Global number field 40.0.438066646008777075750057097905195011045870087641058320988025851411039361.1

• Label: 40.0.43806664600...9361.1
• Degree: $40$
• Signature: $[0, 20]$
• Discriminant: $3^{20}\cdot 11^{36}\cdot 17^{20}$
• Root discriminant: $61.81$
• Ramified primes: $3, 11, 17$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_2^2\times C_{10}$ (as 40T7)

Normalized defining polynomial

\( x^{40} - x^{39} - 4 x^{38} + 13 x^{37} - 13 x^{36} - 52 x^{35} + 233 x^{34} - 233 x^{33} - 932 x^{32} + 3861 x^{31} - 3861 x^{30} + 15577 x^{29} + 34084 x^{28} - 189189 x^{27} + 142853 x^{26} + 690196 x^{25} - 2706561 x^{24} + 2854017 x^{23} + 11153924 x^{22} - 47293389 x^{21} + 46244813 x^{20} + 189173556 x^{19} + 178462784 x^{18} - 182657088 x^{17} - 692879616 x^{16} - 706760704 x^{15} + 585125888 x^{14} + 3099672576 x^{13} + 2233729024 x^{12} - 4083417088 x^{11} - 4048551936 x^{10} - 16194207744 x^{9} - 15636365312 x^{8} + 15636365312 x^{7} + 62545461248 x^{6} + 55834574848 x^{5} - 55834574848 x^{4} - 223338299392 x^{3} - 274877906944 x^{2} + 274877906944 x + 1099511627776 \)

Invariants

Degree:		$40$
Signature:		$[0, 20]$
Discriminant:		\(438066646008777075750057097905195011045870087641058320988025851411039361=3^{20}\cdot 11^{36}\cdot 17^{20}\)
Root discriminant:		$61.81$
Ramified primes:		$3, 11, 17$
This field is Galois and abelian over $\Q$.
Conductor:		\(561=3\cdot 11\cdot 17\)
Dirichlet character group:		$\lbrace$$\chi_{561}(256,·)$, $\chi_{561}(1,·)$, $\chi_{561}(392,·)$, $\chi_{561}(137,·)$, $\chi_{561}(526,·)$, $\chi_{561}(271,·)$, $\chi_{561}(16,·)$, $\chi_{561}(152,·)$, $\chi_{561}(409,·)$, $\chi_{561}(545,·)$, $\chi_{561}(290,·)$, $\chi_{561}(35,·)$, $\chi_{561}(424,·)$, $\chi_{561}(169,·)$, $\chi_{561}(560,·)$, $\chi_{561}(305,·)$, $\chi_{561}(50,·)$, $\chi_{561}(307,·)$, $\chi_{561}(52,·)$, $\chi_{561}(443,·)$, $\chi_{561}(188,·)$, $\chi_{561}(322,·)$, $\chi_{561}(67,·)$, $\chi_{561}(458,·)$, $\chi_{561}(203,·)$, $\chi_{561}(460,·)$, $\chi_{561}(205,·)$, $\chi_{561}(86,·)$, $\chi_{561}(475,·)$, $\chi_{561}(356,·)$, $\chi_{561}(101,·)$, $\chi_{561}(358,·)$, $\chi_{561}(103,·)$, $\chi_{561}(494,·)$, $\chi_{561}(239,·)$, $\chi_{561}(373,·)$, $\chi_{561}(118,·)$, $\chi_{561}(509,·)$, $\chi_{561}(254,·)$, $\chi_{561}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{4} a^{21} - \frac{1}{4} a^{20} + \frac{1}{4} a^{18} - \frac{1}{4} a^{17} + \frac{1}{4} a^{15} - \frac{1}{4} a^{14} + \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{121424} a^{22} - \frac{1}{16} a^{21} - \frac{7}{16} a^{19} + \frac{3}{16} a^{18} + \frac{5}{16} a^{16} + \frac{7}{16} a^{15} + \frac{1}{16} a^{13} - \frac{5}{16} a^{12} + \frac{49661}{121424} a^{11} - \frac{1}{16} a^{9} + \frac{5}{16} a^{8} + \frac{3}{16} a^{6} + \frac{1}{16} a^{5} + \frac{7}{16} a^{3} - \frac{3}{16} a^{2} + \frac{3471}{7589}$, $\frac{1}{485696} a^{23} - \frac{1}{485696} a^{22} - \frac{1}{16} a^{21} + \frac{25}{64} a^{20} - \frac{9}{64} a^{19} + \frac{3}{16} a^{18} + \frac{5}{64} a^{17} + \frac{11}{64} a^{16} + \frac{7}{16} a^{15} + \frac{1}{64} a^{14} + \frac{15}{64} a^{13} - \frac{223543}{485696} a^{12} + \frac{42605}{121424} a^{11} + \frac{15}{64} a^{10} - \frac{31}{64} a^{9} + \frac{5}{16} a^{8} + \frac{3}{64} a^{7} - \frac{19}{64} a^{6} + \frac{1}{16} a^{5} - \frac{25}{64} a^{4} + \frac{9}{64} a^{3} - \frac{3}{16} a^{2} + \frac{3471}{30356} a - \frac{2765}{7589}$, $\frac{1}{1942784} a^{24} - \frac{1}{1942784} a^{23} - \frac{1}{485696} a^{22} + \frac{9}{256} a^{21} - \frac{9}{256} a^{20} - \frac{9}{64} a^{19} - \frac{75}{256} a^{18} + \frac{75}{256} a^{17} + \frac{11}{64} a^{16} + \frac{113}{256} a^{15} - \frac{113}{256} a^{14} - \frac{587815}{1942784} a^{13} - \frac{230599}{485696} a^{12} + \frac{309819}{1942784} a^{11} - \frac{95}{256} a^{10} - \frac{31}{64} a^{9} - \frac{109}{256} a^{8} + \frac{109}{256} a^{7} - \frac{19}{64} a^{6} + \frac{55}{256} a^{5} - \frac{55}{256} a^{4} + \frac{9}{64} a^{3} + \frac{5177}{15178} a^{2} - \frac{5177}{15178} a - \frac{2765}{7589}$, $\frac{1}{7771136} a^{25} - \frac{1}{7771136} a^{24} - \frac{1}{1942784} a^{23} + \frac{13}{7771136} a^{22} - \frac{73}{1024} a^{21} - \frac{73}{256} a^{20} - \frac{267}{1024} a^{19} + \frac{267}{1024} a^{18} + \frac{11}{256} a^{17} + \frac{177}{1024} a^{16} - \frac{177}{1024} a^{15} + \frac{1354969}{7771136} a^{14} - \frac{594871}{1942784} a^{13} - \frac{175877}{7771136} a^{12} + \frac{129541}{7771136} a^{11} - \frac{31}{256} a^{10} - \frac{173}{1024} a^{9} + \frac{173}{1024} a^{8} - \frac{83}{256} a^{7} - \frac{265}{1024} a^{6} + \frac{265}{1024} a^{5} + \frac{9}{256} a^{4} + \frac{33121}{121424} a^{3} - \frac{33121}{121424} a^{2} - \frac{2765}{30356} a - \frac{125}{7589}$, $\frac{1}{31084544} a^{26} - \frac{1}{31084544} a^{25} - \frac{1}{7771136} a^{24} + \frac{13}{31084544} a^{23} - \frac{13}{31084544} a^{22} - \frac{73}{1024} a^{21} - \frac{1291}{4096} a^{20} + \frac{1291}{4096} a^{19} + \frac{267}{1024} a^{18} + \frac{177}{4096} a^{17} - \frac{177}{4096} a^{16} + \frac{1354969}{31084544} a^{15} + \frac{1347913}{7771136} a^{14} - \frac{7947013}{31084544} a^{13} + \frac{7900677}{31084544} a^{12} + \frac{159237}{7771136} a^{11} - \frac{1197}{4096} a^{10} + \frac{1197}{4096} a^{9} + \frac{173}{1024} a^{8} - \frac{1289}{4096} a^{7} + \frac{1289}{4096} a^{6} + \frac{265}{1024} a^{5} - \frac{209727}{485696} a^{4} + \frac{209727}{485696} a^{3} - \frac{33121}{121424} a^{2} + \frac{1866}{7589} a - \frac{1866}{7589}$, $\frac{1}{124338176} a^{27} - \frac{1}{124338176} a^{26} - \frac{1}{31084544} a^{25} + \frac{13}{124338176} a^{24} - \frac{13}{124338176} a^{23} - \frac{13}{31084544} a^{22} - \frac{1291}{16384} a^{21} + \frac{1291}{16384} a^{20} + \frac{1291}{4096} a^{19} + \frac{4273}{16384} a^{18} - \frac{4273}{16384} a^{17} + \frac{32439513}{124338176} a^{16} + \frac{1347913}{31084544} a^{15} + \frac{54222075}{124338176} a^{14} - \frac{54268411}{124338176} a^{13} + \frac{7930373}{31084544} a^{12} - \frac{2772097}{124338176} a^{11} - \frac{2899}{16384} a^{10} + \frac{1197}{4096} a^{9} + \frac{6903}{16384} a^{8} - \frac{6903}{16384} a^{7} + \frac{1289}{4096} a^{6} - \frac{695423}{1942784} a^{5} + \frac{695423}{1942784} a^{4} + \frac{209727}{485696} a^{3} + \frac{9455}{30356} a^{2} - \frac{9455}{30356} a - \frac{1866}{7589}$, $\frac{1}{497352704} a^{28} - \frac{1}{497352704} a^{27} - \frac{1}{124338176} a^{26} + \frac{13}{497352704} a^{25} - \frac{13}{497352704} a^{24} - \frac{13}{124338176} a^{23} + \frac{233}{497352704} a^{22} + \frac{1291}{65536} a^{21} - \frac{6901}{16384} a^{20} + \frac{20657}{65536} a^{19} + \frac{12111}{65536} a^{18} - \frac{91898663}{497352704} a^{17} + \frac{32432457}{124338176} a^{16} + \frac{178560251}{497352704} a^{15} - \frac{178606587}{497352704} a^{14} - \frac{54238715}{124338176} a^{13} + \frac{245904255}{497352704} a^{12} - \frac{245756799}{497352704} a^{11} + \frac{5293}{16384} a^{10} + \frac{6903}{65536} a^{9} + \frac{25865}{65536} a^{8} + \frac{1289}{16384} a^{7} + \frac{1247361}{7771136} a^{6} + \frac{2638207}{7771136} a^{5} - \frac{275969}{1942784} a^{4} - \frac{20901}{121424} a^{3} - \frac{39811}{121424} a^{2} + \frac{5723}{30356} a + \frac{266}{7589}$, $\frac{1}{1989410816} a^{29} - \frac{1}{1989410816} a^{28} - \frac{1}{497352704} a^{27} + \frac{13}{1989410816} a^{26} - \frac{13}{1989410816} a^{25} - \frac{13}{497352704} a^{24} + \frac{233}{1989410816} a^{23} - \frac{233}{1989410816} a^{22} + \frac{1291}{65536} a^{21} - \frac{110415}{262144} a^{20} - \frac{20657}{262144} a^{19} - \frac{837927719}{1989410816} a^{18} - \frac{91905719}{497352704} a^{17} - \frac{567468805}{1989410816} a^{16} - \frac{427282939}{1989410816} a^{15} - \frac{178576891}{497352704} a^{14} + \frac{494580607}{1989410816} a^{13} + \frac{500272257}{1989410816} a^{12} - \frac{245822335}{497352704} a^{11} + \frac{72439}{262144} a^{10} + \frac{58633}{262144} a^{9} + \frac{25865}{65536} a^{8} - \frac{6523775}{31084544} a^{7} - \frac{9018497}{31084544} a^{6} + \frac{2638207}{7771136} a^{5} - \frac{20901}{485696} a^{4} - \frac{221947}{485696} a^{3} - \frac{39811}{121424} a^{2} + \frac{133}{15178} a + \frac{3728}{7589}$, $\frac{1}{39788216320} a^{30} - \frac{1}{7957643264} a^{29} + \frac{29}{39788216320} a^{27} - \frac{13}{7957643264} a^{26} + \frac{441}{39788216320} a^{24} - \frac{233}{7957643264} a^{23} + \frac{1}{5242880} a^{21} + \frac{503631}{1048576} a^{20} - \frac{2230524663}{7957643264} a^{19} - \frac{441}{621690880} a^{18} + \frac{338371}{1048576} a^{17} - \frac{1024069115}{7957643264} a^{16} + \frac{29}{9713920} a^{15} - \frac{68953}{1048576} a^{14} + \frac{3086388353}{7957643264} a^{13} - \frac{1}{151780} a^{12} - \frac{212165}{1048576} a^{11} - \frac{203511}{1048576} a^{10} + \frac{1}{5} a^{9} - \frac{106646005}{497352704} a^{8} + \frac{22066047}{124338176} a^{7} - \frac{1}{5} a^{6} + \frac{1981257}{7771136} a^{5} - \frac{221947}{1942784} a^{4} + \frac{1}{5} a^{3} - \frac{49661}{121424} a^{2} - \frac{3861}{30356} a - \frac{1}{5}$, $\frac{1}{30764145250108702720} a^{31} - \frac{338636829}{30764145250108702720} a^{30} + \frac{105115595}{769103631252717568} a^{29} - \frac{2511439651}{30764145250108702720} a^{28} - \frac{19866226681}{30764145250108702720} a^{27} + \frac{1366502735}{769103631252717568} a^{26} - \frac{32648715399}{30764145250108702720} a^{25} - \frac{279933703909}{30764145250108702720} a^{24} + \frac{24491933635}{769103631252717568} a^{23} - \frac{585165437851}{30764145250108702720} a^{22} - \frac{390342007620749}{4053781163540480} a^{21} - \frac{1188397605538141631}{6152829050021740544} a^{20} + \frac{1600022777342426537}{3845518156263587840} a^{19} + \frac{3018257604518347691}{30764145250108702720} a^{18} - \frac{2525217525105542499}{6152829050021740544} a^{17} - \frac{1251673666199819987}{3845518156263587840} a^{16} + \frac{3470523375914727279}{30764145250108702720} a^{15} + \frac{1865648319578212921}{6152829050021740544} a^{14} + \frac{1528300651518706057}{3845518156263587840} a^{13} - \frac{7827871424803914909}{30764145250108702720} a^{12} - \frac{1998254351759659995}{6152829050021740544} a^{11} + \frac{224764399576657}{506722645442560} a^{10} - \frac{377161327745892581}{1922759078131793920} a^{9} - \frac{2721331013008943}{96137953906589696} a^{8} + \frac{4476764112520823}{60086221191618560} a^{7} + \frac{5568870885515769}{30043110595809280} a^{6} + \frac{702083290588779}{1502155529790464} a^{5} + \frac{334914801454413}{938847206119040} a^{4} - \frac{67732816813981}{469423603059520} a^{3} + \frac{6205150710677}{23471180152976} a^{2} + \frac{11907239886151}{29338975191220} a - \frac{92058807206}{7334743797805}$, $\frac{1}{123056581000434810880} a^{32} - \frac{1}{123056581000434810880} a^{31} + \frac{36334371}{30764145250108702720} a^{30} - \frac{4592674371}{123056581000434810880} a^{29} + \frac{3865986931}{123056581000434810880} a^{28} + \frac{4919683719}{30764145250108702720} a^{27} - \frac{59704766759}{123056581000434810880} a^{26} + \frac{50257830039}{123056581000434810880} a^{25} + \frac{66281288091}{30764145250108702720} a^{24} - \frac{1070093127611}{123056581000434810880} a^{23} + \frac{900774954091}{123056581000434810880} a^{22} + \frac{11769983793122055081}{123056581000434810880} a^{21} + \frac{1293702111056858149}{30764145250108702720} a^{20} + \frac{55455875405155459019}{123056581000434810880} a^{19} + \frac{27533588630638726789}{123056581000434810880} a^{18} + \frac{4948212858070017681}{30764145250108702720} a^{17} - \frac{18434228072490790129}{123056581000434810880} a^{16} + \frac{3706385954201675521}{123056581000434810880} a^{15} - \frac{6697023987994362211}{30764145250108702720} a^{14} - \frac{12996626024170788221}{123056581000434810880} a^{13} - \frac{35516025241021866931}{123056581000434810880} a^{12} + \frac{14275294321793503801}{30764145250108702720} a^{11} - \frac{755588632331144219}{7691036312527175680} a^{10} - \frac{771947769820930951}{1922759078131793920} a^{9} - \frac{6404876312925323}{60086221191618560} a^{8} + \frac{11213070340116231}{120172442383237120} a^{7} + \frac{12186477104172579}{30043110595809280} a^{6} + \frac{117463181906807}{938847206119040} a^{5} - \frac{139945406147619}{1877694412238080} a^{4} - \frac{197833680476591}{469423603059520} a^{3} - \frac{1434522793294}{7334743797805} a^{2} + \frac{82937484481}{29338975191220} a + \frac{3533862449789}{7334743797805}$, $\frac{1}{492226324001739243520} a^{33} - \frac{1}{492226324001739243520} a^{32} - \frac{1}{123056581000434810880} a^{31} - \frac{618267543}{98445264800347848704} a^{30} - \frac{8698470669}{492226324001739243520} a^{29} + \frac{6038789811}{123056581000434810880} a^{28} + \frac{1394368661}{98445264800347848704} a^{27} - \frac{113080118761}{492226324001739243520} a^{26} + \frac{78504267479}{123056581000434810880} a^{25} - \frac{21442330159}{98445264800347848704} a^{24} - \frac{2026743666709}{492226324001739243520} a^{23} + \frac{1857997450129369}{492226324001739243520} a^{22} - \frac{262777685239998987}{24611316200086962176} a^{21} - \frac{228056956198120626181}{492226324001739243520} a^{20} + \frac{211583817428587256581}{492226324001739243520} a^{19} + \frac{10437752528939610177}{24611316200086962176} a^{18} - \frac{125907807230454341249}{492226324001739243520} a^{17} + \frac{131794855208597664129}{492226324001739243520} a^{16} - \frac{4183569979579813747}{24611316200086962176} a^{15} + \frac{11022001381768524339}{492226324001739243520} a^{14} - \frac{4318610912562494259}{492226324001739243520} a^{13} - \frac{1653152884576581783}{24611316200086962176} a^{12} + \frac{2636216680599554809}{7691036312527175680} a^{11} - \frac{3454567028844312269}{7691036312527175680} a^{10} - \frac{367885813281929}{751077764895232} a^{9} + \frac{231626583049253621}{480689769532948480} a^{8} - \frac{6389396068181119}{120172442383237120} a^{7} - \frac{3020766246573}{23471180152976} a^{6} + \frac{1193946546373751}{7510777648952320} a^{5} - \frac{822317947137349}{1877694412238080} a^{4} - \frac{492642721099}{23471180152976} a^{3} - \frac{2764743216071}{29338975191220} a^{2} - \frac{704217464156}{7334743797805} a - \frac{2316535645156}{7334743797805}$, $\frac{1}{1968905296006956974080} a^{34} - \frac{1}{1968905296006956974080} a^{33} - \frac{1}{492226324001739243520} a^{32} + \frac{13}{1968905296006956974080} a^{31} + \frac{22659630067}{1968905296006956974080} a^{30} - \frac{107725511949}{492226324001739243520} a^{29} + \frac{317603897577}{1968905296006956974080} a^{28} + \frac{1927544861463}{1968905296006956974080} a^{27} - \frac{1400431655401}{492226324001739243520} a^{26} + \frac{4128850669333}{1968905296006956974080} a^{25} + \frac{26508299523307}{1968905296006956974080} a^{24} + \frac{1751969120889049}{1968905296006956974080} a^{23} - \frac{444591897470647}{492226324001739243520} a^{22} + \frac{227653408638006178043}{1968905296006956974080} a^{21} + \frac{718547020763290774021}{1968905296006956974080} a^{20} - \frac{155232546482339182843}{492226324001739243520} a^{19} + \frac{284274972727091516287}{1968905296006956974080} a^{18} + \frac{83204205757799413889}{1968905296006956974080} a^{17} + \frac{66547555683079871873}{492226324001739243520} a^{16} + \frac{545245134221972339763}{1968905296006956974080} a^{15} - \frac{185063100464965460019}{1968905296006956974080} a^{14} - \frac{209690918958362984243}{492226324001739243520} a^{13} + \frac{13724673869518217473}{30764145250108702720} a^{12} - \frac{3781553279257102929}{30764145250108702720} a^{11} - \frac{858093287093103773}{7691036312527175680} a^{10} - \frac{185411687830436187}{480689769532948480} a^{9} - \frac{23787748670400759}{480689769532948480} a^{8} + \frac{21196518465944977}{120172442383237120} a^{7} + \frac{2353112582180983}{7510777648952320} a^{6} - \frac{1664725181841389}{7510777648952320} a^{5} + \frac{286307263755727}{1877694412238080} a^{4} + \frac{181569147869707}{469423603059520} a^{3} - \frac{355966934204}{7334743797805} a^{2} - \frac{5302201142369}{29338975191220} a - \frac{3440747286264}{7334743797805}$, $\frac{1}{7875621184027827896320} a^{35} - \frac{1}{7875621184027827896320} a^{34} - \frac{1}{1968905296006956974080} a^{33} + \frac{13}{7875621184027827896320} a^{32} - \frac{13}{7875621184027827896320} a^{31} + \frac{24441466099}{1968905296006956974080} a^{30} - \frac{1590635610903}{7875621184027827896320} a^{29} + \frac{1101806288663}{7875621184027827896320} a^{28} + \frac{1810608805911}{1968905296006956974080} a^{27} - \frac{20678262940907}{7875621184027827896320} a^{26} + \frac{14323481751787}{7875621184027827896320} a^{25} + \frac{1952777971257561}{7875621184027827896320} a^{24} - \frac{555746848839607}{1968905296006956974080} a^{23} - \frac{7152756326867717}{7875621184027827896320} a^{22} - \frac{411189711600065629691}{7875621184027827896320} a^{21} - \frac{620009061837777142523}{1968905296006956974080} a^{20} - \frac{798783741382478588033}{7875621184027827896320} a^{19} - \frac{2237292865782522917759}{7875621184027827896320} a^{18} + \frac{722525492222882913153}{1968905296006956974080} a^{17} - \frac{2471919752058061672397}{7875621184027827896320} a^{16} - \frac{271600509382202956851}{7875621184027827896320} a^{15} - \frac{754169118615904639283}{1968905296006956974080} a^{14} + \frac{40596476240403556753}{123056581000434810880} a^{13} + \frac{57746793502585511039}{123056581000434810880} a^{12} - \frac{3050396002931070693}{30764145250108702720} a^{11} + \frac{1653405281644947657}{7691036312527175680} a^{10} - \frac{355156765454453671}{1922759078131793920} a^{9} - \frac{6557555736737407}{120172442383237120} a^{8} - \frac{1816973831191213}{120172442383237120} a^{7} + \frac{3288502568244099}{30043110595809280} a^{6} - \frac{139296584996983}{469423603059520} a^{5} - \frac{102792131886441}{234711801529760} a^{4} - \frac{23388918182007}{58677950382440} a^{3} - \frac{13579279267351}{29338975191220} a^{2} + \frac{13257217820031}{29338975191220} a + \frac{2018194896763}{7334743797805}$, $\frac{1}{31502484736111311585280} a^{36} - \frac{1}{31502484736111311585280} a^{35} - \frac{1}{7875621184027827896320} a^{34} + \frac{13}{31502484736111311585280} a^{33} - \frac{13}{31502484736111311585280} a^{32} - \frac{13}{7875621184027827896320} a^{31} + \frac{4333680277}{6300496947222262317056} a^{30} - \frac{422359077097}{31502484736111311585280} a^{29} + \frac{78504267543}{7875621184027827896320} a^{28} + \frac{376890384337}{6300496947222262317056} a^{27} - \frac{5490668003093}{31502484736111311585280} a^{26} + \frac{1856451519937753}{31502484736111311585280} a^{25} - \frac{91324232267915}{1575124236805565579264} a^{24} - \frac{7507886857089797}{31502484736111311585280} a^{23} + \frac{24153966851681797}{31502484736111311585280} a^{22} - \frac{175695299837548569087}{1575124236805565579264} a^{21} + \frac{13476357006640693994367}{31502484736111311585280} a^{20} + \frac{3537414254118701046913}{31502484736111311585280} a^{19} + \frac{652528335787147919181}{1575124236805565579264} a^{18} - \frac{15117168220092080288717}{31502484736111311585280} a^{17} + \frac{412157440491481330637}{31502484736111311585280} a^{16} + \frac{388617649240945686313}{1575124236805565579264} a^{15} - \frac{189317227848010547647}{492226324001739243520} a^{14} - \frac{58430222049404136513}{492226324001739243520} a^{13} - \frac{4446495464372454029}{24611316200086962176} a^{12} + \frac{1108245952706213739}{3845518156263587840} a^{11} - \frac{168818213672209559}{3845518156263587840} a^{10} - \frac{3280360042874613}{48068976953294848} a^{9} - \frac{104429546081607589}{240344884766474240} a^{8} - \frac{13898554465877549}{60086221191618560} a^{7} - \frac{250510582582233}{1502155529790464} a^{6} - \frac{2424421347882103}{7510777648952320} a^{5} + \frac{344309544409177}{1877694412238080} a^{4} - \frac{7479534071839}{117355900764880} a^{3} - \frac{55861205335749}{117355900764880} a^{2} - \frac{9012426646599}{29338975191220} a + \frac{3655247083343}{7334743797805}$, $\frac{1}{126009938944445246341120} a^{37} - \frac{1}{126009938944445246341120} a^{36} - \frac{1}{31502484736111311585280} a^{35} + \frac{13}{126009938944445246341120} a^{34} - \frac{13}{126009938944445246341120} a^{33} - \frac{13}{31502484736111311585280} a^{32} + \frac{233}{126009938944445246341120} a^{31} + \frac{619294867223}{126009938944445246341120} a^{30} + \frac{5409123618583}{31502484736111311585280} a^{29} - \frac{24732968808683}{126009938944445246341120} a^{28} - \frac{80972324097813}{126009938944445246341120} a^{27} - \frac{1920137437348647}{126009938944445246341120} a^{26} + \frac{469970817753417}{31502484736111311585280} a^{25} + \frac{7792642120277243}{126009938944445246341120} a^{24} - \frac{23577051039175163}{126009938944445246341120} a^{23} + \frac{5713893129855493}{31502484736111311585280} a^{22} + \frac{14354502791966352700287}{126009938944445246341120} a^{21} + \frac{17957107698540043684993}{126009938944445246341120} a^{20} - \frac{12848529973215554061183}{31502484736111311585280} a^{19} + \frac{39029512169546695334963}{126009938944445246341120} a^{18} + \frac{60097473417462917424077}{126009938944445246341120} a^{17} + \frac{1981099163346380002253}{31502484736111311585280} a^{16} + \frac{432241342233851065473}{1968905296006956974080} a^{15} + \frac{823000399452991585407}{1968905296006956974080} a^{14} + \frac{51711747477807874623}{492226324001739243520} a^{13} - \frac{9940190462082050277}{30764145250108702720} a^{12} - \frac{1314250717166297383}{30764145250108702720} a^{11} - \frac{928291773565024783}{1922759078131793920} a^{10} - \frac{15431567476144889}{961379539065896960} a^{9} + \frac{104062815802045081}{240344884766474240} a^{8} - \frac{13418889857292663}{30043110595809280} a^{7} - \frac{433759034275777}{7510777648952320} a^{6} + \frac{262632333732009}{938847206119040} a^{5} + \frac{23657665700083}{469423603059520} a^{4} + \frac{54731547606629}{117355900764880} a^{3} + \frac{2381024683646}{7334743797805} a^{2} + \frac{1016271327343}{14669487595610} a + \frac{537991038508}{1466948759561}$, $\frac{1}{504039755777780985364480} a^{38} - \frac{1}{504039755777780985364480} a^{37} - \frac{1}{126009938944445246341120} a^{36} + \frac{13}{504039755777780985364480} a^{35} - \frac{13}{504039755777780985364480} a^{34} - \frac{13}{126009938944445246341120} a^{33} + \frac{233}{504039755777780985364480} a^{32} - \frac{233}{504039755777780985364480} a^{31} - \frac{174781874409}{126009938944445246341120} a^{30} - \frac{34827891437803}{504039755777780985364480} a^{29} + \frac{38323528921323}{504039755777780985364480} a^{28} + \frac{1985388716309721}{504039755777780985364480} a^{27} - \frac{576282971676343}{126009938944445246341120} a^{26} - \frac{6911271316153093}{504039755777780985364480} a^{25} + \frac{25765309152177669}{504039755777780985364480} a^{24} - \frac{8048924894797307}{126009938944445246341120} a^{23} - \frac{87393821259025537}{504039755777780985364480} a^{22} + \frac{61419722015939642821761}{504039755777780985364480} a^{21} + \frac{20028780454686983933057}{126009938944445246341120} a^{20} - \frac{28289797480082613248973}{504039755777780985364480} a^{19} - \frac{50735536175918466882611}{504039755777780985364480} a^{18} + \frac{8492303593517494068173}{126009938944445246341120} a^{17} + \frac{3527989936288928923777}{7875621184027827896320} a^{16} - \frac{2280356450237396688001}{7875621184027827896320} a^{15} - \frac{138158322212064423297}{1968905296006956974080} a^{14} + \frac{28481708181528233867}{123056581000434810880} a^{13} - \frac{25374346698849138491}{123056581000434810880} a^{12} + \frac{12336398699059150673}{30764145250108702720} a^{11} + \frac{538289512577096233}{7691036312527175680} a^{10} + \frac{253192240011395627}{961379539065896960} a^{9} - \frac{161780014175359567}{480689769532948480} a^{8} + \frac{18665506473816549}{60086221191618560} a^{7} + \frac{9186326062438249}{30043110595809280} a^{6} + \frac{2707187344143627}{7510777648952320} a^{5} - \frac{245778298275609}{938847206119040} a^{4} - \frac{135031810934157}{469423603059520} a^{3} + \frac{17135020792141}{117355900764880} a^{2} + \frac{14026040572781}{29338975191220} a - \frac{1247892635283}{7334743797805}$, $\frac{1}{2016159023111123941457920} a^{39} - \frac{1}{2016159023111123941457920} a^{38} - \frac{1}{504039755777780985364480} a^{37} + \frac{13}{2016159023111123941457920} a^{36} - \frac{13}{2016159023111123941457920} a^{35} - \frac{13}{504039755777780985364480} a^{34} + \frac{233}{2016159023111123941457920} a^{33} - \frac{233}{2016159023111123941457920} a^{32} - \frac{233}{504039755777780985364480} a^{31} - \frac{7041943597291}{2016159023111123941457920} a^{30} + \frac{156262227243243}{2016159023111123941457920} a^{29} + \frac{1731316788772057}{2016159023111123941457920} a^{28} - \frac{635198924840631}{504039755777780985364480} a^{27} - \frac{5378068237968133}{2016159023111123941457920} a^{26} + \frac{22507118253977093}{2016159023111123941457920} a^{25} - \frac{8370257120484859}{504039755777780985364480} a^{24} - \frac{59914104550018177}{2016159023111123941457920} a^{23} + \frac{403396811783113857}{2016159023111123941457920} a^{22} + \frac{17301518260914245831809}{504039755777780985364480} a^{21} - \frac{128983229512780889826253}{2016159023111123941457920} a^{20} + \frac{1001209325573184186328013}{2016159023111123941457920} a^{19} + \frac{196418640151772783813581}{504039755777780985364480} a^{18} - \frac{7334842706715468792703}{31502484736111311585280} a^{17} + \frac{9272923149694396367743}{31502484736111311585280} a^{16} + \frac{3804631861299491619711}{7875621184027827896320} a^{15} - \frac{59668617926606297413}{492226324001739243520} a^{14} - \frac{192689437199935593147}{492226324001739243520} a^{13} + \frac{27103088184429845061}{123056581000434810880} a^{12} - \frac{229407769271018323}{480689769532948480} a^{11} + \frac{99283257009959469}{240344884766474240} a^{10} - \frac{115573062789083111}{480689769532948480} a^{9} - \frac{117013604779588297}{480689769532948480} a^{8} + \frac{58515738751596023}{120172442383237120} a^{7} + \frac{531717950490215}{1502155529790464} a^{6} + \frac{635157286221677}{7510777648952320} a^{5} - \frac{22220190372563}{1877694412238080} a^{4} + \frac{25111845800881}{117355900764880} a^{3} - \frac{56338640208969}{117355900764880} a^{2} + \frac{286710896441}{29338975191220} a + \frac{3196639296283}{7334743797805}$

Class group and class number

Not computed

Unit group

Rank:		$19$
Torsion generator:		\( \frac{9}{1877694412238080} a^{37} - \frac{315789856455581}{1877694412238080} a^{4} \) (order $66$)
Fundamental units:		Not computed
Regulator:		Not computed

Galois group

$C_2^2\times C_{10}$ (as 40T7):

An abelian group of order 40

The 40 conjugacy class representatives for $C_2^2\times C_{10}$

Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{561}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-187}) \), \(\Q(\sqrt{17}, \sqrt{33})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{-11}, \sqrt{17})\), \(\Q(\sqrt{-11}, \sqrt{-51})\), \(\Q(\sqrt{-3}, \sqrt{-187})\), \(\Q(\sqrt{33}, \sqrt{-51})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\zeta_{11})^+\), 8.0.99049307841.1, 10.10.304358957700017.1, 10.10.813551493932145441.1, \(\Q(\zeta_{33})^+\), 10.0.73959226721104131.3, 10.0.52089208083.1, \(\Q(\zeta_{11})\), 10.0.3347948534700187.1, 20.20.661866033279225680306281949177084481.1, 20.0.5469967217183683308316379745265161.1, 20.0.11208759391001129236841977834969.1, 20.0.661866033279225680306281949177084481.5, 20.0.661866033279225680306281949177084481.4, 20.0.661866033279225680306281949177084481.3, \(\Q(\zeta_{33})\)

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	${\href{/LocalNumberField/2.10.0.1}{10} }^{4}$	R	${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$	R	${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$	R	${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$	${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$	${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

In the table, R denotes a ramified prime. 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
3	Data not computed
11	Data not computed
17	Data not computed