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Properties

Label 32.0.85748808121...0625.1
Degree $32$
Signature $[0, 16]$
Discriminant $5^{24}\cdot 41^{28}$
Root discriminant $86.18$
Ramified primes $5, 41$
Class number $2880$ (GRH)
Class group $[24, 120]$ (GRH)
Galois group $C_4\times C_8$ (as 32T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12117361, 32860640, 57690613, 82119327, 93574213, 191861363, 301645453, 330667302, 198051499, -215674134, -146418413, 24562491, 23039126, -34006613, -1285999, 17827751, 7115533, -2136490, -38117, -304226, -292681, -80653, 51558, 5907, 2405, 1503, 417, -406, -21, -6, -2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - 2*x^30 - 6*x^29 - 21*x^28 - 406*x^27 + 417*x^26 + 1503*x^25 + 2405*x^24 + 5907*x^23 + 51558*x^22 - 80653*x^21 - 292681*x^20 - 304226*x^19 - 38117*x^18 - 2136490*x^17 + 7115533*x^16 + 17827751*x^15 - 1285999*x^14 - 34006613*x^13 + 23039126*x^12 + 24562491*x^11 - 146418413*x^10 - 215674134*x^9 + 198051499*x^8 + 330667302*x^7 + 301645453*x^6 + 191861363*x^5 + 93574213*x^4 + 82119327*x^3 + 57690613*x^2 + 32860640*x + 12117361)
 
gp: K = bnfinit(x^32 - x^31 - 2*x^30 - 6*x^29 - 21*x^28 - 406*x^27 + 417*x^26 + 1503*x^25 + 2405*x^24 + 5907*x^23 + 51558*x^22 - 80653*x^21 - 292681*x^20 - 304226*x^19 - 38117*x^18 - 2136490*x^17 + 7115533*x^16 + 17827751*x^15 - 1285999*x^14 - 34006613*x^13 + 23039126*x^12 + 24562491*x^11 - 146418413*x^10 - 215674134*x^9 + 198051499*x^8 + 330667302*x^7 + 301645453*x^6 + 191861363*x^5 + 93574213*x^4 + 82119327*x^3 + 57690613*x^2 + 32860640*x + 12117361, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} - 2 x^{30} - 6 x^{29} - 21 x^{28} - 406 x^{27} + 417 x^{26} + 1503 x^{25} + 2405 x^{24} + 5907 x^{23} + 51558 x^{22} - 80653 x^{21} - 292681 x^{20} - 304226 x^{19} - 38117 x^{18} - 2136490 x^{17} + 7115533 x^{16} + 17827751 x^{15} - 1285999 x^{14} - 34006613 x^{13} + 23039126 x^{12} + 24562491 x^{11} - 146418413 x^{10} - 215674134 x^{9} + 198051499 x^{8} + 330667302 x^{7} + 301645453 x^{6} + 191861363 x^{5} + 93574213 x^{4} + 82119327 x^{3} + 57690613 x^{2} + 32860640 x + 12117361 \)

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:  \(85748808121891140298253320589821486779348249495029449462890625=5^{24}\cdot 41^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.18$
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:  $5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(205=5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{205}(1,·)$, $\chi_{205}(3,·)$, $\chi_{205}(132,·)$, $\chi_{205}(9,·)$, $\chi_{205}(14,·)$, $\chi_{205}(68,·)$, $\chi_{205}(27,·)$, $\chi_{205}(32,·)$, $\chi_{205}(161,·)$, $\chi_{205}(163,·)$, $\chi_{205}(38,·)$, $\chi_{205}(167,·)$, $\chi_{205}(42,·)$, $\chi_{205}(44,·)$, $\chi_{205}(173,·)$, $\chi_{205}(178,·)$, $\chi_{205}(137,·)$, $\chi_{205}(191,·)$, $\chi_{205}(196,·)$, $\chi_{205}(73,·)$, $\chi_{205}(202,·)$, $\chi_{205}(204,·)$, $\chi_{205}(79,·)$, $\chi_{205}(81,·)$, $\chi_{205}(83,·)$, $\chi_{205}(91,·)$, $\chi_{205}(96,·)$, $\chi_{205}(109,·)$, $\chi_{205}(114,·)$, $\chi_{205}(122,·)$, $\chi_{205}(124,·)$, $\chi_{205}(126,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{20} - \frac{1}{4} a^{10} - \frac{1}{2} a^{5} - \frac{1}{4}$, $\frac{1}{4} a^{21} - \frac{1}{4} a^{11} - \frac{1}{2} a^{6} - \frac{1}{4} a$, $\frac{1}{4} a^{22} - \frac{1}{4} a^{12} - \frac{1}{2} a^{7} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{23} - \frac{1}{4} a^{13} - \frac{1}{2} a^{8} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{24} - \frac{1}{4} a^{14} - \frac{1}{2} a^{9} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{25} - \frac{1}{4} a^{15} + \frac{1}{4} a^{5} - \frac{1}{2}$, $\frac{1}{4} a^{26} - \frac{1}{4} a^{16} + \frac{1}{4} a^{6} - \frac{1}{2} a$, $\frac{1}{4} a^{27} - \frac{1}{4} a^{17} + \frac{1}{4} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{3414952} a^{28} + \frac{192153}{3414952} a^{27} - \frac{48039}{853738} a^{26} + \frac{42555}{3414952} a^{25} - \frac{299207}{3414952} a^{24} + \frac{119055}{3414952} a^{23} + \frac{101105}{853738} a^{22} + \frac{345703}{3414952} a^{21} - \frac{298355}{3414952} a^{20} - \frac{21824}{426869} a^{19} - \frac{168043}{3414952} a^{18} - \frac{640111}{3414952} a^{17} + \frac{200751}{853738} a^{16} - \frac{374569}{3414952} a^{15} - \frac{4063}{3414952} a^{14} - \frac{656189}{3414952} a^{13} + \frac{10450}{426869} a^{12} + \frac{600431}{3414952} a^{11} + \frac{559281}{3414952} a^{10} - \frac{196329}{853738} a^{9} + \frac{1667293}{3414952} a^{8} + \frac{1064049}{3414952} a^{7} + \frac{190521}{426869} a^{6} - \frac{343009}{3414952} a^{5} - \frac{704359}{3414952} a^{4} + \frac{214299}{3414952} a^{3} + \frac{42017}{853738} a^{2} + \frac{404863}{3414952} a - \frac{1089431}{3414952}$, $\frac{1}{7297368284657304606127156104386478989676074888} a^{29} + \frac{30212722593150903779707655125735225385}{1824342071164326151531789026096619747419018722} a^{28} - \frac{139647472052438506379417030583238089378845885}{7297368284657304606127156104386478989676074888} a^{27} - \frac{772523926082395687197322838956933293153368107}{7297368284657304606127156104386478989676074888} a^{26} - \frac{79109632285551130479497675315849714809878743}{912171035582163075765894513048309873709509361} a^{25} - \frac{95997390832400374774922344917113032438761875}{3648684142328652303063578052193239494838037444} a^{24} - \frac{784611028354544220823862330456598958036499671}{7297368284657304606127156104386478989676074888} a^{23} - \frac{168811518605617592381320546844431213678869677}{7297368284657304606127156104386478989676074888} a^{22} - \frac{97384161179726254579115328842198760343129483}{912171035582163075765894513048309873709509361} a^{21} + \frac{403022362115699369197182232913961029305563387}{7297368284657304606127156104386478989676074888} a^{20} + \frac{476311676288054032584174198153392822176709937}{7297368284657304606127156104386478989676074888} a^{19} - \frac{3014863746858485975008477973547473920267001}{15460526026816323318066008695734065656093379} a^{18} - \frac{1080205480624190071001589047257313152654990769}{7297368284657304606127156104386478989676074888} a^{17} - \frac{36448339053668640122302984246007688291889803}{197226169855602827192625840659094026748002024} a^{16} + \frac{101877446178278220672274014374766966570244285}{1824342071164326151531789026096619747419018722} a^{15} + \frac{196651941950621938655623264242508100665561885}{3648684142328652303063578052193239494838037444} a^{14} + \frac{1470124767318456297230725199608541440237913917}{7297368284657304606127156104386478989676074888} a^{13} - \frac{588356761948129573810005921219308077616725677}{7297368284657304606127156104386478989676074888} a^{12} - \frac{134047738555703680789661871925565976853133816}{912171035582163075765894513048309873709509361} a^{11} - \frac{1042700712469440208731721365489331858575490717}{7297368284657304606127156104386478989676074888} a^{10} - \frac{713235795375617350739252552748246554603658003}{7297368284657304606127156104386478989676074888} a^{9} + \frac{772880603570883004869649485285579804955626257}{1824342071164326151531789026096619747419018722} a^{8} + \frac{2341815027216462257451624257826145401879087403}{7297368284657304606127156104386478989676074888} a^{7} - \frac{940627365018347281915807035892873026429516095}{7297368284657304606127156104386478989676074888} a^{6} - \frac{581407292336345719689609488277161821163697721}{1824342071164326151531789026096619747419018722} a^{5} - \frac{1697839318499961251929288042211956465553987309}{3648684142328652303063578052193239494838037444} a^{4} + \frac{941258322443607435800029414256234993204678057}{7297368284657304606127156104386478989676074888} a^{3} - \frac{2002130549414063937874374375557201731299786245}{7297368284657304606127156104386478989676074888} a^{2} + \frac{248785173538149533621234568296118909271118585}{912171035582163075765894513048309873709509361} a + \frac{35559147789526171617068938593723800882685357}{123684208214530586544528069565872525248747032}$, $\frac{1}{430544728794780971761502210158802260390888418392} a^{30} - \frac{1}{430544728794780971761502210158802260390888418392} a^{29} - \frac{9442321548434819063129098770308439718137}{215272364397390485880751105079401130195444209196} a^{28} + \frac{1719564834441427641395266749243090829751565595}{430544728794780971761502210158802260390888418392} a^{27} + \frac{52098582918835484437706888295199813569747795493}{430544728794780971761502210158802260390888418392} a^{26} - \frac{35482670152537016629254675541455293452337216741}{430544728794780971761502210158802260390888418392} a^{25} + \frac{2715851554939248241928370073053061050638120183}{53818091099347621470187776269850282548861052299} a^{24} - \frac{23860425466353701219716789922795897477973240443}{430544728794780971761502210158802260390888418392} a^{23} - \frac{37885587011743228360994160597447981850043766835}{430544728794780971761502210158802260390888418392} a^{22} + \frac{558608260624799008417542880959655603861726075}{215272364397390485880751105079401130195444209196} a^{21} - \frac{20557847825426673165608312260512161701202370137}{430544728794780971761502210158802260390888418392} a^{20} - \frac{142147335218220045236321589341499982425437127}{7297368284657304606127156104386478989676074888} a^{19} + \frac{52294178290434626187873521655583188335658417347}{215272364397390485880751105079401130195444209196} a^{18} + \frac{25315916730077619767652538151263276887094267151}{430544728794780971761502210158802260390888418392} a^{17} + \frac{14604849694700162304851654507175154964708942893}{430544728794780971761502210158802260390888418392} a^{16} - \frac{101653409286540609437707711358517502587073243037}{430544728794780971761502210158802260390888418392} a^{15} - \frac{3821522853381682424832296499164407408530280317}{53818091099347621470187776269850282548861052299} a^{14} + \frac{376491444009279909944233725984884983465882415}{5187285889093746647728942291069906751697450824} a^{13} + \frac{12674014681301285590618971210961693490834940661}{430544728794780971761502210158802260390888418392} a^{12} + \frac{24891771372190063837788019281837428746665868265}{215272364397390485880751105079401130195444209196} a^{11} + \frac{60901692630579140767513770602677101152925007871}{430544728794780971761502210158802260390888418392} a^{10} + \frac{188645786460320870542880427025662821792937787451}{430544728794780971761502210158802260390888418392} a^{9} - \frac{16502143144838434824699074137996410865931515075}{215272364397390485880751105079401130195444209196} a^{8} - \frac{145774719682174255460766640304484810689739771581}{430544728794780971761502210158802260390888418392} a^{7} + \frac{86801214829862915297625840042965748097862925381}{430544728794780971761502210158802260390888418392} a^{6} + \frac{194694949546689405451328617326383679661801534735}{430544728794780971761502210158802260390888418392} a^{5} - \frac{19687804454370743526341459062606179154846871956}{53818091099347621470187776269850282548861052299} a^{4} - \frac{23009688016014415650803276389517166193425438747}{430544728794780971761502210158802260390888418392} a^{3} + \frac{2412697196197200735726603503918509558133093239}{5187285889093746647728942291069906751697450824} a^{2} - \frac{1572159214515708599647873041843133120961186559}{3648684142328652303063578052193239494838037444} a + \frac{19195970201931720796162463736765579517393311}{61842104107265293272264034782936262624373516}$, $\frac{1}{25402138998892077333928630399369333363062416685128} a^{31} - \frac{1}{25402138998892077333928630399369333363062416685128} a^{30} - \frac{1}{12701069499446038666964315199684666681531208342564} a^{29} - \frac{566834810377099814883203904770956345171491}{25402138998892077333928630399369333363062416685128} a^{28} + \frac{2633013870287014505079784182958240983323234927497}{25402138998892077333928630399369333363062416685128} a^{27} - \frac{2288329223647948959792594251566124129340041622685}{25402138998892077333928630399369333363062416685128} a^{26} - \frac{1217719388584335370303857891072239834863532093941}{12701069499446038666964315199684666681531208342564} a^{25} - \frac{611096935143927343053799815198988814432395086873}{25402138998892077333928630399369333363062416685128} a^{24} + \frac{1430096167077716695088767700827979813363376103641}{25402138998892077333928630399369333363062416685128} a^{23} + \frac{168079124136314676828609326004136193143466169337}{3175267374861509666741078799921166670382802085641} a^{22} - \frac{1898304211807539339561763386231600546817277935913}{25402138998892077333928630399369333363062416685128} a^{21} + \frac{6952023928427110506055324682193262416302183531}{430544728794780971761502210158802260390888418392} a^{20} - \frac{3003047299499634437792224855527896872009321506349}{12701069499446038666964315199684666681531208342564} a^{19} + \frac{5789291745850211985930211533113646941397127699025}{25402138998892077333928630399369333363062416685128} a^{18} - \frac{2716128518938096955476712205201044857134395365155}{25402138998892077333928630399369333363062416685128} a^{17} + \frac{27754913164177316330058721629117240839914598041}{182749201430878254200925398556613909086779976152} a^{16} + \frac{33494787251882698811400429269689596016639049553}{12701069499446038666964315199684666681531208342564} a^{15} + \frac{2751324752761488775115145534294408306264131759483}{25402138998892077333928630399369333363062416685128} a^{14} + \frac{4731379225508461074281765609602183972149426149525}{25402138998892077333928630399369333363062416685128} a^{13} + \frac{231362908318483659188360711601815397389716388979}{6350534749723019333482157599842333340765604171282} a^{12} - \frac{1940040392398494849734899862786369346484341894833}{25402138998892077333928630399369333363062416685128} a^{11} - \frac{3147699900530158310257014198811513245198142422943}{25402138998892077333928630399369333363062416685128} a^{10} + \frac{5745268927732501017833574185444010791507731404447}{12701069499446038666964315199684666681531208342564} a^{9} - \frac{5671320600201566868477867122423976316550058605583}{25402138998892077333928630399369333363062416685128} a^{8} + \frac{5331509392067082153681767514228081278135512350313}{25402138998892077333928630399369333363062416685128} a^{7} - \frac{7792716019710219123757062521547811980951701498793}{25402138998892077333928630399369333363062416685128} a^{6} - \frac{3972273193210433972945293538322500090643577437137}{12701069499446038666964315199684666681531208342564} a^{5} - \frac{4777532942932326348990224513536359928318874920101}{25402138998892077333928630399369333363062416685128} a^{4} - \frac{324319000024583742545011134540492652097905248503}{686544297267353441457530551334306307109795045544} a^{3} + \frac{47200508022060019676880763743933424376542269229}{107636182198695242940375552539700565097722104598} a^{2} - \frac{945320339200480901764808032388085497990468441}{3648684142328652303063578052193239494838037444} a - \frac{5891678498765988107846136082504391833261661}{61842104107265293272264034782936262624373516}$

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

Class group and class number

$C_{24}\times C_{120}$, which has order $2880$ (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{3351688928555410103332475260572560665}{29754021724337065158079680650702362273979156} a^{31} + \frac{9125681253553925101559190072863140315}{59508043448674130316159361301404724547958312} a^{30} + \frac{10804301049786676563119830252163405235}{59508043448674130316159361301404724547958312} a^{29} + \frac{35769768776669360540494075849917828335}{59508043448674130316159361301404724547958312} a^{28} + \frac{15719247709336303471531440209523563980}{7438505431084266289519920162675590568494789} a^{27} + \frac{2672631476098600654930321710846353440505}{59508043448674130316159361301404724547958312} a^{26} - \frac{1887963181450120417480060641655046206045}{29754021724337065158079680650702362273979156} a^{25} - \frac{8993802010348008922761568417527869092165}{59508043448674130316159361301404724547958312} a^{24} - \frac{3158173008435643029623647346820705280241}{14877010862168532579039840325351181136989578} a^{23} - \frac{33709466486403738031614715449557773517075}{59508043448674130316159361301404724547958312} a^{22} - \frac{331933440924102812597223129851119610213075}{59508043448674130316159361301404724547958312} a^{21} + \frac{1407678186656370467014184318450646797720}{126076363238716377788473223096196450313471} a^{20} + \frac{1757956179951673180123634937112242891907335}{59508043448674130316159361301404724547958312} a^{19} + \frac{1353245802417454786200230374747520950116687}{59508043448674130316159361301404724547958312} a^{18} - \frac{118799739281401450023285461614234712313985}{14877010862168532579039840325351181136989578} a^{17} + \frac{14303535978651991816473925921192140419532225}{59508043448674130316159361301404724547958312} a^{16} - \frac{26407214063300624971927961814694835713391215}{29754021724337065158079680650702362273979156} a^{15} - \frac{101875224672582246216618106272070296390331785}{59508043448674130316159361301404724547958312} a^{14} + \frac{12497714221068064404135623670301114789089791}{14877010862168532579039840325351181136989578} a^{13} + \frac{224563371162411993435848754397673056942917665}{59508043448674130316159361301404724547958312} a^{12} - \frac{238561814759428046161974492985452444777744755}{59508043448674130316159361301404724547958312} a^{11} - \frac{13623387954110678334557881648743543997788600}{7438505431084266289519920162675590568494789} a^{10} + \frac{1040231377373888666951173455736720436371242695}{59508043448674130316159361301404724547958312} a^{9} + \frac{1105938589058118229104097768440119758782593939}{59508043448674130316159361301404724547958312} a^{8} - \frac{231841766242752510737230587797672268865074860}{7438505431084266289519920162675590568494789} a^{7} - \frac{1721845056251773964006460609914228109686686535}{59508043448674130316159361301404724547958312} a^{6} - \frac{600650296207833401367194359998358086708409975}{29754021724337065158079680650702362273979156} a^{5} - \frac{538780963028975809628517177583439982746957465}{59508043448674130316159361301404724547958312} a^{4} - \frac{45087022749726116917477932672494931891028102}{7438505431084266289519920162675590568494789} a^{3} - \frac{91415668684571027044799246539900346292035}{17095100100164932581487894657111383093352} a^{2} - \frac{892071760559704247130041297320659465175}{289747459324829365787930417917142086328} a - \frac{377261639183494570360475695303218748945}{289747459324829365787930417917142086328} \) (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:  \( 960117309726421.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times C_8$ (as 32T43):

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 $C_4\times C_8$
Character table for $C_4\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{205}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{5}) \), 4.0.8615125.2, \(\Q(\sqrt{5}, \sqrt{41})\), 4.0.8615125.1, 4.4.1723025.1, 4.4.68921.1, 4.0.210125.1, \(\Q(\zeta_{5})\), 8.0.74220378765625.1, 8.8.2968815150625.1, 8.0.44152515625.1, 8.8.3043035529390625.2, 8.8.3043035529390625.1, 8.0.121721421175625.1, 8.0.194754273881.1, 16.0.5508664624112838398681640625.1, 16.16.9260065233133681348183837890625.1, 16.0.14816104373013890157094140625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$