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Properties

Label 34.34.6170141097...1361.1
Degree $34$
Signature $[34, 0]$
Discriminant $7^{17}\cdot 103^{33}$
Root discriminant $237.79$
Ramified primes $7, 103$
Class number Not computed
Class group Not computed
Galois group $C_{34}$ (as 34T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![135848587373, 4464595297619, 6605712907500, -27266505860087, -53392644299570, 46496583550817, 151216828629211, 12814601822229, -193090167454773, -118015374137357, 101582660443885, 129042869503798, 2958831950540, -56071395662326, -22871481227426, 8280597620648, 7959197249784, 517274625625, -1186884493287, -319307014210, 80304743998, 44523867067, -500975171, -3231728723, -311226331, 135988732, 23783876, -3286755, -874103, 40483, 18071, -142, -204, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 - x^33 - 204*x^32 - 142*x^31 + 18071*x^30 + 40483*x^29 - 874103*x^28 - 3286755*x^27 + 23783876*x^26 + 135988732*x^25 - 311226331*x^24 - 3231728723*x^23 - 500975171*x^22 + 44523867067*x^21 + 80304743998*x^20 - 319307014210*x^19 - 1186884493287*x^18 + 517274625625*x^17 + 7959197249784*x^16 + 8280597620648*x^15 - 22871481227426*x^14 - 56071395662326*x^13 + 2958831950540*x^12 + 129042869503798*x^11 + 101582660443885*x^10 - 118015374137357*x^9 - 193090167454773*x^8 + 12814601822229*x^7 + 151216828629211*x^6 + 46496583550817*x^5 - 53392644299570*x^4 - 27266505860087*x^3 + 6605712907500*x^2 + 4464595297619*x + 135848587373)
 
gp: K = bnfinit(x^34 - x^33 - 204*x^32 - 142*x^31 + 18071*x^30 + 40483*x^29 - 874103*x^28 - 3286755*x^27 + 23783876*x^26 + 135988732*x^25 - 311226331*x^24 - 3231728723*x^23 - 500975171*x^22 + 44523867067*x^21 + 80304743998*x^20 - 319307014210*x^19 - 1186884493287*x^18 + 517274625625*x^17 + 7959197249784*x^16 + 8280597620648*x^15 - 22871481227426*x^14 - 56071395662326*x^13 + 2958831950540*x^12 + 129042869503798*x^11 + 101582660443885*x^10 - 118015374137357*x^9 - 193090167454773*x^8 + 12814601822229*x^7 + 151216828629211*x^6 + 46496583550817*x^5 - 53392644299570*x^4 - 27266505860087*x^3 + 6605712907500*x^2 + 4464595297619*x + 135848587373, 1)
 

Normalized defining polynomial

\( x^{34} - x^{33} - 204 x^{32} - 142 x^{31} + 18071 x^{30} + 40483 x^{29} - 874103 x^{28} - 3286755 x^{27} + 23783876 x^{26} + 135988732 x^{25} - 311226331 x^{24} - 3231728723 x^{23} - 500975171 x^{22} + 44523867067 x^{21} + 80304743998 x^{20} - 319307014210 x^{19} - 1186884493287 x^{18} + 517274625625 x^{17} + 7959197249784 x^{16} + 8280597620648 x^{15} - 22871481227426 x^{14} - 56071395662326 x^{13} + 2958831950540 x^{12} + 129042869503798 x^{11} + 101582660443885 x^{10} - 118015374137357 x^{9} - 193090167454773 x^{8} + 12814601822229 x^{7} + 151216828629211 x^{6} + 46496583550817 x^{5} - 53392644299570 x^{4} - 27266505860087 x^{3} + 6605712907500 x^{2} + 4464595297619 x + 135848587373 \)

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

Invariants

Degree:  $34$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[34, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(617014109765059300114299442381532756906016565170482752260957053403874909832001361=7^{17}\cdot 103^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $237.79$
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:  $7, 103$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(721=7\cdot 103\)
Dirichlet character group:    $\lbrace$$\chi_{721}(512,·)$, $\chi_{721}(1,·)$, $\chi_{721}(8,·)$, $\chi_{721}(267,·)$, $\chi_{721}(652,·)$, $\chi_{721}(398,·)$, $\chi_{721}(657,·)$, $\chi_{721}(279,·)$, $\chi_{721}(27,·)$, $\chi_{721}(286,·)$, $\chi_{721}(421,·)$, $\chi_{721}(552,·)$, $\chi_{721}(169,·)$, $\chi_{721}(300,·)$, $\chi_{721}(435,·)$, $\chi_{721}(694,·)$, $\chi_{721}(442,·)$, $\chi_{721}(64,·)$, $\chi_{721}(323,·)$, $\chi_{721}(69,·)$, $\chi_{721}(454,·)$, $\chi_{721}(713,·)$, $\chi_{721}(720,·)$, $\chi_{721}(209,·)$, $\chi_{721}(596,·)$, $\chi_{721}(216,·)$, $\chi_{721}(90,·)$, $\chi_{721}(484,·)$, $\chi_{721}(230,·)$, $\chi_{721}(491,·)$, $\chi_{721}(237,·)$, $\chi_{721}(631,·)$, $\chi_{721}(505,·)$, $\chi_{721}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{47} a^{25} + \frac{8}{47} a^{23} + \frac{5}{47} a^{22} - \frac{13}{47} a^{21} + \frac{9}{47} a^{20} - \frac{20}{47} a^{19} + \frac{21}{47} a^{18} - \frac{18}{47} a^{17} - \frac{8}{47} a^{16} - \frac{10}{47} a^{15} + \frac{10}{47} a^{14} - \frac{18}{47} a^{13} + \frac{8}{47} a^{12} + \frac{18}{47} a^{11} - \frac{10}{47} a^{10} - \frac{2}{47} a^{9} + \frac{3}{47} a^{8} - \frac{11}{47} a^{7} - \frac{6}{47} a^{6} - \frac{5}{47} a^{5} - \frac{20}{47} a^{4} + \frac{9}{47} a^{3} - \frac{17}{47} a^{2} - \frac{13}{47} a + \frac{15}{47}$, $\frac{1}{47} a^{26} + \frac{8}{47} a^{24} + \frac{5}{47} a^{23} - \frac{13}{47} a^{22} + \frac{9}{47} a^{21} - \frac{20}{47} a^{20} + \frac{21}{47} a^{19} - \frac{18}{47} a^{18} - \frac{8}{47} a^{17} - \frac{10}{47} a^{16} + \frac{10}{47} a^{15} - \frac{18}{47} a^{14} + \frac{8}{47} a^{13} + \frac{18}{47} a^{12} - \frac{10}{47} a^{11} - \frac{2}{47} a^{10} + \frac{3}{47} a^{9} - \frac{11}{47} a^{8} - \frac{6}{47} a^{7} - \frac{5}{47} a^{6} - \frac{20}{47} a^{5} + \frac{9}{47} a^{4} - \frac{17}{47} a^{3} - \frac{13}{47} a^{2} + \frac{15}{47} a$, $\frac{1}{47} a^{27} + \frac{5}{47} a^{24} + \frac{17}{47} a^{23} + \frac{16}{47} a^{22} - \frac{10}{47} a^{21} - \frac{4}{47} a^{20} + \frac{1}{47} a^{19} + \frac{12}{47} a^{18} - \frac{7}{47} a^{17} - \frac{20}{47} a^{16} + \frac{15}{47} a^{15} + \frac{22}{47} a^{14} + \frac{21}{47} a^{13} + \frac{20}{47} a^{12} - \frac{5}{47} a^{11} - \frac{11}{47} a^{10} + \frac{5}{47} a^{9} + \frac{17}{47} a^{8} - \frac{11}{47} a^{7} - \frac{19}{47} a^{6} + \frac{2}{47} a^{5} + \frac{2}{47} a^{4} + \frac{9}{47} a^{3} + \frac{10}{47} a^{2} + \frac{10}{47} a + \frac{21}{47}$, $\frac{1}{47} a^{28} + \frac{17}{47} a^{24} + \frac{23}{47} a^{23} + \frac{12}{47} a^{22} + \frac{14}{47} a^{21} + \frac{3}{47} a^{20} + \frac{18}{47} a^{19} - \frac{18}{47} a^{18} + \frac{23}{47} a^{17} + \frac{8}{47} a^{16} - \frac{22}{47} a^{15} + \frac{18}{47} a^{14} + \frac{16}{47} a^{13} + \frac{2}{47} a^{12} - \frac{7}{47} a^{11} + \frac{8}{47} a^{10} - \frac{20}{47} a^{9} + \frac{21}{47} a^{8} - \frac{11}{47} a^{7} - \frac{15}{47} a^{6} - \frac{20}{47} a^{5} + \frac{15}{47} a^{4} + \frac{12}{47} a^{3} + \frac{1}{47} a^{2} - \frac{8}{47} a + \frac{19}{47}$, $\frac{1}{47} a^{29} + \frac{23}{47} a^{24} + \frac{17}{47} a^{23} + \frac{23}{47} a^{22} - \frac{11}{47} a^{21} + \frac{6}{47} a^{20} - \frac{7}{47} a^{19} - \frac{5}{47} a^{18} - \frac{15}{47} a^{17} + \frac{20}{47} a^{16} - \frac{13}{47} a^{14} - \frac{21}{47} a^{13} - \frac{2}{47} a^{12} - \frac{16}{47} a^{11} + \frac{9}{47} a^{10} + \frac{8}{47} a^{9} - \frac{15}{47} a^{8} - \frac{16}{47} a^{7} - \frac{12}{47} a^{6} + \frac{6}{47} a^{5} + \frac{23}{47} a^{4} - \frac{11}{47} a^{3} - \frac{1}{47} a^{2} + \frac{5}{47} a - \frac{20}{47}$, $\frac{1}{47} a^{30} + \frac{17}{47} a^{24} - \frac{20}{47} a^{23} + \frac{15}{47} a^{22} + \frac{23}{47} a^{21} + \frac{21}{47} a^{20} - \frac{15}{47} a^{19} + \frac{19}{47} a^{18} + \frac{11}{47} a^{17} - \frac{4}{47} a^{16} - \frac{18}{47} a^{15} - \frac{16}{47} a^{14} - \frac{11}{47} a^{13} - \frac{12}{47} a^{12} + \frac{18}{47} a^{11} + \frac{3}{47} a^{10} - \frac{16}{47} a^{9} + \frac{9}{47} a^{8} + \frac{6}{47} a^{7} + \frac{3}{47} a^{6} - \frac{3}{47} a^{5} - \frac{21}{47} a^{4} - \frac{20}{47} a^{3} + \frac{20}{47} a^{2} - \frac{3}{47} a - \frac{16}{47}$, $\frac{1}{47} a^{31} - \frac{20}{47} a^{24} + \frac{20}{47} a^{23} - \frac{15}{47} a^{22} + \frac{7}{47} a^{21} + \frac{20}{47} a^{20} - \frac{17}{47} a^{19} - \frac{17}{47} a^{18} + \frac{20}{47} a^{17} - \frac{23}{47} a^{16} + \frac{13}{47} a^{15} + \frac{7}{47} a^{14} + \frac{12}{47} a^{13} + \frac{23}{47} a^{12} - \frac{21}{47} a^{11} + \frac{13}{47} a^{10} - \frac{4}{47} a^{9} + \frac{2}{47} a^{8} + \frac{2}{47} a^{7} + \frac{5}{47} a^{6} + \frac{17}{47} a^{5} - \frac{9}{47} a^{4} + \frac{8}{47} a^{3} + \frac{4}{47} a^{2} + \frac{17}{47} a - \frac{20}{47}$, $\frac{1}{329141} a^{32} - \frac{18}{7003} a^{31} + \frac{3270}{329141} a^{30} - \frac{914}{329141} a^{29} - \frac{2315}{329141} a^{28} + \frac{2203}{329141} a^{27} + \frac{2771}{329141} a^{26} - \frac{1856}{329141} a^{25} - \frac{121561}{329141} a^{24} + \frac{8875}{329141} a^{23} + \frac{141922}{329141} a^{22} - \frac{129875}{329141} a^{21} + \frac{136960}{329141} a^{20} - \frac{77418}{329141} a^{19} + \frac{21482}{329141} a^{18} + \frac{29659}{329141} a^{17} + \frac{3313}{329141} a^{16} - \frac{74132}{329141} a^{15} - \frac{20088}{329141} a^{14} - \frac{60273}{329141} a^{13} + \frac{148765}{329141} a^{12} + \frac{11537}{329141} a^{11} - \frac{95001}{329141} a^{10} + \frac{50231}{329141} a^{9} - \frac{61492}{329141} a^{8} + \frac{111243}{329141} a^{7} + \frac{15665}{329141} a^{6} - \frac{20096}{329141} a^{5} + \frac{154096}{329141} a^{4} + \frac{163534}{329141} a^{3} + \frac{68921}{329141} a^{2} - \frac{139986}{329141} a - \frac{128909}{329141}$, $\frac{1}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{33} + \frac{1010929453277864771508088428535186975565304060828600610952183579622618622627635828118652473965579760258857167682218690062908495657224275868571966305594965991416725335}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{32} - \frac{11567896636431028857131458467476437715034393639832448876293725205281164620741118210639250085909542679810442086921754487636371455691657623832272924896347129054138109470997}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{31} + \frac{2002660608788238850045984033179792046970571155848700235927547950663028433827566644312780213404288987746662957760158327004597766376161507261987927126927967336448058239003}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{30} + \frac{11654681121404999735359171208718096297860351808246381759847509032923295145241356832548665855255672109753057561619620137317224302518562491910976073960223675761816537830594}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{29} + \frac{10826370788981245983374428600745960936179722885201901521698345877422279216926798183298556074882469557514301574770419500938265319993158556694899855336676385606317832627830}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{28} - \frac{6162973412386466520157129001285273579369121625791629380829833042809760359834219331887307414644086027493866211916064034981547895868287515776823066127655985250248825089975}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{27} + \frac{6258668926531989909706153435577261042516049357372989256176389578951489359167652344862026246854132449518693294299869536235272349179525986889239613899467922430165867698859}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{26} - \frac{1253083942172447804986126110596230385762179906906071986181494808642569201829543019338039105696284109118138902095169102480079831796345335578190707163943247781444835540717}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{25} - \frac{87772906781124272036494811989934710679255525787369862371327708173659622706700689589900004029178152812555560824952810407620118942680442320013383490993551981172471104470489}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{24} - \frac{464221707137587107845165536825889545691932664299840573877373107828592985160047856926480651221871361013955755164542548927509694951769607215501116521316214668828602697306368}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{23} + \frac{402438794697034244266604820498424369616979346909150061721345018132839063768840702804869165044485036768809495097422595461186751801893701326717703363943462172985444938048315}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{22} - \frac{364145030346587574979064283357742225515698687079931280678304000750428141046041708075391279812190603313324811848874636299521582462251425059191860036869038246241827493077397}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{21} - \frac{426255860184876625380600299906236748970249070033521259353415232833124634081982901621728320991235259726936895735902970675823379198706592396884706059588858948306266530155421}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{20} + \frac{127543449999454350253664358702033510695580833324329096662376280716414927148914545062142481657467424883428105049598154652706038007570046674368305209425348499496404506770490}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{19} + \frac{253150402851832242077683143765732935403686179691698181325395641465940901363282194674283747231259268308094841095315628179501221534666855502933937975150891216083836277645837}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{18} - \frac{66557962258621028551669592376062359917368835740781183885873335960561458994279885506021718192343987507321985689513989011630021081516771370502578156561880724678332421298189}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{17} - \frac{492900634232885644655182935025815696122494585826164060473213041106376505085720278042867384670447298111260228378993708308220773007282918000650053130250106165028729987679681}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{16} - \frac{281952289316518487830396502183411086646940754412466571281553198278710466732483528323549723192265782330298611371219551956764027953148545482317155035616764542808219372071335}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{15} + \frac{28524109367797464462200680312864238070460021037650873264620327221153451447683983943853390705489451522168854453258661417602601612817368650098006958872986486114269363407797}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{14} - \frac{409973100351618440018169280992809699691411074054810498277512107460861072062699049819260369370491354047517697340330172441904006062142716737182471328820789462858053299199571}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{13} + \frac{110569265850808466397253206367926895921622754514990345639997515833241287419080277720872654407939955797559190244788090031936424764267545083351586806066520370875652828736758}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{12} - \frac{15493396407655790842974320269770686350444681752093177753551086193935807141711808991113044655148048487463162900031485360623388153969474161303819130630169122081927860019380}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{11} + \frac{281087050686823377617023102936877821398220640453968790298812396870602458487828013453418186141662303269299059329274763673663385734335505925847601681251817711918657784113669}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{10} - \frac{84868168212604618768407353105505060934187550393891039864442585936042999261101754697899947934240780259399699939295695235018644235666615404585567357135853989453082170909573}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{9} - \frac{48501935298725298695221426750661829128498535141034451530351717191104772212437906890481995630012291779473416073796121199514071605411051950761237711843949759995253653913792}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{8} - \frac{264059750082338422316830918720068430980282645320221354989925495847073003791374740707388205293944628349311975388158136389332420194249251198541861518269994845740629048246501}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{7} - \frac{210484035676367239134706413589968695628197793126704665918104976761493194022287449257603571967813528474558356446374436866191073359187392650442666616236937837848511154803421}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{6} - \frac{211678068600498723308154921591729240021433985576870962233400816772229653870279511124784272301408031183067572094599595418069783119118927239570412053578781748080478464714335}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{5} + \frac{471372646632461054333933119212572402013382615108138882053059255431502395384337499216162496620225884796213514893170095965072594557900106244808158463574267698512913661998480}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{4} - \frac{294713278632700313453939977707754885735700551981515199811615726373447553432856223301468548831978689774412484648512361781292319537643506777715445667606256965695428565421416}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{3} + \frac{221127472881808979402013894177528295149539581820176009618905187707730305215272102166209994477876181235900456327345840193787065923179072048327041773392698602993121487219423}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a^{2} - \frac{351048775888962123377125125739272041621062479226290121425735740322987773907543398388945522586975779701335234017949215862678546977509127793793855519959141471045425308805413}{1137325075196830834525297436004264180270489208161614799011223934431065877344533740864011579601115511243568756556810456348541075316438689754152304604095639154460420368545131} a + \frac{26358456102943466055383525822219295648003360373428150057972915160969906693397980720888666659109916972923263821696796153369982554424781040610277409684846021613559320574}{218842615970142550418567911488217082984508217849069616896521826906112348921403452157785564672140756444789062258381846516940749531737288773167655301923347922736274844823}$

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:  $33$
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

$C_{34}$ (as 34T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 34
The 34 conjugacy class representatives for $C_{34}$
Character table for $C_{34}$ is not computed

Intermediate fields

\(\Q(\sqrt{721}) \), 17.17.160470643909878751793805444097921.1

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 $17^{2}$ $17^{2}$ $17^{2}$ R $34$ $34$ $34$ $34$ $17^{2}$ $17^{2}$ $17^{2}$ $34$ $34$ $34$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{34}$ $34$ $34$

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
7Data not computed
103Data not computed