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Properties

Label 34.34.3425230050...4149.1
Degree $34$
Signature $[34, 0]$
Discriminant $3^{17}\cdot 103^{33}$
Root discriminant $155.67$
Ramified primes $3, 103$
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![10497967, 51273995, -160716846, -1123034742, -122533277, 7567081746, 10323419848, -14514531077, -42024711035, -9898232087, 58215655753, 57474448334, -18084925848, -57678181587, -20555590855, 20594193249, 18112177824, -597495179, -5837837426, -1610027003, 872682200, 511384499, -34999669, -76649821, -8038724, 6473545, 1428035, -308922, -107886, 7317, 4475, -39, -101, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 - x^33 - 101*x^32 - 39*x^31 + 4475*x^30 + 7317*x^29 - 107886*x^28 - 308922*x^27 + 1428035*x^26 + 6473545*x^25 - 8038724*x^24 - 76649821*x^23 - 34999669*x^22 + 511384499*x^21 + 872682200*x^20 - 1610027003*x^19 - 5837837426*x^18 - 597495179*x^17 + 18112177824*x^16 + 20594193249*x^15 - 20555590855*x^14 - 57678181587*x^13 - 18084925848*x^12 + 57474448334*x^11 + 58215655753*x^10 - 9898232087*x^9 - 42024711035*x^8 - 14514531077*x^7 + 10323419848*x^6 + 7567081746*x^5 - 122533277*x^4 - 1123034742*x^3 - 160716846*x^2 + 51273995*x + 10497967)
 
gp: K = bnfinit(x^34 - x^33 - 101*x^32 - 39*x^31 + 4475*x^30 + 7317*x^29 - 107886*x^28 - 308922*x^27 + 1428035*x^26 + 6473545*x^25 - 8038724*x^24 - 76649821*x^23 - 34999669*x^22 + 511384499*x^21 + 872682200*x^20 - 1610027003*x^19 - 5837837426*x^18 - 597495179*x^17 + 18112177824*x^16 + 20594193249*x^15 - 20555590855*x^14 - 57678181587*x^13 - 18084925848*x^12 + 57474448334*x^11 + 58215655753*x^10 - 9898232087*x^9 - 42024711035*x^8 - 14514531077*x^7 + 10323419848*x^6 + 7567081746*x^5 - 122533277*x^4 - 1123034742*x^3 - 160716846*x^2 + 51273995*x + 10497967, 1)
 

Normalized defining polynomial

\( x^{34} - x^{33} - 101 x^{32} - 39 x^{31} + 4475 x^{30} + 7317 x^{29} - 107886 x^{28} - 308922 x^{27} + 1428035 x^{26} + 6473545 x^{25} - 8038724 x^{24} - 76649821 x^{23} - 34999669 x^{22} + 511384499 x^{21} + 872682200 x^{20} - 1610027003 x^{19} - 5837837426 x^{18} - 597495179 x^{17} + 18112177824 x^{16} + 20594193249 x^{15} - 20555590855 x^{14} - 57678181587 x^{13} - 18084925848 x^{12} + 57474448334 x^{11} + 58215655753 x^{10} - 9898232087 x^{9} - 42024711035 x^{8} - 14514531077 x^{7} + 10323419848 x^{6} + 7567081746 x^{5} - 122533277 x^{4} - 1123034742 x^{3} - 160716846 x^{2} + 51273995 x + 10497967 \)

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:  \(342523005011894297428856269332610453116457630461733441736562419892654124149=3^{17}\cdot 103^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $155.67$
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:  $3, 103$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(309=3\cdot 103\)
Dirichlet character group:    $\lbrace$$\chi_{309}(1,·)$, $\chi_{309}(133,·)$, $\chi_{309}(134,·)$, $\chi_{309}(140,·)$, $\chi_{309}(13,·)$, $\chi_{309}(275,·)$, $\chi_{309}(34,·)$, $\chi_{309}(296,·)$, $\chi_{309}(169,·)$, $\chi_{309}(175,·)$, $\chi_{309}(176,·)$, $\chi_{309}(308,·)$, $\chi_{309}(184,·)$, $\chi_{309}(61,·)$, $\chi_{309}(64,·)$, $\chi_{309}(196,·)$, $\chi_{309}(197,·)$, $\chi_{309}(76,·)$, $\chi_{309}(79,·)$, $\chi_{309}(80,·)$, $\chi_{309}(209,·)$, $\chi_{309}(214,·)$, $\chi_{309}(89,·)$, $\chi_{309}(220,·)$, $\chi_{309}(95,·)$, $\chi_{309}(100,·)$, $\chi_{309}(229,·)$, $\chi_{309}(230,·)$, $\chi_{309}(233,·)$, $\chi_{309}(112,·)$, $\chi_{309}(113,·)$, $\chi_{309}(245,·)$, $\chi_{309}(248,·)$, $\chi_{309}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{47} a^{24} + \frac{16}{47} a^{23} + \frac{7}{47} a^{22} + \frac{22}{47} a^{21} - \frac{11}{47} a^{20} - \frac{13}{47} a^{17} - \frac{19}{47} a^{16} - \frac{17}{47} a^{15} + \frac{20}{47} a^{14} - \frac{9}{47} a^{13} + \frac{22}{47} a^{12} + \frac{5}{47} a^{11} + \frac{11}{47} a^{10} + \frac{6}{47} a^{9} - \frac{4}{47} a^{8} - \frac{17}{47} a^{7} - \frac{4}{47} a^{6} - \frac{22}{47} a^{5} + \frac{4}{47} a^{4} - \frac{19}{47} a^{3} + \frac{18}{47} a^{2} - \frac{1}{47} a$, $\frac{1}{47} a^{25} - \frac{14}{47} a^{23} + \frac{4}{47} a^{22} + \frac{13}{47} a^{21} - \frac{12}{47} a^{20} - \frac{13}{47} a^{18} + \frac{1}{47} a^{17} + \frac{5}{47} a^{16} + \frac{10}{47} a^{15} - \frac{22}{47} a^{13} - \frac{18}{47} a^{12} - \frac{22}{47} a^{11} + \frac{18}{47} a^{10} - \frac{6}{47} a^{9} - \frac{14}{47} a^{7} - \frac{5}{47} a^{6} - \frac{20}{47} a^{5} + \frac{11}{47} a^{4} - \frac{7}{47} a^{3} - \frac{7}{47} a^{2} + \frac{16}{47} a$, $\frac{1}{47} a^{26} - \frac{7}{47} a^{23} + \frac{17}{47} a^{22} + \frac{14}{47} a^{21} - \frac{13}{47} a^{20} - \frac{13}{47} a^{19} + \frac{1}{47} a^{18} + \frac{11}{47} a^{17} - \frac{21}{47} a^{16} - \frac{3}{47} a^{15} + \frac{23}{47} a^{14} - \frac{3}{47} a^{13} + \frac{4}{47} a^{12} - \frac{6}{47} a^{11} + \frac{7}{47} a^{10} - \frac{10}{47} a^{9} - \frac{23}{47} a^{8} - \frac{8}{47} a^{7} + \frac{18}{47} a^{6} - \frac{15}{47} a^{5} + \frac{2}{47} a^{4} + \frac{9}{47} a^{3} - \frac{14}{47} a^{2} - \frac{14}{47} a$, $\frac{1}{47} a^{27} - \frac{12}{47} a^{23} + \frac{16}{47} a^{22} + \frac{4}{47} a^{20} + \frac{1}{47} a^{19} + \frac{11}{47} a^{18} - \frac{18}{47} a^{17} + \frac{5}{47} a^{16} - \frac{2}{47} a^{15} - \frac{4}{47} a^{14} - \frac{12}{47} a^{13} + \frac{7}{47} a^{12} - \frac{5}{47} a^{11} + \frac{20}{47} a^{10} + \frac{19}{47} a^{9} + \frac{11}{47} a^{8} - \frac{7}{47} a^{7} + \frac{4}{47} a^{6} - \frac{11}{47} a^{5} - \frac{10}{47} a^{4} - \frac{6}{47} a^{3} + \frac{18}{47} a^{2} - \frac{7}{47} a$, $\frac{1}{47} a^{28} + \frac{20}{47} a^{23} - \frac{10}{47} a^{22} - \frac{14}{47} a^{21} + \frac{10}{47} a^{20} + \frac{11}{47} a^{19} - \frac{18}{47} a^{18} - \frac{10}{47} a^{17} + \frac{5}{47} a^{16} - \frac{20}{47} a^{15} - \frac{7}{47} a^{14} - \frac{7}{47} a^{13} - \frac{23}{47} a^{12} - \frac{14}{47} a^{11} + \frac{10}{47} a^{10} - \frac{11}{47} a^{9} - \frac{8}{47} a^{8} - \frac{12}{47} a^{7} - \frac{12}{47} a^{6} + \frac{8}{47} a^{5} - \frac{5}{47} a^{4} - \frac{22}{47} a^{3} + \frac{21}{47} a^{2} - \frac{12}{47} a$, $\frac{1}{12361} a^{29} - \frac{93}{12361} a^{28} - \frac{59}{12361} a^{27} + \frac{19}{12361} a^{26} - \frac{120}{12361} a^{25} + \frac{124}{12361} a^{24} + \frac{122}{12361} a^{23} + \frac{308}{12361} a^{22} + \frac{2494}{12361} a^{21} + \frac{1338}{12361} a^{20} - \frac{3321}{12361} a^{19} + \frac{1654}{12361} a^{18} - \frac{65}{12361} a^{17} + \frac{240}{12361} a^{16} + \frac{1061}{12361} a^{15} - \frac{1726}{12361} a^{14} - \frac{6135}{12361} a^{13} + \frac{3933}{12361} a^{12} + \frac{43}{263} a^{11} + \frac{5597}{12361} a^{10} - \frac{5485}{12361} a^{9} - \frac{4436}{12361} a^{8} - \frac{3564}{12361} a^{7} - \frac{5307}{12361} a^{6} + \frac{1090}{12361} a^{5} + \frac{3316}{12361} a^{4} - \frac{4748}{12361} a^{3} - \frac{2837}{12361} a^{2} - \frac{2359}{12361} a + \frac{76}{263}$, $\frac{1}{12361} a^{30} - \frac{29}{12361} a^{28} + \frac{55}{12361} a^{27} + \frac{69}{12361} a^{26} + \frac{10}{12361} a^{25} + \frac{82}{12361} a^{24} + \frac{345}{12361} a^{23} + \frac{6153}{12361} a^{22} + \frac{3418}{12361} a^{21} - \frac{1971}{12361} a^{20} - \frac{278}{12361} a^{19} - \frac{361}{12361} a^{18} + \frac{1559}{12361} a^{17} - \frac{5286}{12361} a^{16} + \frac{1741}{12361} a^{15} + \frac{1930}{12361} a^{14} + \frac{406}{12361} a^{13} - \frac{5670}{12361} a^{12} + \frac{245}{12361} a^{11} - \frac{5967}{12361} a^{10} + \frac{5410}{12361} a^{9} + \frac{3899}{12361} a^{8} - \frac{4590}{12361} a^{7} - \frac{6174}{12361} a^{6} - \frac{2355}{12361} a^{5} + \frac{5135}{12361} a^{4} + \frac{3751}{12361} a^{3} - \frac{833}{12361} a^{2} + \frac{108}{12361} a - \frac{33}{263}$, $\frac{1}{12361} a^{31} - \frac{12}{12361} a^{28} - \frac{64}{12361} a^{27} + \frac{35}{12361} a^{26} + \frac{21}{12361} a^{25} - \frac{4}{12361} a^{24} - \frac{2144}{12361} a^{23} + \frac{778}{12361} a^{22} - \frac{3811}{12361} a^{21} + \frac{6175}{12361} a^{20} + \frac{2481}{12361} a^{19} - \frac{708}{12361} a^{18} - \frac{596}{12361} a^{17} - \frac{3134}{12361} a^{16} + \frac{5610}{12361} a^{15} - \frac{4675}{12361} a^{14} + \frac{515}{12361} a^{13} + \frac{949}{12361} a^{12} + \frac{2672}{12361} a^{11} + \frac{5189}{12361} a^{10} + \frac{4738}{12361} a^{9} + \frac{2211}{12361} a^{8} - \frac{5119}{12361} a^{7} + \frac{5487}{12361} a^{6} + \frac{4922}{12361} a^{5} + \frac{5235}{12361} a^{4} + \frac{24}{263} a^{3} + \frac{417}{12361} a^{2} + \frac{2889}{12361} a + \frac{100}{263}$, $\frac{1}{580967} a^{32} - \frac{8}{580967} a^{31} - \frac{18}{580967} a^{30} + \frac{12}{580967} a^{29} - \frac{889}{580967} a^{28} + \frac{2349}{580967} a^{27} - \frac{67}{12361} a^{26} - \frac{2180}{580967} a^{25} - \frac{2190}{580967} a^{24} - \frac{70038}{580967} a^{23} - \frac{121287}{580967} a^{22} - \frac{150157}{580967} a^{21} - \frac{145019}{580967} a^{20} - \frac{289876}{580967} a^{19} + \frac{252983}{580967} a^{18} - \frac{162644}{580967} a^{17} - \frac{208995}{580967} a^{16} - \frac{221382}{580967} a^{15} - \frac{78751}{580967} a^{14} - \frac{30953}{580967} a^{13} + \frac{227300}{580967} a^{12} + \frac{2133}{580967} a^{11} + \frac{61362}{580967} a^{10} + \frac{219733}{580967} a^{9} + \frac{253696}{580967} a^{8} + \frac{21168}{580967} a^{7} + \frac{211998}{580967} a^{6} + \frac{126196}{580967} a^{5} - \frac{36766}{580967} a^{4} - \frac{253723}{580967} a^{3} - \frac{91150}{580967} a^{2} + \frac{142107}{580967} a - \frac{1012}{12361}$, $\frac{1}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{33} + \frac{28147849274504564282850485175274696372704394313074953559153140614}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{32} - \frac{726859925131443812965171860628781474517440759420704111806033548235}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{31} + \frac{1291156989786732955711316336476840500925812821365307632989147689179}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{30} - \frac{736051492622354966166590768235063872178935647002563840260680106}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{29} - \frac{201199826908408723913388870182793465650735550766058487930560212190729}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{28} - \frac{77920977941511055720538126636089762751255927372727374568113547983815}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{27} + \frac{70378961579886711342072012809309513298252124232549417400029948446971}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{26} - \frac{99422467650819604869510864540921905122234045150910532971693663557579}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{25} - \frac{276043386786124164625473120308987032178283822860106251339694016407359}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{24} + \frac{7935610451465536995654200768657738912935586829850003821721277601318631}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{23} + \frac{15332086900441003850454308075176422269679946066432964625895265702958799}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{22} - \frac{10449743092077074136748979350454316203419004270797878057725011096135456}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{21} - \frac{1943218725712087090901012921791278916611726204427610368896276030274420}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{20} + \frac{7973696707624377833956499693861400766893151303930176084254217575379574}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{19} - \frac{10535170781691739061643479220841137079235159778710592864431962354734612}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{18} + \frac{2532001846643168522148588149140060975945398566885301902962315709191515}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{17} - \frac{12160665588375503765172461897605679820888628397685906717696972381408690}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{16} - \frac{9155106835391425896145557806639769633602742954437991617098047105991253}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{15} - \frac{3703354564434250027302620777353337482844589801103012117766485587916539}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{14} - \frac{6780642569864623880653050181403417145209285559482588615375701712317915}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{13} - \frac{3622247028563824252508798254573175131602043349622846524906472736477240}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{12} + \frac{2306190201649533298293420308923998559864304254074383950837076125905304}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{11} - \frac{17064738782422920314416834614211659766810487248731298169228336935991149}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{10} - \frac{13294247590400334785766414449644371355023983793912337482893456465327954}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{9} + \frac{11561629042742205162471313582093633792028029214956937915828122036822858}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{8} + \frac{2972776227137794355675300817401928884933344310511420408190113371069752}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{7} - \frac{14961021604992506869996189391952711939918211924930716531672319110891806}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{6} + \frac{13104056420534761557633704025720561232372125574536558222810564661206273}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{5} + \frac{8098425821884282013183676897369203226751154906542961524262721569110213}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{4} + \frac{15353306579703956539386179733376347951742327502078213785737215913771860}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{3} - \frac{14912710981325633284540175295016060217848932245154346481959900408073999}{34680544774373147733790654086415134952014413980881388529166540959231347} a^{2} - \frac{484479671369451903012153518273356005538084389580180403575595046816285}{34680544774373147733790654086415134952014413980881388529166540959231347} a + \frac{11934129206914490119012115837810975818671521516589237981984683948790}{737883931369641441144482001838619892596051361295348692109926403387901}$

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

\(\Q(\sqrt{309}) \), 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 $34$ R $17^{2}$ $17^{2}$ $17^{2}$ $17^{2}$ $34$ $17^{2}$ $34$ $34$ $34$ $34$ $34$ $34$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{34}$ $17^{2}$ $34$

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
3Data not computed
103Data not computed