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Properties

Label 22.14.1801251752...6529.1
Degree $22$
Signature $[14, 4]$
Discriminant $7^{10}\cdot 251^{10}\cdot 33893^{10}\cdot 1792166448977^{2}$
Root discriminant $44{,}461.15$
Ramified primes $7, 251, 33893, 1792166448977$
Class number Not computed
Class group Not computed
Galois group 22T51

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-133708001088324569741903664154721856901221813847387342588501211986724999, -529892375938497944209490322543314025268468196224103590775254275058, 529891435435819882897472854176851909714543741204188875533041914071, 1881004369773644332011928098109604490847972220298952944948509, -940499719015049358045656409945459020782100574483826871424882, -2959044233311684659640186168114291609441101078849350201, 986344921008399892498102559525711856850940930222963887, 2705794236189661878727800809354360542269940615953, -676446181077317936754989616553711670587334903597, -1585312256016172990286816098491706527517901, 317061308864468722415113561065019172613923, 623093432219287495095768869183695022, -103848538639555020655359200328639754, -169260037585124322611844491980, 24179924667975031656547634456, 32280512415301802940118, -4035052200728025834218, -4182769221227550, 464751108360962, 324420235, -32441985, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 - 32441985*x^20 + 324420235*x^19 + 464751108360962*x^18 - 4182769221227550*x^17 - 4035052200728025834218*x^16 + 32280512415301802940118*x^15 + 24179924667975031656547634456*x^14 - 169260037585124322611844491980*x^13 - 103848538639555020655359200328639754*x^12 + 623093432219287495095768869183695022*x^11 + 317061308864468722415113561065019172613923*x^10 - 1585312256016172990286816098491706527517901*x^9 - 676446181077317936754989616553711670587334903597*x^8 + 2705794236189661878727800809354360542269940615953*x^7 + 986344921008399892498102559525711856850940930222963887*x^6 - 2959044233311684659640186168114291609441101078849350201*x^5 - 940499719015049358045656409945459020782100574483826871424882*x^4 + 1881004369773644332011928098109604490847972220298952944948509*x^3 + 529891435435819882897472854176851909714543741204188875533041914071*x^2 - 529892375938497944209490322543314025268468196224103590775254275058*x - 133708001088324569741903664154721856901221813847387342588501211986724999)
 
gp: K = bnfinit(x^22 - 11*x^21 - 32441985*x^20 + 324420235*x^19 + 464751108360962*x^18 - 4182769221227550*x^17 - 4035052200728025834218*x^16 + 32280512415301802940118*x^15 + 24179924667975031656547634456*x^14 - 169260037585124322611844491980*x^13 - 103848538639555020655359200328639754*x^12 + 623093432219287495095768869183695022*x^11 + 317061308864468722415113561065019172613923*x^10 - 1585312256016172990286816098491706527517901*x^9 - 676446181077317936754989616553711670587334903597*x^8 + 2705794236189661878727800809354360542269940615953*x^7 + 986344921008399892498102559525711856850940930222963887*x^6 - 2959044233311684659640186168114291609441101078849350201*x^5 - 940499719015049358045656409945459020782100574483826871424882*x^4 + 1881004369773644332011928098109604490847972220298952944948509*x^3 + 529891435435819882897472854176851909714543741204188875533041914071*x^2 - 529892375938497944209490322543314025268468196224103590775254275058*x - 133708001088324569741903664154721856901221813847387342588501211986724999, 1)
 

Normalized defining polynomial

\( x^{22} - 11 x^{21} - 32441985 x^{20} + 324420235 x^{19} + 464751108360962 x^{18} - 4182769221227550 x^{17} - 4035052200728025834218 x^{16} + 32280512415301802940118 x^{15} + 24179924667975031656547634456 x^{14} - 169260037585124322611844491980 x^{13} - 103848538639555020655359200328639754 x^{12} + 623093432219287495095768869183695022 x^{11} + 317061308864468722415113561065019172613923 x^{10} - 1585312256016172990286816098491706527517901 x^{9} - 676446181077317936754989616553711670587334903597 x^{8} + 2705794236189661878727800809354360542269940615953 x^{7} + 986344921008399892498102559525711856850940930222963887 x^{6} - 2959044233311684659640186168114291609441101078849350201 x^{5} - 940499719015049358045656409945459020782100574483826871424882 x^{4} + 1881004369773644332011928098109604490847972220298952944948509 x^{3} + 529891435435819882897472854176851909714543741204188875533041914071 x^{2} - 529892375938497944209490322543314025268468196224103590775254275058 x - 133708001088324569741903664154721856901221813847387342588501211986724999 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1801251752870747612751801736092368693324216289918084803333433041508210634924445873691187972700463846529=7^{10}\cdot 251^{10}\cdot 33893^{10}\cdot 1792166448977^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44{,}461.15$
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:  $7, 251, 33893, 1792166448977$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{59550001} a^{4} - \frac{2}{59550001} a^{3} - \frac{17554541}{59550001} a^{2} + \frac{17554542}{59550001} a + \frac{4158320}{8507143}$, $\frac{1}{59550001} a^{5} - \frac{17554545}{59550001} a^{3} - \frac{17554540}{59550001} a^{2} + \frac{4667323}{59550001} a - \frac{190503}{8507143}$, $\frac{1}{3546202619100001} a^{6} - \frac{3}{3546202619100001} a^{5} + \frac{3872566}{506600374157143} a^{4} - \frac{54215919}{3546202619100001} a^{3} - \frac{1467170988185069}{3546202619100001} a^{2} + \frac{1467171015293028}{3546202619100001} a - \frac{91977119304956}{506600374157143}$, $\frac{1}{3546202619100001} a^{7} + \frac{27107953}{3546202619100001} a^{5} + \frac{27107967}{3546202619100001} a^{4} - \frac{1467171150832826}{3546202619100001} a^{3} + \frac{611860669837822}{3546202619100001} a^{2} + \frac{211470591644391}{3546202619100001} a + \frac{230669016242275}{506600374157143}$, $\frac{1}{211176369513607678650001} a^{8} - \frac{4}{211176369513607678650001} a^{7} + \frac{12220465}{211176369513607678650001} a^{6} - \frac{36661381}{211176369513607678650001} a^{5} + \frac{1675461900855851}{211176369513607678650001} a^{4} - \frac{3350923740609405}{211176369513607678650001} a^{3} - \frac{3569979167569582056332}{211176369513607678650001} a^{2} + \frac{3569980843031446250805}{211176369513607678650001} a + \frac{9889236946370117381121}{30168052787658239807143}$, $\frac{1}{211176369513607678650001} a^{9} + \frac{12220449}{211176369513607678650001} a^{7} + \frac{12220479}{211176369513607678650001} a^{6} + \frac{1675461754210327}{211176369513607678650001} a^{5} - \frac{195278756286002}{211176369513607678650001} a^{4} - \frac{3569985478859306293950}{211176369513607678650001} a^{3} + \frac{51542023444051468680018}{211176369513607678650001} a^{2} + \frac{21252619179215636916525}{211176369513607678650001} a - \frac{5357350277233686440979}{30168052787658239807143}$, $\frac{1}{12575553015711706777215238200001} a^{10} - \frac{5}{12575553015711706777215238200001} a^{9} - \frac{2667031}{12575553015711706777215238200001} a^{8} + \frac{1524022}{1796507573673100968173605457143} a^{7} + \frac{213361394975676}{1796507573673100968173605457143} a^{6} - \frac{640084190261108}{1796507573673100968173605457143} a^{5} - \frac{581906425829333126723}{256643939096157281167657922449} a^{4} + \frac{57026837198923545244637}{12575553015711706777215238200001} a^{3} + \frac{2234391596510727324098700846055}{12575553015711706777215238200001} a^{2} - \frac{2234391625024146670325363884360}{12575553015711706777215238200001} a - \frac{518451356100258314923565866821}{1796507573673100968173605457143}$, $\frac{1}{12575553015711706777215238200001} a^{11} - \frac{381008}{1796507573673100968173605457143} a^{9} - \frac{2667001}{12575553015711706777215238200001} a^{8} + \frac{213361402595786}{1796507573673100968173605457143} a^{7} - \frac{79877589539871}{1796507573673100968173605457143} a^{6} - \frac{4073346661425160721172}{1796507573673100968173605457143} a^{5} + \frac{29505806541481306539541}{12575553015711706777215238200001} a^{4} + \frac{319198807364686892316368126451}{1796507573673100968173605457143} a^{3} - \frac{306029327724366367048308424794}{1796507573673100968173605457143} a^{2} - \frac{3721345851176432727233246528032}{12575553015711706777215238200001} a + \frac{408559215424142706627027186314}{1796507573673100968173605457143}$, $\frac{1}{748874194661185154294874212025297750001} a^{12} - \frac{6}{748874194661185154294874212025297750001} a^{11} - \frac{17554526}{748874194661185154294874212025297750001} a^{10} + \frac{12538955}{106982027808740736327839173146471107143} a^{9} + \frac{1533235178174078}{748874194661185154294874212025297750001} a^{8} - \frac{876134462761784}{106982027808740736327839173146471107143} a^{7} - \frac{7249762108421518075953}{106982027808740736327839173146471107143} a^{6} + \frac{152245025742146585904064}{748874194661185154294874212025297750001} a^{5} + \frac{2658885003296065774455500063918}{748874194661185154294874212025297750001} a^{4} - \frac{5317770260333848274253598067915}{748874194661185154294874212025297750001} a^{3} - \frac{15787624916746842083404277301525905524}{106982027808740736327839173146471107143} a^{2} + \frac{110513377076113050124926057418810910528}{748874194661185154294874212025297750001} a + \frac{31705063368648973535789773053006830615}{106982027808740736327839173146471107143}$, $\frac{1}{748874194661185154294874212025297750001} a^{13} - \frac{17554562}{748874194661185154294874212025297750001} a^{11} - \frac{17554471}{748874194661185154294874212025297750001} a^{10} + \frac{1533235704810188}{748874194661185154294874212025297750001} a^{9} - \frac{479732789388021}{748874194661185154294874212025297750001} a^{8} - \frac{7249765338826798018085}{106982027808740736327839173146471107143} a^{7} + \frac{15595141712430611643574}{748874194661185154294874212025297750001} a^{6} + \frac{379840773320828144076427614240}{106982027808740736327839173146471107143} a^{5} - \frac{2156979637213609992204652432297}{748874194661185154294874212025297750001} a^{4} - \frac{110513380738809824073884943651900802670}{748874194661185154294874212025297750001} a^{3} + \frac{119893412979468274098237124444362604325}{748874194661185154294874212025297750001} a^{2} + \frac{212555410837978623847203820102339315153}{748874194661185154294874212025297750001} a - \frac{23537453306399972542969518181895467850}{106982027808740736327839173146471107143}$, $\frac{1}{44595459040947770599444913620980693037857300001} a^{14} - \frac{1}{6370779862992538657063559088711527576836757143} a^{13} + \frac{27107981}{44595459040947770599444913620980693037857300001} a^{12} - \frac{162647795}{44595459040947770599444913620980693037857300001} a^{11} + \frac{749206055371867}{44595459040947770599444913620980693037857300001} a^{10} - \frac{3746028785921381}{44595459040947770599444913620980693037857300001} a^{9} + \frac{17729789052425549649895}{44595459040947770599444913620980693037857300001} a^{8} - \frac{70919133733531272196753}{44595459040947770599444913620980693037857300001} a^{7} + \frac{56048185590153273698218798669}{6370779862992538657063559088711527576836757143} a^{6} - \frac{1177011649176266413622253827769}{44595459040947770599444913620980693037857300001} a^{5} + \frac{8239085019137906697702879203154468165}{44595459040947770599444913620980693037857300001} a^{4} - \frac{16478168076589814173943599432359506219}{44595459040947770599444913620980693037857300001} a^{3} + \frac{166417841448696665556705034671033905327578298}{44595459040947770599444913620980693037857300001} a^{2} - \frac{166417833209612823430394323889138012811666966}{44595459040947770599444913620980693037857300001} a + \frac{5554652425265107075478064654466815141206086}{6370779862992538657063559088711527576836757143}$, $\frac{1}{44595459040947770599444913620980693037857300001} a^{15} + \frac{27107932}{44595459040947770599444913620980693037857300001} a^{13} + \frac{27108072}{44595459040947770599444913620980693037857300001} a^{12} + \frac{749204916837302}{44595459040947770599444913620980693037857300001} a^{11} + \frac{1498413601681688}{44595459040947770599444913620980693037857300001} a^{10} + \frac{17729762830224048200228}{44595459040947770599444913620980693037857300001} a^{9} + \frac{53189389633447575352512}{44595459040947770599444913620980693037857300001} a^{8} + \frac{56048114671019540166946601916}{6370779862992538657063559088711527576836757143} a^{7} + \frac{1569349444741243997590467307012}{44595459040947770599444913620980693037857300001} a^{6} + \frac{8239076780056362463837983847377673782}{44595459040947770599444913620980693037857300001} a^{5} + \frac{41195427057375532709976554989721770936}{44595459040947770599444913620980693037857300001} a^{4} + \frac{166417726101520129428005817065837878811034765}{44595459040947770599444913620980693037857300001} a^{3} + \frac{998507056931263835466540918808099324481381120}{44595459040947770599444913620980693037857300001} a^{2} - \frac{160863180784347716354916259234671197670460880}{6370779862992538657063559088711527576836757143} a + \frac{5554652425265107075478064654466815141206086}{910111408998934093866222726958789653833822449}$, $\frac{1}{2655659630483898780144715205574313891385095252916850001} a^{16} - \frac{8}{2655659630483898780144715205574313891385095252916850001} a^{15} + \frac{1745784}{379379947211985540020673600796330555912156464702407143} a^{14} - \frac{12220468}{379379947211985540020673600796330555912156464702407143} a^{13} + \frac{49376592983166}{379379947211985540020673600796330555912156464702407143} a^{12} - \frac{296259399032964}{379379947211985540020673600796330555912156464702407143} a^{11} + \frac{134203829573585012424}{54197135315997934288667657256618650844593780671772449} a^{10} - \frac{32879919235552262209148}{2655659630483898780144715205574313891385095252916850001} a^{9} + \frac{128385135532221279005991974936}{2655659630483898780144715205574313891385095252916850001} a^{8} - \frac{73362906407056073525803590136}{379379947211985540020673600796330555912156464702407143} a^{7} + \frac{342594950762243976956238038445019062}{379379947211985540020673600796330555912156464702407143} a^{6} - \frac{1027784595516579234120739253514712412}{379379947211985540020673600796330555912156464702407143} a^{5} + \frac{6251211386349879454923942068238741840092431}{379379947211985540020673600796330555912156464702407143} a^{4} - \frac{1786060151389359805561510653284076924569036}{54197135315997934288667657256618650844593780671772449} a^{3} + \frac{782739410298538190977345500078415711453068377107676}{2655659630483898780144715205574313891385095252916850001} a^{2} - \frac{782739366540065681019972776449566922145685520590710}{2655659630483898780144715205574313891385095252916850001} a + \frac{33743855574780347448970981921005879107139804977611}{379379947211985540020673600796330555912156464702407143}$, $\frac{1}{2655659630483898780144715205574313891385095252916850001} a^{17} + \frac{12220424}{2655659630483898780144715205574313891385095252916850001} a^{15} + \frac{1745804}{379379947211985540020673600796330555912156464702407143} a^{14} + \frac{7053785031346}{54197135315997934288667657256618650844593780671772449} a^{13} + \frac{98753344832364}{379379947211985540020673600796330555912156464702407143} a^{12} + \frac{939424436939902823256}{379379947211985540020673600796330555912156464702407143} a^{11} + \frac{19727981957293062661060}{2655659630483898780144715205574313891385095252916850001} a^{10} + \frac{18340696070409627798270614536}{379379947211985540020673600796330555912156464702407143} a^{9} + \frac{513540739408377717367310668536}{2655659630483898780144715205574313891385095252916850001} a^{8} + \frac{342594363858992720507649832016297974}{379379947211985540020673600796330555912156464702407143} a^{7} + \frac{34958673685334134316921735796845716}{7742447902285419184095379608088378692084825810253207} a^{6} + \frac{6251203164073115322290069102324713722393135}{379379947211985540020673600796330555912156464702407143} a^{5} + \frac{37507270031073517000460961972921396248756196}{379379947211985540020673600796330555912156464702407143} a^{4} + \frac{782738710162958846348301719966239624094913946045564}{2655659630483898780144715205574313891385095252916850001} a^{3} + \frac{782739416549748549542684460596822681354123070895814}{379379947211985540020673600796330555912156464702407143} a^{2} - \frac{6025707943297063016016985338149494223415505529882403}{2655659630483898780144715205574313891385095252916850001} a + \frac{269950844598242779591767855368047032857118439820888}{379379947211985540020673600796330555912156464702407143}$, $\frac{1}{158144533650975802841516570636665597806296313696293670476400001} a^{18} - \frac{9}{158144533650975802841516570636665597806296313696293670476400001} a^{17} - \frac{2667004}{158144533650975802841516570636665597806296313696293670476400001} a^{16} + \frac{21336236}{158144533650975802841516570636665597806296313696293670476400001} a^{15} + \frac{23386245903634}{22592076235853686120216652948095085400899473385184810068057143} a^{14} - \frac{163703774666130}{22592076235853686120216652948095085400899473385184810068057143} a^{13} + \frac{204333075237610294932}{22592076235853686120216652948095085400899473385184810068057143} a^{12} - \frac{8581974262935166967924}{158144533650975802841516570636665597806296313696293670476400001} a^{11} + \frac{30485152286079917807601484084}{158144533650975802841516570636665597806296313696293670476400001} a^{10} - \frac{152425682762329490047926255888}{158144533650975802841516570636665597806296313696293670476400001} a^{9} + \frac{486831463640108396884475422880564386}{158144533650975802841516570636665597806296313696293670476400001} a^{8} - \frac{278189277143775916461576343810031474}{22592076235853686120216652948095085400899473385184810068057143} a^{7} + \frac{1150830084661567764567182391627774069279843}{22592076235853686120216652948095085400899473385184810068057143} a^{6} - \frac{3452489280322324745882011004526999278425683}{22592076235853686120216652948095085400899473385184810068057143} a^{5} + \frac{131285131215498796359714659806107838362205215402940}{158144533650975802841516570636665597806296313696293670476400001} a^{4} - \frac{262570222151958260837765773840414568087199248648386}{158144533650975802841516570636665597806296313696293670476400001} a^{3} + \frac{2124530211077335116743664516871776460856394603606246789175}{158144533650975802841516570636665597806296313696293670476400001} a^{2} - \frac{2124530079792228068668369919923486362103134734317057523465}{158144533650975802841516570636665597806296313696293670476400001} a - \frac{10305858335597177239741833116134294189476911885161698029}{22592076235853686120216652948095085400899473385184810068057143}$, $\frac{1}{158144533650975802841516570636665597806296313696293670476400001} a^{19} - \frac{2667085}{158144533650975802841516570636665597806296313696293670476400001} a^{17} - \frac{2666800}{158144533650975802841516570636665597806296313696293670476400001} a^{16} + \frac{163703913351562}{158144533650975802841516570636665597806296313696293670476400001} a^{15} + \frac{46772438466576}{22592076235853686120216652948095085400899473385184810068057143} a^{14} + \frac{29190228843376899966}{3227439462264812302888093278299297914414210483597830009722449} a^{13} + \frac{4291009477034281612792}{158144533650975802841516570636665597806296313696293670476400001} a^{12} + \frac{4355010721187364484442681824}{22592076235853686120216652948095085400899473385184810068057143} a^{11} + \frac{121940687812389770220487100868}{158144533650975802841516570636665597806296313696293670476400001} a^{10} + \frac{486830091808963535919064991544261394}{158144533650975802841516570636665597806296313696293670476400001} a^{9} + \frac{2434158232754544156729244399254859156}{158144533650975802841516570636665597806296313696293670476400001} a^{8} + \frac{164403940136867638654847748205811396999511}{3227439462264812302888093278299297914414210483597830009722449} a^{7} + \frac{986425925947397876460375788588995335013272}{3227439462264812302888093278299297914414210483597830009722449} a^{6} + \frac{131284913708674136053255669239414553161250674584911}{158144533650975802841516570636665597806296313696293670476400001} a^{5} + \frac{131285136969647272342809452059222282453235384282582}{22592076235853686120216652948095085400899473385184810068057143} a^{4} + \frac{2124527847945335749119316976979811897125281818813008953701}{158144533650975802841516570636665597806296313696293670476400001} a^{3} + \frac{16996241819903787982024610731922501785604416698139163579110}{158144533650975802841516570636665597806296313696293670476400001} a^{2} - \frac{19192911726479232858693522111124317318254550992049649597388}{158144533650975802841516570636665597806296313696293670476400001} a - \frac{92752725020374595157676498045208647705292206966455282261}{22592076235853686120216652948095085400899473385184810068057143}$, $\frac{1}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{20} - \frac{10}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{19} - \frac{358255}{192194023205309034901798257610816469102664148712461260676801847672449} a^{18} + \frac{157990740}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{17} + \frac{203408722039202}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{16} - \frac{1627273357438348}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{15} - \frac{20547070992709459134}{192194023205309034901798257610816469102664148712461260676801847672449} a^{14} + \frac{7047673827795633592552}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{13} + \frac{9191100264854717882167402008}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{12} - \frac{7878099029851729945794169996}{1345358162437163244312587803275715283718649040987228824737612933707143} a^{11} + \frac{32983911406776382853298376506270274}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{10} - \frac{164919051522359529027007695018184704}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{9} + \frac{808109125013800599784064114294422858604219}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{8} - \frac{461776501505928554127475577682929778530526}{1345358162437163244312587803275715283718649040987228824737612933707143} a^{7} + \frac{11354275185130127951030385992675464856374525745221}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{6} - \frac{34062814241866789617251361837252494146467974413826}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{5} + \frac{24289011810974133985239050967731408236233220927746598223}{1345358162437163244312587803275715283718649040987228824737612933707143} a^{4} - \frac{340046108582284577189796478746090940561350523670457737640}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{3} + \frac{2530277803586018306294101752768285175706731874007784869985224978}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{2} - \frac{2530277633562969692286981396466034097409526074915855746185326513}{9417507137060142710188114622930006986030543286910601773163290535950001} a + \frac{1373400582707302026648355799713583916394409088490324523081823222}{1345358162437163244312587803275715283718649040987228824737612933707143}$, $\frac{1}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{21} - \frac{17554595}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{19} - \frac{17554210}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{18} + \frac{203410301946602}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{17} + \frac{406813862953672}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{16} - \frac{1006822751376337881046}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{15} - \frac{3020390958632001383108}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{14} + \frac{1313024391656142262643332504}{1345358162437163244312587803275715283718649040987228824737612933707143} a^{13} + \frac{36764309439585069201114830108}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{12} + \frac{32983359939844293232202170914370554}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{11} + \frac{164920062545404299505976070044518036}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{10} + \frac{808107475823285376188773844217472676757179}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{9} + \frac{4848655739596506118948312099163720136328508}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{8} + \frac{11354242860775022536031597069385027051290028608401}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{7} + \frac{11354276801347784270436071155643164916753897576912}{1345358162437163244312587803275715283718649040987228824737612933707143} a^{6} + \frac{170022742048676519228777184260501485128691081814482049301}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{5} + \frac{1360184718185904801776937088995107635974974941271804137970}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{4} + \frac{2530274403124932483448329854803497714797326260502548165407848578}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{3} + \frac{22772500402297213370654036131216817659657792665161992953666923267}{9417507137060142710188114622930006986030543286910601773163290535950001} a^{2} - \frac{15688972256678582736331323366665253559334397129726285800280502576}{9417507137060142710188114622930006986030543286910601773163290535950001} a + \frac{13734005827073020266483557997135839163944090884903245230818232220}{1345358162437163244312587803275715283718649040987228824737612933707143}$

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:  $17$
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

22T51:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 40874803200
The 376 conjugacy class representatives for t22n51 are not computed
Character table for t22n51 is not computed

Intermediate fields

11.7.1792166448977.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 22 sibling: data not computed
Degree 44 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.10.0.1}{10} }$ R ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ $20{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.8.0.1}{8} }$

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
251Data not computed
33893Data not computed
1792166448977Data not computed