LMFDB - Number field 39.39.8320218171789251028069979650899790239310515771557947367834077287023540263376505733290557577514801.1

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Normalized defining polynomial

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(y^39 - 237*y^37 - 158*y^36 + 24885*y^35 + 31758*y^34 - 1521777*y^33 - 2797074*y^32 + 59981382*y^31 + 142319290*y^30 - 1593020385*y^29 - 4645069176*y^28 + 28911926408*y^27 + 102131256564*y^26 - 354430642254*y^25 - 1546711782640*y^24 + 2789344797921*y^23 + 16230796918608*y^22 - 11865055676615*y^21 - 117384643087056*y^20 + 1203314824617*y^19 + 577307993368452*y^18 + 284377372208856*y^17 - 1897069485800418*y^16 - 1634584478920060*y^15 + 4091661940360473*y^14 + 4742867841110748*y^13 - 5697404621999031*y^12 - 8186462453659176*y^11 + 5033143416507438*y^10 + 8729302726558491*y^9 - 2808785515834836*y^8 - 5719188124989093*y^7 + 1072119182609331*y^6 + 2216792963213496*y^5 - 346349054137671*y^4 - 464961302461611*y^3 + 86786523385245*y^2 + 39491507056968*y - 9364853836691, 1)

magma: R<x> := PolynomialRing(Rationals()); 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);

oscar: Qx, x = PolynomialRing(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)

$ x^{39} - 237 x^{37} - 158 x^{36} + 24885 x^{35} + 31758 x^{34} - 1521777 x^{33} - 2797074 x^{32} + \cdots - 9364853836691 $ 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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$39$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[39, 0]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$832\!\cdots\!801$ 8320218171789251028069979650899790239310515771557947367834077287023540263376505733290557577514801 $\medspace = 3^{52}\cdot 79^{38}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$305.58$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$3^{4/3}79^{38/39}\approx 305.58473738571735$
Ramified primes:	$3$, $79$ 3, 79	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$39$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{23}a^{21}-\frac{2}{23}a^{20}-\frac{4}{23}a^{19}+\frac{1}{23}a^{18}+\frac{7}{23}a^{17}+\frac{1}{23}a^{16}+\frac{11}{23}a^{15}-\frac{7}{23}a^{14}-\frac{5}{23}a^{13}-\frac{3}{23}a^{12}+\frac{1}{23}a^{10}-\frac{2}{23}a^{9}-\frac{4}{23}a^{8}+\frac{1}{23}a^{7}+\frac{7}{23}a^{6}+\frac{1}{23}a^{5}+\frac{11}{23}a^{4}-\frac{7}{23}a^{3}-\frac{5}{23}a^{2}-\frac{3}{23}a$, $\frac{1}{23}a^{22}-\frac{8}{23}a^{20}-\frac{7}{23}a^{19}+\frac{9}{23}a^{18}-\frac{8}{23}a^{17}-\frac{10}{23}a^{16}-\frac{8}{23}a^{15}+\frac{4}{23}a^{14}+\frac{10}{23}a^{13}-\frac{6}{23}a^{12}+\frac{1}{23}a^{11}-\frac{8}{23}a^{9}-\frac{7}{23}a^{8}+\frac{9}{23}a^{7}-\frac{8}{23}a^{6}-\frac{10}{23}a^{5}-\frac{8}{23}a^{4}+\frac{4}{23}a^{3}+\frac{10}{23}a^{2}-\frac{6}{23}a$, $\frac{1}{23}a^{23}-\frac{1}{23}a$, $\frac{1}{23}a^{24}-\frac{1}{23}a^{2}$, $\frac{1}{23}a^{25}-\frac{1}{23}a^{3}$, $\frac{1}{23}a^{26}-\frac{1}{23}a^{4}$, $\frac{1}{23}a^{27}-\frac{1}{23}a^{5}$, $\frac{1}{23}a^{28}-\frac{1}{23}a^{6}$, $\frac{1}{23}a^{29}-\frac{1}{23}a^{7}$, $\frac{1}{23}a^{30}-\frac{1}{23}a^{8}$, $\frac{1}{529}a^{31}+\frac{9}{529}a^{30}-\frac{5}{529}a^{29}+\frac{1}{529}a^{28}+\frac{7}{529}a^{27}+\frac{11}{529}a^{26}-\frac{4}{529}a^{25}-\frac{2}{529}a^{24}+\frac{7}{529}a^{23}-\frac{6}{529}a^{22}-\frac{4}{529}a^{21}+\frac{217}{529}a^{20}+\frac{173}{529}a^{19}-\frac{219}{529}a^{18}+\frac{158}{529}a^{17}+\frac{79}{529}a^{16}-\frac{203}{529}a^{15}-\frac{157}{529}a^{14}+\frac{52}{529}a^{13}-\frac{251}{529}a^{12}-\frac{52}{529}a^{11}-\frac{257}{529}a^{10}+\frac{170}{529}a^{9}-\frac{158}{529}a^{8}-\frac{7}{529}a^{7}+\frac{134}{529}a^{6}+\frac{141}{529}a^{5}-\frac{76}{529}a^{4}-\frac{176}{529}a^{3}+\frac{215}{529}a^{2}-\frac{5}{529}a+\frac{9}{23}$, $\frac{1}{529}a^{32}+\frac{6}{529}a^{30}-\frac{2}{529}a^{28}-\frac{6}{529}a^{27}-\frac{11}{529}a^{26}+\frac{11}{529}a^{25}+\frac{2}{529}a^{24}+\frac{4}{529}a^{22}+\frac{152}{529}a^{20}+\frac{87}{529}a^{19}-\frac{125}{529}a^{18}-\frac{101}{529}a^{17}-\frac{178}{529}a^{16}-\frac{216}{529}a^{15}-\frac{122}{529}a^{14}+\frac{86}{529}a^{13}+\frac{68}{529}a^{12}+\frac{165}{529}a^{11}+\frac{114}{529}a^{10}+\frac{244}{529}a^{9}+\frac{12}{529}a^{8}+\frac{105}{529}a^{7}+\frac{177}{529}a^{6}-\frac{126}{529}a^{5}+\frac{117}{529}a^{4}+\frac{235}{529}a^{3}-\frac{54}{529}a^{2}+\frac{160}{529}a+\frac{11}{23}$, $\frac{1}{12167}a^{33}+\frac{7}{12167}a^{32}+\frac{8}{12167}a^{31}+\frac{60}{12167}a^{30}+\frac{34}{12167}a^{29}-\frac{179}{12167}a^{28}-\frac{16}{12167}a^{27}+\frac{140}{12167}a^{26}+\frac{94}{12167}a^{25}+\frac{240}{12167}a^{24}+\frac{41}{12167}a^{23}-\frac{237}{12167}a^{22}-\frac{201}{12167}a^{21}+\frac{3770}{12167}a^{20}+\frac{1336}{12167}a^{19}-\frac{4565}{12167}a^{18}+\frac{3272}{12167}a^{17}-\frac{4938}{12167}a^{16}-\frac{3282}{12167}a^{15}+\frac{1908}{12167}a^{14}-\frac{4792}{12167}a^{13}-\frac{5772}{12167}a^{12}-\frac{5436}{12167}a^{11}+\frac{6002}{12167}a^{10}+\frac{4774}{12167}a^{9}-\frac{3853}{12167}a^{8}+\frac{4049}{12167}a^{7}+\frac{4325}{12167}a^{6}+\frac{119}{529}a^{5}-\frac{2111}{12167}a^{4}-\frac{5845}{12167}a^{3}-\frac{5584}{12167}a^{2}+\frac{5480}{12167}a-\frac{158}{529}$, $\frac{1}{12167}a^{34}+\frac{5}{12167}a^{32}+\frac{4}{12167}a^{31}-\frac{110}{12167}a^{30}+\frac{112}{12167}a^{29}+\frac{87}{12167}a^{28}-\frac{24}{12167}a^{27}+\frac{195}{12167}a^{26}+\frac{88}{12167}a^{25}+\frac{40}{12167}a^{24}+\frac{5}{12167}a^{23}+\frac{55}{12167}a^{22}-\frac{113}{12167}a^{21}+\frac{5214}{12167}a^{20}-\frac{1980}{12167}a^{19}-\frac{2263}{12167}a^{18}+\frac{4013}{12167}a^{17}-\frac{2825}{12167}a^{16}+\frac{5953}{12167}a^{15}-\frac{5245}{12167}a^{14}+\frac{5807}{12167}a^{13}+\frac{2653}{12167}a^{12}+\frac{1389}{12167}a^{11}-\frac{785}{12167}a^{10}-\frac{2771}{12167}a^{9}+\frac{3006}{12167}a^{8}-\frac{2789}{12167}a^{7}+\frac{5996}{12167}a^{6}-\frac{4319}{12167}a^{5}+\frac{3734}{12167}a^{4}+\frac{3821}{12167}a^{3}+\frac{2409}{12167}a^{2}+\frac{2396}{12167}a+\frac{25}{529}$, $\frac{1}{7677377}a^{35}+\frac{123}{7677377}a^{34}+\frac{125}{7677377}a^{33}-\frac{6683}{7677377}a^{32}+\frac{2745}{7677377}a^{31}+\frac{19979}{7677377}a^{30}-\frac{92227}{7677377}a^{29}-\frac{128011}{7677377}a^{28}-\frac{104175}{7677377}a^{27}+\frac{110954}{7677377}a^{26}-\frac{142329}{7677377}a^{25}+\frac{140008}{7677377}a^{24}+\frac{95819}{7677377}a^{23}+\frac{19221}{7677377}a^{22}-\frac{54287}{7677377}a^{21}+\frac{811924}{7677377}a^{20}+\frac{301101}{7677377}a^{19}+\frac{1328295}{7677377}a^{18}-\frac{1443358}{7677377}a^{17}-\frac{3069034}{7677377}a^{16}-\frac{69895}{7677377}a^{15}+\frac{3746169}{7677377}a^{14}-\frac{36261}{7677377}a^{13}+\frac{2032335}{7677377}a^{12}-\frac{1524641}{7677377}a^{11}-\frac{2971111}{7677377}a^{10}+\frac{3544684}{7677377}a^{9}+\frac{2857076}{7677377}a^{8}-\frac{3443426}{7677377}a^{7}-\frac{3192}{333799}a^{6}+\frac{1928713}{7677377}a^{5}-\frac{85279}{333799}a^{4}+\frac{759302}{7677377}a^{3}+\frac{669350}{7677377}a^{2}-\frac{370606}{7677377}a-\frac{138314}{333799}$, $\frac{1}{176579671}a^{36}-\frac{9}{176579671}a^{35}-\frac{6646}{176579671}a^{34}+\frac{164}{176579671}a^{33}+\frac{166823}{176579671}a^{32}+\frac{56431}{176579671}a^{31}-\frac{700790}{176579671}a^{30}+\frac{3682679}{176579671}a^{29}+\frac{448484}{176579671}a^{28}-\frac{1643509}{176579671}a^{27}+\frac{122273}{176579671}a^{26}-\frac{2978360}{176579671}a^{25}+\frac{1079220}{176579671}a^{24}+\frac{3614315}{176579671}a^{23}+\frac{319975}{176579671}a^{22}+\frac{2854088}{176579671}a^{21}+\frac{7216981}{176579671}a^{20}+\frac{74899205}{176579671}a^{19}-\frac{57576719}{176579671}a^{18}+\frac{39288481}{176579671}a^{17}+\frac{86277168}{176579671}a^{16}+\frac{78856281}{176579671}a^{15}+\frac{34638651}{176579671}a^{14}-\frac{54494221}{176579671}a^{13}+\frac{86150453}{176579671}a^{12}-\frac{88147067}{176579671}a^{11}-\frac{58490228}{176579671}a^{10}+\frac{25334759}{176579671}a^{9}+\frac{56672099}{176579671}a^{8}+\frac{57335180}{176579671}a^{7}+\frac{4915250}{176579671}a^{6}-\frac{19173119}{176579671}a^{5}+\frac{85427054}{176579671}a^{4}+\frac{21206052}{176579671}a^{3}-\frac{65632730}{176579671}a^{2}-\frac{31901994}{176579671}a+\frac{147117}{7677377}$, $\frac{1}{1189970402869}a^{37}-\frac{2651}{1189970402869}a^{36}+\frac{31001}{1189970402869}a^{35}+\frac{38886359}{1189970402869}a^{34}-\frac{11115611}{1189970402869}a^{33}-\frac{161843662}{1189970402869}a^{32}+\frac{727972067}{1189970402869}a^{31}+\frac{14373059791}{1189970402869}a^{30}+\frac{2259890754}{1189970402869}a^{29}-\frac{527065190}{51737843603}a^{28}+\frac{16257557256}{1189970402869}a^{27}-\frac{3241343891}{1189970402869}a^{26}+\frac{14185669448}{1189970402869}a^{25}-\frac{8417944364}{1189970402869}a^{24}-\frac{11901792696}{1189970402869}a^{23}-\frac{12193513235}{1189970402869}a^{22}-\frac{5556251079}{1189970402869}a^{21}+\frac{69823373792}{1189970402869}a^{20}-\frac{498490732946}{1189970402869}a^{19}-\frac{2211258452}{51737843603}a^{18}-\frac{575905482177}{1189970402869}a^{17}-\frac{362725858141}{1189970402869}a^{16}+\frac{200090386717}{1189970402869}a^{15}+\frac{211630611389}{1189970402869}a^{14}+\frac{19980835178}{51737843603}a^{13}+\frac{497931299883}{1189970402869}a^{12}+\frac{514691733375}{1189970402869}a^{11}+\frac{85854937351}{1189970402869}a^{10}+\frac{393769621385}{1189970402869}a^{9}-\frac{432592910181}{1189970402869}a^{8}+\frac{219048886588}{1189970402869}a^{7}-\frac{237279314742}{1189970402869}a^{6}+\frac{294711467268}{1189970402869}a^{5}+\frac{42709660514}{1189970402869}a^{4}-\frac{44327396442}{1189970402869}a^{3}+\frac{460520901123}{1189970402869}a^{2}+\frac{439031891768}{1189970402869}a+\frac{53628003}{176579671}$, $\frac{1}{30\!\cdots\!79}a^{38}+\frac{10\!\cdots\!90}{30\!\cdots\!79}a^{37}-\frac{30\!\cdots\!80}{30\!\cdots\!79}a^{36}-\frac{63\!\cdots\!78}{30\!\cdots\!79}a^{35}-\frac{62\!\cdots\!78}{30\!\cdots\!79}a^{34}-\frac{20\!\cdots\!74}{30\!\cdots\!79}a^{33}+\frac{14\!\cdots\!93}{30\!\cdots\!79}a^{32}-\frac{68\!\cdots\!25}{30\!\cdots\!79}a^{31}+\frac{83\!\cdots\!30}{13\!\cdots\!73}a^{30}+\frac{53\!\cdots\!60}{30\!\cdots\!79}a^{29}-\frac{17\!\cdots\!59}{30\!\cdots\!79}a^{28}+\frac{43\!\cdots\!20}{30\!\cdots\!79}a^{27}-\frac{17\!\cdots\!38}{30\!\cdots\!79}a^{26}-\frac{64\!\cdots\!79}{30\!\cdots\!79}a^{25}+\frac{21\!\cdots\!01}{30\!\cdots\!79}a^{24}-\frac{29\!\cdots\!98}{30\!\cdots\!79}a^{23}+\frac{21\!\cdots\!36}{30\!\cdots\!79}a^{22}+\frac{40\!\cdots\!42}{30\!\cdots\!79}a^{21}-\frac{15\!\cdots\!62}{30\!\cdots\!79}a^{20}+\frac{10\!\cdots\!60}{30\!\cdots\!79}a^{19}+\frac{31\!\cdots\!10}{30\!\cdots\!79}a^{18}-\frac{86\!\cdots\!85}{30\!\cdots\!79}a^{17}-\frac{31\!\cdots\!07}{13\!\cdots\!73}a^{16}-\frac{10\!\cdots\!36}{30\!\cdots\!79}a^{15}-\frac{68\!\cdots\!07}{30\!\cdots\!79}a^{14}-\frac{14\!\cdots\!39}{30\!\cdots\!79}a^{13}-\frac{73\!\cdots\!15}{30\!\cdots\!79}a^{12}-\frac{70\!\cdots\!26}{30\!\cdots\!79}a^{11}+\frac{10\!\cdots\!43}{30\!\cdots\!79}a^{10}-\frac{13\!\cdots\!29}{30\!\cdots\!79}a^{9}-\frac{71\!\cdots\!74}{30\!\cdots\!79}a^{8}-\frac{68\!\cdots\!09}{30\!\cdots\!79}a^{7}+\frac{90\!\cdots\!03}{30\!\cdots\!79}a^{6}+\frac{12\!\cdots\!47}{30\!\cdots\!79}a^{5}-\frac{20\!\cdots\!95}{30\!\cdots\!79}a^{4}-\frac{92\!\cdots\!10}{30\!\cdots\!79}a^{3}-\frac{11\!\cdots\!44}{30\!\cdots\!79}a^{2}+\frac{11\!\cdots\!10}{30\!\cdots\!79}a-\frac{78\!\cdots\!48}{26\!\cdots\!51}$ 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, 1/23*a^21 - 2/23*a^20 - 4/23*a^19 + 1/23*a^18 + 7/23*a^17 + 1/23*a^16 + 11/23*a^15 - 7/23*a^14 - 5/23*a^13 - 3/23*a^12 + 1/23*a^10 - 2/23*a^9 - 4/23*a^8 + 1/23*a^7 + 7/23*a^6 + 1/23*a^5 + 11/23*a^4 - 7/23*a^3 - 5/23*a^2 - 3/23*a, 1/23*a^22 - 8/23*a^20 - 7/23*a^19 + 9/23*a^18 - 8/23*a^17 - 10/23*a^16 - 8/23*a^15 + 4/23*a^14 + 10/23*a^13 - 6/23*a^12 + 1/23*a^11 - 8/23*a^9 - 7/23*a^8 + 9/23*a^7 - 8/23*a^6 - 10/23*a^5 - 8/23*a^4 + 4/23*a^3 + 10/23*a^2 - 6/23*a, 1/23*a^23 - 1/23*a, 1/23*a^24 - 1/23*a^2, 1/23*a^25 - 1/23*a^3, 1/23*a^26 - 1/23*a^4, 1/23*a^27 - 1/23*a^5, 1/23*a^28 - 1/23*a^6, 1/23*a^29 - 1/23*a^7, 1/23*a^30 - 1/23*a^8, 1/529*a^31 + 9/529*a^30 - 5/529*a^29 + 1/529*a^28 + 7/529*a^27 + 11/529*a^26 - 4/529*a^25 - 2/529*a^24 + 7/529*a^23 - 6/529*a^22 - 4/529*a^21 + 217/529*a^20 + 173/529*a^19 - 219/529*a^18 + 158/529*a^17 + 79/529*a^16 - 203/529*a^15 - 157/529*a^14 + 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86389135888030387050247417647212360908034390814594499381997874307887556861911603495066059080686032799395411835212154192944950996768092934410973687503776978724240717982280365685457924880529357342460739517149292323069929643185/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a^17 - 31149650216923162283237518979165374654949480495013834875190133795661080674785942475064015026900554183737200017229720837970569950352894702418616280421860651633792720064274131159794037173832290019211450350254428012960485338107/134226392885163617582702935575841475713708399661546975837480127234421198724855253452498061683690060933603098334989506530671713004216476424563027695233749310347400912819495693396595463104551215529373536488285627096591056496173*a^16 - 1068224290426485516458176647954009425565941815822550254686558674924110796245997347303778608757417935399049955189676552582059141502706665809376692472086774558277668216505994614248061648386156144685770907381016635418894768515636/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a^15 - 689143215367638842410408857647067085167824420876950152585970751270650304924738829882677631629117992946700736093526501031249501184447835959557570070515777900782755086443451600596719985016930695202241574219874959501856902115607/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a^14 - 1493773499156497933113357830704914493054655506046928954313272952267624798201512259718744309150882867155249490473690186421120311795686257506290173842452951479325058041986874144068571897665816741823424061763211164802455898730339/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a^13 - 732346592232726262935209021196515298350669089182329307911759228713585232224782012084702675221639359689271436777978309760451318097876364219183045731340320793785040091330056338611348381615234953805508394290602640745306175724815/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a^12 - 706782301857847160117442969590833466563756217480522732670100353995412655315564918053248537551014509389709934312439530250291539210678780762311888596760695645516514375271388034066448418158187820771305579459789006948815092671526/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a^11 + 1037935806332880097163641118647751832850788122447475160709510690054747970171929717035770634081388709599925806204562114654252465093192979981091265172530694635263528015004551920628199812873643153085229120510703690007364093635743/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a^10 - 1308281724928372899221294484016800847419737060915953232116899369187591892955108718663344084164712365959802930088766941495650028726662137362329686116175573873255021956182656485362103288395018911483259964177915138131941719117429/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a^9 - 712487668483214130092575968737813745611863526442435565108502866040961262197256903862008949782777973289079343247905937906020997908028922380437065135663662058770684342943841157677519427306692002221512520763090260383893551070774/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a^8 - 687698850999770364960357592656759029933329137145131605582967409871436967155676240862783142846303502265728001305648418812562143611784938804769880487409072111183938060013004605352820360726086998714432751215469921924243390676909/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a^7 + 903691409276243298370979484383757622857416618365586405037906900610780996136748421696465595493991775745387214408344516343094123370271311461549366098582365356432543605191412956634303903220860433605525713642713681446797535071403/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a^6 + 1281218295205307363811058852620989609058139900345240623209590446206655616381848164985082357939590346526177775540228252063361736015921680121923512772254499600530883811277916041125610353830844831637035202875766963523908943226347/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a^5 - 202361583714592355549400339739338584118974156417193799894893699828026423861922255056877736202137444055564814342018015291158204206354664538945015526347001161340842301079246744946646554008249543563222637013558799726762004232295/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a^4 - 927873527573694890401403037635517314748230839085242641727074137837100049782009187411099618792687993932078699570785976983513217300779308272307149072136473165879082073321225242604896229878691562215703456370806781101358990336810/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a^3 - 1165274318529781890009028712482600006363516102436806583543747730374494989590181290013107261462981330331765725923858296426283434467035483228320038859279229150715226786007219952220762330628504937968017731949984394902944174825044/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a^2 + 1137381881206516907948713074279945281008869909785598336975552031611828150190230968571115604554732025943449861529038602136554927734367773214721641857652223197397492502736420869029706498845120239505864904068940881181064042086010/3087207036358763204402167518244353941415293192215580444262042926391687570671670829407455418724871401472871261704758650205449399096978957764949636990376234137990220994848400948121695651404677957175591339230569423221594299411979*a - 7803248678639818429288977520308222325037899128854478055398309317513991211991460714617399185742191700009953929409951494305250138593379745505787181927047951688328932877234133295736896680621353757932581794090240842048/26043049910378392547471230461925315320278296044376036594666350529683682920756113189692980475171934248650351611196096024460578689992816006257845609574076771600938034054840906047614568490961751705615957823971568666151

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$23$

Class group and class number

not computed

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$38$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	not computed	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	not computed	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula

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)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

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);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(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);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(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);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

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

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

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.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.13.0.1}{13} }^{3}$	R	${\href{/padicField/5.13.0.1}{13} }^{3}$	$39$	$39$	$39$	${\href{/padicField/17.13.0.1}{13} }^{3}$	$39$	${\href{/padicField/23.1.0.1}{1} }^{39}$	$39$	${\href{/padicField/31.13.0.1}{13} }^{3}$	$39$	$39$	${\href{/padicField/43.13.0.1}{13} }^{3}$	$39$	$39$	$39$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$3$ 3		Deg $39$	$3$	$13$	$52$
$79$ 79		Deg $39$	$39$	$1$	$38$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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