdots\!26}{18\!\cdots\!63}a-\frac{20\!\cdots\!03}{74\!\cdots\!29}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $18$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{55586702350005621504291281964092074614416948600587231593409749281227979927475190247138328655833600263530749265940217853457441455390515598843752704544912916686512301189016303734837}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{37} + \frac{63700541561348811960024422120999064114133038771851583451704540620600374000450095703912101574165296287279194191156937647832647901688364509029788328086771538288325128085118566889641}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{36} - \frac{6070819822933056473503886154389670069356098202703290008848628967832853324932858339511607302830613606744887250952016072041916982673864867941110062450799508607387595574488728226088672}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{35} + \frac{18564203147199774337614281694790314258865012223421982681411950494020379334564313642066056804052135809332061544238775063093006620679636614308019963002753782108666179548024365668235414}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{34} - \frac{425837226986037138732426954853364631925562334633674103686349076369521915703719166680340435191356154208476152273059261811126852627356983291316578805261425804322048117953821495174833122}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{33} + \frac{1512671597827720088152144829082814574299961014736379240628640959351846835141622804362837348694612499772161261940638082716320383526051969937775495816337174382195610278150704043933414561}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{32} - \frac{19488105892474924058705822550740742469561314146254220436633150102952515257951423726111708656755007087613411655285812221427649310506818955361389652940097188386490024326083710062234860487}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{31} + \frac{73921262186799719057734050272954795346224044306939581949040048999183391320660238477753002843644520574902367594172669623599884660707689077763502774924703427601043788739739976226894288242}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{30} - \frac{657873586336819324756868399845377046431535608143240668962810226746683604587644215409590141792827286564584530311942966684356551164227502784186231592541660112322398617935369776088762325256}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{29} + \frac{2369533525954685274953389757987952260692087053685137447809773441034622607783023860566658851685985086588259185397805965557140461352974580627190182827682246140058670466428552007181178470815}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{28} - \frac{15847312046161114868038930371186706967477593055800020304339176290208439328377184897198175358311060870067698064918519463798748853023381445571160365645943425181424258273201975077018795206295}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{27} + \frac{51884131812680846330231293819171201609730326465745754585733766739611508920223338973605047293893917791753919076164642489286493993357434997919924040637568053112980801916337515296314081821666}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{26} - \frac{279297842312329571199923219802776644255566504466822540353631380405836499949406652308743217789779042377712535235528363996191578329612646422346850212466678146645007113192220706030959546773006}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{25} + \frac{786412300654298897831050094778161613738949227976368104938028520401722826792892933912142806114633918187453026185374793601568226244480642596139523524563052667406813697708357062585973381497883}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{24} - \frac{3417939882271392226549951691969351019484319902473889292515838321500645348569243233452774464586783020531811634121833754482820417019399028753762626188659878851397081844043154804667642395291768}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{23} + \frac{7740229280145840374681445275648517696790151473691252339012741052481637909182459604166977484634987228468971471233690082234572941351870645668443489141370901963879442542950630516450633231411242}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{22} - \frac{28936869664166308547500262162405349580765832343390966341843977155179727348365678692143763599239469452989099539071463907834955348502877362717756713056138672676212154129772337189140255481859908}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{21} + \frac{52560084508956731199951449121722126464800568804396979373630628976855592605819598236966972657609924298697696786720378848689730759106033277882329768289420657287352152147204284303907212384853685}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{20} - \frac{179562662633980906880558452898222049702744037300233331363820472995480709013788522146588209141650489392900358830967166443251819022575154921216841479219365092765274857127187207543715158130310028}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{19} + \frac{246903546409439295470277912169294937662995201134879957064422665061163197992790195233196443758448743771429014751967643462347290056142349488967014698483577850758544927596292870275384766471387081}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{18} - \frac{801142124812751242754238510786518258104256659347768178959103514508257069096716481804500884071312595284793085897912502318909557178405462088744479576746762204794464418064841040349733008390886741}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{17} + \frac{785839873312853709044777092688549161172643677676010188430123270927684072876461519166795937256914103984758612484982731101805963960623683426440765377337458379540004164953746543133958629567332962}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{16} - \frac{2671961778366496406163462809401730919044742959102219088137784964344193837636320495374231248381170118976311374223225642222058772875603450328959501360161111834607884173430358573794261489592462621}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{15} + \frac{1694023808784086245982980402449456170559577125009265431253333653408860150735951007699576245863020911486803179253339039181465335359607537856056091130518701165708984889216897184715403871987360735}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{14} - \frac{6162691363796572187849071339542078360945148913245913165030836444595120416193496140392322041845557433006596533740175275189006441636962335419657571188363455676888421498063095118352119990407231625}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{13} + \frac{2071198457819411350471021762780594321376820819137725913825292082253230823662278548025656625178950355257767322959050967811606523969307945032161824250382048368853183852808372313845391966241038165}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{12} - \frac{10154715170421044384822424324200129653281923633039501667726677157074535966964078554027146724908263533138059808170686798053789965315545875606301264652515363319188111796397788352495179799459118626}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{11} + \frac{643863972360458091204798763832020441360751021563638053823371337954743725195171727882193738444398495910399566127146165184105236260535693338040325950220621623012627642086471430760447682909670137}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{10} - \frac{9862448368625450692456790621943673256343299009746876006660644932489500786442287120957196238512880308150834794660025658427005391382156796004021197820471917989082226008486790760308840333240120508}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{9} - \frac{2598506373164270683580826524567806397806509762551835527079324906084823243919879642923573065417458834543476375451375601508207440139368946359985296002595967175545155971557052618973370733139401278}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{8} - \frac{7067715651129055330156846294771592674275055078835039267612119143771175835411124399050132144899039928274150280362260977831828897097260178405603036137922213254935633709626317864485038015411739850}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{7} - \frac{2758122686902895109583131857905490766391650414459763311635414317112555305697199978316485549379836913530207318280404642774445113347542788810542784734597498679276202588814701633539112527188473677}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{6} - \frac{3556018575747735727503120034330073352854963624465203968017178746777221342551598861433530154324875341797561669969189899759970582099594013805075409569783080571127592146932933801044964655696207319}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{5} - \frac{1720802742515326211000589941844385774220478057299458075446987722375630444263436734515440366748912157179379483242553760563440998135177209568059877347187567655066075699317670860515589453188996977}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{4} - \frac{1184329357613728391129018891594043373642503075069972604436530632767572372157769903650145519192623899296284915561348578297624391337317935035182062055789491358674814377258446558920351835912388658}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{3} - \frac{369497569692221659111910146366820843668648641260304164371770378327945985743526249726284872078356452965120995776339237812293616441820063770585383756200718048324326930344479490419398577186435586}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a^{2} - \frac{156016599382682360339657862327879627403167422756902892397837240313344581856487360829339210286992941160937068059736647282311412233950530268128650173721285146027959415476136440926710558663385225}{20340933385902240731646379675455089751676681287599109834103462891193948062988891695061146633135847887625575523150568555689889054307663399522084402820382456034933847248846864298439239741533267} a - \frac{31839754352465456150775555172886917743439940746842430972138237844239717301446209058348518139691950756011933980774773757106789156128717844205852484219985273674146455412256945128689953994}{80927695122290064061422573873788387176599208614382148321258908565425280838796133214484941666842444459753152109038773312153672231248685679646402793032669799261315421504322169345324351361} \) (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 38 |
The 38 conjugacy class representatives for $C_{38}$ |
Character table for $C_{38}$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 19.19.2999429662895796650415561622892044448017561.1 |
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 | $38$ | R | $38$ | $19^{2}$ | $38$ | $19^{2}$ | $38$ | $19^{2}$ | $38$ | $38$ | $19^{2}$ | $19^{2}$ | $38$ | $19^{2}$ | $38$ | $38$ | $38$ |
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 $38$ | $2$ | $19$ | $19$ | |||
\(229\) | Deg $19$ | $19$ | $1$ | $18$ | |||
Deg $19$ | $19$ | $1$ | $18$ |