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Properties

Label 36.0.12295831229...0625.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{18}\cdot 5^{18}\cdot 19^{32}$
Root discriminant $53.05$
Ramified primes $3, 5, 19$
Class number $14661$ (GRH)
Class group $[9, 1629]$ (GRH)
Galois group $C_2\times C_{18}$ (as 36T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 80, -215, 2670, -3269, 24598, -9682, 158238, -2642, 567447, 75846, 1437594, 259210, 2498777, 424487, 3227734, 471070, 3064502, 335012, 2210278, 171615, 1180619, 48836, 472642, 7347, 134511, -2298, 28017, -789, 4027, -173, 413, -15, 26, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 26*x^34 - 15*x^33 + 413*x^32 - 173*x^31 + 4027*x^30 - 789*x^29 + 28017*x^28 - 2298*x^27 + 134511*x^26 + 7347*x^25 + 472642*x^24 + 48836*x^23 + 1180619*x^22 + 171615*x^21 + 2210278*x^20 + 335012*x^19 + 3064502*x^18 + 471070*x^17 + 3227734*x^16 + 424487*x^15 + 2498777*x^14 + 259210*x^13 + 1437594*x^12 + 75846*x^11 + 567447*x^10 - 2642*x^9 + 158238*x^8 - 9682*x^7 + 24598*x^6 - 3269*x^5 + 2670*x^4 - 215*x^3 + 80*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^36 - x^35 + 26*x^34 - 15*x^33 + 413*x^32 - 173*x^31 + 4027*x^30 - 789*x^29 + 28017*x^28 - 2298*x^27 + 134511*x^26 + 7347*x^25 + 472642*x^24 + 48836*x^23 + 1180619*x^22 + 171615*x^21 + 2210278*x^20 + 335012*x^19 + 3064502*x^18 + 471070*x^17 + 3227734*x^16 + 424487*x^15 + 2498777*x^14 + 259210*x^13 + 1437594*x^12 + 75846*x^11 + 567447*x^10 - 2642*x^9 + 158238*x^8 - 9682*x^7 + 24598*x^6 - 3269*x^5 + 2670*x^4 - 215*x^3 + 80*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} + 26 x^{34} - 15 x^{33} + 413 x^{32} - 173 x^{31} + 4027 x^{30} - 789 x^{29} + 28017 x^{28} - 2298 x^{27} + 134511 x^{26} + 7347 x^{25} + 472642 x^{24} + 48836 x^{23} + 1180619 x^{22} + 171615 x^{21} + 2210278 x^{20} + 335012 x^{19} + 3064502 x^{18} + 471070 x^{17} + 3227734 x^{16} + 424487 x^{15} + 2498777 x^{14} + 259210 x^{13} + 1437594 x^{12} + 75846 x^{11} + 567447 x^{10} - 2642 x^{9} + 158238 x^{8} - 9682 x^{7} + 24598 x^{6} - 3269 x^{5} + 2670 x^{4} - 215 x^{3} + 80 x^{2} + 5 x + 1 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(122958312298624478991860638998897557972401876087680816650390625=3^{18}\cdot 5^{18}\cdot 19^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.05$
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:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(285=3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{285}(256,·)$, $\chi_{285}(1,·)$, $\chi_{285}(131,·)$, $\chi_{285}(4,·)$, $\chi_{285}(134,·)$, $\chi_{285}(11,·)$, $\chi_{285}(271,·)$, $\chi_{285}(16,·)$, $\chi_{285}(149,·)$, $\chi_{285}(26,·)$, $\chi_{285}(161,·)$, $\chi_{285}(169,·)$, $\chi_{285}(44,·)$, $\chi_{285}(176,·)$, $\chi_{285}(49,·)$, $\chi_{285}(61,·)$, $\chi_{285}(191,·)$, $\chi_{285}(64,·)$, $\chi_{285}(194,·)$, $\chi_{285}(139,·)$, $\chi_{285}(196,·)$, $\chi_{285}(199,·)$, $\chi_{285}(74,·)$, $\chi_{285}(206,·)$, $\chi_{285}(214,·)$, $\chi_{285}(101,·)$, $\chi_{285}(226,·)$, $\chi_{285}(229,·)$, $\chi_{285}(104,·)$, $\chi_{285}(106,·)$, $\chi_{285}(239,·)$, $\chi_{285}(244,·)$, $\chi_{285}(119,·)$, $\chi_{285}(121,·)$, $\chi_{285}(251,·)$, $\chi_{285}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{21} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{28} - \frac{1}{2} a^{22} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{29} - \frac{1}{2} a^{23} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{30} - \frac{1}{2} a^{24} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{31} - \frac{1}{2} a^{25} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{74} a^{32} - \frac{2}{37} a^{31} - \frac{15}{74} a^{30} + \frac{4}{37} a^{29} - \frac{5}{74} a^{28} - \frac{5}{74} a^{27} + \frac{11}{74} a^{26} + \frac{2}{37} a^{25} - \frac{3}{74} a^{24} + \frac{25}{74} a^{22} + \frac{29}{74} a^{21} - \frac{14}{37} a^{20} - \frac{9}{37} a^{19} + \frac{18}{37} a^{18} + \frac{12}{37} a^{17} - \frac{6}{37} a^{16} - \frac{17}{37} a^{15} + \frac{18}{37} a^{14} + \frac{17}{37} a^{13} - \frac{1}{37} a^{12} - \frac{33}{74} a^{11} - \frac{10}{37} a^{10} + \frac{19}{74} a^{9} + \frac{21}{74} a^{8} + \frac{27}{74} a^{7} - \frac{7}{37} a^{6} + \frac{1}{74} a^{5} + \frac{31}{74} a^{4} - \frac{14}{37} a^{3} - \frac{11}{37} a^{2} - \frac{35}{74} a + \frac{7}{74}$, $\frac{1}{74} a^{33} + \frac{3}{37} a^{31} - \frac{15}{74} a^{30} - \frac{5}{37} a^{29} + \frac{6}{37} a^{28} - \frac{9}{74} a^{27} - \frac{13}{37} a^{26} - \frac{12}{37} a^{25} + \frac{25}{74} a^{24} - \frac{6}{37} a^{23} + \frac{9}{37} a^{22} + \frac{7}{37} a^{21} + \frac{9}{37} a^{20} - \frac{18}{37} a^{19} + \frac{10}{37} a^{18} + \frac{5}{37} a^{17} - \frac{4}{37} a^{16} - \frac{13}{37} a^{15} + \frac{15}{37} a^{14} - \frac{7}{37} a^{13} + \frac{33}{74} a^{12} - \frac{2}{37} a^{11} - \frac{12}{37} a^{10} - \frac{7}{37} a^{9} + \frac{10}{37} a^{7} - \frac{9}{37} a^{6} - \frac{1}{37} a^{5} + \frac{11}{37} a^{4} - \frac{23}{74} a^{3} - \frac{6}{37} a^{2} - \frac{11}{37} a + \frac{14}{37}$, $\frac{1}{193033005698565382397562129218} a^{34} - \frac{1789260471868834311397239}{2608554131061694356723812557} a^{33} + \frac{365880686594699037551401001}{96516502849282691198781064609} a^{32} - \frac{18780639902075150542254363604}{96516502849282691198781064609} a^{31} + \frac{20934294653000890732567554562}{96516502849282691198781064609} a^{30} + \frac{11510269802061495921186785461}{96516502849282691198781064609} a^{29} - \frac{12231605796511427375051060990}{96516502849282691198781064609} a^{28} + \frac{2249709132527648219860585849}{193033005698565382397562129218} a^{27} + \frac{30292377825403254927383743598}{96516502849282691198781064609} a^{26} + \frac{39476530361007266434265375136}{96516502849282691198781064609} a^{25} - \frac{6681698161309959676598601883}{96516502849282691198781064609} a^{24} - \frac{4454329795778124967066627805}{96516502849282691198781064609} a^{23} + \frac{38217252452688415985028411655}{193033005698565382397562129218} a^{22} + \frac{68271338298828097305103670667}{193033005698565382397562129218} a^{21} - \frac{37011511557965618107737673638}{96516502849282691198781064609} a^{20} - \frac{29853130132459057234737565923}{96516502849282691198781064609} a^{19} - \frac{20918907077782068632063233950}{96516502849282691198781064609} a^{18} + \frac{29127837191070805512467408378}{96516502849282691198781064609} a^{17} - \frac{34009778109147018041064628062}{96516502849282691198781064609} a^{16} + \frac{9155312614897231429966464653}{96516502849282691198781064609} a^{15} - \frac{402161783837056094518452342}{2608554131061694356723812557} a^{14} + \frac{87947776278834616643248018179}{193033005698565382397562129218} a^{13} + \frac{42114018741182835629194422646}{96516502849282691198781064609} a^{12} + \frac{15316874530206549865778030385}{96516502849282691198781064609} a^{11} + \frac{87939504841514802558450934483}{193033005698565382397562129218} a^{10} - \frac{3516550040157151289736402175}{96516502849282691198781064609} a^{9} - \frac{29953054954232863687476977203}{96516502849282691198781064609} a^{8} + \frac{3267719813403138071327775319}{96516502849282691198781064609} a^{7} - \frac{406859819666608636082026335}{1278364276149439618526901518} a^{6} + \frac{34177031824980674560897571732}{96516502849282691198781064609} a^{5} + \frac{10690693468103348694573887775}{193033005698565382397562129218} a^{4} - \frac{83789944280504708936813444261}{193033005698565382397562129218} a^{3} - \frac{19611676600004794765769794190}{96516502849282691198781064609} a^{2} + \frac{575490407703038727198138337}{5217108262123388713447625114} a + \frac{70423063594626803580967838007}{193033005698565382397562129218}$, $\frac{1}{313428206632692104435144019046939460400325801796023599188214221676182} a^{35} - \frac{639037658617816635677644927221977527663}{313428206632692104435144019046939460400325801796023599188214221676182} a^{34} - \frac{273040144521659866274568533731527077239490273278279776374068497608}{156714103316346052217572009523469730200162900898011799594107110838091} a^{33} - \frac{1615716890336831989096683946230001054490505698556377187981988989255}{313428206632692104435144019046939460400325801796023599188214221676182} a^{32} + \frac{13675016675418809233285562623849924766235506906107331379369061942162}{156714103316346052217572009523469730200162900898011799594107110838091} a^{31} + \frac{69442590891879605091520088440208281616055624274309290203347828627315}{313428206632692104435144019046939460400325801796023599188214221676182} a^{30} + \frac{47490107077846741156502597073212464709187463931801496523959609903595}{313428206632692104435144019046939460400325801796023599188214221676182} a^{29} + \frac{55577035142543500653933508200150306396153761447988890609496639662321}{313428206632692104435144019046939460400325801796023599188214221676182} a^{28} + \frac{8556401970553764972310156012135998749420869040916208617834050951924}{156714103316346052217572009523469730200162900898011799594107110838091} a^{27} + \frac{142436331768564740299052893150043822025064667051210159343509850208175}{313428206632692104435144019046939460400325801796023599188214221676182} a^{26} - \frac{66872372935335049799375847820688122670452289562631738685123850398480}{156714103316346052217572009523469730200162900898011799594107110838091} a^{25} + \frac{46198462908997405812271776875477366249547903820672432902833936766879}{313428206632692104435144019046939460400325801796023599188214221676182} a^{24} + \frac{19965605330319620126245372519374130455935363948065388354380643992239}{156714103316346052217572009523469730200162900898011799594107110838091} a^{23} - \frac{7003057420721740649598705792079903774653446942844297178007874145400}{156714103316346052217572009523469730200162900898011799594107110838091} a^{22} + \frac{7014197300649534751188040103076296655830113359825011359929345520758}{156714103316346052217572009523469730200162900898011799594107110838091} a^{21} - \frac{45076673681890106306831709181786208407430368145209488191736574553335}{156714103316346052217572009523469730200162900898011799594107110838091} a^{20} + \frac{5614485689590691182379553349993687235568882368295000576072695213670}{156714103316346052217572009523469730200162900898011799594107110838091} a^{19} - \frac{9802754100481932740593442711253487016446135528593555402773352322964}{156714103316346052217572009523469730200162900898011799594107110838091} a^{18} + \frac{53301420310819808877880131367801729675672710531125055845083144159417}{156714103316346052217572009523469730200162900898011799594107110838091} a^{17} + \frac{58556084517006570419441794225320025420255432539388908577397817428688}{156714103316346052217572009523469730200162900898011799594107110838091} a^{16} - \frac{13245853643812373243325510254822451086897635447388495910214370733700}{156714103316346052217572009523469730200162900898011799594107110838091} a^{15} - \frac{140300175706196498145062282170369654931897055279751416780946302601837}{313428206632692104435144019046939460400325801796023599188214221676182} a^{14} - \frac{89695470163466567199276530921541685192786564581981908312318336720659}{313428206632692104435144019046939460400325801796023599188214221676182} a^{13} - \frac{34593442732642934769869958778196652254600299482462914079243859976502}{156714103316346052217572009523469730200162900898011799594107110838091} a^{12} - \frac{1537607254731686432768585816848374399617629776061646428877482486526}{4235516305847190600474919176309992708112510835081399989029921914543} a^{11} + \frac{87014909605542003791061575953690463563516589274093040052069725991577}{313428206632692104435144019046939460400325801796023599188214221676182} a^{10} - \frac{18521633335103732358211811422226127579677376282115376222672960936143}{313428206632692104435144019046939460400325801796023599188214221676182} a^{9} - \frac{73672555300465985462911955202662684008142222297290370746027447822340}{156714103316346052217572009523469730200162900898011799594107110838091} a^{8} - \frac{28426588683968722865503326578763482201688541225641847374827235285631}{313428206632692104435144019046939460400325801796023599188214221676182} a^{7} - \frac{123463783294483942268318965200172282499086559208377305523379455133867}{313428206632692104435144019046939460400325801796023599188214221676182} a^{6} - \frac{6968612397201322050144939941866738771348649266727271637564447409643}{313428206632692104435144019046939460400325801796023599188214221676182} a^{5} + \frac{53266495113563457242020063111837586487270401604077632694190612209693}{156714103316346052217572009523469730200162900898011799594107110838091} a^{4} + \frac{83828189517523354410601343675098060964406729392488347570237667597671}{313428206632692104435144019046939460400325801796023599188214221676182} a^{3} + \frac{67671924170192010549923302335898702488156919514254015122384760830171}{156714103316346052217572009523469730200162900898011799594107110838091} a^{2} + \frac{4921908536169215276441957360847391151148115102368956287750944158821}{156714103316346052217572009523469730200162900898011799594107110838091} a + \frac{49659803325180987295307008243704714082056266882849798778343183092933}{156714103316346052217572009523469730200162900898011799594107110838091}$

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

Class group and class number

$C_{9}\times C_{1629}$, which has order $14661$ (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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{6355253221129489500946454320832515795239537530518066091}{75890162852489880758613834906298611400603788341297007038} a^{35} - \frac{3770012955844936853836908891589672595247933467724199065}{37945081426244940379306917453149305700301894170648503519} a^{34} + \frac{83149155625423813594374889417142447996583870058115048117}{37945081426244940379306917453149305700301894170648503519} a^{33} - \frac{125960682773862194736481725374244992645495706767803681243}{75890162852489880758613834906298611400603788341297007038} a^{32} + \frac{1319622627216924824531971063335417027776041254408171986050}{37945081426244940379306917453149305700301894170648503519} a^{31} - \frac{792829009373877854787506487757191685194006672572950189488}{37945081426244940379306917453149305700301894170648503519} a^{30} + \frac{25746087862050570043047053993153508937285157833711753606517}{75890162852489880758613834906298611400603788341297007038} a^{29} - \frac{9744065977820939624630813345895279191716765256500816226269}{75890162852489880758613834906298611400603788341297007038} a^{28} + \frac{178487050153458216796936802525034284089765270565054934304879}{75890162852489880758613834906298611400603788341297007038} a^{27} - \frac{47507390064491807241947005517081277956144489055374612021561}{75890162852489880758613834906298611400603788341297007038} a^{26} + \frac{427049667171990026076534616706136472239933627374982773704883}{37945081426244940379306917453149305700301894170648503519} a^{25} - \frac{55526087310167660864164592307450938282670010347726676646021}{37945081426244940379306917453149305700301894170648503519} a^{24} + \frac{1489233281148114830252179235802283496215415441214473063192363}{37945081426244940379306917453149305700301894170648503519} a^{23} - \frac{244163139203608994889912121898659942285490230426072698835299}{75890162852489880758613834906298611400603788341297007038} a^{22} + \frac{7387827038259523949618329519053459161325984820508220652228741}{75890162852489880758613834906298611400603788341297007038} a^{21} - \frac{146042440210709256919920579457766598763138465773038392844826}{37945081426244940379306917453149305700301894170648503519} a^{20} + \frac{6850884054893245096022342927832138043283854261711155454936118}{37945081426244940379306917453149305700301894170648503519} a^{19} - \frac{228383083042706870005257540017638071169201830326063462104049}{37945081426244940379306917453149305700301894170648503519} a^{18} + \frac{9408856064975430034284217199906680443771549126674381083633571}{37945081426244940379306917453149305700301894170648503519} a^{17} - \frac{289694443682129423578953536701717088377216378478118269673865}{37945081426244940379306917453149305700301894170648503519} a^{16} + \frac{9799154351730193571901192739486326987955894563638565018738573}{37945081426244940379306917453149305700301894170648503519} a^{15} - \frac{1051630372599183794738169628090840217190322184938671232866063}{75890162852489880758613834906298611400603788341297007038} a^{14} + \frac{7504132365733125615987759479979619785890268820838675881680684}{37945081426244940379306917453149305700301894170648503519} a^{13} - \frac{615287389376944212501467290355839468907417713043677201932770}{37945081426244940379306917453149305700301894170648503519} a^{12} + \frac{4273425452850423843628626663086398099425041546193324541385337}{37945081426244940379306917453149305700301894170648503519} a^{11} - \frac{577280391959947983589959840713070328566503096658948988087249}{37945081426244940379306917453149305700301894170648503519} a^{10} + \frac{1677706071739338583956588715336999490501737337931141777693634}{37945081426244940379306917453149305700301894170648503519} a^{9} - \frac{322807770385142221194254713980599918360484467870170842423723}{37945081426244940379306917453149305700301894170648503519} a^{8} + \frac{944548021384264087683118185650095858067956441308271174372983}{75890162852489880758613834906298611400603788341297007038} a^{7} - \frac{230637200939241439860385280859455343528503604652113306558765}{75890162852489880758613834906298611400603788341297007038} a^{6} + \frac{149288361426748381931533306150693314017417206429108933541779}{75890162852489880758613834906298611400603788341297007038} a^{5} - \frac{44493602609240577206248095498277008610205704333539503937585}{75890162852489880758613834906298611400603788341297007038} a^{4} + \frac{17810301601453616584149889386511012026859424650139986645019}{75890162852489880758613834906298611400603788341297007038} a^{3} - \frac{1967038885408407577725075046420857343322998307991979101920}{37945081426244940379306917453149305700301894170648503519} a^{2} + \frac{404303415201056017939940620469411046013902096637029216781}{75890162852489880758613834906298611400603788341297007038} a + \frac{17862988947419591443044311891354748597506305695594109047}{75890162852489880758613834906298611400603788341297007038} \) (order $6$)
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:  \( 3431432369157.3267 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{18}$ (as 36T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_2\times C_{18}$
Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), 3.3.361.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 6.0.3518667.1, 6.0.439833375.1, 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.0.193453397763890625.1, 18.0.5677392343251487443465123.1, 18.0.11088656920413061413017818359375.2, 18.18.563362135874260093126953125.1

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 $18^{2}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ R $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$

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
3Data not computed
5Data not computed
$19$19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$