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Properties

Label 33.33.1945350369...5281.1
Degree $33$
Signature $[33, 0]$
Discriminant $13^{22}\cdot 67^{30}$
Root discriminant $252.75$
Ramified primes $13, 67$
Class number Not computed
Class group Not computed
Galois group $C_{33}$ (as 33T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8672509, 227068790, 1212304894, -3948105037, -22741173483, 52579210647, 77322457297, -188148415319, -130110920902, 323210228406, 142179324189, -320812557227, -111827183164, 199922687695, 63440275941, -82027847149, -25350973354, 22835380012, 7038388451, -4397140947, -1351731560, 592568182, 178946863, -56209081, -16170687, 3753002, 974941, -174850, -37410, 5517, 826, -108, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^33 - 8*x^32 - 108*x^31 + 826*x^30 + 5517*x^29 - 37410*x^28 - 174850*x^27 + 974941*x^26 + 3753002*x^25 - 16170687*x^24 - 56209081*x^23 + 178946863*x^22 + 592568182*x^21 - 1351731560*x^20 - 4397140947*x^19 + 7038388451*x^18 + 22835380012*x^17 - 25350973354*x^16 - 82027847149*x^15 + 63440275941*x^14 + 199922687695*x^13 - 111827183164*x^12 - 320812557227*x^11 + 142179324189*x^10 + 323210228406*x^9 - 130110920902*x^8 - 188148415319*x^7 + 77322457297*x^6 + 52579210647*x^5 - 22741173483*x^4 - 3948105037*x^3 + 1212304894*x^2 + 227068790*x + 8672509)
 
gp: K = bnfinit(x^33 - 8*x^32 - 108*x^31 + 826*x^30 + 5517*x^29 - 37410*x^28 - 174850*x^27 + 974941*x^26 + 3753002*x^25 - 16170687*x^24 - 56209081*x^23 + 178946863*x^22 + 592568182*x^21 - 1351731560*x^20 - 4397140947*x^19 + 7038388451*x^18 + 22835380012*x^17 - 25350973354*x^16 - 82027847149*x^15 + 63440275941*x^14 + 199922687695*x^13 - 111827183164*x^12 - 320812557227*x^11 + 142179324189*x^10 + 323210228406*x^9 - 130110920902*x^8 - 188148415319*x^7 + 77322457297*x^6 + 52579210647*x^5 - 22741173483*x^4 - 3948105037*x^3 + 1212304894*x^2 + 227068790*x + 8672509, 1)
 

Normalized defining polynomial

\( x^{33} - 8 x^{32} - 108 x^{31} + 826 x^{30} + 5517 x^{29} - 37410 x^{28} - 174850 x^{27} + 974941 x^{26} + 3753002 x^{25} - 16170687 x^{24} - 56209081 x^{23} + 178946863 x^{22} + 592568182 x^{21} - 1351731560 x^{20} - 4397140947 x^{19} + 7038388451 x^{18} + 22835380012 x^{17} - 25350973354 x^{16} - 82027847149 x^{15} + 63440275941 x^{14} + 199922687695 x^{13} - 111827183164 x^{12} - 320812557227 x^{11} + 142179324189 x^{10} + 323210228406 x^{9} - 130110920902 x^{8} - 188148415319 x^{7} + 77322457297 x^{6} + 52579210647 x^{5} - 22741173483 x^{4} - 3948105037 x^{3} + 1212304894 x^{2} + 227068790 x + 8672509 \)

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

Invariants

Degree:  $33$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[33, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(19453503693937104587930129781385805482525288081047228454446873116938739522375281=13^{22}\cdot 67^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $252.75$
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$$\chi_{871}(1,·)$, $\chi_{871}(131,·)$, $\chi_{871}(263,·)$, $\chi_{871}(9,·)$, $\chi_{871}(269,·)$, $\chi_{871}(14,·)$, $\chi_{871}(399,·)$, $\chi_{871}(22,·)$, $\chi_{871}(68,·)$, $\chi_{871}(282,·)$, $\chi_{871}(796,·)$, $\chi_{871}(159,·)$, $\chi_{871}(417,·)$, $\chi_{871}(679,·)$, $\chi_{871}(40,·)$, $\chi_{871}(685,·)$, $\chi_{871}(560,·)$, $\chi_{871}(308,·)$, $\chi_{871}(692,·)$, $\chi_{871}(828,·)$, $\chi_{871}(196,·)$, $\chi_{871}(198,·)$, $\chi_{871}(484,·)$, $\chi_{871}(464,·)$, $\chi_{871}(81,·)$, $\chi_{871}(729,·)$, $\chi_{871}(92,·)$, $\chi_{871}(612,·)$, $\chi_{871}(360,·)$, $\chi_{871}(107,·)$, $\chi_{871}(625,·)$, $\chi_{871}(627,·)$, $\chi_{871}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{239} a^{28} + \frac{52}{239} a^{27} - \frac{3}{239} a^{26} - \frac{75}{239} a^{25} - \frac{89}{239} a^{24} - \frac{19}{239} a^{23} + \frac{38}{239} a^{22} + \frac{111}{239} a^{21} - \frac{111}{239} a^{20} + \frac{29}{239} a^{19} + \frac{117}{239} a^{18} - \frac{51}{239} a^{17} - \frac{21}{239} a^{16} - \frac{56}{239} a^{15} + \frac{15}{239} a^{14} - \frac{65}{239} a^{13} + \frac{80}{239} a^{12} - \frac{2}{239} a^{11} + \frac{70}{239} a^{10} + \frac{88}{239} a^{9} + \frac{101}{239} a^{8} + \frac{87}{239} a^{7} + \frac{77}{239} a^{6} + \frac{84}{239} a^{5} - \frac{63}{239} a^{4} - \frac{27}{239} a^{3} - \frac{36}{239} a^{2} + \frac{17}{239} a + \frac{32}{239}$, $\frac{1}{239} a^{29} - \frac{78}{239} a^{27} + \frac{81}{239} a^{26} - \frac{13}{239} a^{25} + \frac{68}{239} a^{24} + \frac{70}{239} a^{23} + \frac{47}{239} a^{22} + \frac{92}{239} a^{21} + \frac{65}{239} a^{20} + \frac{43}{239} a^{19} + \frac{79}{239} a^{18} + \frac{2}{239} a^{17} + \frac{80}{239} a^{16} + \frac{59}{239} a^{15} + \frac{111}{239} a^{14} + \frac{114}{239} a^{13} - \frac{99}{239} a^{12} - \frac{65}{239} a^{11} + \frac{33}{239} a^{10} + \frac{66}{239} a^{9} + \frac{93}{239} a^{8} + \frac{94}{239} a^{7} - \frac{96}{239} a^{6} + \frac{110}{239} a^{5} - \frac{97}{239} a^{4} - \frac{66}{239} a^{3} - \frac{23}{239} a^{2} + \frac{104}{239} a + \frac{9}{239}$, $\frac{1}{256447} a^{30} + \frac{423}{256447} a^{29} + \frac{113}{256447} a^{28} + \frac{58757}{256447} a^{27} - \frac{4802}{256447} a^{26} + \frac{18962}{256447} a^{25} + \frac{36691}{256447} a^{24} - \frac{125498}{256447} a^{23} - \frac{92030}{256447} a^{22} + \frac{41301}{256447} a^{21} + \frac{91660}{256447} a^{20} - \frac{102624}{256447} a^{19} + \frac{16092}{256447} a^{18} + \frac{111880}{256447} a^{17} + \frac{94657}{256447} a^{16} - \frac{110386}{256447} a^{15} + \frac{2958}{8843} a^{14} - \frac{101000}{256447} a^{13} - \frac{102425}{256447} a^{12} - \frac{9441}{256447} a^{11} - \frac{113615}{256447} a^{10} - \frac{85675}{256447} a^{9} - \frac{114551}{256447} a^{8} - \frac{2277}{6931} a^{7} + \frac{4562}{256447} a^{6} - \frac{30255}{256447} a^{5} - \frac{1267}{256447} a^{4} + \frac{3947}{256447} a^{3} + \frac{69778}{256447} a^{2} + \frac{74255}{256447} a - \frac{35491}{256447}$, $\frac{1}{256447} a^{31} + \frac{375}{256447} a^{29} + \frac{228}{256447} a^{28} + \frac{98967}{256447} a^{27} - \frac{72186}{256447} a^{26} - \frac{20429}{256447} a^{25} + \frac{58637}{256447} a^{24} + \frac{90432}{256447} a^{23} + \frac{1471}{6931} a^{22} - \frac{32545}{256447} a^{21} + \frac{121235}{256447} a^{20} + \frac{43581}{256447} a^{19} + \frac{50915}{256447} a^{18} + \frac{91936}{256447} a^{17} + \frac{1897}{8843} a^{16} - \frac{4813}{256447} a^{15} + \frac{11520}{256447} a^{14} - \frac{3776}{8843} a^{13} + \frac{99113}{256447} a^{12} - \frac{52617}{256447} a^{11} + \frac{51144}{256447} a^{10} + \frac{78539}{256447} a^{9} + \frac{96554}{256447} a^{8} + \frac{6486}{256447} a^{7} + \frac{14339}{256447} a^{6} + \frac{59}{239} a^{5} - \frac{9488}{256447} a^{4} - \frac{57902}{256447} a^{3} - \frac{95289}{256447} a^{2} + \frac{86895}{256447} a + \frac{125891}{256447}$, $\frac{1}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{32} - \frac{11231026513664647198361789549981398379104240065194468545477253153576315860787967883221825922804179208288682234841651444110713359}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{31} - \frac{6462564463596229866619247050445673668741302787825997394954675712585226903435587162507639530854683157533253159258474856970855462}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{30} - \frac{30536329803869690787295491934063078937156967453770638143123500997348036140931919971701069483662757064951232749976936083958320766341}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{29} + \frac{16994991318575602820167719603635251874184080375688212193604206352341524132581162326415326828621170935647792844897907087346378268317}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{28} + \frac{3500621592143191011296359500946980685930581900366517672922122728487190046551904928572194314583520191862531665831007711706260704676780}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{27} - \frac{701614832589626272909029982508384200202104760813839309558890638140985699102932192916073449201467246599749347245630846590898098475092}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{26} + \frac{4170622582271597265678753114878393924609140903047951304757391957816708974808973134140291188341985434438391073702778031944928473586967}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{25} + \frac{7494202875153019285823327080916404881966421712086302704915811183146322323813813418834342582692222088819728598341196684984194827857687}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{24} + \frac{60921994824386858371279977372619351973598762886033908554475484107261896068960846572494475616225675675459835112734918812159369901269}{458194656562471833358159148662580241398276400483675731637930316519395930993248798564443452621308579358410289838443371630573189978507} a^{23} + \frac{5414461205017763844712375694211458645540491634415528462003810723243695551876457795037712637853533921444103258367925736734749303645757}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{22} + \frac{1202128915249481752352932617788214136291537117661770168443746623320910168451927343253265356857928707960533705210554982906536055811516}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{21} - \frac{3227033169801970497430038372853200015155284117257582848379441226319191111984167254229067056511062247843528260502102013789909293537531}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{20} - \frac{228345371782733274320925625942873780617075009187919403589813598253800266367484682200831400969404917104152011890205647782481666031335}{584593182510739925319030637948809273508145752341241450710462817628194808508627777478772680930635084009006231862841543114869242386371} a^{19} - \frac{5040600141446754374609120913336970698713947581211195489139293315993594171311717626126793303448171795864046270266731264059607068763900}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{18} - \frac{1826568398877270213062519613876177287101726424689241492381018843427280486638479694021920788681170015883527972292924447659233110547472}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{17} - \frac{7261610472112294884086383617916963385264986027442487309349406451585698313594842051046977808563109606820257769354923977922095971545346}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{16} - \frac{3937717945889909104362368672339991703874229739887103657880292065611379649966559507840115455336639285681575790370949844014206351335491}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{15} - \frac{7956424053162553549851355542923655779599125843027066088178496003350617246548841193475983483085117782432570555227225591864681688134580}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{14} + \frac{7802193788212798889877418242804144360248218400858566449374879891709973141241259470244841432911853996680812275474095781285622357776061}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{13} - \frac{4905409124832309462840207088431539121200029876377085447420520825864141993797795103351619747025010427472867904802398158890661409146331}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{12} - \frac{8224859590084627773449614738629996452509802471451719518292925231792209319843729863728789296404388174421251952446044602805378409266847}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{11} - \frac{2326988668181593367740545200360261454269686835741937992892627853041463663973727239336034356730137276709005517090038889612704825245985}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{10} - \frac{1574206708522172829129714811600099055025754953149526484893766359253014031906765171935305114657426330069838559862439719654759760366062}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{9} - \frac{4764181889846411535210112237657379813081776774262660956995669821212030756767915300476943157433753203313904457044910831503891032043253}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{8} + \frac{1497878211843615413111707598097281972901480569764942032313867195467037460644289201680463491673752902377072559323623105496344790292333}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{7} - \frac{7577678847390928040880853833699681886165664225823788697575858911172627132345043398792510310417690987236291091482648483613816653956952}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{6} + \frac{742670475845867471941477691567946887476527112028442472258561299315557384127103620499219669312285266632828390290915663118859946758369}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{5} + \frac{5498836782608169469468301852072516288345980179908980305651625636220382677332946484183907418413814408516549931274315918367969367283333}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{4} + \frac{1592986331884272260860212816112582736123861468973477046815034209590079365587277187074104964180106169281388668865258320021920033244410}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{3} - \frac{5967401801706030002598131616290339548709125708265737102736028161630108297154226562215717164289960118501899813390031572960697799714534}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a^{2} + \frac{1418875837138700959999979153267439484726215362400926444740958895232820780970547886003790056417727239902264180840082665664449008375550}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759} a + \frac{1311417060330778672815778772038676883806460802980258996348242104845966244658589763160213875528099292550341424409310958361983156514712}{16953202292811457834251888500515468931736226817896002070603421711217649446750205546884407746988417436261180724022404750331208029204759}$

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:  $32$
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

$C_{33}$ (as 33T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 33
The 33 conjugacy class representatives for $C_{33}$
Character table for $C_{33}$ is not computed

Intermediate fields

3.3.169.1, 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 $33$ $33$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{3}$ $33$ $33$ R $33$ $33$ $33$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{11}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{11}$ $33$ $33$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{3}$ $33$

In the table, R denotes a ramified prime. 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
13Data not computed
67Data not computed