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Properties

Label 34.0.41960441237...4411.1
Degree $34$
Signature $[0, 17]$
Discriminant $-\,3^{17}\cdot 137^{33}$
Root discriminant $205.32$
Ramified primes $3, 137$
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![195083006077, -392425151665, 1087761544277, -162805628636, 82957823229, 276611418393, 400126458226, 161406767274, 509311176532, 298994470956, 233471643412, 189166843484, 100414605637, 53970569062, 36044872838, 9927882579, 8865827065, 1737857598, 1307903253, 490618713, 117900834, 73008077, 17638446, 5106957, 2758416, -63617, 363437, -34952, 33367, -2429, 2028, -76, 71, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 - x^33 + 71*x^32 - 76*x^31 + 2028*x^30 - 2429*x^29 + 33367*x^28 - 34952*x^27 + 363437*x^26 - 63617*x^25 + 2758416*x^24 + 5106957*x^23 + 17638446*x^22 + 73008077*x^21 + 117900834*x^20 + 490618713*x^19 + 1307903253*x^18 + 1737857598*x^17 + 8865827065*x^16 + 9927882579*x^15 + 36044872838*x^14 + 53970569062*x^13 + 100414605637*x^12 + 189166843484*x^11 + 233471643412*x^10 + 298994470956*x^9 + 509311176532*x^8 + 161406767274*x^7 + 400126458226*x^6 + 276611418393*x^5 + 82957823229*x^4 - 162805628636*x^3 + 1087761544277*x^2 - 392425151665*x + 195083006077)
 
gp: K = bnfinit(x^34 - x^33 + 71*x^32 - 76*x^31 + 2028*x^30 - 2429*x^29 + 33367*x^28 - 34952*x^27 + 363437*x^26 - 63617*x^25 + 2758416*x^24 + 5106957*x^23 + 17638446*x^22 + 73008077*x^21 + 117900834*x^20 + 490618713*x^19 + 1307903253*x^18 + 1737857598*x^17 + 8865827065*x^16 + 9927882579*x^15 + 36044872838*x^14 + 53970569062*x^13 + 100414605637*x^12 + 189166843484*x^11 + 233471643412*x^10 + 298994470956*x^9 + 509311176532*x^8 + 161406767274*x^7 + 400126458226*x^6 + 276611418393*x^5 + 82957823229*x^4 - 162805628636*x^3 + 1087761544277*x^2 - 392425151665*x + 195083006077, 1)
 

Normalized defining polynomial

\( x^{34} - x^{33} + 71 x^{32} - 76 x^{31} + 2028 x^{30} - 2429 x^{29} + 33367 x^{28} - 34952 x^{27} + 363437 x^{26} - 63617 x^{25} + 2758416 x^{24} + 5106957 x^{23} + 17638446 x^{22} + 73008077 x^{21} + 117900834 x^{20} + 490618713 x^{19} + 1307903253 x^{18} + 1737857598 x^{17} + 8865827065 x^{16} + 9927882579 x^{15} + 36044872838 x^{14} + 53970569062 x^{13} + 100414605637 x^{12} + 189166843484 x^{11} + 233471643412 x^{10} + 298994470956 x^{9} + 509311176532 x^{8} + 161406767274 x^{7} + 400126458226 x^{6} + 276611418393 x^{5} + 82957823229 x^{4} - 162805628636 x^{3} + 1087761544277 x^{2} - 392425151665 x + 195083006077 \)

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

Invariants

Degree:  $34$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 17]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-4196044123758294387691867377481129713297026828522826841087263264798772158264411=-\,3^{17}\cdot 137^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $205.32$
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:  $3, 137$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(411=3\cdot 137\)
Dirichlet character group:    $\lbrace$$\chi_{411}(256,·)$, $\chi_{411}(1,·)$, $\chi_{411}(259,·)$, $\chi_{411}(133,·)$, $\chi_{411}(65,·)$, $\chi_{411}(395,·)$, $\chi_{411}(397,·)$, $\chi_{411}(14,·)$, $\chi_{411}(16,·)$, $\chi_{411}(278,·)$, $\chi_{411}(152,·)$, $\chi_{411}(410,·)$, $\chi_{411}(155,·)$, $\chi_{411}(218,·)$, $\chi_{411}(34,·)$, $\chi_{411}(296,·)$, $\chi_{411}(175,·)$, $\chi_{411}(187,·)$, $\chi_{411}(193,·)$, $\chi_{411}(323,·)$, $\chi_{411}(196,·)$, $\chi_{411}(200,·)$, $\chi_{411}(73,·)$, $\chi_{411}(77,·)$, $\chi_{411}(334,·)$, $\chi_{411}(338,·)$, $\chi_{411}(211,·)$, $\chi_{411}(215,·)$, $\chi_{411}(88,·)$, $\chi_{411}(346,·)$, $\chi_{411}(224,·)$, $\chi_{411}(236,·)$, $\chi_{411}(115,·)$, $\chi_{411}(377,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{41} a^{21} + \frac{19}{41} a^{20} + \frac{20}{41} a^{19} - \frac{5}{41} a^{18} + \frac{9}{41} a^{17} + \frac{2}{41} a^{16} - \frac{13}{41} a^{15} - \frac{2}{41} a^{14} + \frac{1}{41} a^{13} + \frac{18}{41} a^{12} - \frac{14}{41} a^{11} + \frac{11}{41} a^{10} - \frac{14}{41} a^{9} + \frac{18}{41} a^{8} + \frac{4}{41} a^{7} - \frac{12}{41} a^{6} - \frac{6}{41} a^{5} - \frac{2}{41} a^{4} - \frac{8}{41} a^{3} - \frac{5}{41} a - \frac{8}{41}$, $\frac{1}{1517} a^{22} + \frac{12}{1517} a^{21} - \frac{113}{1517} a^{20} - \frac{309}{1517} a^{19} - \frac{612}{1517} a^{18} + \frac{677}{1517} a^{17} - \frac{601}{1517} a^{16} + \frac{7}{1517} a^{15} - \frac{395}{1517} a^{14} - \frac{276}{1517} a^{13} + \frac{434}{1517} a^{12} - \frac{670}{1517} a^{11} - \frac{337}{1517} a^{10} - \frac{212}{1517} a^{9} + \frac{206}{1517} a^{8} + \frac{247}{1517} a^{7} + \frac{78}{1517} a^{6} - \frac{124}{1517} a^{5} - \frac{650}{1517} a^{4} - \frac{682}{1517} a^{3} + \frac{200}{1517} a^{2} + \frac{68}{1517} a - \frac{723}{1517}$, $\frac{1}{1517} a^{23} + \frac{2}{1517} a^{21} - \frac{100}{1517} a^{20} + \frac{691}{1517} a^{19} + \frac{658}{1517} a^{18} - \frac{326}{1517} a^{17} + \frac{152}{1517} a^{16} + \frac{705}{1517} a^{15} - \frac{605}{1517} a^{14} - \frac{546}{1517} a^{13} + \frac{301}{1517} a^{12} - \frac{474}{1517} a^{11} + \frac{613}{1517} a^{10} + \frac{641}{1517} a^{9} - \frac{597}{1517} a^{8} - \frac{9}{41} a^{7} + \frac{383}{1517} a^{6} - \frac{716}{1517} a^{5} + \frac{532}{1517} a^{4} + \frac{244}{1517} a^{3} + \frac{702}{1517} a^{2} + \frac{200}{1517} a + \frac{536}{1517}$, $\frac{1}{1517} a^{24} - \frac{13}{1517} a^{21} - \frac{8}{1517} a^{20} + \frac{462}{1517} a^{19} + \frac{343}{1517} a^{18} - \frac{203}{1517} a^{17} + \frac{612}{1517} a^{16} - \frac{545}{1517} a^{15} + \frac{22}{1517} a^{14} - \frac{553}{1517} a^{13} + \frac{16}{37} a^{12} + \frac{399}{1517} a^{11} - \frac{498}{1517} a^{10} - \frac{210}{1517} a^{9} - \frac{264}{1517} a^{8} + \frac{9}{41} a^{7} - \frac{687}{1517} a^{6} + \frac{114}{1517} a^{5} - \frac{195}{1517} a^{4} - \frac{339}{1517} a^{3} - \frac{200}{1517} a^{2} - \frac{155}{1517} a + \frac{558}{1517}$, $\frac{1}{1517} a^{25} + \frac{732}{1517} a^{20} - \frac{566}{1517} a^{19} + \frac{166}{1517} a^{18} + \frac{496}{1517} a^{17} + \frac{448}{1517} a^{16} + \frac{520}{1517} a^{15} + \frac{676}{1517} a^{14} - \frac{46}{1517} a^{13} + \frac{343}{1517} a^{12} + \frac{449}{1517} a^{11} - \frac{151}{1517} a^{10} + \frac{569}{1517} a^{9} + \frac{347}{1517} a^{8} + \frac{415}{1517} a^{7} - \frac{130}{1517} a^{6} + \frac{598}{1517} a^{5} + \frac{609}{1517} a^{4} - \frac{297}{1517} a^{3} - \frac{589}{1517} a^{2} + \frac{665}{1517} a - \frac{630}{1517}$, $\frac{1}{1517} a^{26} - \frac{8}{1517} a^{21} + \frac{544}{1517} a^{20} + \frac{536}{1517} a^{19} - \frac{355}{1517} a^{18} - \frac{144}{1517} a^{17} + \frac{557}{1517} a^{16} - \frac{323}{1517} a^{15} - \frac{83}{1517} a^{14} - \frac{397}{1517} a^{13} - \frac{735}{1517} a^{12} - \frac{10}{37} a^{11} + \frac{14}{1517} a^{10} + \frac{88}{1517} a^{9} + \frac{748}{1517} a^{8} - \frac{56}{1517} a^{7} + \frac{376}{1517} a^{6} + \frac{498}{1517} a^{5} - \frac{334}{1517} a^{4} - \frac{737}{1517} a^{3} + \frac{665}{1517} a^{2} + \frac{36}{1517} a - \frac{4}{41}$, $\frac{1}{1517} a^{27} + \frac{11}{1517} a^{21} - \frac{183}{1517} a^{20} - \frac{237}{1517} a^{19} - \frac{378}{1517} a^{18} + \frac{312}{1517} a^{17} - \frac{321}{1517} a^{16} + \frac{565}{1517} a^{15} + \frac{735}{1517} a^{14} - \frac{538}{1517} a^{13} - \frac{675}{1517} a^{12} + \frac{426}{1517} a^{11} - \frac{425}{1517} a^{10} + \frac{273}{1517} a^{9} - \frac{628}{1517} a^{8} - \frac{4}{37} a^{7} - \frac{432}{1517} a^{6} - \frac{586}{1517} a^{5} - \frac{128}{1517} a^{4} + \frac{241}{1517} a^{3} + \frac{119}{1517} a^{2} + \frac{507}{1517} a - \frac{752}{1517}$, $\frac{1}{56129} a^{28} - \frac{18}{56129} a^{27} + \frac{1}{56129} a^{26} + \frac{14}{56129} a^{25} + \frac{5}{56129} a^{24} - \frac{8}{56129} a^{23} + \frac{4}{56129} a^{22} - \frac{480}{56129} a^{21} + \frac{13772}{56129} a^{20} + \frac{221}{1369} a^{19} + \frac{17035}{56129} a^{18} + \frac{10519}{56129} a^{17} + \frac{13303}{56129} a^{16} - \frac{19439}{56129} a^{15} - \frac{27160}{56129} a^{14} - \frac{3593}{56129} a^{13} + \frac{23394}{56129} a^{12} + \frac{13292}{56129} a^{11} - \frac{9017}{56129} a^{10} - \frac{9286}{56129} a^{9} - \frac{13282}{56129} a^{8} - \frac{26755}{56129} a^{7} - \frac{3704}{56129} a^{6} + \frac{6291}{56129} a^{5} - \frac{23466}{56129} a^{4} + \frac{523}{56129} a^{3} + \frac{972}{56129} a^{2} - \frac{24991}{56129} a - \frac{19767}{56129}$, $\frac{1}{56129} a^{29} + \frac{10}{56129} a^{27} - \frac{5}{56129} a^{26} - \frac{2}{56129} a^{25} + \frac{8}{56129} a^{24} + \frac{8}{56129} a^{23} - \frac{1}{56129} a^{22} + \frac{174}{56129} a^{21} - \frac{23244}{56129} a^{20} - \frac{18631}{56129} a^{19} + \frac{3500}{56129} a^{18} + \frac{9542}{56129} a^{17} + \frac{15738}{56129} a^{16} - \frac{24489}{56129} a^{15} - \frac{24127}{56129} a^{14} - \frac{24334}{56129} a^{13} - \frac{6582}{56129} a^{12} - \frac{2824}{56129} a^{11} - \frac{726}{56129} a^{10} - \frac{5198}{56129} a^{9} - \frac{6609}{56129} a^{8} - \frac{5404}{56129} a^{7} + \frac{23239}{56129} a^{6} - \frac{16418}{56129} a^{5} - \frac{21451}{56129} a^{4} - \frac{13331}{56129} a^{3} + \frac{23400}{56129} a^{2} + \frac{6474}{56129} a - \frac{12039}{56129}$, $\frac{1}{56129} a^{30} - \frac{10}{56129} a^{27} - \frac{12}{56129} a^{26} + \frac{16}{56129} a^{25} - \frac{5}{56129} a^{24} + \frac{5}{56129} a^{23} - \frac{14}{56129} a^{22} + \frac{389}{56129} a^{21} + \frac{2823}{56129} a^{20} + \frac{13345}{56129} a^{19} - \frac{15842}{56129} a^{18} - \frac{4019}{56129} a^{17} + \frac{2802}{56129} a^{16} - \frac{8706}{56129} a^{15} - \frac{288}{1369} a^{14} - \frac{18382}{56129} a^{13} + \frac{14910}{56129} a^{12} + \frac{13947}{56129} a^{11} - \frac{9563}{56129} a^{10} - \frac{5176}{56129} a^{9} - \frac{11889}{56129} a^{8} + \frac{27016}{56129} a^{7} + \frac{17366}{56129} a^{6} - \frac{7660}{56129} a^{5} + \frac{1438}{56129} a^{4} - \frac{17831}{56129} a^{3} + \frac{23135}{56129} a^{2} - \frac{16726}{56129} a - \frac{23035}{56129}$, $\frac{1}{19839412469} a^{31} + \frac{151189}{19839412469} a^{30} + \frac{106605}{19839412469} a^{29} - \frac{63965}{19839412469} a^{28} + \frac{2301481}{19839412469} a^{27} - \frac{1327724}{19839412469} a^{26} - \frac{4726925}{19839412469} a^{25} + \frac{3794622}{19839412469} a^{24} + \frac{4913689}{19839412469} a^{23} - \frac{1185569}{19839412469} a^{22} - \frac{1973963}{536200337} a^{21} - \frac{6902161636}{19839412469} a^{20} - \frac{1303018703}{19839412469} a^{19} - \frac{3828925453}{19839412469} a^{18} - \frac{4256281833}{19839412469} a^{17} + \frac{464182}{11802149} a^{16} - \frac{8671759965}{19839412469} a^{15} + \frac{6457903063}{19839412469} a^{14} + \frac{2849500238}{19839412469} a^{13} - \frac{39290615}{19839412469} a^{12} + \frac{4215081179}{19839412469} a^{11} + \frac{7306205713}{19839412469} a^{10} - \frac{7755137173}{19839412469} a^{9} + \frac{173242597}{483888109} a^{8} - \frac{4183452393}{19839412469} a^{7} + \frac{2327225228}{19839412469} a^{6} + \frac{2078713847}{19839412469} a^{5} - \frac{4852456013}{19839412469} a^{4} - \frac{1406132427}{19839412469} a^{3} - \frac{5914995855}{19839412469} a^{2} - \frac{6447443434}{19839412469} a + \frac{9223181202}{19839412469}$, $\frac{1}{366184388332982449} a^{32} - \frac{2212873}{366184388332982449} a^{31} + \frac{2231240996020}{366184388332982449} a^{30} - \frac{1764768936651}{366184388332982449} a^{29} - \frac{2526287488782}{366184388332982449} a^{28} - \frac{79290117047148}{366184388332982449} a^{27} + \frac{112750753463895}{366184388332982449} a^{26} - \frac{12352863252975}{366184388332982449} a^{25} - \frac{4715285613972}{366184388332982449} a^{24} - \frac{13078537261514}{366184388332982449} a^{23} + \frac{116557087285343}{366184388332982449} a^{22} - \frac{4061102555087452}{366184388332982449} a^{21} + \frac{3774100012992664}{9896875360350877} a^{20} + \frac{166355821287255449}{366184388332982449} a^{19} - \frac{87047000449701321}{366184388332982449} a^{18} - \frac{2050145877207673}{366184388332982449} a^{17} + \frac{121358275799350231}{366184388332982449} a^{16} + \frac{175752935574119016}{366184388332982449} a^{15} + \frac{82270423338160166}{366184388332982449} a^{14} - \frac{60373475648104278}{366184388332982449} a^{13} - \frac{166644385533777909}{366184388332982449} a^{12} - \frac{73393684041390735}{366184388332982449} a^{11} - \frac{114838547062905296}{366184388332982449} a^{10} - \frac{142150747990300563}{366184388332982449} a^{9} - \frac{91607063317690208}{366184388332982449} a^{8} - \frac{165669126979510590}{366184388332982449} a^{7} + \frac{88365137228596232}{366184388332982449} a^{6} - \frac{3094309606245801}{366184388332982449} a^{5} - \frac{139939364820607020}{366184388332982449} a^{4} - \frac{32650631254906360}{366184388332982449} a^{3} - \frac{115915204037527879}{366184388332982449} a^{2} - \frac{60207555605503864}{366184388332982449} a - \frac{147602549468668861}{366184388332982449}$, $\frac{1}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{33} - \frac{129175688162004380740609916568152606093535255246504178971015537216530599843361966103156412468016797263739002486607805913878429418833718171346103757352066956}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{32} - \frac{1974168467975733346845764174601217428551491876756574982890036814293771829556821400714855906568764490732958014955808667979610579063488114539637813063178018737546981}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{31} - \frac{568207339622065517195840145012308060984419712618060811675639301432418282010922610116490287875055142593915969345489440963055557669682984932086278939620062423251634062559}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{30} - \frac{11796735565018766153076300766246551253548619587114778329767134679586600486626834397467669952381154351184603010094275116740650094274505625624053421742068449708252855289}{2308922518089361220915968299690696619161061327629347404573464864725297649801435781968563232766259643069559901482172019287938942315988619503772019148303512071689264355560447} a^{29} + \frac{662171412288432446959292842296221399046838091053698331445562935664284199349119091123670503623487917814274736617264695141588117112755550898493737508399252779743982899041}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{28} - \frac{371939391821546011537731153943408633437223280624818179089105868101938622888084029708966790771165855936840793937392482815747677363537831380617988288598776652906695044313}{2558535763288211082636613521278880037448743633319006583446271877128032530861050461100299798470720145023025836777541967319067476620419821612287913110282270133493509150756171} a^{27} - \frac{4843659342972623711919713169504892841325539772463837913959406814990360517216112850097276174116693765621146058115521106501735898429612455125854772624383108970001955792826}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{26} + \frac{7790253861327309428232812733293748795593885437145077499661990050637271922885519543294824275597237641900506252929756687971820368607123365474691027207033882751057699818316}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{25} - \frac{5527312857132161977297635460093111486012906252002447492444191721192022937245497109282993467155265253351160328090751774866555200788110874679541791531680542470518921207363}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{24} - \frac{29450599674261531932342132373562662095898497699406898413059964792394607246117122713469792062659690911601672321563952241523116391913009448285744654338824710567729957239395}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{23} - \frac{26858055074954078860643591531780308049485725779820421721646396545647551352087209623548908713115930665947714025196390128729508814698526155396349285706217858758973288039343}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{22} + \frac{419710540059576565365016891920108684498018734762777216990445853466326170005966828884966296491196724075094126852873772169892620004360439702120615564779523843290838924582425}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{21} - \frac{13996822901515219146750105513623273665611017970616241796283213355953213926455276908257034010886742331360912322977227787635320545747050383617529785151971207664618116456961524}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{20} - \frac{21663106367424828730844470594278453559952037906182760859371983799740646132808904143262193727427891219044656039514976167520929476412444274700693427041498630764406822477729344}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{19} - \frac{41550772932752886093754069340058479931871108237804494854701134084927040071657335623261802622751929100996120702589700017725620815831708777295813691264475316866840001010523641}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{18} - \frac{15141785525886004514165226220668676256401652807268350008225959158740900638680270528761847177849018138932155111709234571230047693429958109943341427192033880318191683386085057}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{17} - \frac{31451109594034493437534060118468166306685514599409030299978470020540541886853228902063948606565944229476906488932917116949597836504637705107237612102984310636367149767984010}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{16} - \frac{34772705326696250544993957458155207738719056994125055218308665288370360207316053348667388773879924604776461445497145558562648198117087208607263784198429455430437545183351593}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{15} + \frac{290282998206274221480399042412940792390649940683158448079356553108564941074176960243130968340626510422171817411166642454999485559905560212125100072088915100858440694906222}{2558535763288211082636613521278880037448743633319006583446271877128032530861050461100299798470720145023025836777541967319067476620419821612287913110282270133493509150756171} a^{14} + \frac{8995920095804209380151713053650481697287397513497040515638806080320177083301192954914696859360276785211120423880553515651845981822909789018407670488637436385145867547907969}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{13} - \frac{29927468676342264536173767363743400632603091021344457672405812595182111919474669070751412219351855416707888680814102929968010960183601724316567733347084563966098784339570020}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{12} - \frac{45248110032245462803751691298486185813272152523475809893319190294106008627109579429614877054553249661497559585746277637275241354631515827730350757453997388590829660547657136}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{11} + \frac{207292148470627527602818812162315666750552350045950685998180133484948320788563100587422223428525096769643418986848536414392522042896195256155843806442204875459396856371717}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{10} - \frac{2348693003414627438702297347435013784658635995668644789486465118065915622945062677659296220579438449550317430555563619379059290868960139087015075930115248958949311173800403}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{9} - \frac{45746066557101219103230151824821072698927984591988124768764111185942747201091871356947487083146653958892320605981778353307713054733610052106859743566051590461224215926695803}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{8} - \frac{8387650007843390372965886445052387006896849785687278583472494333816763177890085802610010672666844147751280026009878698921940803135585249143222856437023101568095507957218581}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{7} + \frac{44524576472148899386719783079805502024396344438863220023264616188484636567057826000731402406051598291113574396324548737742427237537113544314712990059323130602283089286082622}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{6} - \frac{45003639460740260311974256206099346097223462172935598364454285515616032723395483949804326487713355228402562853495200652657987884042386553023224635286028970155406411868233337}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{5} - \frac{1553917141283305786800216585155610264058789214101424452603041615199576217288572400002714947523885442435117519257813986411494233353828391501733745677510950453321540991716032}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{4} - \frac{33930917114804414847094211596473758337533973872559550788493265468193132650936341093165634939529933022389670614237092113775511541854711930595912565743760599641770895411672661}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{3} - \frac{17451982379049607446303721918207244883161969892702977481415485135476437897772247221865119058936652160020260206309620503913146557376545318210632142159943321854874171606485372}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a^{2} + \frac{17052904898823676215369961493256166789316258281989807557965215390425848974304944915689244669888748036717897473456262023438806248959411877612783584636715080709087738059436543}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327} a - \frac{24130666023094902864963527393731276452576033991308216463393499324386772511082911151301948475225499330097576822308452592038442396921192298616763015677490626670040434522757673}{94665823241663810057554700287318561385603514432803243587512059453737203641858867060711092543416645365851955960769052790805496634955533399654652785080443994939259838577978327}$

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:  $16$
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

\(\Q(\sqrt{-411}) \), 17.17.15400296222263289476715621650663041.1

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 $17^{2}$ $17^{2}$ $34$ $34$ $34$ $17^{2}$ $17^{2}$ $17^{2}$ $34$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{34}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{34}$ $34$ $17^{2}$ $17^{2}$ $34$

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
137Data not computed