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Properties

Label 39.39.8320218171...4801.1
Degree $39$
Signature $[39, 0]$
Discriminant $3^{52}\cdot 79^{38}$
Root discriminant $305.58$
Ramified primes $3, 79$
Class number Not computed
Class group Not computed
Galois group $C_{39}$ (as 39T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9364853836691, 39491507056968, 86786523385245, -464961302461611, -346349054137671, 2216792963213496, 1072119182609331, -5719188124989093, -2808785515834836, 8729302726558491, 5033143416507438, -8186462453659176, -5697404621999031, 4742867841110748, 4091661940360473, -1634584478920060, -1897069485800418, 284377372208856, 577307993368452, 1203314824617, -117384643087056, -11865055676615, 16230796918608, 2789344797921, -1546711782640, -354430642254, 102131256564, 28911926408, -4645069176, -1593020385, 142319290, 59981382, -2797074, -1521777, 31758, 24885, -158, -237, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^39 - 237*x^37 - 158*x^36 + 24885*x^35 + 31758*x^34 - 1521777*x^33 - 2797074*x^32 + 59981382*x^31 + 142319290*x^30 - 1593020385*x^29 - 4645069176*x^28 + 28911926408*x^27 + 102131256564*x^26 - 354430642254*x^25 - 1546711782640*x^24 + 2789344797921*x^23 + 16230796918608*x^22 - 11865055676615*x^21 - 117384643087056*x^20 + 1203314824617*x^19 + 577307993368452*x^18 + 284377372208856*x^17 - 1897069485800418*x^16 - 1634584478920060*x^15 + 4091661940360473*x^14 + 4742867841110748*x^13 - 5697404621999031*x^12 - 8186462453659176*x^11 + 5033143416507438*x^10 + 8729302726558491*x^9 - 2808785515834836*x^8 - 5719188124989093*x^7 + 1072119182609331*x^6 + 2216792963213496*x^5 - 346349054137671*x^4 - 464961302461611*x^3 + 86786523385245*x^2 + 39491507056968*x - 9364853836691)
 
gp: K = bnfinit(x^39 - 237*x^37 - 158*x^36 + 24885*x^35 + 31758*x^34 - 1521777*x^33 - 2797074*x^32 + 59981382*x^31 + 142319290*x^30 - 1593020385*x^29 - 4645069176*x^28 + 28911926408*x^27 + 102131256564*x^26 - 354430642254*x^25 - 1546711782640*x^24 + 2789344797921*x^23 + 16230796918608*x^22 - 11865055676615*x^21 - 117384643087056*x^20 + 1203314824617*x^19 + 577307993368452*x^18 + 284377372208856*x^17 - 1897069485800418*x^16 - 1634584478920060*x^15 + 4091661940360473*x^14 + 4742867841110748*x^13 - 5697404621999031*x^12 - 8186462453659176*x^11 + 5033143416507438*x^10 + 8729302726558491*x^9 - 2808785515834836*x^8 - 5719188124989093*x^7 + 1072119182609331*x^6 + 2216792963213496*x^5 - 346349054137671*x^4 - 464961302461611*x^3 + 86786523385245*x^2 + 39491507056968*x - 9364853836691, 1)
 

Normalized defining polynomial

\( x^{39} - 237 x^{37} - 158 x^{36} + 24885 x^{35} + 31758 x^{34} - 1521777 x^{33} - 2797074 x^{32} + 59981382 x^{31} + 142319290 x^{30} - 1593020385 x^{29} - 4645069176 x^{28} + 28911926408 x^{27} + 102131256564 x^{26} - 354430642254 x^{25} - 1546711782640 x^{24} + 2789344797921 x^{23} + 16230796918608 x^{22} - 11865055676615 x^{21} - 117384643087056 x^{20} + 1203314824617 x^{19} + 577307993368452 x^{18} + 284377372208856 x^{17} - 1897069485800418 x^{16} - 1634584478920060 x^{15} + 4091661940360473 x^{14} + 4742867841110748 x^{13} - 5697404621999031 x^{12} - 8186462453659176 x^{11} + 5033143416507438 x^{10} + 8729302726558491 x^{9} - 2808785515834836 x^{8} - 5719188124989093 x^{7} + 1072119182609331 x^{6} + 2216792963213496 x^{5} - 346349054137671 x^{4} - 464961302461611 x^{3} + 86786523385245 x^{2} + 39491507056968 x - 9364853836691 \)

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

Invariants

Degree:  $39$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[39, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8320218171789251028069979650899790239310515771557947367834077287023540263376505733290557577514801=3^{52}\cdot 79^{38}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $305.58$
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:  $3, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(711=3^{2}\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{711}(640,·)$, $\chi_{711}(1,·)$, $\chi_{711}(130,·)$, $\chi_{711}(10,·)$, $\chi_{711}(268,·)$, $\chi_{711}(13,·)$, $\chi_{711}(655,·)$, $\chi_{711}(277,·)$, $\chi_{711}(151,·)$, $\chi_{711}(664,·)$, $\chi_{711}(541,·)$, $\chi_{711}(289,·)$, $\chi_{711}(547,·)$, $\chi_{711}(292,·)$, $\chi_{711}(421,·)$, $\chi_{711}(49,·)$, $\chi_{711}(169,·)$, $\chi_{711}(682,·)$, $\chi_{711}(46,·)$, $\chi_{711}(433,·)$, $\chi_{711}(694,·)$, $\chi_{711}(64,·)$, $\chi_{711}(652,·)$, $\chi_{711}(202,·)$, $\chi_{711}(76,·)$, $\chi_{711}(589,·)$, $\chi_{711}(334,·)$, $\chi_{711}(598,·)$, $\chi_{711}(88,·)$, $\chi_{711}(100,·)$, $\chi_{711}(490,·)$, $\chi_{711}(493,·)$, $\chi_{711}(496,·)$, $\chi_{711}(241,·)$, $\chi_{711}(499,·)$, $\chi_{711}(121,·)$, $\chi_{711}(634,·)$, $\chi_{711}(460,·)$, $\chi_{711}(637,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{23} a^{21} - \frac{2}{23} a^{20} - \frac{4}{23} a^{19} + \frac{1}{23} a^{18} + \frac{7}{23} a^{17} + \frac{1}{23} a^{16} + \frac{11}{23} a^{15} - \frac{7}{23} a^{14} - \frac{5}{23} a^{13} - \frac{3}{23} a^{12} + \frac{1}{23} a^{10} - \frac{2}{23} a^{9} - \frac{4}{23} a^{8} + \frac{1}{23} a^{7} + \frac{7}{23} a^{6} + \frac{1}{23} a^{5} + \frac{11}{23} a^{4} - \frac{7}{23} a^{3} - \frac{5}{23} a^{2} - \frac{3}{23} a$, $\frac{1}{23} a^{22} - \frac{8}{23} a^{20} - \frac{7}{23} a^{19} + \frac{9}{23} a^{18} - \frac{8}{23} a^{17} - \frac{10}{23} a^{16} - \frac{8}{23} a^{15} + \frac{4}{23} a^{14} + \frac{10}{23} a^{13} - \frac{6}{23} a^{12} + \frac{1}{23} a^{11} - \frac{8}{23} a^{9} - \frac{7}{23} a^{8} + \frac{9}{23} a^{7} - \frac{8}{23} a^{6} - \frac{10}{23} a^{5} - \frac{8}{23} a^{4} + \frac{4}{23} a^{3} + \frac{10}{23} a^{2} - \frac{6}{23} a$, $\frac{1}{23} a^{23} - \frac{1}{23} a$, $\frac{1}{23} a^{24} - \frac{1}{23} a^{2}$, $\frac{1}{23} a^{25} - \frac{1}{23} a^{3}$, $\frac{1}{23} a^{26} - \frac{1}{23} a^{4}$, $\frac{1}{23} a^{27} - \frac{1}{23} a^{5}$, $\frac{1}{23} a^{28} - \frac{1}{23} a^{6}$, $\frac{1}{23} a^{29} - \frac{1}{23} a^{7}$, $\frac{1}{23} a^{30} - \frac{1}{23} a^{8}$, $\frac{1}{529} a^{31} + \frac{9}{529} a^{30} - \frac{5}{529} a^{29} + \frac{1}{529} a^{28} + \frac{7}{529} a^{27} + \frac{11}{529} a^{26} - \frac{4}{529} a^{25} - \frac{2}{529} a^{24} + \frac{7}{529} a^{23} - \frac{6}{529} a^{22} - \frac{4}{529} a^{21} + \frac{217}{529} a^{20} + \frac{173}{529} a^{19} - \frac{219}{529} a^{18} + \frac{158}{529} a^{17} + \frac{79}{529} a^{16} - \frac{203}{529} a^{15} - \frac{157}{529} a^{14} + \frac{52}{529} a^{13} - \frac{251}{529} a^{12} - \frac{52}{529} a^{11} - \frac{257}{529} a^{10} + \frac{170}{529} a^{9} - \frac{158}{529} a^{8} - \frac{7}{529} a^{7} + \frac{134}{529} a^{6} + \frac{141}{529} a^{5} - \frac{76}{529} a^{4} - \frac{176}{529} a^{3} + \frac{215}{529} a^{2} - \frac{5}{529} a + \frac{9}{23}$, $\frac{1}{529} a^{32} + \frac{6}{529} a^{30} - \frac{2}{529} a^{28} - \frac{6}{529} a^{27} - \frac{11}{529} a^{26} + \frac{11}{529} a^{25} + \frac{2}{529} a^{24} + \frac{4}{529} a^{22} + \frac{152}{529} a^{20} + \frac{87}{529} a^{19} - \frac{125}{529} a^{18} - \frac{101}{529} a^{17} - \frac{178}{529} a^{16} - \frac{216}{529} a^{15} - \frac{122}{529} a^{14} + \frac{86}{529} a^{13} + \frac{68}{529} a^{12} + \frac{165}{529} a^{11} + \frac{114}{529} a^{10} + \frac{244}{529} a^{9} + \frac{12}{529} a^{8} + \frac{105}{529} a^{7} + \frac{177}{529} a^{6} - \frac{126}{529} a^{5} + \frac{117}{529} a^{4} + \frac{235}{529} a^{3} - \frac{54}{529} a^{2} + \frac{160}{529} a + \frac{11}{23}$, $\frac{1}{12167} a^{33} + \frac{7}{12167} a^{32} + \frac{8}{12167} a^{31} + \frac{60}{12167} a^{30} + \frac{34}{12167} a^{29} - \frac{179}{12167} a^{28} - \frac{16}{12167} a^{27} + \frac{140}{12167} a^{26} + \frac{94}{12167} a^{25} + \frac{240}{12167} a^{24} + \frac{41}{12167} a^{23} - \frac{237}{12167} a^{22} - \frac{201}{12167} a^{21} + \frac{3770}{12167} a^{20} + \frac{1336}{12167} a^{19} - \frac{4565}{12167} a^{18} + \frac{3272}{12167} a^{17} - \frac{4938}{12167} a^{16} - \frac{3282}{12167} a^{15} + \frac{1908}{12167} a^{14} - \frac{4792}{12167} a^{13} - \frac{5772}{12167} a^{12} - \frac{5436}{12167} a^{11} + \frac{6002}{12167} a^{10} + \frac{4774}{12167} a^{9} - \frac{3853}{12167} a^{8} + \frac{4049}{12167} a^{7} + \frac{4325}{12167} a^{6} + \frac{119}{529} a^{5} - \frac{2111}{12167} a^{4} - \frac{5845}{12167} a^{3} - \frac{5584}{12167} a^{2} + \frac{5480}{12167} a - \frac{158}{529}$, $\frac{1}{12167} a^{34} + \frac{5}{12167} a^{32} + \frac{4}{12167} a^{31} - \frac{110}{12167} a^{30} + \frac{112}{12167} a^{29} + \frac{87}{12167} a^{28} - \frac{24}{12167} a^{27} + \frac{195}{12167} a^{26} + \frac{88}{12167} a^{25} + \frac{40}{12167} a^{24} + \frac{5}{12167} a^{23} + \frac{55}{12167} a^{22} - \frac{113}{12167} a^{21} + \frac{5214}{12167} a^{20} - \frac{1980}{12167} a^{19} - \frac{2263}{12167} a^{18} + \frac{4013}{12167} a^{17} - \frac{2825}{12167} a^{16} + \frac{5953}{12167} a^{15} - \frac{5245}{12167} a^{14} + \frac{5807}{12167} a^{13} + \frac{2653}{12167} a^{12} + \frac{1389}{12167} a^{11} - \frac{785}{12167} a^{10} - \frac{2771}{12167} a^{9} + \frac{3006}{12167} a^{8} - \frac{2789}{12167} a^{7} + \frac{5996}{12167} a^{6} - \frac{4319}{12167} a^{5} + \frac{3734}{12167} a^{4} + \frac{3821}{12167} a^{3} + \frac{2409}{12167} a^{2} + \frac{2396}{12167} a + \frac{25}{529}$, $\frac{1}{7677377} a^{35} + \frac{123}{7677377} a^{34} + \frac{125}{7677377} a^{33} - \frac{6683}{7677377} a^{32} + \frac{2745}{7677377} a^{31} + \frac{19979}{7677377} a^{30} - \frac{92227}{7677377} a^{29} - \frac{128011}{7677377} a^{28} - \frac{104175}{7677377} a^{27} + \frac{110954}{7677377} a^{26} - \frac{142329}{7677377} a^{25} + \frac{140008}{7677377} a^{24} + \frac{95819}{7677377} a^{23} + \frac{19221}{7677377} a^{22} - \frac{54287}{7677377} a^{21} + \frac{811924}{7677377} a^{20} + \frac{301101}{7677377} a^{19} + \frac{1328295}{7677377} a^{18} - \frac{1443358}{7677377} a^{17} - \frac{3069034}{7677377} a^{16} - \frac{69895}{7677377} a^{15} + \frac{3746169}{7677377} a^{14} - \frac{36261}{7677377} a^{13} + \frac{2032335}{7677377} a^{12} - \frac{1524641}{7677377} a^{11} - \frac{2971111}{7677377} a^{10} + \frac{3544684}{7677377} a^{9} + \frac{2857076}{7677377} a^{8} - \frac{3443426}{7677377} a^{7} - \frac{3192}{333799} a^{6} + \frac{1928713}{7677377} a^{5} - \frac{85279}{333799} a^{4} + \frac{759302}{7677377} a^{3} + \frac{669350}{7677377} a^{2} - \frac{370606}{7677377} a - \frac{138314}{333799}$, $\frac{1}{176579671} a^{36} - \frac{9}{176579671} a^{35} - \frac{6646}{176579671} a^{34} + \frac{164}{176579671} a^{33} + \frac{166823}{176579671} a^{32} + \frac{56431}{176579671} a^{31} - \frac{700790}{176579671} a^{30} + \frac{3682679}{176579671} a^{29} + \frac{448484}{176579671} a^{28} - \frac{1643509}{176579671} a^{27} + \frac{122273}{176579671} a^{26} - \frac{2978360}{176579671} a^{25} + \frac{1079220}{176579671} a^{24} + \frac{3614315}{176579671} a^{23} + \frac{319975}{176579671} a^{22} + \frac{2854088}{176579671} a^{21} + \frac{7216981}{176579671} a^{20} + \frac{74899205}{176579671} a^{19} - \frac{57576719}{176579671} a^{18} + \frac{39288481}{176579671} a^{17} + \frac{86277168}{176579671} a^{16} + \frac{78856281}{176579671} a^{15} + \frac{34638651}{176579671} a^{14} - \frac{54494221}{176579671} a^{13} + \frac{86150453}{176579671} a^{12} - \frac{88147067}{176579671} a^{11} - \frac{58490228}{176579671} a^{10} + \frac{25334759}{176579671} a^{9} + \frac{56672099}{176579671} a^{8} + \frac{57335180}{176579671} a^{7} + \frac{4915250}{176579671} a^{6} - \frac{19173119}{176579671} a^{5} + \frac{85427054}{176579671} a^{4} + \frac{21206052}{176579671} a^{3} - \frac{65632730}{176579671} a^{2} - \frac{31901994}{176579671} a + \frac{147117}{7677377}$, $\frac{1}{1189970402869} a^{37} - \frac{2651}{1189970402869} a^{36} + \frac{31001}{1189970402869} a^{35} + \frac{38886359}{1189970402869} a^{34} - \frac{11115611}{1189970402869} a^{33} - \frac{161843662}{1189970402869} a^{32} + \frac{727972067}{1189970402869} a^{31} + \frac{14373059791}{1189970402869} a^{30} + \frac{2259890754}{1189970402869} a^{29} - \frac{527065190}{51737843603} a^{28} + \frac{16257557256}{1189970402869} a^{27} - \frac{3241343891}{1189970402869} a^{26} + \frac{14185669448}{1189970402869} a^{25} - \frac{8417944364}{1189970402869} a^{24} - \frac{11901792696}{1189970402869} a^{23} - \frac{12193513235}{1189970402869} a^{22} - \frac{5556251079}{1189970402869} a^{21} + \frac{69823373792}{1189970402869} a^{20} - \frac{498490732946}{1189970402869} a^{19} - \frac{2211258452}{51737843603} a^{18} - \frac{575905482177}{1189970402869} a^{17} - \frac{362725858141}{1189970402869} a^{16} + \frac{200090386717}{1189970402869} a^{15} + \frac{211630611389}{1189970402869} a^{14} + \frac{19980835178}{51737843603} a^{13} + \frac{497931299883}{1189970402869} a^{12} + \frac{514691733375}{1189970402869} a^{11} + \frac{85854937351}{1189970402869} a^{10} + \frac{393769621385}{1189970402869} a^{9} - \frac{432592910181}{1189970402869} a^{8} + \frac{219048886588}{1189970402869} a^{7} - \frac{237279314742}{1189970402869} a^{6} + \frac{294711467268}{1189970402869} a^{5} + \frac{42709660514}{1189970402869} a^{4} - \frac{44327396442}{1189970402869} a^{3} + \frac{460520901123}{1189970402869} a^{2} + \frac{439031891768}{1189970402869} a + \frac{53628003}{176579671}$, $\frac{1}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{38} + \frac{103586559223131619067110383147348641388553751036376916013924410323952638320239444511341681193410444282778611998185200041844659013823570631691875602365636799012357604278629749709998070373342088878695281128128249590}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{37} - \frac{3090219015107243512434313251105215128200039640398296706879300230183690013105680223382996519883331680389779037108095355230136689085582770190713979045682841181350761910082431940713185797755113937795333130368512187126980}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{36} - \frac{63420022600025413017170665735809468388344407177034805405353223600307306671231434683513365771683485565585628837979362405925185442782761157770143107792065959953846438473259026810782660229937796670246154756571429198488078}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{35} - \frac{62447582342276883986869443956179922957641918765872924941335270182208864608325244148600476587064887832496796352333601694269393362169431232034181924151306256179981844082680529982350267829759428503639082650563116808202736178}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{34} - \frac{20082660307239394804352959909634063788569864089453877895229331817433199987761851823990776541272367450515037385130498765665815206302654731069874674710434464486991337511173352546202674413473665794562539057742972206050933474}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{33} + \frac{1414770450795021264530403029077057047916892696874078908629438681200306281495639647342380085156983992775469562302088316408376904650762440087808686942067506097437341490724209471690299291047896467066077254809303734854363613593}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{32} - \frac{685551337889188225517143229250725374340779857889928643069564211565937960958019682709751182104896407984404628021928076245575864821916678460381317083812837689798130248370869660478910774946039863191283984522554211560356159425}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{31} + \frac{838969120678138242495646102255329979325832745489391070391741989579016715146276484373001868275870309895404225841686544057414520173209361748620328372287635760553691387011981788839558632042904964768233203261787416344426828230}{134226392885163617582702935575841475713708399661546975837480127234421198724855253452498061683690060933603098334989506530671713004216476424563027695233749310347400912819495693396595463104551215529373536488285627096591056496173} a^{30} + \frac{53158933856917458033889444531285566926374165859775595509453912929704231110245853962363749865741125812414030640546657080037433948490741516389783453184209365041373128945233707260992305301857024669199367592658527668429112001260}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{29} - \frac{17777590928591915645105716561791769414564174510568695084608658967424849947542128739803081606736382401178139409824545204184073565680202861953224974138382360892906447584484346199664059249938418628412417253113874390033079942859}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{28} + \frac{43824743579879142188972484104111358305401030178420009126874791030625785424960250167579125773935504460519576064305466151968329709016227404597568534875228720836074241577478641651633699733492608056165026576070847852433569386620}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{27} - \frac{17515183216828186381661111568228194687566030330588585062543568589613883658457389488827096465521325879294150238306062032178112645486518168936880385311141135503676639801863308571310587147951278697330711276306134883225630612438}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{26} - \frac{64175996829198653863508828819923122295256839347448188721692244237355720841341341259461481985504848100420252727524029968644000388657432927918796829002332035997098131929441511219155351701428061314641856563132509038995360833879}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{25} + \frac{21823623258516930320434526959967278132544816841266060724001700566235086947368641141566726691006017895091944756471879474273111530881757227348713294413490683030296351120308029798698148549326740985990473804981138392146221202301}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{24} - \frac{29857194744359154248694026447964182201338552086241563616642721124413406359021444967979055215720212761078195762671483028533326180612545561817428648508040028349830345234701910168059568444412852247756489919973912134113068479898}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{23} + \frac{21703014484496764899722414377925742901707382986180061954313096995504002603152317401149514465487485316282267300683043857448140471401380188238548272725269140738098557823889386821703093250450951823181636601739877476186216924036}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{22} + \frac{40684113109623401320116651100792901906474218489152901412906000241792805463997492085400191095847008578686894549393537734986492242934130766242102147880277499571620567408773001626117289168234506050542785128263576818482434583042}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{21} - \frac{1543475121414037583444061976036384167898374224830829517423165971216536095101842734286188283763852009460307768144722463294362831566093853559781920671370417905058029720443439408649173440778277540911771943856397461322100096060662}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{20} + \frac{1083175845418651676518895078390834812317752492615496969772642042237461574876834363254085428437715421054872710516034120182677617086206891521604745315115409735912088619728776455699018627722424644454338034332864691537508496661160}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{19} + \frac{318558974905733231242125988907961548353671456272828110024175582646317556552576643005462049396426522983987791173021320352586491973142994815299253818456591597501814953027867525998592183120527755473963360696351932287788623130310}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{18} - \frac{86389135888030387050247417647212360908034390814594499381997874307887556861911603495066059080686032799395411835212154192944950996768092934410973687503776978724240717982280365685457924880529357342460739517149292323069929643185}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{17} - \frac{31149650216923162283237518979165374654949480495013834875190133795661080674785942475064015026900554183737200017229720837970569950352894702418616280421860651633792720064274131159794037173832290019211450350254428012960485338107}{134226392885163617582702935575841475713708399661546975837480127234421198724855253452498061683690060933603098334989506530671713004216476424563027695233749310347400912819495693396595463104551215529373536488285627096591056496173} a^{16} - \frac{1068224290426485516458176647954009425565941815822550254686558674924110796245997347303778608757417935399049955189676552582059141502706665809376692472086774558277668216505994614248061648386156144685770907381016635418894768515636}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{15} - \frac{689143215367638842410408857647067085167824420876950152585970751270650304924738829882677631629117992946700736093526501031249501184447835959557570070515777900782755086443451600596719985016930695202241574219874959501856902115607}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{14} - \frac{1493773499156497933113357830704914493054655506046928954313272952267624798201512259718744309150882867155249490473690186421120311795686257506290173842452951479325058041986874144068571897665816741823424061763211164802455898730339}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{13} - \frac{732346592232726262935209021196515298350669089182329307911759228713585232224782012084702675221639359689271436777978309760451318097876364219183045731340320793785040091330056338611348381615234953805508394290602640745306175724815}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{12} - \frac{706782301857847160117442969590833466563756217480522732670100353995412655315564918053248537551014509389709934312439530250291539210678780762311888596760695645516514375271388034066448418158187820771305579459789006948815092671526}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{11} + \frac{1037935806332880097163641118647751832850788122447475160709510690054747970171929717035770634081388709599925806204562114654252465093192979981091265172530694635263528015004551920628199812873643153085229120510703690007364093635743}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{10} - \frac{1308281724928372899221294484016800847419737060915953232116899369187591892955108718663344084164712365959802930088766941495650028726662137362329686116175573873255021956182656485362103288395018911483259964177915138131941719117429}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{9} - \frac{712487668483214130092575968737813745611863526442435565108502866040961262197256903862008949782777973289079343247905937906020997908028922380437065135663662058770684342943841157677519427306692002221512520763090260383893551070774}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{8} - \frac{687698850999770364960357592656759029933329137145131605582967409871436967155676240862783142846303502265728001305648418812562143611784938804769880487409072111183938060013004605352820360726086998714432751215469921924243390676909}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{7} + \frac{903691409276243298370979484383757622857416618365586405037906900610780996136748421696465595493991775745387214408344516343094123370271311461549366098582365356432543605191412956634303903220860433605525713642713681446797535071403}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{6} + \frac{1281218295205307363811058852620989609058139900345240623209590446206655616381848164985082357939590346526177775540228252063361736015921680121923512772254499600530883811277916041125610353830844831637035202875766963523908943226347}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{5} - \frac{202361583714592355549400339739338584118974156417193799894893699828026423861922255056877736202137444055564814342018015291158204206354664538945015526347001161340842301079246744946646554008249543563222637013558799726762004232295}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{4} - \frac{927873527573694890401403037635517314748230839085242641727074137837100049782009187411099618792687993932078699570785976983513217300779308272307149072136473165879082073321225242604896229878691562215703456370806781101358990336810}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{3} - \frac{1165274318529781890009028712482600006363516102436806583543747730374494989590181290013107261462981330331765725923858296426283434467035483228320038859279229150715226786007219952220762330628504937968017731949984394902944174825044}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a^{2} + \frac{1137381881206516907948713074279945281008869909785598336975552031611828150190230968571115604554732025943449861529038602136554927734367773214721641857652223197397492502736420869029706498845120239505864904068940881181064042086010}{3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979} a - \frac{7803248678639818429288977520308222325037899128854478055398309317513991211991460714617399185742191700009953929409951494305250138593379745505787181927047951688328932877234133295736896680621353757932581794090240842048}{26043049910378392547471230461925315320278296044376036594666350529683682920756113189692980475171934248650351611196096024460578689992816006257845609574076771600938034054840906047614568490961751705615957823971568666151}$

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:  $38$
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

$C_{39}$ (as 39T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 39
The 39 conjugacy class representatives for $C_{39}$
Character table for $C_{39}$ is not computed

Intermediate fields

3.3.505521.1, 13.13.59091511031674153381441.1

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 ${\href{/LocalNumberField/2.13.0.1}{13} }^{3}$ R ${\href{/LocalNumberField/5.13.0.1}{13} }^{3}$ $39$ $39$ $39$ ${\href{/LocalNumberField/17.13.0.1}{13} }^{3}$ $39$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{39}$ $39$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{3}$ $39$ $39$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{3}$ $39$ $39$ $39$

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
79Data not computed