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Properties

Label 32.0.56158893576...0625.2
Degree $32$
Signature $[0, 16]$
Discriminant $5^{24}\cdot 7^{16}\cdot 17^{28}$
Root discriminant $105.54$
Ramified primes $5, 7, 17$
Class number Not computed
Class group Not computed
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![18479977261, 5111337362, -79785142736, -98707729302, 32016319033, 207815025774, 331005058977, 270487093719, 154455067964, 45573354048, 651843825, -14252605152, -6297437270, -4980292302, 77390757, 90690253, 148502093, 115457045, 32840498, -12009033, 81720, -1199025, -710547, 71120, 57950, -4187, 4802, 556, -546, 5, -6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - 6*x^30 + 5*x^29 - 546*x^28 + 556*x^27 + 4802*x^26 - 4187*x^25 + 57950*x^24 + 71120*x^23 - 710547*x^22 - 1199025*x^21 + 81720*x^20 - 12009033*x^19 + 32840498*x^18 + 115457045*x^17 + 148502093*x^16 + 90690253*x^15 + 77390757*x^14 - 4980292302*x^13 - 6297437270*x^12 - 14252605152*x^11 + 651843825*x^10 + 45573354048*x^9 + 154455067964*x^8 + 270487093719*x^7 + 331005058977*x^6 + 207815025774*x^5 + 32016319033*x^4 - 98707729302*x^3 - 79785142736*x^2 + 5111337362*x + 18479977261)
 
gp: K = bnfinit(x^32 - x^31 - 6*x^30 + 5*x^29 - 546*x^28 + 556*x^27 + 4802*x^26 - 4187*x^25 + 57950*x^24 + 71120*x^23 - 710547*x^22 - 1199025*x^21 + 81720*x^20 - 12009033*x^19 + 32840498*x^18 + 115457045*x^17 + 148502093*x^16 + 90690253*x^15 + 77390757*x^14 - 4980292302*x^13 - 6297437270*x^12 - 14252605152*x^11 + 651843825*x^10 + 45573354048*x^9 + 154455067964*x^8 + 270487093719*x^7 + 331005058977*x^6 + 207815025774*x^5 + 32016319033*x^4 - 98707729302*x^3 - 79785142736*x^2 + 5111337362*x + 18479977261, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} - 6 x^{30} + 5 x^{29} - 546 x^{28} + 556 x^{27} + 4802 x^{26} - 4187 x^{25} + 57950 x^{24} + 71120 x^{23} - 710547 x^{22} - 1199025 x^{21} + 81720 x^{20} - 12009033 x^{19} + 32840498 x^{18} + 115457045 x^{17} + 148502093 x^{16} + 90690253 x^{15} + 77390757 x^{14} - 4980292302 x^{13} - 6297437270 x^{12} - 14252605152 x^{11} + 651843825 x^{10} + 45573354048 x^{9} + 154455067964 x^{8} + 270487093719 x^{7} + 331005058977 x^{6} + 207815025774 x^{5} + 32016319033 x^{4} - 98707729302 x^{3} - 79785142736 x^{2} + 5111337362 x + 18479977261 \)

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:  \(56158893576626869963359344621945280012905849018156528472900390625=5^{24}\cdot 7^{16}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.54$
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:  $5, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(595=5\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{595}(512,·)$, $\chi_{595}(1,·)$, $\chi_{595}(132,·)$, $\chi_{595}(8,·)$, $\chi_{595}(393,·)$, $\chi_{595}(13,·)$, $\chi_{595}(526,·)$, $\chi_{595}(111,·)$, $\chi_{595}(412,·)$, $\chi_{595}(162,·)$, $\chi_{595}(293,·)$, $\chi_{595}(169,·)$, $\chi_{595}(43,·)$, $\chi_{595}(559,·)$, $\chi_{595}(307,·)$, $\chi_{595}(314,·)$, $\chi_{595}(188,·)$, $\chi_{595}(64,·)$, $\chi_{595}(321,·)$, $\chi_{595}(76,·)$, $\chi_{595}(461,·)$, $\chi_{595}(344,·)$, $\chi_{595}(349,·)$, $\chi_{595}(421,·)$, $\chi_{595}(104,·)$, $\chi_{595}(106,·)$, $\chi_{595}(237,·)$, $\chi_{595}(239,·)$, $\chi_{595}(372,·)$, $\chi_{595}(118,·)$, $\chi_{595}(253,·)$, $\chi_{595}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{10}$, $\frac{1}{356} a^{26} + \frac{19}{89} a^{25} - \frac{22}{89} a^{24} - \frac{69}{356} a^{23} - \frac{61}{356} a^{22} + \frac{3}{178} a^{21} + \frac{45}{356} a^{20} + \frac{1}{178} a^{19} + \frac{13}{356} a^{18} + \frac{83}{356} a^{17} - \frac{85}{356} a^{16} - \frac{41}{356} a^{15} - \frac{25}{89} a^{14} + \frac{7}{178} a^{13} + \frac{85}{178} a^{12} - \frac{151}{356} a^{11} - \frac{31}{178} a^{10} + \frac{6}{89} a^{9} + \frac{161}{356} a^{8} + \frac{81}{356} a^{7} + \frac{23}{89} a^{6} + \frac{107}{356} a^{5} + \frac{59}{178} a^{4} - \frac{147}{356} a^{3} - \frac{85}{356} a^{2} - \frac{147}{356} a + \frac{151}{356}$, $\frac{1}{356} a^{27} + \frac{5}{178} a^{25} + \frac{33}{356} a^{24} + \frac{21}{356} a^{23} + \frac{7}{178} a^{22} - \frac{55}{356} a^{21} - \frac{9}{89} a^{20} + \frac{39}{356} a^{19} - \frac{15}{356} a^{18} + \frac{15}{356} a^{17} + \frac{11}{356} a^{16} - \frac{5}{178} a^{15} + \frac{69}{178} a^{14} + \frac{87}{178} a^{13} + \frac{101}{356} a^{12} + \frac{11}{178} a^{11} - \frac{35}{178} a^{10} - \frac{61}{356} a^{9} + \frac{127}{356} a^{8} - \frac{3}{89} a^{7} - \frac{121}{356} a^{6} - \frac{1}{89} a^{5} - \frac{37}{356} a^{4} - \frac{127}{356} a^{3} - \frac{95}{356} a^{2} - \frac{69}{356} a + \frac{47}{178}$, $\frac{1}{356} a^{28} - \frac{15}{356} a^{25} + \frac{11}{356} a^{24} - \frac{2}{89} a^{23} + \frac{21}{356} a^{22} + \frac{41}{178} a^{21} - \frac{55}{356} a^{20} - \frac{35}{356} a^{19} + \frac{63}{356} a^{18} + \frac{71}{356} a^{17} - \frac{25}{178} a^{16} + \frac{7}{178} a^{15} + \frac{53}{178} a^{14} - \frac{39}{356} a^{13} + \frac{51}{178} a^{12} + \frac{4}{89} a^{11} - \frac{153}{356} a^{10} + \frac{65}{356} a^{9} + \frac{79}{178} a^{8} - \frac{41}{356} a^{7} - \frac{17}{178} a^{6} - \frac{39}{356} a^{5} + \frac{117}{356} a^{4} + \frac{129}{356} a^{3} - \frac{109}{356} a^{2} - \frac{19}{178} a + \frac{23}{89}$, $\frac{1}{356} a^{29} + \frac{83}{356} a^{25} - \frac{41}{178} a^{24} + \frac{27}{178} a^{23} + \frac{57}{356} a^{22} + \frac{35}{356} a^{21} - \frac{18}{89} a^{20} - \frac{85}{356} a^{19} + \frac{22}{89} a^{18} - \frac{51}{356} a^{17} - \frac{15}{356} a^{16} + \frac{25}{356} a^{15} - \frac{115}{356} a^{14} - \frac{11}{89} a^{13} + \frac{37}{178} a^{12} + \frac{37}{178} a^{11} - \frac{153}{356} a^{10} - \frac{4}{89} a^{9} - \frac{59}{178} a^{8} - \frac{65}{356} a^{7} - \frac{83}{356} a^{6} - \frac{29}{178} a^{5} - \frac{59}{356} a^{4} - \frac{67}{356} a^{2} - \frac{155}{356} a - \frac{49}{356}$, $\frac{1}{12937846165941988} a^{30} - \frac{9368416985863}{12937846165941988} a^{29} - \frac{7083748366323}{6468923082970994} a^{28} - \frac{6669658117377}{6468923082970994} a^{27} + \frac{13101177230207}{12937846165941988} a^{26} + \frac{170604909677953}{12937846165941988} a^{25} - \frac{353724466882565}{3234461541485497} a^{24} - \frac{2595872302672641}{12937846165941988} a^{23} + \frac{1608942345737045}{6468923082970994} a^{22} + \frac{2848255376389195}{12937846165941988} a^{21} + \frac{2845423117457877}{12937846165941988} a^{20} - \frac{697118288640151}{12937846165941988} a^{19} + \frac{503809443061471}{12937846165941988} a^{18} + \frac{422119679701235}{6468923082970994} a^{17} + \frac{150753757622054}{3234461541485497} a^{16} + \frac{7404355066724}{36342264511073} a^{15} - \frac{2793034318537079}{12937846165941988} a^{14} - \frac{219381512679396}{3234461541485497} a^{13} - \frac{2925656942487923}{6468923082970994} a^{12} + \frac{5965790584204561}{12937846165941988} a^{11} - \frac{2305669027820825}{12937846165941988} a^{10} - \frac{2253922278584297}{6468923082970994} a^{9} - \frac{3741460417054555}{12937846165941988} a^{8} + \frac{730645299019651}{6468923082970994} a^{7} - \frac{1705922114317997}{12937846165941988} a^{6} - \frac{3045388022769511}{12937846165941988} a^{5} - \frac{4224892405781905}{12937846165941988} a^{4} + \frac{3152655078901223}{12937846165941988} a^{3} - \frac{918528327638689}{6468923082970994} a^{2} + \frac{583260892680125}{6468923082970994} a - \frac{6457792153793647}{12937846165941988}$, $\frac{1}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{31} + \frac{35693184673952381802750406386092071106159661013249263226388631094485434135963152781438425297071090188667769631123568480250516527492022865662254279482445}{5510523829006567453141886421418777961268213880213453888016993072304492253109051247771009364182520660809860072272377198592358956061056313702783846668864106355233973798292} a^{30} - \frac{6663941060155738610360674416877557749920706942200612123056281127404645578475523238454455641912228126830833453207423152894822480894890247509494485709398185707893787073}{5510523829006567453141886421418777961268213880213453888016993072304492253109051247771009364182520660809860072272377198592358956061056313702783846668864106355233973798292} a^{29} - \frac{12374075476537263051805517378942032759080040883906206959428127944827184885757857976109739256842720861731370038935456819769128096180242673717962630575409784435868286685}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{28} + \frac{6235979001360577302379194969548693622903233451138663799253832569051934979431991679545474261576588221090878993266914829590133092702282753182461920789756913353637943909}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{27} + \frac{3884604649180740957971739769717795216269863540775528240192448855756519013595780519684380561062312146079171837897218077450096053506757397490823366313907747777877925597}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{26} - \frac{2354190444085305962372524647983377533287080404572870798784987935675879999503978540831191060939376761217117542831371190316713356256752310262685896099466789729637292651761}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{25} - \frac{31761129597192866926869651746981963183122427492558954364505315504989933212322286891002952886043720544286455793982276449102547383125226017229119277235681902266245206227}{2755261914503283726570943210709388980634106940106726944008496536152246126554525623885504682091260330404930036136188599296179478030528156851391923334432053177616986899146} a^{24} + \frac{1178602681245835302506467450475036467594485446331421067369527649244498807048353311447121367563004925850765689895370378730684173516027692768713958040054257113861149711031}{5510523829006567453141886421418777961268213880213453888016993072304492253109051247771009364182520660809860072272377198592358956061056313702783846668864106355233973798292} a^{23} + \frac{1146285464322302892859884794748288659264732565200146961769569641240450211458455014156222131296642978498239759684337942976730416917081523408388167872449155668550586707371}{5510523829006567453141886421418777961268213880213453888016993072304492253109051247771009364182520660809860072272377198592358956061056313702783846668864106355233973798292} a^{22} + \frac{975768301190583509013069673495165094881363247355034900181758104081557406277397798574202519062285819434633435682832811553204701638826758973439897305678825227802632903795}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{21} - \frac{646473360762423637303891435769560148541931781073677173514778528123327748642983293875157827541547542875354636494037875241418570942959563838410548482237386149288279419329}{5510523829006567453141886421418777961268213880213453888016993072304492253109051247771009364182520660809860072272377198592358956061056313702783846668864106355233973798292} a^{20} + \frac{1367564052627321817810437023545674054272000430226569703877756419027952763356439951203113444277998307431221852647577312481663242581705293718768348525940224624204178516109}{5510523829006567453141886421418777961268213880213453888016993072304492253109051247771009364182520660809860072272377198592358956061056313702783846668864106355233973798292} a^{19} - \frac{1933053560222732673284697623747924976944962644406443999565411785434337931946948482187854818962173724925577215947793070544767004152094005570843055612453014662051477393795}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{18} - \frac{731184567632262180329053158697418389578527349010446985701914565262021785386151054941197748392153896641210332645830616205838337757467408419321310968646399949432986096725}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{17} - \frac{594404514808605660794386781443056259968776246219637727030537929970647924898785710783830353895111514383700434178725071639752870431689694404472721495659310308449939653685}{2755261914503283726570943210709388980634106940106726944008496536152246126554525623885504682091260330404930036136188599296179478030528156851391923334432053177616986899146} a^{16} - \frac{307781531162169917703422889347036131644699723172149929921840386765952926225836679221388101145827987393123576456990730901679533402773254542529329705726076504436888900399}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{15} + \frac{2210825333533288082613546795531321292471704006342971497409650195847964473241811627650445195259474890405202715471187343308943025888461272156466851470551090505116529227347}{5510523829006567453141886421418777961268213880213453888016993072304492253109051247771009364182520660809860072272377198592358956061056313702783846668864106355233973798292} a^{14} + \frac{4713761846144992081353724392268024325617668625287931569817159028584795901050757207033401407251430890335770903664233189741426556579244415238405455446819381102574109522379}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{13} + \frac{3334093265977893167147123871502909953632781769406005811598608402832204209590812233209938733718314981112221209462868196167611425921185812310391220758758192289925066205873}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{12} - \frac{1011521159431459985712934350406584177940711229927145154191432479264867631260407550023983030366419170107907897558153092649662593499505572423033280755099833154073738487577}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{11} - \frac{1698822649453121144463751405858404461526628545317890453554292840738877735914387025350856649664215190770777537977069637357998894944910863788821058701557863038828244422585}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{10} + \frac{382626097384930383224146782046347967046486788398873654461322397335825372577669484917275856642557105380262254544092481002777356158413043686961418936240792843687648170536}{1377630957251641863285471605354694490317053470053363472004248268076123063277262811942752341045630165202465018068094299648089739015264078425695961667216026588808493449573} a^{9} + \frac{172023485127980452191764514978496820745213691119516690176852374053961195530033650514577752137128230264084924020818389724037995039969196610210054995732371924771297348911}{2755261914503283726570943210709388980634106940106726944008496536152246126554525623885504682091260330404930036136188599296179478030528156851391923334432053177616986899146} a^{8} - \frac{581249672711291012853875660781839121503226479104050102068666764597414610478219912534498498714572931118742535049351202322585376661579679266634441293135231653196269011913}{1377630957251641863285471605354694490317053470053363472004248268076123063277262811942752341045630165202465018068094299648089739015264078425695961667216026588808493449573} a^{7} - \frac{2560932555657024931263687587447146914657582254311765567069794620609495935486187630323072069515244107067635326453890618245421276025654187512649899727839010804756287409305}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{6} - \frac{237786522289790631263073540302657436523405318518817383649208260208973005211474334802106507994520749844902135515784480046523479896509089410077796912645832842625887338433}{2755261914503283726570943210709388980634106940106726944008496536152246126554525623885504682091260330404930036136188599296179478030528156851391923334432053177616986899146} a^{5} + \frac{99159730349672177419001863555942473618925444629371323560428409930005955079595285252376693996743434857579254465820451419433742821985876223954837330234413737273811955047}{5510523829006567453141886421418777961268213880213453888016993072304492253109051247771009364182520660809860072272377198592358956061056313702783846668864106355233973798292} a^{4} - \frac{3841617102902979259354588585245543274671686521803630155537988376938914723471425955650267987010430465487936726065474029334745450151955854766643177179020756993104659621389}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{3} - \frac{2888949283666682456634583345496518471857900499616552189567429732403567297207646693685655648131281195607595219131250847028923051526139936206454190934443351493779465522767}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a^{2} - \frac{3275036723883744583461224697117100043171754511819360565791057142192248465293660652140779089391980839434193778292579084057954183166303099320828966821109892984167124590115}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584} a + \frac{3094591509979270111062587093532021295918696812521474162949772429677756077665086806835476368916328520239416095350192063312494614123474364483499337100504380884501776761493}{11021047658013134906283772842837555922536427760426907776033986144608984506218102495542018728365041321619720144544754397184717912122112627405567693337728212710467947596584}$

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:  $15$
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:  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

$C_4\times C_8$ (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

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), 4.4.30092125.1, \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.30092125.2, 4.4.4913.1, 4.4.122825.1, 4.4.6125.1, 4.4.1770125.1, 8.8.905535987015625.1, 8.8.15085980625.1, 8.8.3133342515625.1, 8.0.615764471170625.1, 8.0.985223153873.1, 8.0.6411541765625.1, 8.0.6411541765625.2, 16.16.819995423780362168593994140625.1, 16.0.379165883956039466757862890625.2, 16.0.41107867812353742431640625.1

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 R ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
7Data not computed
17Data not computed