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Properties

Label 34.0.11900532391...4307.1
Degree $34$
Signature $[0, 17]$
Discriminant $-\,307^{33}$
Root discriminant $259.41$
Ramified prime $307$
Class number Not computed
Class group Not computed
Galois group $C_{34}$ (as 34T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1529204967919, 2831508795674, -1703289189460, 2105471029067, 3366043453003, -6932505875168, 8308097316560, -5216066869146, 4320866108408, -1353081034662, 699331611291, -322848882778, -247987829208, -51790646711, -63184882750, 20964157639, 6322928754, 7261880290, 3218823280, 753601139, 355208564, 27920994, 9900617, 1954599, -903587, 146749, 36049, -12987, 11996, 611, 17, 173, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 - x^33 + 5*x^32 + 173*x^31 + 17*x^30 + 611*x^29 + 11996*x^28 - 12987*x^27 + 36049*x^26 + 146749*x^25 - 903587*x^24 + 1954599*x^23 + 9900617*x^22 + 27920994*x^21 + 355208564*x^20 + 753601139*x^19 + 3218823280*x^18 + 7261880290*x^17 + 6322928754*x^16 + 20964157639*x^15 - 63184882750*x^14 - 51790646711*x^13 - 247987829208*x^12 - 322848882778*x^11 + 699331611291*x^10 - 1353081034662*x^9 + 4320866108408*x^8 - 5216066869146*x^7 + 8308097316560*x^6 - 6932505875168*x^5 + 3366043453003*x^4 + 2105471029067*x^3 - 1703289189460*x^2 + 2831508795674*x + 1529204967919)
 
gp: K = bnfinit(x^34 - x^33 + 5*x^32 + 173*x^31 + 17*x^30 + 611*x^29 + 11996*x^28 - 12987*x^27 + 36049*x^26 + 146749*x^25 - 903587*x^24 + 1954599*x^23 + 9900617*x^22 + 27920994*x^21 + 355208564*x^20 + 753601139*x^19 + 3218823280*x^18 + 7261880290*x^17 + 6322928754*x^16 + 20964157639*x^15 - 63184882750*x^14 - 51790646711*x^13 - 247987829208*x^12 - 322848882778*x^11 + 699331611291*x^10 - 1353081034662*x^9 + 4320866108408*x^8 - 5216066869146*x^7 + 8308097316560*x^6 - 6932505875168*x^5 + 3366043453003*x^4 + 2105471029067*x^3 - 1703289189460*x^2 + 2831508795674*x + 1529204967919, 1)
 

Normalized defining polynomial

\( x^{34} - x^{33} + 5 x^{32} + 173 x^{31} + 17 x^{30} + 611 x^{29} + 11996 x^{28} - 12987 x^{27} + 36049 x^{26} + 146749 x^{25} - 903587 x^{24} + 1954599 x^{23} + 9900617 x^{22} + 27920994 x^{21} + 355208564 x^{20} + 753601139 x^{19} + 3218823280 x^{18} + 7261880290 x^{17} + 6322928754 x^{16} + 20964157639 x^{15} - 63184882750 x^{14} - 51790646711 x^{13} - 247987829208 x^{12} - 322848882778 x^{11} + 699331611291 x^{10} - 1353081034662 x^{9} + 4320866108408 x^{8} - 5216066869146 x^{7} + 8308097316560 x^{6} - 6932505875168 x^{5} + 3366043453003 x^{4} + 2105471029067 x^{3} - 1703289189460 x^{2} + 2831508795674 x + 1529204967919 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $34$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 17]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-11900532391042529187998394459246062870229883563431367936702497007412902558434544307=-\,307^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $259.41$
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:  $307$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(307\)
Dirichlet character group:    $\lbrace$$\chi_{307}(1,·)$, $\chi_{307}(3,·)$, $\chi_{307}(8,·)$, $\chi_{307}(9,·)$, $\chi_{307}(269,·)$, $\chi_{307}(272,·)$, $\chi_{307}(273,·)$, $\chi_{307}(24,·)$, $\chi_{307}(283,·)$, $\chi_{307}(34,·)$, $\chi_{307}(35,·)$, $\chi_{307}(38,·)$, $\chi_{307}(64,·)$, $\chi_{307}(298,·)$, $\chi_{307}(299,·)$, $\chi_{307}(304,·)$, $\chi_{307}(306,·)$, $\chi_{307}(115,·)$, $\chi_{307}(192,·)$, $\chi_{307}(193,·)$, $\chi_{307}(72,·)$, $\chi_{307}(202,·)$, $\chi_{307}(205,·)$, $\chi_{307}(81,·)$, $\chi_{307}(216,·)$, $\chi_{307}(91,·)$, $\chi_{307}(280,·)$, $\chi_{307}(226,·)$, $\chi_{307}(102,·)$, $\chi_{307}(105,·)$, $\chi_{307}(235,·)$, $\chi_{307}(27,·)$, $\chi_{307}(114,·)$, $\chi_{307}(243,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} - \frac{1}{17} a^{13} - \frac{1}{17} a^{12} + \frac{1}{17} a^{11} + \frac{3}{17} a^{10} - \frac{3}{17} a^{9} - \frac{5}{17} a^{8} + \frac{5}{17} a^{7} - \frac{6}{17} a^{6} + \frac{6}{17} a^{5} - \frac{4}{17} a^{4} + \frac{4}{17} a^{3} - \frac{8}{17} a^{2} + \frac{8}{17} a$, $\frac{1}{17} a^{15} - \frac{2}{17} a^{13} + \frac{4}{17} a^{11} - \frac{8}{17} a^{9} - \frac{1}{17} a^{7} + \frac{2}{17} a^{5} - \frac{4}{17} a^{3} + \frac{8}{17} a$, $\frac{1}{17} a^{16} - \frac{2}{17} a^{13} + \frac{2}{17} a^{12} + \frac{2}{17} a^{11} - \frac{2}{17} a^{10} - \frac{6}{17} a^{9} + \frac{6}{17} a^{8} - \frac{7}{17} a^{7} + \frac{7}{17} a^{6} - \frac{5}{17} a^{5} + \frac{5}{17} a^{4} + \frac{8}{17} a^{3} - \frac{8}{17} a^{2} - \frac{1}{17} a$, $\frac{1}{17} a^{17} - \frac{1}{17} a$, $\frac{1}{17} a^{18} - \frac{1}{17} a^{2}$, $\frac{1}{17} a^{19} - \frac{1}{17} a^{3}$, $\frac{1}{17} a^{20} - \frac{1}{17} a^{4}$, $\frac{1}{17} a^{21} - \frac{1}{17} a^{5}$, $\frac{1}{17} a^{22} - \frac{1}{17} a^{6}$, $\frac{1}{17} a^{23} - \frac{1}{17} a^{7}$, $\frac{1}{289} a^{24} - \frac{3}{289} a^{23} - \frac{7}{289} a^{22} + \frac{6}{289} a^{21} - \frac{7}{289} a^{20} - \frac{5}{289} a^{19} + \frac{2}{289} a^{18} + \frac{6}{289} a^{17} + \frac{7}{289} a^{16} + \frac{6}{289} a^{15} + \frac{2}{289} a^{14} + \frac{142}{289} a^{13} + \frac{131}{289} a^{12} - \frac{96}{289} a^{11} - \frac{8}{289} a^{10} - \frac{45}{289} a^{9} - \frac{88}{289} a^{8} - \frac{127}{289} a^{7} + \frac{10}{289} a^{6} - \frac{6}{17} a^{5} - \frac{4}{17} a^{4} - \frac{40}{289} a^{3} - \frac{23}{289} a^{2} + \frac{1}{17} a$, $\frac{1}{15317} a^{25} + \frac{4}{15317} a^{24} + \frac{23}{15317} a^{23} - \frac{230}{15317} a^{22} + \frac{52}{15317} a^{21} + \frac{14}{15317} a^{20} - \frac{203}{15317} a^{19} - \frac{405}{15317} a^{18} + \frac{202}{15317} a^{17} + \frac{361}{15317} a^{16} + \frac{44}{15317} a^{15} - \frac{354}{15317} a^{14} - \frac{1578}{15317} a^{13} + \frac{2521}{15317} a^{12} - \frac{14}{53} a^{11} - \frac{5422}{15317} a^{10} - \frac{131}{15317} a^{9} + \frac{6533}{15317} a^{8} + \frac{2759}{15317} a^{7} + \frac{155}{15317} a^{6} - \frac{368}{901} a^{5} - \frac{7418}{15317} a^{4} - \frac{2615}{15317} a^{3} - \frac{4734}{15317} a^{2} - \frac{158}{901} a$, $\frac{1}{15317} a^{26} + \frac{7}{15317} a^{24} - \frac{322}{15317} a^{23} + \frac{71}{15317} a^{22} - \frac{194}{15317} a^{21} - \frac{259}{15317} a^{20} + \frac{407}{15317} a^{19} + \frac{20}{15317} a^{18} - \frac{447}{15317} a^{17} + \frac{402}{15317} a^{16} + \frac{7}{289} a^{15} - \frac{162}{15317} a^{14} + \frac{3427}{15317} a^{13} + \frac{4791}{15317} a^{12} + \frac{2653}{15317} a^{11} + \frac{2636}{15317} a^{10} + \frac{4354}{15317} a^{9} + \frac{52}{289} a^{8} + \frac{6238}{15317} a^{7} + \frac{6639}{15317} a^{6} - \frac{4919}{15317} a^{5} + \frac{5433}{15317} a^{4} + \frac{1221}{15317} a^{3} + \frac{3636}{15317} a^{2} + \frac{49}{901} a$, $\frac{1}{15317} a^{27} + \frac{21}{15317} a^{24} - \frac{302}{15317} a^{23} - \frac{280}{15317} a^{22} - \frac{199}{15317} a^{21} + \frac{415}{15317} a^{20} - \frac{414}{15317} a^{19} + \frac{427}{15317} a^{18} + \frac{313}{15317} a^{17} + \frac{441}{15317} a^{16} - \frac{46}{15317} a^{15} + \frac{20}{901} a^{14} + \frac{1845}{15317} a^{13} - \frac{6037}{15317} a^{12} - \frac{168}{901} a^{11} + \frac{5102}{15317} a^{10} + \frac{294}{901} a^{9} + \frac{5345}{15317} a^{8} + \frac{1477}{15317} a^{7} + \frac{4013}{15317} a^{6} + \frac{2373}{15317} a^{5} + \frac{4493}{15317} a^{4} + \frac{4398}{15317} a^{3} + \frac{2012}{15317} a^{2} + \frac{417}{901} a$, $\frac{1}{15317} a^{28} - \frac{15}{15317} a^{24} - \frac{74}{15317} a^{23} + \frac{232}{15317} a^{22} - \frac{253}{15317} a^{21} + \frac{299}{15317} a^{20} + \frac{132}{15317} a^{19} - \frac{351}{15317} a^{18} + \frac{227}{15317} a^{17} + \frac{376}{15317} a^{16} - \frac{160}{15317} a^{15} + \frac{110}{15317} a^{14} + \frac{5901}{15317} a^{13} - \frac{1790}{15317} a^{12} + \frac{2194}{15317} a^{11} - \frac{1238}{15317} a^{10} + \frac{3114}{15317} a^{9} + \frac{5529}{15317} a^{8} - \frac{4636}{15317} a^{7} - \frac{5281}{15317} a^{6} - \frac{5588}{15317} a^{5} - \frac{1103}{15317} a^{4} - \frac{5666}{15317} a^{3} + \frac{6068}{15317} a^{2} - \frac{445}{901} a$, $\frac{1}{15317} a^{29} - \frac{14}{15317} a^{24} - \frac{324}{15317} a^{23} - \frac{99}{15317} a^{22} + \frac{178}{15317} a^{21} + \frac{342}{15317} a^{20} + \frac{208}{15317} a^{19} - \frac{26}{901} a^{18} - \frac{198}{15317} a^{17} - \frac{151}{15317} a^{16} - \frac{131}{15317} a^{15} - \frac{310}{15317} a^{14} + \frac{3372}{15317} a^{13} - \frac{536}{15317} a^{12} - \frac{660}{15317} a^{11} + \frac{6478}{15317} a^{10} - \frac{40}{15317} a^{9} + \frac{4160}{15317} a^{8} - \frac{5342}{15317} a^{7} + \frac{6648}{15317} a^{6} + \frac{2365}{15317} a^{5} - \frac{2509}{15317} a^{4} - \frac{3424}{15317} a^{3} - \frac{2891}{15317} a^{2} + \frac{15}{901} a$, $\frac{1}{15317} a^{30} - \frac{3}{15317} a^{24} + \frac{329}{15317} a^{23} - \frac{392}{15317} a^{22} - \frac{43}{15317} a^{21} + \frac{351}{15317} a^{20} - \frac{104}{15317} a^{19} + \frac{4}{901} a^{18} - \frac{14}{901} a^{17} - \frac{430}{15317} a^{16} + \frac{94}{15317} a^{15} - \frac{9}{901} a^{14} + \frac{1487}{15317} a^{13} - \frac{7236}{15317} a^{12} - \frac{4427}{15317} a^{11} + \frac{319}{15317} a^{10} - \frac{589}{15317} a^{9} - \frac{270}{15317} a^{8} + \frac{6213}{15317} a^{7} - \frac{7231}{15317} a^{6} + \frac{7}{15317} a^{5} + \frac{1745}{15317} a^{4} + \frac{5761}{15317} a^{3} + \frac{3568}{15317} a^{2} + \frac{120}{901} a$, $\frac{1}{15317} a^{31} + \frac{23}{15317} a^{24} - \frac{270}{15317} a^{23} - \frac{309}{15317} a^{22} + \frac{401}{15317} a^{21} + \frac{362}{15317} a^{20} + \frac{148}{15317} a^{19} - \frac{287}{15317} a^{18} + \frac{70}{15317} a^{17} - \frac{148}{15317} a^{16} - \frac{127}{15317} a^{15} - \frac{211}{15317} a^{14} - \frac{1264}{15317} a^{13} - \frac{358}{901} a^{12} - \frac{55}{289} a^{11} - \frac{796}{15317} a^{10} - \frac{6175}{15317} a^{9} - \frac{2066}{15317} a^{8} + \frac{3590}{15317} a^{7} + \frac{5401}{15317} a^{6} - \frac{2607}{15317} a^{5} - \frac{3879}{15317} a^{4} - \frac{5973}{15317} a^{3} + \frac{1459}{15317} a^{2} - \frac{103}{901} a$, $\frac{1}{13800617} a^{32} + \frac{38}{13800617} a^{31} - \frac{92}{13800617} a^{30} - \frac{347}{13800617} a^{29} + \frac{206}{13800617} a^{28} - \frac{260}{13800617} a^{27} + \frac{41}{13800617} a^{26} + \frac{9}{13800617} a^{25} - \frac{1}{2809} a^{24} + \frac{294123}{13800617} a^{23} + \frac{358897}{13800617} a^{22} + \frac{133859}{13800617} a^{21} - \frac{332831}{13800617} a^{20} + \frac{204256}{13800617} a^{19} - \frac{284884}{13800617} a^{18} - \frac{143336}{13800617} a^{17} + \frac{326119}{13800617} a^{16} + \frac{266358}{13800617} a^{15} + \frac{91631}{13800617} a^{14} - \frac{374172}{811801} a^{13} + \frac{4805922}{13800617} a^{12} - \frac{2796493}{13800617} a^{11} + \frac{6450913}{13800617} a^{10} - \frac{2420842}{13800617} a^{9} - \frac{4141286}{13800617} a^{8} - \frac{4419510}{13800617} a^{7} - \frac{2028793}{13800617} a^{6} + \frac{1449877}{13800617} a^{5} + \frac{1251172}{13800617} a^{4} + \frac{5269309}{13800617} a^{3} - \frac{6446681}{13800617} a^{2} + \frac{6122}{15317} a$, $\frac{1}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{33} - \frac{926176672247040297315075369881054803156035148523182470159464996394944641717917351722096229848262017773630795570198972445440421822488179086613026548332047501116247672351932805000012734762}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{32} + \frac{156204379293992322310471921192202122134146448036344440162063375428483798238525568032690745141297949297543982116677640415597300364749478040460947272783921853648470453728792666967289024750909}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{31} - \frac{775525642901653369928776280963555902394348118061884264466299465055326002441820011142805960081193891580780314539648755141122171101931649779261833292642195458262881554370439915293383045340400}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{30} - \frac{845640596931080839758122668154151559298603645594703249989905685730950886795854706988389154163651719535063332953566741413154652828815133510868367974760617641853795437485551208650903987622055}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{29} - \frac{830380296913648400258428770614037395823157540106645272975053458497145070772751288985725668725043887135798747747961912832750466725746146109976483433494368740922077332611664262251938466069168}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{28} + \frac{611062763382187092643451985242319607806097986008824978452429275978810369517133924129602795363399856393966722662437056161068058353646948542965303900390841000149205469532557007645482532446488}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{27} + \frac{398030900171522895022397475519886020647740671399284547955323766013031533277857619223889057667316067397146564056834375216415045415803784915015994801812614066615666378206359514204015833989595}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{26} - \frac{834913418999105543818477678750621779132531618665420414391093950036229392039012606758207192358796496313124905180434196408998833414297397316885283898509271324607348739353824188513521957562434}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{25} - \frac{26092085307286916955623406325688678129949856331201151765555449745159241405489327365118826987642409598745354403337809155578900794126817398779555468733771850182529762212583075252697041329300339}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{24} + \frac{10820475678909651463576376215913752279547852647803706386104299198395215815718192270111828727309205822672220370822576802233718852323941372668997378882941286049454041763137203874570311261045668}{1561416722457878111730297827893920367945881636633974596850496250775983949850008178329570937701868532062708925173527004848623083335420557682579872347930934327760322867278717104869926976664670971} a^{23} - \frac{291942777833781584158778163404810030011037234031504280110264567666912799088002401422101637396373548631310751392187993789303820685038161314764011808366118901813023024445868171932136562881328715}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{22} + \frac{469530374768524791255903653175777646304821350613638152662207550292193518327376693413058742104769702933889209756808909432787096978323419456896800840704507823071687172869006598875751052757959592}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{21} + \frac{123094946462818877801175602155828811334266542140316518982416652159672634993194666978850294525145396344661528651608076161843247395754141189312495540793980039152364951578964578298549189298183932}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{20} - \frac{574078473433866370408130651859725918962966122953358651998473049872412857485347012512433039473631158335449361805489748719617004511210172467942177259713976055242881588255320996506928181827870370}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{19} - \frac{495966380917737685470875449870729388517785290568273502206171225993651658603323948171286274166047254726482679807900723287002140986263003224908616185598046499618234350639962062158861247256057002}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{18} + \frac{218518436558471254153758982383920206019984011635132747099533140937829064191756932795296758551288305547658696039422159751839698220899330299938805247575787180849484873901527440703800537039332854}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{17} + \frac{22492067620722261900100346953255457357797477750977509691173266420049455051292062497319904706026864960655483516294713894138577752092577115752184751543847276127117391360278414257339139704220224}{1561416722457878111730297827893920367945881636633974596850496250775983949850008178329570937701868532062708925173527004848623083335420557682579872347930934327760322867278717104869926976664670971} a^{16} - \frac{253357178961248275395405488600264495819591726113013124277238935031458863228787560410017604327059690089488378974622791632296350249491375944505409869031404227261080040782991857000954261747051164}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{15} - \frac{550943380261702374361015222628546077399106479699732631655515221926208366295552891664784755131925011825734612806917985063063346297760183786407246041029747680972535633267478838942001182662656067}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{14} - \frac{8165400940864044097746774730150570020629710162543125793530722091469389824035225788892955754343872057131928169136205257616131325440622734790729033299957746128446654896543952973450156720999022271}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{13} + \frac{9911869579628198491891076662581329392867082019308900989060756905748165035919837119699458151404154754564309033987339926811309436839382487016895158963616900108668170798152213979267314805733190508}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{12} + \frac{1535529043574485873455622303949320336633549484190138182059960235237429898591369313601381139686368613519067986398003672189121926766277357183381213251470464755377385050136802995799726004363968951}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{11} + \frac{12662959292845489168730376600301140143846061519599911561994713086875161496094077233470866238846170691965061447005913419306352883815610651444663763051952970688301467046084960333230744526290250095}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{10} - \frac{8410639698594057928912400122014869649348323562065748910292153504395361551002077618198364244623394733182153269411396620154386301834607196464800622712112191504606863696207780328423267867435799144}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{9} + \frac{8275912595188669351830218295179376008869516563723740482886548703443121509602755168849815430297042451345097913455473398756216698232993233237270391529537153374485260807439538642245050388008375246}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{8} - \frac{12142242747072255986469959364308996516840607642786108639131657701259425589445973849458458927983051631924892998156407679446770902375151956627591797045600736798112256649556837601174481762868588383}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{7} + \frac{10698044767634652938098559821534864292118425122758433852727305798639506246129655062954686733090079487176237944205776520040819757979567177009171938328004051329170000543612064976440560530760542288}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{6} - \frac{7077515600388995921319172544256583388573108066720275260728195681758724619114333282879781714658698302745456350672760793937574179430296349040657645136014031151185133261452492443347008081867166753}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{5} - \frac{9005398460542927292188492877598485534191691086210414448116471613016516354366880080351759072936101086963176815282245681988788078342757960950909861982734249661607998263391615242911889362516389704}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{4} - \frac{5708250277384150424289413510799175120302959860332482285566612244261001605949352764767132345896573274035471387506514832693374738350202656446779758123160878773790527075303350688792015515630479181}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{3} + \frac{7926207799149975612946183866151568964195477472963617295760598185965094107193277357817739375014992701356025780149896580262583740588065657673914450844689184942743177999982148474907644811844119676}{26544084281783927899415063074196646255079987822777568146458436263191727147450139031602705940931765045066051727949959082426592416702149480603857829914825883571925488743738190782788758603299406507} a^{2} - \frac{3526958157776184772075235962402166851592001637819209340826757963148572947128199320479428614050973226228605282833944048317009993148379304006036584667291649355565377994308517784311343865693787}{29460692876563737957175430714979629583884559181773105600952759448603470751886946760935300711356010038919036324028811412238171383687180333633582497130772345806798544665636171789998622201220207} a + \frac{446236956178451655309110814022108387792884679625802747233774039224430256038327211465919131716941967498610026088051786092126014085497116018822178816591298324050682104545105675200757591936}{1923398372825209764129753262060431519480613643779663485078850913925930061492912891619462082088921462356795477184096847439979851386510435048219788283004005079767483493219048886204780453171}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $16$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{34}$ (as 34T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 34
The 34 conjugacy class representatives for $C_{34}$
Character table for $C_{34}$ is not computed

Intermediate fields

\(\Q(\sqrt{-307}) \), 17.17.6226070121392010397563990173530787496001.1

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 $34$ $34$ $34$ $17^{2}$ $17^{2}$ $34$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{34}$ $17^{2}$ $34$ $34$ $34$ $17^{2}$ $17^{2}$ $34$ $34$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{34}$ $34$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
307Data not computed