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Properties

Label 32.0.10751998612...0625.1
Degree $32$
Signature $[0, 16]$
Discriminant $5^{24}\cdot 7^{16}\cdot 13^{24}$
Root discriminant $60.57$
Ramified primes $5, 7, 13$
Class number $800$ (GRH)
Class group $[20, 40]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7890481, 5508449, 7865200, 4426189, -4091466, 38717700, 9826086, 18427375, -1032078, -37188866, 29606989, -31103793, 73936, 4910425, 4479186, -6441701, 5274231, -487441, -1121097, 584480, -62330, -62094, 47407, -13702, -1932, 2813, 387, -486, 63, 14, -8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 8*x^30 + 14*x^29 + 63*x^28 - 486*x^27 + 387*x^26 + 2813*x^25 - 1932*x^24 - 13702*x^23 + 47407*x^22 - 62094*x^21 - 62330*x^20 + 584480*x^19 - 1121097*x^18 - 487441*x^17 + 5274231*x^16 - 6441701*x^15 + 4479186*x^14 + 4910425*x^13 + 73936*x^12 - 31103793*x^11 + 29606989*x^10 - 37188866*x^9 - 1032078*x^8 + 18427375*x^7 + 9826086*x^6 + 38717700*x^5 - 4091466*x^4 + 4426189*x^3 + 7865200*x^2 + 5508449*x + 7890481)
 
gp: K = bnfinit(x^32 - 2*x^31 - 8*x^30 + 14*x^29 + 63*x^28 - 486*x^27 + 387*x^26 + 2813*x^25 - 1932*x^24 - 13702*x^23 + 47407*x^22 - 62094*x^21 - 62330*x^20 + 584480*x^19 - 1121097*x^18 - 487441*x^17 + 5274231*x^16 - 6441701*x^15 + 4479186*x^14 + 4910425*x^13 + 73936*x^12 - 31103793*x^11 + 29606989*x^10 - 37188866*x^9 - 1032078*x^8 + 18427375*x^7 + 9826086*x^6 + 38717700*x^5 - 4091466*x^4 + 4426189*x^3 + 7865200*x^2 + 5508449*x + 7890481, 1)
 

Normalized defining polynomial

\( x^{32} - 2 x^{31} - 8 x^{30} + 14 x^{29} + 63 x^{28} - 486 x^{27} + 387 x^{26} + 2813 x^{25} - 1932 x^{24} - 13702 x^{23} + 47407 x^{22} - 62094 x^{21} - 62330 x^{20} + 584480 x^{19} - 1121097 x^{18} - 487441 x^{17} + 5274231 x^{16} - 6441701 x^{15} + 4479186 x^{14} + 4910425 x^{13} + 73936 x^{12} - 31103793 x^{11} + 29606989 x^{10} - 37188866 x^{9} - 1032078 x^{8} + 18427375 x^{7} + 9826086 x^{6} + 38717700 x^{5} - 4091466 x^{4} + 4426189 x^{3} + 7865200 x^{2} + 5508449 x + 7890481 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1075199861227118411720654551127625657377302646636962890625=5^{24}\cdot 7^{16}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.57$
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:  $5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(455=5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{455}(1,·)$, $\chi_{455}(391,·)$, $\chi_{455}(8,·)$, $\chi_{455}(398,·)$, $\chi_{455}(272,·)$, $\chi_{455}(274,·)$, $\chi_{455}(148,·)$, $\chi_{455}(281,·)$, $\chi_{455}(27,·)$, $\chi_{455}(34,·)$, $\chi_{455}(421,·)$, $\chi_{455}(428,·)$, $\chi_{455}(174,·)$, $\chi_{455}(307,·)$, $\chi_{455}(181,·)$, $\chi_{455}(183,·)$, $\chi_{455}(57,·)$, $\chi_{455}(447,·)$, $\chi_{455}(64,·)$, $\chi_{455}(118,·)$, $\chi_{455}(454,·)$, $\chi_{455}(209,·)$, $\chi_{455}(83,·)$, $\chi_{455}(216,·)$, $\chi_{455}(92,·)$, $\chi_{455}(99,·)$, $\chi_{455}(356,·)$, $\chi_{455}(337,·)$, $\chi_{455}(363,·)$, $\chi_{455}(239,·)$, $\chi_{455}(372,·)$, $\chi_{455}(246,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{8}$, $\frac{1}{6} a^{24} - \frac{1}{6} a^{22} + \frac{1}{6} a^{16} + \frac{1}{3} a^{12} - \frac{1}{2} a^{9} - \frac{1}{3} a^{8} - \frac{1}{2} a^{7} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{25} - \frac{1}{6} a^{23} + \frac{1}{6} a^{17} + \frac{1}{3} a^{13} - \frac{1}{2} a^{10} - \frac{1}{3} a^{9} - \frac{1}{2} a^{8} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{12} a^{26} - \frac{1}{12} a^{25} - \frac{1}{12} a^{24} + \frac{1}{12} a^{23} - \frac{1}{4} a^{21} - \frac{1}{4} a^{19} + \frac{1}{12} a^{18} + \frac{1}{6} a^{17} - \frac{1}{4} a^{15} - \frac{1}{3} a^{14} - \frac{1}{6} a^{13} - \frac{1}{2} a^{12} + \frac{1}{4} a^{11} + \frac{1}{12} a^{10} - \frac{1}{12} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{5}{12} a^{4} + \frac{1}{12} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{4}$, $\frac{1}{12} a^{27} - \frac{1}{12} a^{23} - \frac{1}{4} a^{22} - \frac{1}{4} a^{21} - \frac{1}{4} a^{20} - \frac{1}{6} a^{19} - \frac{1}{4} a^{18} - \frac{1}{6} a^{17} - \frac{1}{4} a^{16} - \frac{1}{12} a^{15} - \frac{1}{2} a^{14} - \frac{1}{3} a^{13} - \frac{1}{4} a^{12} + \frac{1}{3} a^{11} - \frac{1}{2} a^{10} - \frac{1}{6} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{12} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{12} a + \frac{1}{4}$, $\frac{1}{3106596} a^{28} - \frac{4175}{1035532} a^{27} + \frac{4173}{517766} a^{26} + \frac{50099}{1553298} a^{25} - \frac{175259}{3106596} a^{24} - \frac{237737}{1553298} a^{23} - \frac{27691}{1553298} a^{22} + \frac{94363}{517766} a^{21} - \frac{701159}{3106596} a^{20} - \frac{205605}{1035532} a^{19} - \frac{331217}{3106596} a^{18} - \frac{550409}{3106596} a^{17} - \frac{360115}{1553298} a^{16} - \frac{82489}{1035532} a^{15} + \frac{592753}{1553298} a^{14} - \frac{794131}{3106596} a^{13} - \frac{109645}{3106596} a^{12} - \frac{108929}{258883} a^{11} - \frac{673537}{1553298} a^{10} - \frac{1104695}{3106596} a^{9} - \frac{134651}{1553298} a^{8} + \frac{109748}{258883} a^{7} + \frac{355346}{776649} a^{6} + \frac{182669}{1035532} a^{5} + \frac{430661}{1035532} a^{4} - \frac{147253}{3106596} a^{3} - \frac{852631}{3106596} a^{2} + \frac{682175}{1553298} a - \frac{34045}{3106596}$, $\frac{1}{279939575726732661387902501618315976947153772} a^{29} + \frac{11353646175056890683536698160474210617}{279939575726732661387902501618315976947153772} a^{28} + \frac{5341524619539482797942445173709052880696133}{139969787863366330693951250809157988473576886} a^{27} + \frac{1962063255315689907867225332978860865617519}{279939575726732661387902501618315976947153772} a^{26} - \frac{1289960224020075794490405370680181066897577}{23328297977227721782325208468192998078929481} a^{25} - \frac{23080148600878931356408239129666681430529477}{279939575726732661387902501618315976947153772} a^{24} - \frac{45905548640004849862134914753888121515191865}{279939575726732661387902501618315976947153772} a^{23} - \frac{6625327182272461606827175431819820366757905}{46656595954455443564650416936385996157858962} a^{22} - \frac{7317530513870803720568804757485630979593072}{69984893931683165346975625404578994236788443} a^{21} - \frac{23762006158986685817794264461306417799602215}{279939575726732661387902501618315976947153772} a^{20} - \frac{556651951267454739133899469028049158003002}{69984893931683165346975625404578994236788443} a^{19} - \frac{12598600933600424320457425760137323052297333}{69984893931683165346975625404578994236788443} a^{18} + \frac{9166976004019265853453666560204199460828855}{69984893931683165346975625404578994236788443} a^{17} - \frac{10228962005834740453936597529140723894332289}{279939575726732661387902501618315976947153772} a^{16} + \frac{34353879624239269754220212856077861012365819}{279939575726732661387902501618315976947153772} a^{15} - \frac{1251120150376113957577136496686550544747891}{5281878787296842667696273615439924093342524} a^{14} + \frac{15251062231176589335145326547864871648989451}{279939575726732661387902501618315976947153772} a^{13} + \frac{68631961108480627028257823403344126382317747}{139969787863366330693951250809157988473576886} a^{12} - \frac{62506163582842707788388046883527511792000203}{279939575726732661387902501618315976947153772} a^{11} - \frac{19894084193136818593450042332241504531182311}{139969787863366330693951250809157988473576886} a^{10} + \frac{19195938454380925014486837271235277817118267}{279939575726732661387902501618315976947153772} a^{9} - \frac{24754877759698161276251264621757089185130599}{93313191908910887129300833872771992315717924} a^{8} - \frac{21215732281465063024850768393035361636207458}{69984893931683165346975625404578994236788443} a^{7} - \frac{31073328538560697738324143393879430580489017}{69984893931683165346975625404578994236788443} a^{6} + \frac{28833829813556967835229640046225964920069325}{279939575726732661387902501618315976947153772} a^{5} - \frac{6457734141928186520415012930322700479953622}{69984893931683165346975625404578994236788443} a^{4} - \frac{3538312740960501144168029418685363823146781}{46656595954455443564650416936385996157858962} a^{3} - \frac{43404150315015177399926026443795321150390353}{139969787863366330693951250809157988473576886} a^{2} + \frac{130846778784560121513790534393293140982464621}{279939575726732661387902501618315976947153772} a + \frac{11868033444440650245574350787012776630127}{60711250428699341008003145005056598774052}$, $\frac{1}{14836797513516831053558832585770746778199149916} a^{30} - \frac{1}{7418398756758415526779416292885373389099574958} a^{29} + \frac{352716956033757704505512460343679256653}{2472799585586138508926472097628457796366524986} a^{28} + \frac{120812546194097890251654635442841851218690465}{4945599171172277017852944195256915592733049972} a^{27} + \frac{255749560003823741199291919630016522982320341}{7418398756758415526779416292885373389099574958} a^{26} - \frac{33195916815228801608759488904615890174430984}{3709199378379207763389708146442686694549787479} a^{25} + \frac{64772997366780162278761335408200359403211715}{1236399792793069254463236048814228898183262493} a^{24} + \frac{2070118439328404600160074067295198905612149947}{14836797513516831053558832585770746778199149916} a^{23} + \frac{2969901422171530443919668300151333272517956673}{14836797513516831053558832585770746778199149916} a^{22} - \frac{2424420295609332731028854953763375201387806685}{14836797513516831053558832585770746778199149916} a^{21} - \frac{786388298018334130876919530529803085324206037}{14836797513516831053558832585770746778199149916} a^{20} - \frac{1583230277464266726717569121627109653144317143}{7418398756758415526779416292885373389099574958} a^{19} - \frac{517442439491656551225030654991212323907765061}{14836797513516831053558832585770746778199149916} a^{18} - \frac{58867904571050298325555461775961884356181327}{7418398756758415526779416292885373389099574958} a^{17} - \frac{1827637047715835587149555054766582365971830357}{14836797513516831053558832585770746778199149916} a^{16} - \frac{33330263286321060168232367033664113944185245}{279939575726732661387902501618315976947153772} a^{15} + \frac{766386168683715752328164471741060222846459854}{3709199378379207763389708146442686694549787479} a^{14} + \frac{1523479776998185671412880796926841219638892745}{7418398756758415526779416292885373389099574958} a^{13} + \frac{5425906646603848357771063741179272390120929759}{14836797513516831053558832585770746778199149916} a^{12} - \frac{2017599978971117813116168225754664971384800789}{7418398756758415526779416292885373389099574958} a^{11} - \frac{371675110272761535805648480641585359771732501}{2472799585586138508926472097628457796366524986} a^{10} - \frac{2773091176097393368394775862871396117435426597}{7418398756758415526779416292885373389099574958} a^{9} + \frac{2600412555815479773929679984901935307925403581}{14836797513516831053558832585770746778199149916} a^{8} + \frac{629378907121357867056427007896417158257325995}{14836797513516831053558832585770746778199149916} a^{7} + \frac{123302751235934974906952583605887146272358173}{4945599171172277017852944195256915592733049972} a^{6} - \frac{668134206922487287887598704434478973273277077}{4945599171172277017852944195256915592733049972} a^{5} - \frac{544212551541516718800214853272708693631503705}{7418398756758415526779416292885373389099574958} a^{4} + \frac{3785834462770157359102226581694912690824588421}{14836797513516831053558832585770746778199149916} a^{3} - \frac{199954867362923807730221685275387699185648273}{1236399792793069254463236048814228898183262493} a^{2} + \frac{579110043450876866848313653955278869012793}{3543538933249780523897500020485012366419668} a + \frac{58639370620994923894454348290017248683547}{880313131216140444616045602573320682223754}$, $\frac{1}{1572700536432784091677236254091699158489109891096} a^{31} + \frac{17}{524233512144261363892412084697233052829703297032} a^{30} + \frac{2695}{1572700536432784091677236254091699158489109891096} a^{29} - \frac{81825348214578484519584002875555645710485}{524233512144261363892412084697233052829703297032} a^{28} - \frac{12142731746220145715529298113931999727415935327}{786350268216392045838618127045849579244554945548} a^{27} - \frac{4165348558050398685805239460947951373617066097}{393175134108196022919309063522924789622277472774} a^{26} + \frac{12686658346666778304024774005538021174495983623}{1572700536432784091677236254091699158489109891096} a^{25} - \frac{550260171887953588753236043119545513417198829}{131058378036065340973103021174308263207425824258} a^{24} + \frac{134381779526177978294927701732071550781490978313}{786350268216392045838618127045849579244554945548} a^{23} + \frac{56680831422205866014604356098297637616648058883}{786350268216392045838618127045849579244554945548} a^{22} + \frac{62989029993841154190546285662248621903190689763}{1572700536432784091677236254091699158489109891096} a^{21} - \frac{95275777490420882503446824867588447295008155427}{524233512144261363892412084697233052829703297032} a^{20} - \frac{82931053474032788563075953310817813838335292693}{524233512144261363892412084697233052829703297032} a^{19} + \frac{85639835417381107626864975391831332586350227039}{1572700536432784091677236254091699158489109891096} a^{18} - \frac{36249654440807868913222597388418770063052205497}{262116756072130681946206042348616526414851648516} a^{17} + \frac{6120423277956090726071775514400769884509930327}{29673595027033662107117665171541493556398299832} a^{16} - \frac{15742648838685233166758603149959534931130010534}{65529189018032670486551510587154131603712912129} a^{15} + \frac{161351282062775134042247586036769768120606669903}{1572700536432784091677236254091699158489109891096} a^{14} + \frac{104431691201739349323108305163227763757304714599}{524233512144261363892412084697233052829703297032} a^{13} + \frac{670597423351063266761149051169487667675635574}{1739712982779628419996942758950994644346360499} a^{12} + \frac{63431805765342604645276201704089759552874898496}{196587567054098011459654531761462394811138736387} a^{11} - \frac{381925617839264221610955283079497094356753989613}{1572700536432784091677236254091699158489109891096} a^{10} + \frac{170884843060083797889273639577330512263711587021}{393175134108196022919309063522924789622277472774} a^{9} - \frac{24376888664980738270654016544340952442196491926}{196587567054098011459654531761462394811138736387} a^{8} - \frac{16486023699463276557549589395686190947639690384}{196587567054098011459654531761462394811138736387} a^{7} - \frac{85220749684957760108856954419921312019576103905}{524233512144261363892412084697233052829703297032} a^{6} + \frac{774982770520056733923704979382418887517056217281}{1572700536432784091677236254091699158489109891096} a^{5} - \frac{223164673109216641443448193416722147776132888159}{1572700536432784091677236254091699158489109891096} a^{4} - \frac{749936460904447244581942452827783274293220698527}{1572700536432784091677236254091699158489109891096} a^{3} + \frac{2088437721340623934317003266414437482126585405}{4945599171172277017852944195256915592733049972} a^{2} + \frac{14957233218977732821549422987519112526247676}{69984893931683165346975625404578994236788443} a - \frac{2861881685571961328815662244775271558920633}{10563757574593685335392547230879848186685048}$

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

Class group and class number

$C_{20}\times C_{40}$, which has order $800$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1217119931911967502527970860480693}{19103607309758759817573746423121315778684} a^{31} + \frac{529949397406022444320176909569853}{19103607309758759817573746423121315778684} a^{30} + \frac{15170704205450168021556094376403765}{19103607309758759817573746423121315778684} a^{29} - \frac{6510871315720762181790475295058899}{19103607309758759817573746423121315778684} a^{28} - \frac{28539628480119096136977290268832520}{4775901827439689954393436605780328944671} a^{27} + \frac{253980313954092778064719791661071947}{9551803654879379908786873211560657889342} a^{26} + \frac{541348593906996899195996233372651811}{19103607309758759817573746423121315778684} a^{25} - \frac{1265068675720934973273197163556798656}{4775901827439689954393436605780328944671} a^{24} - \frac{426537845054919703715610210810159108}{4775901827439689954393436605780328944671} a^{23} + \frac{6196494182384986802364796023099411725}{4775901827439689954393436605780328944671} a^{22} - \frac{39391990162572868439326035989945188193}{19103607309758759817573746423121315778684} a^{21} - \frac{36302167205796201180117709683351887297}{19103607309758759817573746423121315778684} a^{20} + \frac{295335784347757856244291424951051124059}{19103607309758759817573746423121315778684} a^{19} - \frac{752690108351001170922908049475019585683}{19103607309758759817573746423121315778684} a^{18} + \frac{47299558648570761856631072512326735879}{4775901827439689954393436605780328944671} a^{17} + \frac{3832311177664687297854193781613613178809}{19103607309758759817573746423121315778684} a^{16} - \frac{2025436391216772173150129789688779959411}{4775901827439689954393436605780328944671} a^{15} - \frac{1917832439017570824998255556360538653815}{19103607309758759817573746423121315778684} a^{14} + \frac{17268144577871543651279087494541643651453}{19103607309758759817573746423121315778684} a^{13} - \frac{16277171580455818487678143019442237262263}{9551803654879379908786873211560657889342} a^{12} + \frac{762869465357261723789326072052338932901}{9551803654879379908786873211560657889342} a^{11} + \frac{48146259113421891675557748866783612739491}{19103607309758759817573746423121315778684} a^{10} + \frac{3599527763879936773784403916923860321412}{4775901827439689954393436605780328944671} a^{9} - \frac{37583223052364503208132322197699388237497}{9551803654879379908786873211560657889342} a^{8} + \frac{84455173517099426581332845742554336685861}{9551803654879379908786873211560657889342} a^{7} - \frac{104710245712381482495586943663507382753619}{19103607309758759817573746423121315778684} a^{6} - \frac{42277470675529955995927849656555932618371}{19103607309758759817573746423121315778684} a^{5} - \frac{18774044300485757411729456801842636937425}{19103607309758759817573746423121315778684} a^{4} - \frac{650815289346223872457060854741942954361}{360445420938844524859882007983421052428} a^{3} + \frac{5456859331257156905695572126963148642717}{4775901827439689954393436605780328944671} a^{2} - \frac{2266690083313810696203952458534633434}{1700214249711530777640952867846325719} a - \frac{3082438277908978956521356705095186307}{6800856998846123110563811471385302876} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 15664142477642.164 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-91}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-455}) \), \(\Q(\sqrt{-35}, \sqrt{65})\), 4.0.13456625.1, 4.4.274625.1, \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{13}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-7}, \sqrt{65})\), 4.0.13456625.2, 4.4.274625.2, \(\Q(\sqrt{5}, \sqrt{-91})\), \(\Q(\sqrt{-7}, \sqrt{13})\), 4.4.1035125.1, 4.0.21125.1, 4.4.6125.1, \(\Q(\zeta_{5})\), 4.0.2197.1, 4.4.2691325.1, 4.0.54925.1, 4.4.107653.1, 8.0.181080756390625.11, 8.0.42859350625.1, 8.0.181080756390625.8, 8.0.181080756390625.7, 8.0.181080756390625.2, 8.8.75418890625.1, 8.0.181080756390625.3, 8.0.1071483765625.1, 8.0.37515625.1, 8.0.7243230255625.2, 8.0.7243230255625.3, 8.8.1071483765625.1, 8.0.446265625.1, 8.0.3016755625.1, 8.8.7243230255625.1, 8.0.1071483765625.3, 8.0.1071483765625.2, 8.0.11589168409.1, 8.0.7243230255625.1, 16.0.32790240335000876777587890625.2, 16.0.1148077459997929931640625.1, 16.0.52464384536001402844140625.1, 16.0.32790240335000876777587890625.1, 16.0.32790240335000876777587890625.3, 16.16.32790240335000876777587890625.1, 16.0.5688009063105712890625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
5Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13Data not computed