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Properties

Label 33.33.1185378592...7249.1
Degree $33$
Signature $[33, 0]$
Discriminant $7^{22}\cdot 89^{30}$
Root discriminant $216.56$
Ramified primes $7, 89$
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![-88092733, 2900591056, -15544432294, 8651377149, 99441822253, -148659543549, -229613689017, 487414494739, 249662858736, -778209959832, -105270881315, 727374495587, -45802829578, -427869988653, 81953781823, 162998575211, -47126502322, -40385752376, 15227509843, 6410478765, -3064646220, -620888184, 397774975, 30987203, -33533077, -22554, 1807185, -90148, -59420, 5007, 1072, -116, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^33 - 8*x^32 - 116*x^31 + 1072*x^30 + 5007*x^29 - 59420*x^28 - 90148*x^27 + 1807185*x^26 - 22554*x^25 - 33533077*x^24 + 30987203*x^23 + 397774975*x^22 - 620888184*x^21 - 3064646220*x^20 + 6410478765*x^19 + 15227509843*x^18 - 40385752376*x^17 - 47126502322*x^16 + 162998575211*x^15 + 81953781823*x^14 - 427869988653*x^13 - 45802829578*x^12 + 727374495587*x^11 - 105270881315*x^10 - 778209959832*x^9 + 249662858736*x^8 + 487414494739*x^7 - 229613689017*x^6 - 148659543549*x^5 + 99441822253*x^4 + 8651377149*x^3 - 15544432294*x^2 + 2900591056*x - 88092733)
 
gp: K = bnfinit(x^33 - 8*x^32 - 116*x^31 + 1072*x^30 + 5007*x^29 - 59420*x^28 - 90148*x^27 + 1807185*x^26 - 22554*x^25 - 33533077*x^24 + 30987203*x^23 + 397774975*x^22 - 620888184*x^21 - 3064646220*x^20 + 6410478765*x^19 + 15227509843*x^18 - 40385752376*x^17 - 47126502322*x^16 + 162998575211*x^15 + 81953781823*x^14 - 427869988653*x^13 - 45802829578*x^12 + 727374495587*x^11 - 105270881315*x^10 - 778209959832*x^9 + 249662858736*x^8 + 487414494739*x^7 - 229613689017*x^6 - 148659543549*x^5 + 99441822253*x^4 + 8651377149*x^3 - 15544432294*x^2 + 2900591056*x - 88092733, 1)
 

Normalized defining polynomial

\( x^{33} - 8 x^{32} - 116 x^{31} + 1072 x^{30} + 5007 x^{29} - 59420 x^{28} - 90148 x^{27} + 1807185 x^{26} - 22554 x^{25} - 33533077 x^{24} + 30987203 x^{23} + 397774975 x^{22} - 620888184 x^{21} - 3064646220 x^{20} + 6410478765 x^{19} + 15227509843 x^{18} - 40385752376 x^{17} - 47126502322 x^{16} + 162998575211 x^{15} + 81953781823 x^{14} - 427869988653 x^{13} - 45802829578 x^{12} + 727374495587 x^{11} - 105270881315 x^{10} - 778209959832 x^{9} + 249662858736 x^{8} + 487414494739 x^{7} - 229613689017 x^{6} - 148659543549 x^{5} + 99441822253 x^{4} + 8651377149 x^{3} - 15544432294 x^{2} + 2900591056 x - 88092733 \)

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:  \(118537859218696126011390650866708479108541231162187324696804896914030203497249=7^{22}\cdot 89^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $216.56$
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:  $7, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(623=7\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{623}(256,·)$, $\chi_{623}(1,·)$, $\chi_{623}(2,·)$, $\chi_{623}(4,·)$, $\chi_{623}(134,·)$, $\chi_{623}(8,·)$, $\chi_{623}(128,·)$, $\chi_{623}(268,·)$, $\chi_{623}(16,·)$, $\chi_{623}(401,·)$, $\chi_{623}(275,·)$, $\chi_{623}(512,·)$, $\chi_{623}(536,·)$, $\chi_{623}(156,·)$, $\chi_{623}(32,·)$, $\chi_{623}(550,·)$, $\chi_{623}(39,·)$, $\chi_{623}(93,·)$, $\chi_{623}(179,·)$, $\chi_{623}(312,·)$, $\chi_{623}(186,·)$, $\chi_{623}(64,·)$, $\chi_{623}(449,·)$, $\chi_{623}(67,·)$, $\chi_{623}(331,·)$, $\chi_{623}(78,·)$, $\chi_{623}(345,·)$, $\chi_{623}(477,·)$, $\chi_{623}(484,·)$, $\chi_{623}(358,·)$, $\chi_{623}(242,·)$, $\chi_{623}(372,·)$, $\chi_{623}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{37} a^{27} + \frac{18}{37} a^{26} - \frac{2}{37} a^{25} - \frac{17}{37} a^{24} + \frac{6}{37} a^{22} + \frac{14}{37} a^{21} + \frac{2}{37} a^{20} + \frac{10}{37} a^{19} - \frac{4}{37} a^{18} + \frac{10}{37} a^{17} - \frac{14}{37} a^{16} - \frac{6}{37} a^{15} + \frac{7}{37} a^{14} + \frac{16}{37} a^{13} + \frac{8}{37} a^{12} - \frac{10}{37} a^{11} - \frac{17}{37} a^{10} - \frac{7}{37} a^{9} - \frac{1}{37} a^{8} + \frac{18}{37} a^{7} + \frac{4}{37} a^{6} - \frac{8}{37} a^{5} - \frac{16}{37} a^{4} - \frac{3}{37} a^{3} + \frac{5}{37} a^{2} - \frac{13}{37} a - \frac{2}{37}$, $\frac{1}{37} a^{28} + \frac{7}{37} a^{26} - \frac{18}{37} a^{25} + \frac{10}{37} a^{24} + \frac{6}{37} a^{23} + \frac{17}{37} a^{22} + \frac{9}{37} a^{21} + \frac{11}{37} a^{20} + \frac{1}{37} a^{19} + \frac{8}{37} a^{18} - \frac{9}{37} a^{17} - \frac{13}{37} a^{16} + \frac{4}{37} a^{15} + \frac{1}{37} a^{14} + \frac{16}{37} a^{13} - \frac{6}{37} a^{12} + \frac{15}{37} a^{11} + \frac{3}{37} a^{10} + \frac{14}{37} a^{9} - \frac{1}{37} a^{8} + \frac{13}{37} a^{7} - \frac{6}{37} a^{6} + \frac{17}{37} a^{5} - \frac{11}{37} a^{4} - \frac{15}{37} a^{3} + \frac{8}{37} a^{2} + \frac{10}{37} a - \frac{1}{37}$, $\frac{1}{37} a^{29} + \frac{4}{37} a^{26} - \frac{13}{37} a^{25} + \frac{14}{37} a^{24} + \frac{17}{37} a^{23} + \frac{4}{37} a^{22} - \frac{13}{37} a^{21} - \frac{13}{37} a^{20} + \frac{12}{37} a^{19} - \frac{18}{37} a^{18} - \frac{9}{37} a^{17} - \frac{9}{37} a^{16} + \frac{6}{37} a^{15} + \frac{4}{37} a^{14} - \frac{7}{37} a^{13} - \frac{4}{37} a^{12} - \frac{1}{37} a^{11} - \frac{15}{37} a^{10} + \frac{11}{37} a^{9} - \frac{17}{37} a^{8} + \frac{16}{37} a^{7} - \frac{11}{37} a^{6} + \frac{8}{37} a^{5} - \frac{14}{37} a^{4} - \frac{8}{37} a^{3} + \frac{12}{37} a^{2} + \frac{16}{37} a + \frac{14}{37}$, $\frac{1}{677618999} a^{30} - \frac{7861178}{677618999} a^{29} - \frac{9153949}{677618999} a^{28} - \frac{8245500}{677618999} a^{27} + \frac{119465900}{677618999} a^{26} - \frac{293689154}{677618999} a^{25} - \frac{197109426}{677618999} a^{24} - \frac{200665269}{677618999} a^{23} - \frac{135273985}{677618999} a^{22} + \frac{40280142}{677618999} a^{21} + \frac{66957628}{677618999} a^{20} + \frac{278498913}{677618999} a^{19} + \frac{60805106}{677618999} a^{18} + \frac{46067436}{677618999} a^{17} + \frac{324151871}{677618999} a^{16} + \frac{126102110}{677618999} a^{15} + \frac{146691735}{677618999} a^{14} + \frac{6382060}{18314027} a^{13} - \frac{70438397}{677618999} a^{12} - \frac{202600602}{677618999} a^{11} - \frac{247678478}{677618999} a^{10} + \frac{160188592}{677618999} a^{9} - \frac{260919060}{677618999} a^{8} + \frac{4431999}{677618999} a^{7} + \frac{202966479}{677618999} a^{6} + \frac{205544793}{677618999} a^{5} - \frac{288548673}{677618999} a^{4} - \frac{47356549}{677618999} a^{3} - \frac{170558283}{677618999} a^{2} + \frac{288664408}{677618999} a - \frac{213635774}{677618999}$, $\frac{1}{677618999} a^{31} - \frac{8553913}{677618999} a^{29} - \frac{611024}{677618999} a^{28} - \frac{3377433}{677618999} a^{27} + \frac{205422666}{677618999} a^{26} - \frac{318574275}{677618999} a^{25} - \frac{239842501}{677618999} a^{24} - \frac{223523631}{677618999} a^{23} + \frac{226938204}{677618999} a^{22} + \frac{71081147}{677618999} a^{21} - \frac{31504272}{677618999} a^{20} + \frac{108980400}{677618999} a^{19} - \frac{143003366}{677618999} a^{18} - \frac{282813462}{677618999} a^{17} - \frac{66570849}{677618999} a^{16} + \frac{47885194}{677618999} a^{15} - \frac{214879612}{677618999} a^{14} + \frac{182606491}{677618999} a^{13} - \frac{179195085}{677618999} a^{12} - \frac{46770283}{677618999} a^{11} - \frac{107900367}{677618999} a^{10} - \frac{639146}{677618999} a^{9} + \frac{167682810}{677618999} a^{8} - \frac{16930012}{677618999} a^{7} - \frac{7394673}{677618999} a^{6} - \frac{13247093}{677618999} a^{5} + \frac{56865558}{677618999} a^{4} - \frac{139440640}{677618999} a^{3} - \frac{8197237}{677618999} a^{2} + \frac{20997129}{677618999} a + \frac{119856423}{677618999}$, $\frac{1}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{32} - \frac{2382392532544774466022270718311748977574967322086681767604244245667430767343588760896951393294852101513357074801541025972739688}{21463840274701032538650535867245917362577759883356923401166935321337818618260156092526183070171877418841864766622447569920263320819831681} a^{31} + \frac{60264871534740171303148318589750798974116098034957160127315452462267162726680219992059125934962606977342316402525568917939749867}{103838578626256346605903943790189708321659973489753764562402200608634311693745079474653696474615298864126859276362651757181814443966212727} a^{30} - \frac{1872182286025295702588323034721205514201877726255245105996992075047202797420170016148412108622156701092101295196897494643096533998932926}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{29} - \frac{31676642289538644756819149886102015164768862858063155058961257950715608767756219602918255332902472763093809116609131088585648746774757415}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{28} - \frac{32274049002343676455803635563918071864056168959884296086844028949815762043223823849782876135255169472103636410245165172742127399198605335}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{27} + \frac{1439856463610014563428902349178295997022405379128903757753143542501465075215180785896442138091692553487976754643435250445139419364026872911}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{26} + \frac{1720051891343587827715008307188835700804079603817091540222136074887105465985337736326677219703142069063040971126175568326603972142502794416}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{25} - \frac{1875164534842400536478209704227621922297171366584647284117899096029936798807908327483854151507560238742589948111866572086891166435410260631}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{24} + \frac{576389826567647058513822294050191602182660024485081123233246271330809360614192451317283072492641979726312012317001579499852864002763375418}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{23} - \frac{118973931734201902278494020279149571538379655860103392524195280685369658139195333143363441867532604465147618500052574639143735432080873595}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{22} - \frac{105430576467271938816815939032047296865010520326142833583801481187286980941620367510981326578569827436024435261556694163397114473176373584}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{21} - \frac{88148194781341398367794509838868775221132773434724170164618390312881912801294151073506293985887272684421901677962421937795072954508406060}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{20} - \frac{1883056440976056347814303254891128382206751925708969369592759871531601640541081405270981668544683340584235304829899533111071094127146136852}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{19} - \frac{1866803898245633273261747269475158914225122109759197419027870751680561925042692726973467442235126214165182585875630020932781209474171166352}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{18} + \frac{862116362454652730731510951676276204438358451105418696016446910548424023963006222286200611407460458327511561290100372812232013582965671327}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{17} - \frac{3083101113579929855918907834146445929656728453432555617838444754333097661707566133591728629008437067944895325313912472245237969234035602}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{16} + \frac{1481365204648940413402505099682865508953140912908900112106027002425464551241734605651446945380525695957433895673547877614808963853775304637}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{15} + \frac{12315251608109032502286261076178834091372665214621355341743001158803152131082325005391072806700196152047630869979583653765299770187254492}{38039875338331532915034118022148705028726920981394943453553281411083856759094732084774126431294713445274195972528892227878486479472770999} a^{14} - \frac{1016946778349032191033244380833671985399551666692014726510208253528359842506471679972359910225232824087930645986935697063092180123599787456}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{13} - \frac{1584121703010897921200010262925069731105923103483160566596032297940673181622258149838873039358283056016017273170349296758164370107994484582}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{12} - \frac{1878914362865340558769048115672163447518951594440561619134125270287379272441595176944552096252581802945653778573007731083995292872380965377}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{11} - \frac{34385154942548484193970387956952617695436929165377986920263073438632296500320073874658907947860906161532743447067942534106177618320141435}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{10} + \frac{1884472513812997819045351812025084829104615391091553640922954681131339956919462160983211976882041294657176315896125821244095394583835648244}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{9} + \frac{271106623332635271968482226537793632953193612716405827266845247169711020608158998607109771782529322292235360514754921570436363141353749}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{8} - \frac{1068811178162196379749682608841404431828706907867138816693410790229636070353756977400559732715046261376996214207802349027036545060409050081}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{7} + \frac{1666328010108868676535592870454353638107512622606652247260096264664632125969641740809205679084035361618444081823649930947901014425088732825}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{6} - \frac{1783165125039352248093133148403854945063317889131837727627795071836867045543634512502633336098612646498793139243997250183677948168639261511}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{5} + \frac{815947844668758990663826094209155630233887741326077336737938831669249819146608074798243494638580284484782560012395097904590558027865803}{103838578626256346605903943790189708321659973489753764562402200608634311693745079474653696474615298864126859276362651757181814443966212727} a^{4} - \frac{1834089283105998165066631631172502543045383589335681045873817321481571409650905840296253652527085436165439093106591761266789364975559822972}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{3} - \frac{254523169846733648202806233579020794510160644561228389224942686152639479322285346731086500977639796123093452182238895900087571084279968842}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a^{2} - \frac{9397264409587534242397741011426216867754587395115706562369525067164233704943260763676198151977127254608093428652873960192639900599951329}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899} a - \frac{519216121130870351989758775058048609845601845914699424654723390141086774800894763230385416203915422476381261064215409703722630707017371017}{3842027409171484824418445920237019207901419019120889288808881422519469532668567940562186769560766057972693793225418115015727134426749870899}$

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

\(\Q(\zeta_{7})^+\), 11.11.31181719929966183601.1

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 $33$ $33$ $33$ R $33$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{3}$ $33$ $33$ $33$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{3}$ $33$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{11}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{3}$ $33$ $33$ $33$

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
7Data not computed
89Data not computed