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Properties

Label 32.0.59816430901...0000.7
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}$
Root discriminant $79.30$
Ramified primes $2, 3, 5, 13$
Class number $106496$ (GRH)
Class group $[8, 8, 8, 208]$ (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![13791376, -6471264, 53648656, -23248712, 88122560, -66043536, -52870050, 118490302, 42474577, -58786556, 31876680, -2351046, -10518084, 10131622, -929450, 243430, 2465482, -937946, 496440, 132756, -98673, 78166, -20336, 5560, 5491, -5330, 1774, 378, -389, 74, 16, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 + 16*x^30 + 74*x^29 - 389*x^28 + 378*x^27 + 1774*x^26 - 5330*x^25 + 5491*x^24 + 5560*x^23 - 20336*x^22 + 78166*x^21 - 98673*x^20 + 132756*x^19 + 496440*x^18 - 937946*x^17 + 2465482*x^16 + 243430*x^15 - 929450*x^14 + 10131622*x^13 - 10518084*x^12 - 2351046*x^11 + 31876680*x^10 - 58786556*x^9 + 42474577*x^8 + 118490302*x^7 - 52870050*x^6 - 66043536*x^5 + 88122560*x^4 - 23248712*x^3 + 53648656*x^2 - 6471264*x + 13791376)
 
gp: K = bnfinit(x^32 - 8*x^31 + 16*x^30 + 74*x^29 - 389*x^28 + 378*x^27 + 1774*x^26 - 5330*x^25 + 5491*x^24 + 5560*x^23 - 20336*x^22 + 78166*x^21 - 98673*x^20 + 132756*x^19 + 496440*x^18 - 937946*x^17 + 2465482*x^16 + 243430*x^15 - 929450*x^14 + 10131622*x^13 - 10518084*x^12 - 2351046*x^11 + 31876680*x^10 - 58786556*x^9 + 42474577*x^8 + 118490302*x^7 - 52870050*x^6 - 66043536*x^5 + 88122560*x^4 - 23248712*x^3 + 53648656*x^2 - 6471264*x + 13791376, 1)
 

Normalized defining polynomial

\( x^{32} - 8 x^{31} + 16 x^{30} + 74 x^{29} - 389 x^{28} + 378 x^{27} + 1774 x^{26} - 5330 x^{25} + 5491 x^{24} + 5560 x^{23} - 20336 x^{22} + 78166 x^{21} - 98673 x^{20} + 132756 x^{19} + 496440 x^{18} - 937946 x^{17} + 2465482 x^{16} + 243430 x^{15} - 929450 x^{14} + 10131622 x^{13} - 10518084 x^{12} - 2351046 x^{11} + 31876680 x^{10} - 58786556 x^{9} + 42474577 x^{8} + 118490302 x^{7} - 52870050 x^{6} - 66043536 x^{5} + 88122560 x^{4} - 23248712 x^{3} + 53648656 x^{2} - 6471264 x + 13791376 \)

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:  \(5981643090147991811559885370844487936000000000000000000000000=2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.30$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(780=2^{2}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{780}(1,·)$, $\chi_{780}(131,·)$, $\chi_{780}(649,·)$, $\chi_{780}(779,·)$, $\chi_{780}(541,·)$, $\chi_{780}(671,·)$, $\chi_{780}(547,·)$, $\chi_{780}(421,·)$, $\chi_{780}(551,·)$, $\chi_{780}(307,·)$, $\chi_{780}(181,·)$, $\chi_{780}(311,·)$, $\chi_{780}(187,·)$, $\chi_{780}(317,·)$, $\chi_{780}(53,·)$, $\chi_{780}(343,·)$, $\chi_{780}(437,·)$, $\chi_{780}(599,·)$, $\chi_{780}(77,·)$, $\chi_{780}(463,·)$, $\chi_{780}(593,·)$, $\chi_{780}(469,·)$, $\chi_{780}(727,·)$, $\chi_{780}(473,·)$, $\chi_{780}(677,·)$, $\chi_{780}(229,·)$, $\chi_{780}(103,·)$, $\chi_{780}(233,·)$, $\chi_{780}(359,·)$, $\chi_{780}(109,·)$, $\chi_{780}(239,·)$, $\chi_{780}(703,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{12} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{13} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{24} + \frac{1}{6} a^{22} + \frac{1}{6} a^{20} - \frac{1}{6} a^{18} + \frac{1}{6} a^{16} + \frac{1}{6} a^{14} - \frac{1}{3} a^{12} - \frac{1}{6} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{6} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{25} + \frac{1}{6} a^{23} + \frac{1}{6} a^{21} - \frac{1}{6} a^{19} + \frac{1}{6} a^{17} + \frac{1}{6} a^{15} - \frac{1}{3} a^{13} - \frac{1}{6} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{6} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{6} a^{26} + \frac{1}{6} a^{20} - \frac{1}{6} a^{18} + \frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{2} a^{6} - \frac{1}{3} a^{4} - \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{27} + \frac{1}{6} a^{21} - \frac{1}{6} a^{19} + \frac{1}{6} a^{13} - \frac{1}{6} a^{11} - \frac{1}{2} a^{7} - \frac{1}{3} a^{5} - \frac{1}{6} a^{3} + \frac{1}{3} a$, $\frac{1}{36} a^{28} + \frac{1}{18} a^{27} + \frac{1}{18} a^{26} + \frac{1}{18} a^{25} - \frac{1}{12} a^{24} + \frac{2}{9} a^{23} + \frac{1}{9} a^{22} + \frac{1}{9} a^{21} - \frac{5}{36} a^{20} + \frac{1}{18} a^{19} - \frac{2}{9} a^{18} - \frac{1}{9} a^{17} + \frac{1}{12} a^{16} + \frac{1}{18} a^{15} - \frac{1}{18} a^{14} - \frac{2}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{18} a^{11} + \frac{5}{18} a^{10} + \frac{7}{18} a^{9} - \frac{1}{2} a^{8} + \frac{2}{9} a^{7} + \frac{5}{18} a^{6} - \frac{7}{18} a^{5} - \frac{11}{36} a^{4} - \frac{5}{18} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{36} a^{29} - \frac{1}{18} a^{27} - \frac{1}{18} a^{26} - \frac{1}{36} a^{25} + \frac{1}{18} a^{24} - \frac{1}{6} a^{23} + \frac{1}{18} a^{22} - \frac{7}{36} a^{21} + \frac{1}{6} a^{18} - \frac{1}{36} a^{17} + \frac{1}{18} a^{16} + \frac{1}{18} a^{14} + \frac{2}{9} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} + \frac{1}{6} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{2}{9} a^{6} - \frac{13}{36} a^{5} + \frac{2}{9} a^{3} - \frac{1}{2} a^{2} + \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{123313590142538967144980274630246194647355040843745317357528} a^{30} - \frac{515219487727469852305240509318948897349330298920745490895}{61656795071269483572490137315123097323677520421872658678764} a^{29} - \frac{166291585729599438057867341137325301709293056513752999}{36119973679712644154944427249632745942400422039761370052} a^{28} + \frac{221233488658415811034685601787810943964225320734330059145}{6850755007918831508054459701680344147075280046874739853196} a^{27} + \frac{4802101019042885120329308390034767115536030352425770835295}{123313590142538967144980274630246194647355040843745317357528} a^{26} - \frac{1711459251957987167167205797748411835853931088132508080205}{30828397535634741786245068657561548661838760210936329339382} a^{25} + \frac{1817565649223919531851148941045531505614804407622211016403}{30828397535634741786245068657561548661838760210936329339382} a^{24} - \frac{1383892810118323081671201999088794120891586281301871590483}{20552265023756494524163379105041032441225840140624219559588} a^{23} - \frac{1002733279964149645135841259823516295482561488211794349085}{13701510015837663016108919403360688294150560093749479706392} a^{22} + \frac{14881708195605354540738727643329034356893015157802429375789}{61656795071269483572490137315123097323677520421872658678764} a^{21} + \frac{13075068304956019599511898076462378918802878911888351604295}{61656795071269483572490137315123097323677520421872658678764} a^{20} + \frac{6522170327471314609243071841918523885889982728182243339375}{61656795071269483572490137315123097323677520421872658678764} a^{19} - \frac{9322169277900507690355972063610085379939075455380427424589}{123313590142538967144980274630246194647355040843745317357528} a^{18} - \frac{459657904284497370250852220723731527401514673279355559647}{61656795071269483572490137315123097323677520421872658678764} a^{17} + \frac{8360023080612933636239649569072622014993529833970602640601}{61656795071269483572490137315123097323677520421872658678764} a^{16} - \frac{3251233756581144585252513861404626117938120061013841143411}{20552265023756494524163379105041032441225840140624219559588} a^{15} - \frac{1041728290648775629417719627031766825793439806012402899755}{20552265023756494524163379105041032441225840140624219559588} a^{14} + \frac{8358827877575959093286318273425179635639793811585348391303}{61656795071269483572490137315123097323677520421872658678764} a^{13} - \frac{2366548090899109679254085187172714696261082240726028806339}{61656795071269483572490137315123097323677520421872658678764} a^{12} - \frac{12570264696269996111691160399058965804270979862699964003381}{61656795071269483572490137315123097323677520421872658678764} a^{11} + \frac{13128988171057175223141838119547971057515583797605158057271}{30828397535634741786245068657561548661838760210936329339382} a^{10} - \frac{2668268042128391898272947328131432022656102672572930087491}{61656795071269483572490137315123097323677520421872658678764} a^{9} - \frac{132669449986553778660667139917714216495270402461374427857}{15414198767817370893122534328780774330919380105468164669691} a^{8} + \frac{1784452544457363610847442113007274256283165668494648213384}{5138066255939123631040844776260258110306460035156054889897} a^{7} + \frac{19944289756224824149097083014307884064289200294131679740175}{41104530047512989048326758210082064882451680281248439119176} a^{6} - \frac{13235635212204478405633938947009567612804387538756824546555}{30828397535634741786245068657561548661838760210936329339382} a^{5} + \frac{70954008963852344002279016123810051581870607283751776209}{15414198767817370893122534328780774330919380105468164669691} a^{4} - \frac{3202817119636959606154299991685299816106843164204960988737}{10276132511878247262081689552520516220612920070312109779794} a^{3} + \frac{14505041447884448601641134856957995933252667341695746782831}{30828397535634741786245068657561548661838760210936329339382} a^{2} - \frac{5742338569550362567803047816265745439451748052025624339299}{15414198767817370893122534328780774330919380105468164669691} a + \frac{625520068198638031739536206675957142793102936307764013326}{5138066255939123631040844776260258110306460035156054889897}$, $\frac{1}{34728580099512671826657715456906431831380291945050735198657259798695616134941282648} a^{31} - \frac{53171109134979390283997}{34728580099512671826657715456906431831380291945050735198657259798695616134941282648} a^{30} + \frac{67138485513927100701580971346937413001611739438630491976238489588451601240848829}{5788096683252111971109619242817738638563381990841789199776209966449269355823547108} a^{29} + \frac{78812738203641949700700450780709182871705788690977183680112677366318417135613149}{8682145024878167956664428864226607957845072986262683799664314949673904033735320662} a^{28} - \frac{265915488986037848313716745617202262961181501591314636556883160536940359630589643}{3858731122168074647406412828545159092375587993894526133184139977632846237215698072} a^{27} - \frac{2596146107061923983754258682002786203100605424050950931454754046281327491145266681}{34728580099512671826657715456906431831380291945050735198657259798695616134941282648} a^{26} + \frac{1048829398474626372566863613928400504011582765617172711886877140003030717070762597}{17364290049756335913328857728453215915690145972525367599328629899347808067470641324} a^{25} - \frac{387386339228039416991397512571473928113948467994412654007548152543990908838658335}{5788096683252111971109619242817738638563381990841789199776209966449269355823547108} a^{24} - \frac{4023895786036020512500664122425872011948305144649397422732346671074658198480720039}{34728580099512671826657715456906431831380291945050735198657259798695616134941282648} a^{23} - \frac{2381072615870394733462860269684888952866146408180856578185786377473950316413201763}{34728580099512671826657715456906431831380291945050735198657259798695616134941282648} a^{22} + \frac{659048487987229921207531764333068100967682056716460077441191422929474756724221107}{5788096683252111971109619242817738638563381990841789199776209966449269355823547108} a^{21} + \frac{258203805105639245355866745915446253179637116276621535015399308410034235219369935}{8682145024878167956664428864226607957845072986262683799664314949673904033735320662} a^{20} + \frac{1934903323515813898459871646250864144553749675370359120638447905075390799100827131}{11576193366504223942219238485635477277126763981683578399552419932898538711647094216} a^{19} + \frac{1049509137423453294058368901676241712227687816259577317913357378699346461749750123}{11576193366504223942219238485635477277126763981683578399552419932898538711647094216} a^{18} - \frac{2301281795576378373776418556048854073048196414127848025864301734777943353628706389}{17364290049756335913328857728453215915690145972525367599328629899347808067470641324} a^{17} + \frac{30535566924741367986730241666337036168382460907235548787119564451719866814269287}{1447024170813027992777404810704434659640845497710447299944052491612317338955886777} a^{16} - \frac{1701039049029516248926929359373160700501594259626326956332368895911586213990533441}{8682145024878167956664428864226607957845072986262683799664314949673904033735320662} a^{15} - \frac{737715926803639705504302487974584580803766800751992768673250523771269797841675655}{4341072512439083978332214432113303978922536493131341899832157474836952016867660331} a^{14} + \frac{143436065101004536929954134509005274923252972561861318209998818137618857035249482}{482341390271009330925801603568144886546948499236815766648017497204105779651962259} a^{13} + \frac{98726906890828392148725919137172616932356572191862525123028848511553847478047893}{4341072512439083978332214432113303978922536493131341899832157474836952016867660331} a^{12} + \frac{962301685270479348220445812097455013446347776273470728211068800129784765462404909}{5788096683252111971109619242817738638563381990841789199776209966449269355823547108} a^{11} + \frac{1411718649172445305149276558248089554630202889111917066301622337249417851727576529}{5788096683252111971109619242817738638563381990841789199776209966449269355823547108} a^{10} + \frac{143243011112863071278843937905010765903717594062625503252386493011900360575926003}{17364290049756335913328857728453215915690145972525367599328629899347808067470641324} a^{9} - \frac{444874941695140329001373300665401294952820419317076859909388771391864234285561660}{1447024170813027992777404810704434659640845497710447299944052491612317338955886777} a^{8} + \frac{10372914066966214568095002080575446579304453740749695294449848766401537796829349345}{34728580099512671826657715456906431831380291945050735198657259798695616134941282648} a^{7} + \frac{4439061984185874086988042369487074918470738701744469049897383750175893658344874581}{34728580099512671826657715456906431831380291945050735198657259798695616134941282648} a^{6} - \frac{2036311014083867875297704909239981418139869231342176793995610563354102281957257317}{17364290049756335913328857728453215915690145972525367599328629899347808067470641324} a^{5} - \frac{2248108400998946672091297269239842966010544496172581474038037421048128281774977835}{8682145024878167956664428864226607957845072986262683799664314949673904033735320662} a^{4} + \frac{95202961737571500009567232134229220507156347093334085972686941046950185627101251}{2894048341626055985554809621408869319281690995420894599888104983224634677911773554} a^{3} - \frac{1783867038284914219083370800793730354083118487212537816772912952632381528931169762}{4341072512439083978332214432113303978922536493131341899832157474836952016867660331} a^{2} - \frac{13118755643757050253842394090542433038595403051568178283493932560425615339284978}{482341390271009330925801603568144886546948499236815766648017497204105779651962259} a + \frac{1398943366778663587776184036931220066664755593839571492618768967487022297767615721}{4341072512439083978332214432113303978922536493131341899832157474836952016867660331}$

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

Class group and class number

$C_{8}\times C_{8}\times C_{8}\times C_{208}$, which has order $106496$ (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:  \( -1 \) (order $2$)
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:  \( 1832903051577.0957 \) (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

\(\Q(\sqrt{195}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{39}) \), \(\Q(\sqrt{3}, \sqrt{65})\), 4.0.4394000.2, 4.0.2471625.2, \(\Q(\sqrt{13}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{39})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{15}, \sqrt{39})\), 4.0.4394000.1, 4.0.2471625.1, \(\Q(\sqrt{3}, \sqrt{13})\), \(\Q(\sqrt{3}, \sqrt{5})\), 4.0.54925.1, 4.0.316368.2, 4.0.2197.1, 4.0.7909200.1, 4.4.338000.1, \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{20})^+\), 4.4.190125.1, 8.0.1563886116000000.2, 8.8.370150560000.1, 8.0.1563886116000000.62, 8.0.19307236000000.5, 8.0.1563886116000000.26, 8.0.6108930140625.4, 8.0.1563886116000000.24, 8.0.62555444640000.61, 8.0.62555444640000.66, 8.8.9253764000000.3, 8.8.9253764000000.2, 8.0.3016755625.1, 8.0.62555444640000.63, 8.8.114244000000.1, 8.8.36147515625.1, 8.0.62555444640000.41, 8.0.100088711424.3, 8.8.9253764000000.1, \(\Q(\zeta_{60})^+\), 16.0.2445739783817565456000000000000.3, 16.0.3913183654108104729600000000.2, 16.16.85632148167696000000000000.1, 16.0.372769361959696000000000000.2, 16.0.2445739783817565456000000000000.15, 16.0.37319027463036582275390625.2, 16.0.2445739783817565456000000000000.5

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ 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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
13Data not computed