Global number field 39.39.110539730308983252644692019573737637228828219789892386533857010792412648335681998371651796263456819412889.1

• Label: 39.39.1105397303...2889.1
• Degree: $39$
• Signature: $[39, 0]$
• Discriminant: $547^{38}$
• Root discriminant: $465.35$
• Ramified prime: $547$
• Class number: $4$ (GRH)
• Class group: $[2, 2]$ (GRH)
• Galois group: $C_{39}$ (as 39T1)

Normalized defining polynomial

\( x^{39} - x^{38} - 266 x^{37} + 201 x^{36} + 30249 x^{35} - 20577 x^{34} - 1955719 x^{33} + 1451515 x^{32} + 80733464 x^{31} - 73319589 x^{30} - 2263189338 x^{29} + 2590988751 x^{28} + 44569388886 x^{27} - 63687593067 x^{26} - 626428894665 x^{25} + 1097818462656 x^{24} + 6291493806807 x^{23} - 13394812097448 x^{22} - 44496937680997 x^{21} + 116206956876955 x^{20} + 212799596796731 x^{19} - 715024587078660 x^{18} - 615613322642382 x^{17} + 3085798869034590 x^{16} + 605869572443830 x^{15} - 9130534820399395 x^{14} + 2644115229153457 x^{13} + 17760841353343332 x^{12} - 12225754395203670 x^{11} - 20865350159842410 x^{10} + 23224133468755773 x^{9} + 11651476867778808 x^{8} - 23071484201964822 x^{7} + 1000224610088400 x^{6} + 11129065111375902 x^{5} - 4226325744669057 x^{4} - 1602614683727208 x^{3} + 1322681130702459 x^{2} - 264907416527640 x + 15539564953953 \)

Invariants

Degree:		$39$
Signature:		$[39, 0]$
Discriminant:		\(110539730308983252644692019573737637228828219789892386533857010792412648335681998371651796263456819412889=547^{38}\)
Root discriminant:		$465.35$
Ramified primes:		$547$
This field is Galois and abelian over $\Q$.
Conductor:		\(547\)
Dirichlet character group:		$\lbrace$$\chi_{547}(1,·)$, $\chi_{547}(261,·)$, $\chi_{547}(519,·)$, $\chi_{547}(136,·)$, $\chi_{547}(521,·)$, $\chi_{547}(11,·)$, $\chi_{547}(402,·)$, $\chi_{547}(46,·)$, $\chi_{547}(517,·)$, $\chi_{547}(419,·)$, $\chi_{547}(293,·)$, $\chi_{547}(129,·)$, $\chi_{547}(302,·)$, $\chi_{547}(47,·)$, $\chi_{547}(181,·)$, $\chi_{547}(54,·)$, $\chi_{547}(440,·)$, $\chi_{547}(441,·)$, $\chi_{547}(445,·)$, $\chi_{547}(296,·)$, $\chi_{547}(325,·)$, $\chi_{547}(199,·)$, $\chi_{547}(464,·)$, $\chi_{547}(217,·)$, $\chi_{547}(475,·)$, $\chi_{547}(350,·)$, $\chi_{547}(96,·)$, $\chi_{547}(353,·)$, $\chi_{547}(231,·)$, $\chi_{547}(488,·)$, $\chi_{547}(233,·)$, $\chi_{547}(237,·)$, $\chi_{547}(239,·)$, $\chi_{547}(40,·)$, $\chi_{547}(375,·)$, $\chi_{547}(121,·)$, $\chi_{547}(506,·)$, $\chi_{547}(509,·)$, $\chi_{547}(21,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{2}$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{10} - \frac{1}{27} a^{8} + \frac{1}{27} a^{6} - \frac{10}{81} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{81} a^{11} + \frac{1}{27} a^{7} + \frac{8}{81} a^{5} - \frac{4}{27} a^{3}$, $\frac{1}{243} a^{12} - \frac{1}{243} a^{10} - \frac{1}{81} a^{8} - \frac{4}{243} a^{6} - \frac{11}{243} a^{4} + \frac{2}{27} a^{2}$, $\frac{1}{243} a^{13} - \frac{1}{243} a^{11} - \frac{13}{243} a^{7} - \frac{2}{243} a^{5} - \frac{4}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{243} a^{14} - \frac{1}{243} a^{10} + \frac{11}{243} a^{8} - \frac{2}{81} a^{6} - \frac{23}{243} a^{4} + \frac{2}{27} a^{2}$, $\frac{1}{729} a^{15} + \frac{1}{729} a^{13} + \frac{4}{729} a^{11} - \frac{1}{729} a^{9} - \frac{19}{729} a^{7} + \frac{95}{729} a^{5} + \frac{1}{9} a^{3} - \frac{2}{9} a$, $\frac{1}{729} a^{16} + \frac{1}{729} a^{14} + \frac{1}{729} a^{12} + \frac{2}{729} a^{10} - \frac{10}{729} a^{8} + \frac{26}{729} a^{6} + \frac{11}{243} a^{4} - \frac{2}{27} a^{2}$, $\frac{1}{2187} a^{17} + \frac{1}{2187} a^{16} + \frac{1}{2187} a^{15} + \frac{4}{2187} a^{14} - \frac{2}{2187} a^{13} + \frac{1}{2187} a^{12} - \frac{13}{2187} a^{11} - \frac{1}{2187} a^{10} - \frac{1}{2187} a^{9} + \frac{23}{2187} a^{8} + \frac{65}{2187} a^{7} - \frac{73}{2187} a^{6} - \frac{80}{729} a^{5} + \frac{14}{243} a^{4} - \frac{2}{81} a^{3} + \frac{8}{27} a^{2} + \frac{4}{9} a$, $\frac{1}{6561} a^{18} + \frac{1}{2187} a^{16} - \frac{1}{2187} a^{14} - \frac{2}{6561} a^{12} + \frac{8}{2187} a^{10} - \frac{113}{2187} a^{8} + \frac{199}{6561} a^{6} - \frac{104}{729} a^{4} + \frac{13}{81} a^{2}$, $\frac{1}{6561} a^{19} - \frac{1}{2187} a^{16} + \frac{1}{2187} a^{15} - \frac{4}{2187} a^{14} + \frac{13}{6561} a^{13} - \frac{1}{2187} a^{12} + \frac{2}{729} a^{11} + \frac{1}{2187} a^{10} - \frac{7}{2187} a^{9} - \frac{23}{2187} a^{8} + \frac{76}{6561} a^{7} + \frac{73}{2187} a^{6} + \frac{26}{729} a^{5} - \frac{14}{243} a^{4} - \frac{13}{81} a^{3} - \frac{8}{27} a^{2} - \frac{2}{9} a$, $\frac{1}{6561} a^{20} - \frac{1}{2187} a^{16} - \frac{11}{6561} a^{14} + \frac{4}{2187} a^{12} - \frac{5}{2187} a^{10} - \frac{62}{6561} a^{8} - \frac{19}{2187} a^{6} - \frac{13}{243} a^{4} + \frac{2}{27} a^{2}$, $\frac{1}{19683} a^{21} - \frac{1}{19683} a^{19} + \frac{1}{6561} a^{17} - \frac{1}{2187} a^{16} + \frac{1}{19683} a^{15} - \frac{4}{2187} a^{14} - \frac{31}{19683} a^{13} - \frac{1}{2187} a^{12} - \frac{16}{6561} a^{11} + \frac{1}{2187} a^{10} + \frac{106}{19683} a^{9} - \frac{23}{2187} a^{8} - \frac{49}{19683} a^{7} + \frac{73}{2187} a^{6} - \frac{70}{2187} a^{5} - \frac{14}{243} a^{4} - \frac{19}{243} a^{3} - \frac{8}{27} a^{2} - \frac{2}{9} a$, $\frac{1}{59049} a^{22} + \frac{1}{59049} a^{21} - \frac{4}{59049} a^{20} + \frac{2}{59049} a^{19} + \frac{4}{19683} a^{17} - \frac{26}{59049} a^{16} + \frac{19}{59049} a^{15} - \frac{97}{59049} a^{14} + \frac{71}{59049} a^{13} - \frac{8}{19683} a^{12} + \frac{98}{19683} a^{11} + \frac{268}{59049} a^{10} + \frac{34}{59049} a^{9} - \frac{439}{59049} a^{8} + \frac{1169}{59049} a^{7} + \frac{440}{19683} a^{6} + \frac{182}{6561} a^{5} - \frac{163}{2187} a^{4} + \frac{95}{729} a^{3} - \frac{94}{243} a^{2} + \frac{10}{27} a$, $\frac{1}{177147} a^{23} + \frac{1}{177147} a^{22} - \frac{1}{177147} a^{21} + \frac{11}{177147} a^{20} - \frac{1}{59049} a^{19} + \frac{1}{59049} a^{18} + \frac{10}{177147} a^{17} - \frac{35}{177147} a^{16} + \frac{95}{177147} a^{15} - \frac{1}{177147} a^{14} - \frac{28}{19683} a^{13} - \frac{22}{59049} a^{12} - \frac{65}{177147} a^{11} + \frac{169}{177147} a^{10} + \frac{419}{177147} a^{9} + \frac{5120}{177147} a^{8} + \frac{1732}{59049} a^{7} + \frac{142}{19683} a^{6} - \frac{506}{6561} a^{5} + \frac{226}{2187} a^{4} - \frac{20}{729} a^{3} + \frac{92}{243} a^{2} - \frac{11}{27} a$, $\frac{1}{531441} a^{24} + \frac{4}{531441} a^{22} - \frac{11}{531441} a^{20} - \frac{2}{59049} a^{19} + \frac{7}{531441} a^{18} + \frac{2}{19683} a^{17} - \frac{188}{531441} a^{16} + \frac{2}{6561} a^{15} + \frac{4}{531441} a^{14} + \frac{61}{59049} a^{13} + \frac{100}{531441} a^{12} - \frac{14}{19683} a^{11} + \frac{2263}{531441} a^{10} - \frac{34}{6561} a^{9} + \frac{3382}{531441} a^{8} - \frac{2858}{59049} a^{7} + \frac{550}{19683} a^{6} - \frac{400}{6561} a^{5} + \frac{2}{81} a^{4} + \frac{47}{729} a^{3} + \frac{143}{729} a^{2} - \frac{2}{81} a - \frac{1}{3}$, $\frac{1}{531441} a^{25} + \frac{1}{531441} a^{23} - \frac{1}{177147} a^{22} - \frac{8}{531441} a^{21} + \frac{10}{177147} a^{20} + \frac{16}{531441} a^{19} - \frac{4}{59049} a^{18} + \frac{25}{531441} a^{17} + \frac{8}{177147} a^{16} - \frac{38}{531441} a^{15} + \frac{292}{177147} a^{14} + \frac{370}{531441} a^{13} - \frac{110}{59049} a^{12} - \frac{701}{531441} a^{11} + \frac{695}{177147} a^{10} + \frac{1882}{531441} a^{9} - \frac{8726}{177147} a^{8} + \frac{1673}{59049} a^{7} + \frac{473}{19683} a^{6} - \frac{52}{6561} a^{5} + \frac{308}{2187} a^{4} - \frac{98}{729} a^{3} - \frac{92}{243} a^{2} - \frac{4}{27} a$, $\frac{1}{1594323} a^{26} + \frac{1}{1594323} a^{25} + \frac{1}{1594323} a^{24} - \frac{2}{1594323} a^{23} + \frac{7}{1594323} a^{22} - \frac{14}{1594323} a^{21} - \frac{26}{1594323} a^{20} - \frac{92}{1594323} a^{19} + \frac{70}{1594323} a^{18} - \frac{140}{1594323} a^{17} + \frac{490}{1594323} a^{16} + \frac{397}{1594323} a^{15} - \frac{2201}{1594323} a^{14} + \frac{712}{1594323} a^{13} + \frac{2818}{1594323} a^{12} + \frac{2950}{1594323} a^{11} - \frac{3116}{1594323} a^{10} + \frac{7042}{1594323} a^{9} - \frac{17195}{531441} a^{8} - \frac{440}{19683} a^{7} - \frac{437}{19683} a^{6} - \frac{130}{2187} a^{5} + \frac{170}{2187} a^{4} + \frac{164}{2187} a^{3} - \frac{358}{729} a^{2} + \frac{16}{81} a - \frac{1}{3}$, $\frac{1}{1594323} a^{27} + \frac{1}{177147} a^{22} + \frac{8}{531441} a^{21} - \frac{7}{177147} a^{20} - \frac{2}{59049} a^{19} + \frac{1}{59049} a^{18} + \frac{11}{59049} a^{17} - \frac{116}{177147} a^{16} + \frac{154}{531441} a^{15} - \frac{343}{177147} a^{14} - \frac{31}{19683} a^{13} + \frac{104}{59049} a^{12} - \frac{86}{59049} a^{11} + \frac{574}{177147} a^{10} - \frac{9235}{1594323} a^{9} - \frac{7804}{177147} a^{8} - \frac{424}{59049} a^{7} - \frac{188}{19683} a^{6} + \frac{1}{729} a^{5} + \frac{229}{2187} a^{4} - \frac{50}{2187} a^{3} - \frac{22}{243} a^{2}$, $\frac{1}{1594323} a^{28} - \frac{4}{531441} a^{22} - \frac{5}{177147} a^{20} - \frac{1}{19683} a^{19} + \frac{1}{59049} a^{18} + \frac{1}{6561} a^{17} - \frac{236}{531441} a^{16} - \frac{1}{2187} a^{15} + \frac{175}{177147} a^{14} - \frac{1}{19683} a^{13} - \frac{76}{59049} a^{12} + \frac{14}{6561} a^{11} - \frac{9244}{1594323} a^{10} + \frac{4}{729} a^{9} + \frac{1810}{177147} a^{8} - \frac{547}{19683} a^{7} + \frac{5}{243} a^{6} + \frac{14}{243} a^{5} + \frac{199}{2187} a^{4} - \frac{2}{27} a^{3} + \frac{26}{243} a^{2} + \frac{7}{27} a$, $\frac{1}{4782969} a^{29} + \frac{1}{4782969} a^{28} + \frac{2}{1594323} a^{23} - \frac{7}{1594323} a^{22} - \frac{1}{531441} a^{21} - \frac{34}{531441} a^{20} + \frac{1}{19683} a^{19} - \frac{2}{59049} a^{18} + \frac{364}{1594323} a^{17} + \frac{31}{1594323} a^{16} - \frac{169}{531441} a^{15} - \frac{328}{531441} a^{14} - \frac{14}{59049} a^{13} - \frac{32}{59049} a^{12} + \frac{10322}{4782969} a^{11} + \frac{9161}{4782969} a^{10} + \frac{1718}{531441} a^{9} + \frac{9767}{531441} a^{8} + \frac{3091}{59049} a^{7} + \frac{25}{2187} a^{6} - \frac{818}{6561} a^{5} + \frac{196}{6561} a^{4} + \frac{94}{729} a^{3} - \frac{143}{729} a^{2} + \frac{11}{81} a + \frac{1}{3}$, $\frac{1}{43046721} a^{30} + \frac{5}{43046721} a^{28} - \frac{1}{4782969} a^{26} + \frac{1}{1594323} a^{25} - \frac{7}{14348907} a^{24} + \frac{2}{1594323} a^{23} - \frac{38}{14348907} a^{22} + \frac{2}{531441} a^{21} + \frac{322}{4782969} a^{20} + \frac{76}{1594323} a^{19} + \frac{58}{14348907} a^{18} - \frac{289}{1594323} a^{17} - \frac{7372}{14348907} a^{16} + \frac{68}{177147} a^{15} - \frac{9461}{4782969} a^{14} - \frac{2780}{1594323} a^{13} + \frac{77201}{43046721} a^{12} + \frac{8387}{1594323} a^{11} - \frac{229685}{43046721} a^{10} + \frac{1702}{531441} a^{9} - \frac{28705}{4782969} a^{8} - \frac{19664}{531441} a^{7} - \frac{7849}{177147} a^{6} + \frac{416}{6561} a^{5} - \frac{3409}{59049} a^{4} + \frac{146}{2187} a^{3} + \frac{2881}{6561} a^{2} - \frac{52}{729} a + \frac{10}{27}$, $\frac{1}{43046721} a^{31} - \frac{4}{43046721} a^{29} - \frac{1}{4782969} a^{28} - \frac{1}{4782969} a^{27} + \frac{11}{14348907} a^{25} + \frac{1}{1594323} a^{24} - \frac{11}{14348907} a^{23} - \frac{1}{531441} a^{22} + \frac{58}{4782969} a^{21} + \frac{17}{531441} a^{20} - \frac{869}{14348907} a^{19} + \frac{73}{1594323} a^{18} + \frac{2222}{14348907} a^{17} + \frac{892}{1594323} a^{16} - \frac{968}{4782969} a^{15} - \frac{176}{177147} a^{14} + \frac{60974}{43046721} a^{13} - \frac{1883}{1594323} a^{12} - \frac{216257}{43046721} a^{11} + \frac{25522}{4782969} a^{10} - \frac{10690}{4782969} a^{9} - \frac{20756}{531441} a^{8} - \frac{3841}{177147} a^{7} - \frac{491}{19683} a^{6} + \frac{9533}{59049} a^{5} - \frac{82}{6561} a^{4} + \frac{850}{6561} a^{3} - \frac{160}{729} a^{2} + \frac{1}{9} a$, $\frac{1}{43046721} a^{32} - \frac{7}{43046721} a^{28} - \frac{1}{14348907} a^{26} - \frac{1}{1594323} a^{25} - \frac{4}{4782969} a^{24} + \frac{1}{1594323} a^{23} + \frac{94}{14348907} a^{22} + \frac{4}{531441} a^{21} - \frac{650}{14348907} a^{20} - \frac{70}{1594323} a^{19} - \frac{172}{4782969} a^{18} + \frac{31}{1594323} a^{17} - \frac{6112}{14348907} a^{16} + \frac{53}{177147} a^{15} + \frac{6875}{43046721} a^{14} - \frac{1372}{1594323} a^{13} - \frac{1939}{4782969} a^{12} + \frac{3208}{1594323} a^{11} + \frac{89179}{43046721} a^{10} - \frac{1990}{531441} a^{9} - \frac{200707}{4782969} a^{8} - \frac{19022}{531441} a^{7} - \frac{6163}{177147} a^{6} + \frac{226}{6561} a^{5} - \frac{9109}{59049} a^{4} + \frac{191}{2187} a^{3} + \frac{670}{6561} a^{2} - \frac{46}{729} a + \frac{13}{27}$, $\frac{1}{129140163} a^{33} + \frac{1}{129140163} a^{31} - \frac{2}{129140163} a^{29} - \frac{1}{4782969} a^{28} + \frac{5}{43046721} a^{27} + \frac{1}{4782969} a^{26} + \frac{17}{43046721} a^{25} + \frac{1}{4782969} a^{24} - \frac{97}{43046721} a^{23} + \frac{34}{4782969} a^{22} - \frac{134}{43046721} a^{21} + \frac{265}{4782969} a^{20} - \frac{2798}{43046721} a^{19} - \frac{128}{4782969} a^{18} + \frac{3751}{43046721} a^{17} - \frac{1109}{4782969} a^{16} - \frac{7372}{129140163} a^{15} - \frac{6140}{4782969} a^{14} + \frac{188405}{129140163} a^{13} + \frac{19}{4782969} a^{12} + \frac{758864}{129140163} a^{11} - \frac{24547}{4782969} a^{10} + \frac{46600}{14348907} a^{9} - \frac{72004}{1594323} a^{8} - \frac{22697}{531441} a^{7} - \frac{1039}{19683} a^{6} - \frac{11276}{177147} a^{5} - \frac{289}{6561} a^{4} + \frac{2048}{19683} a^{3} - \frac{170}{2187} a^{2} + \frac{8}{81} a$, $\frac{1}{387420489} a^{34} + \frac{4}{387420489} a^{32} - \frac{1}{129140163} a^{31} + \frac{4}{387420489} a^{30} + \frac{13}{129140163} a^{29} + \frac{8}{129140163} a^{28} - \frac{4}{14348907} a^{27} - \frac{4}{129140163} a^{26} - \frac{26}{43046721} a^{25} + \frac{41}{129140163} a^{24} + \frac{5}{43046721} a^{23} - \frac{539}{129140163} a^{22} + \frac{2}{177147} a^{21} - \frac{2192}{129140163} a^{20} + \frac{2492}{43046721} a^{19} - \frac{7925}{129140163} a^{18} + \frac{832}{43046721} a^{17} - \frac{70822}{387420489} a^{16} + \frac{1762}{4782969} a^{15} - \frac{173317}{387420489} a^{14} - \frac{153278}{129140163} a^{13} + \frac{686552}{387420489} a^{12} - \frac{624928}{129140163} a^{11} + \frac{187}{1594323} a^{10} + \frac{35128}{14348907} a^{9} + \frac{10480}{4782969} a^{8} - \frac{1675}{59049} a^{7} - \frac{9701}{531441} a^{6} + \frac{10096}{177147} a^{5} + \frac{1583}{19683} a^{4} + \frac{659}{19683} a^{3} + \frac{911}{6561} a^{2} + \frac{160}{729} a + \frac{8}{27}$, $\frac{1}{1162261467} a^{35} + \frac{1}{1162261467} a^{34} - \frac{2}{1162261467} a^{33} + \frac{1}{1162261467} a^{32} - \frac{5}{1162261467} a^{31} - \frac{2}{1162261467} a^{30} - \frac{29}{387420489} a^{29} - \frac{22}{387420489} a^{28} + \frac{11}{387420489} a^{27} + \frac{80}{387420489} a^{26} - \frac{220}{387420489} a^{25} - \frac{88}{387420489} a^{24} - \frac{995}{387420489} a^{23} - \frac{422}{387420489} a^{22} + \frac{3553}{387420489} a^{21} - \frac{14507}{387420489} a^{20} + \frac{19174}{387420489} a^{19} + \frac{5515}{387420489} a^{18} - \frac{215173}{1162261467} a^{17} - \frac{554824}{1162261467} a^{16} + \frac{543863}{1162261467} a^{15} - \frac{1030537}{1162261467} a^{14} - \frac{1242940}{1162261467} a^{13} - \frac{1941568}{1162261467} a^{12} + \frac{84331}{387420489} a^{11} - \frac{371744}{129140163} a^{10} - \frac{177400}{43046721} a^{9} + \frac{226387}{4782969} a^{8} - \frac{21761}{531441} a^{7} - \frac{50786}{1594323} a^{6} + \frac{21407}{531441} a^{5} - \frac{15946}{177147} a^{4} - \frac{2252}{59049} a^{3} - \frac{301}{729} a^{2} - \frac{139}{729} a + \frac{1}{27}$, $\frac{1}{3486784401} a^{36} + \frac{1}{3486784401} a^{35} + \frac{4}{3486784401} a^{34} - \frac{8}{3486784401} a^{33} - \frac{35}{3486784401} a^{32} - \frac{29}{3486784401} a^{31} - \frac{4}{387420489} a^{30} - \frac{100}{1162261467} a^{29} - \frac{337}{1162261467} a^{28} + \frac{305}{1162261467} a^{27} + \frac{134}{1162261467} a^{26} - \frac{952}{1162261467} a^{25} + \frac{844}{1162261467} a^{24} - \frac{674}{1162261467} a^{23} - \frac{8294}{1162261467} a^{22} + \frac{307}{1162261467} a^{21} - \frac{25946}{1162261467} a^{20} + \frac{952}{1162261467} a^{19} - \frac{219475}{3486784401} a^{18} + \frac{440774}{3486784401} a^{17} - \frac{1989499}{3486784401} a^{16} - \frac{2378692}{3486784401} a^{15} - \frac{6099346}{3486784401} a^{14} - \frac{4936030}{3486784401} a^{13} + \frac{351209}{1162261467} a^{12} + \frac{215615}{43046721} a^{11} + \frac{416014}{129140163} a^{10} + \frac{134089}{43046721} a^{9} - \frac{251102}{4782969} a^{8} + \frac{185737}{4782969} a^{7} - \frac{29249}{1594323} a^{6} - \frac{8047}{177147} a^{5} - \frac{19966}{177147} a^{4} - \frac{8605}{59049} a^{3} + \frac{3100}{6561} a^{2} + \frac{52}{243} a + \frac{1}{9}$, $\frac{1}{31381059609} a^{37} + \frac{2}{10460353203} a^{35} + \frac{4}{3486784401} a^{34} - \frac{29}{10460353203} a^{33} - \frac{8}{1162261467} a^{32} + \frac{275}{31381059609} a^{31} + \frac{26}{3486784401} a^{30} - \frac{4}{387420489} a^{29} + \frac{350}{1162261467} a^{28} - \frac{190}{3486784401} a^{27} - \frac{104}{387420489} a^{26} - \frac{9610}{10460353203} a^{25} + \frac{646}{1162261467} a^{24} - \frac{9664}{3486784401} a^{23} + \frac{5104}{1162261467} a^{22} + \frac{28075}{3486784401} a^{21} + \frac{599}{43046721} a^{20} - \frac{125419}{31381059609} a^{19} + \frac{46508}{1162261467} a^{18} + \frac{1274407}{10460353203} a^{17} + \frac{1728743}{3486784401} a^{16} + \frac{3519956}{10460353203} a^{15} + \frac{1792418}{1162261467} a^{14} - \frac{61034885}{31381059609} a^{13} - \frac{534743}{3486784401} a^{12} - \frac{12288110}{3486784401} a^{11} - \frac{782591}{387420489} a^{10} + \frac{999802}{387420489} a^{9} - \frac{1034714}{43046721} a^{8} + \frac{1830905}{43046721} a^{7} + \frac{103949}{4782969} a^{6} + \frac{267062}{4782969} a^{5} + \frac{17960}{531441} a^{4} - \frac{739}{531441} a^{3} + \frac{17219}{59049} a^{2} - \frac{2686}{6561} a - \frac{23}{243}$, $\frac{1}{29037239010391606948874972935916312312612760997839392394438952618503334456900158843833310591394880094372998675696994797166998913} a^{38} + \frac{276251759957528387222543973289109745877480296247704582118769153289622810550450095039466965825817893190852376299853373}{29037239010391606948874972935916312312612760997839392394438952618503334456900158843833310591394880094372998675696994797166998913} a^{37} + \frac{984751429186391992581839873528988140080181363585754775877547304168127517814576971946872862393103270451528272647727558}{9679079670130535649624990978638770770870920332613130798146317539501111485633386281277770197131626698124332891898998265722332971} a^{36} - \frac{3536091017594663164639467169606801359223874019733288332725282463850352889684440020815595190969113095163698809349704020}{9679079670130535649624990978638770770870920332613130798146317539501111485633386281277770197131626698124332891898998265722332971} a^{35} - \frac{11507918555560747602602784906678969438507042316043840546867488983081916378765928913921869275743412896190987612630461281}{9679079670130535649624990978638770770870920332613130798146317539501111485633386281277770197131626698124332891898998265722332971} a^{34} - \frac{8145086208174077655348825492687678683217553741633996193442795635938901899703445092179336699177131340558918993100608577}{9679079670130535649624990978638770770870920332613130798146317539501111485633386281277770197131626698124332891898998265722332971} a^{33} - \frac{337089896282393856930008689000882448381780844366410419449015324104987346460809856975934382338537727289752932715578754817}{29037239010391606948874972935916312312612760997839392394438952618503334456900158843833310591394880094372998675696994797166998913} a^{32} + \frac{84814379407269699816897534141467531573870264309063971611869998900571903625172029434544571134333873218015761688095182323}{29037239010391606948874972935916312312612760997839392394438952618503334456900158843833310591394880094372998675696994797166998913} a^{31} - \frac{29368415650056087431704271314544785408018455389437835938115242234270053316245492449687518145021690083285613735827509895}{3226359890043511883208330326212923590290306777537710266048772513167037161877795427092590065710542232708110963966332755240777657} a^{30} + \frac{2525923114355027789872076749617535855739782913534415937525416340200144059168441589351914397831382971712013196676252555}{358484432227056875912036702912547065587811864170856696227641390351893017986421714121398896190060248078678995996259195026753073} a^{29} + \frac{636763314670590739259361051650908971172089471502696135388258690270716377027566691698484719110863300745963436941090137488}{3226359890043511883208330326212923590290306777537710266048772513167037161877795427092590065710542232708110963966332755240777657} a^{28} - \frac{234518753770289639018737216002717471035665583611052989441253856189893457472610093836068844993457679666947936818946052850}{3226359890043511883208330326212923590290306777537710266048772513167037161877795427092590065710542232708110963966332755240777657} a^{27} - \frac{1159474044513014587456803194123478130900590073346212639947622335190929956010148168511555947923831714667743877549976353093}{9679079670130535649624990978638770770870920332613130798146317539501111485633386281277770197131626698124332891898998265722332971} a^{26} - \frac{2974038899766200882739764727798510041659525461610392389789621956579445912924982151351297316669081633523755262373733537218}{9679079670130535649624990978638770770870920332613130798146317539501111485633386281277770197131626698124332891898998265722332971} a^{25} + \frac{1893196896969963836461487698213465304933308401184461078297856486381287148297718909205047237211860384898554788788470392877}{3226359890043511883208330326212923590290306777537710266048772513167037161877795427092590065710542232708110963966332755240777657} a^{24} + \frac{6450830346425965153619820969810320106804216781626450304197801298252949589291468321351100465124809821070439574705902959529}{3226359890043511883208330326212923590290306777537710266048772513167037161877795427092590065710542232708110963966332755240777657} a^{23} - \frac{521485036663938528660230892835882568066281856810899319316650872037603687948673850733883502934013223737530071680168418184}{3226359890043511883208330326212923590290306777537710266048772513167037161877795427092590065710542232708110963966332755240777657} a^{22} + \frac{45748082588588758493581132669235886150070485943313522855426328277613696316805216835010047715387295347789456359502117150355}{3226359890043511883208330326212923590290306777537710266048772513167037161877795427092590065710542232708110963966332755240777657} a^{21} + \frac{229500566142624365529926129692266247347141636416065170893516832446156270669565827564802298226284288016540065932337818090365}{29037239010391606948874972935916312312612760997839392394438952618503334456900158843833310591394880094372998675696994797166998913} a^{20} + \frac{293793260621843547526895772822228754891923044414158912701774241917421532107055159283151821973692260841273847949623457200349}{29037239010391606948874972935916312312612760997839392394438952618503334456900158843833310591394880094372998675696994797166998913} a^{19} + \frac{561772156119082699892237909568654108109566878368386541108669980486977525696267361534839663073431874411844226836156385613286}{9679079670130535649624990978638770770870920332613130798146317539501111485633386281277770197131626698124332891898998265722332971} a^{18} - \frac{1682948270779120466881621289022080287647636281935093164283372642811397113952931767538985578478527952348923150697474242328378}{9679079670130535649624990978638770770870920332613130798146317539501111485633386281277770197131626698124332891898998265722332971} a^{17} + \frac{1820036457955304265360039967896686336224356900518465122872283067511603042871571358433181999812151726733277474092321303842177}{9679079670130535649624990978638770770870920332613130798146317539501111485633386281277770197131626698124332891898998265722332971} a^{16} + \frac{1503684194567829711019041163144729931310036604170363225064411730021614481837617627199787253066030387947142522614525605937704}{9679079670130535649624990978638770770870920332613130798146317539501111485633386281277770197131626698124332891898998265722332971} a^{15} + \frac{50305921089389216264715481376112558689778377615409758757607037675409072744831492900221853643039116885339588338937978185079154}{29037239010391606948874972935916312312612760997839392394438952618503334456900158843833310591394880094372998675696994797166998913} a^{14} + \frac{33016525698196136816347110276814398404129262459050869371104393243701586434686435274505395108754065153095302616009400745255173}{29037239010391606948874972935916312312612760997839392394438952618503334456900158843833310591394880094372998675696994797166998913} a^{13} - \frac{507271916707477520331387115285657795825571958422089387291946756753963066733139756878101258764625606518640161554103934069644}{3226359890043511883208330326212923590290306777537710266048772513167037161877795427092590065710542232708110963966332755240777657} a^{12} + \frac{17658129818551978737621299273834307405416483814003254433511510746055737515216667949475034987671809584522267965014428616336684}{3226359890043511883208330326212923590290306777537710266048772513167037161877795427092590065710542232708110963966332755240777657} a^{11} + \frac{1002486170606996179845826520271218353829336857500467787490062581254635575813741482523590978199202848171307325693240196114318}{358484432227056875912036702912547065587811864170856696227641390351893017986421714121398896190060248078678995996259195026753073} a^{10} - \frac{1020227035825526153237111297716887588575874535997551353398456937701573632259031126168551952728968714521722940032897787784571}{358484432227056875912036702912547065587811864170856696227641390351893017986421714121398896190060248078678995996259195026753073} a^{9} - \frac{137313326375242627449755002193855035216207651798100441699465542622608195948242258080347708313345338049300951683988695572533}{4425733731198233035951070406327741550466813137911811064538782596936950839338539680511097483827904297267641925879743148478433} a^{8} - \frac{1901676665625489571425025440205656114870865523318944829947225144044061318697883852038177220340615744341102858468585695667361}{39831603580784097323559633656949673954201318241206299580849043372432557554046857124599877354451138675408777332917688336305897} a^{7} - \frac{173795014669921799343509694194683207827436363944671334419400092820972664251163002625546878943955658608151805942990391277971}{4425733731198233035951070406327741550466813137911811064538782596936950839338539680511097483827904297267641925879743148478433} a^{6} - \frac{184329291943095485506181821272088372657313335444646057225509167671094319814983206157983720630354293095697504881478387351713}{4425733731198233035951070406327741550466813137911811064538782596936950839338539680511097483827904297267641925879743148478433} a^{5} + \frac{30685167870882402376332715796268147514963501654286549518123415221129446944567578819206627919317627901255917341932008656392}{491748192355359226216785600703082394496312570879090118282086955215216759926504408945677498203100477474182436208860349830937} a^{4} - \frac{47464634096131911162946613950361715975752995322023616021875248531236325195542345934945402662871306319812927104909659362100}{491748192355359226216785600703082394496312570879090118282086955215216759926504408945677498203100477474182436208860349830937} a^{3} - \frac{5417522417672702746651036019859766221352645840550712776535792534812531636610113021512288253489114244757565472090658982047}{54638688039484358468531733411453599388479174542121124253565217246135195547389378771741944244788941941575826245428927758993} a^{2} + \frac{2847932615368155151217264708607838138342925275680997728834154119359788401780400454883122412028869592574130284838942417398}{6070965337720484274281303712383733265386574949124569361507246360681688394154375419082438249420993549063980693936547528777} a - \frac{6174125270464501718522860843912100209228242230019981421387299811908387625535799370483862739951592171914450459386075009}{224850568063721639788196433791990120940243516634243309685453568914136607190902793299349564793370131446814099775427686251}$

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)

Unit group

Rank:		$38$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 6525765843490048000000000000000000000000000000 \) (assuming GRH)

Galois group

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

A cyclic group of order 39

The 39 conjugacy class representatives for $C_{39}$

Character table for $C_{39}$ is not computed

Intermediate fields

3.3.299209.1, 13.13.717542973516054083971838830896241.1

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	$39$	${\href{/LocalNumberField/3.1.0.1}{1} }^{39}$	$39$	$39$	$39$	${\href{/LocalNumberField/13.3.0.1}{3} }^{13}$	$39$	$39$	$39$	${\href{/LocalNumberField/29.13.0.1}{13} }^{3}$	${\href{/LocalNumberField/31.13.0.1}{13} }^{3}$	$39$	${\href{/LocalNumberField/41.3.0.1}{3} }^{13}$	$39$	$39$	$39$	$39$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
547	Data not computed