Normalized defining polynomial
\( x^{44} - x^{43} - 245 x^{42} + 238 x^{41} + 27464 x^{40} - 25798 x^{39} - 1870216 x^{38} + \cdots - 669015421113803 \)
Invariants
| Degree: | $44$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[44, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(282\!\cdots\!781\)
\(\medspace = 23^{42}\cdot 29^{33}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(249.25\) | 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: | $23^{21/22}29^{3/4}\approx 249.24681341311788$ | ||
| Ramified primes: |
\(23\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{29}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $44$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(667=23\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{667}(1,·)$, $\chi_{667}(389,·)$, $\chi_{667}(262,·)$, $\chi_{667}(655,·)$, $\chi_{667}(144,·)$, $\chi_{667}(17,·)$, $\chi_{667}(146,·)$, $\chi_{667}(534,·)$, $\chi_{667}(407,·)$, $\chi_{667}(539,·)$, $\chi_{667}(157,·)$, $\chi_{667}(289,·)$, $\chi_{667}(173,·)$, $\chi_{667}(563,·)$, $\chi_{667}(59,·)$, $\chi_{667}(191,·)$, $\chi_{667}(579,·)$, $\chi_{667}(452,·)$, $\chi_{667}(581,·)$, $\chi_{667}(202,·)$, $\chi_{667}(463,·)$, $\chi_{667}(336,·)$, $\chi_{667}(597,·)$, $\chi_{667}(249,·)$, $\chi_{667}(592,·)$, $\chi_{667}(347,·)$, $\chi_{667}(349,·)$, $\chi_{667}(481,·)$, $\chi_{667}(610,·)$, $\chi_{667}(99,·)$, $\chi_{667}(273,·)$, $\chi_{667}(360,·)$, $\chi_{667}(233,·)$, $\chi_{667}(231,·)$, $\chi_{667}(492,·)$, $\chi_{667}(365,·)$, $\chi_{667}(626,·)$, $\chi_{667}(244,·)$, $\chi_{667}(117,·)$, $\chi_{667}(376,·)$, $\chi_{667}(505,·)$, $\chi_{667}(447,·)$, $\chi_{667}(637,·)$, $\chi_{667}(639,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{139}a^{33}+\frac{63}{139}a^{32}+\frac{55}{139}a^{30}-\frac{7}{139}a^{29}+\frac{36}{139}a^{28}+\frac{8}{139}a^{27}-\frac{39}{139}a^{26}+\frac{59}{139}a^{25}+\frac{53}{139}a^{24}+\frac{37}{139}a^{23}-\frac{61}{139}a^{22}+\frac{58}{139}a^{21}+\frac{6}{139}a^{20}+\frac{51}{139}a^{19}+\frac{50}{139}a^{18}+\frac{21}{139}a^{17}-\frac{51}{139}a^{16}-\frac{20}{139}a^{15}+\frac{69}{139}a^{14}+\frac{67}{139}a^{13}-\frac{7}{139}a^{12}+\frac{30}{139}a^{11}+\frac{42}{139}a^{10}+\frac{13}{139}a^{9}-\frac{32}{139}a^{8}-\frac{56}{139}a^{7}+\frac{39}{139}a^{6}-\frac{31}{139}a^{5}+\frac{6}{139}a^{4}-\frac{37}{139}a^{3}+\frac{41}{139}a^{2}-\frac{23}{139}a-\frac{26}{139}$, $\frac{1}{22\!\cdots\!67}a^{34}+\frac{3445049006556}{22\!\cdots\!67}a^{33}+\frac{294151439301228}{22\!\cdots\!67}a^{32}+\frac{27560169769239}{22\!\cdots\!67}a^{31}+\frac{10\!\cdots\!31}{22\!\cdots\!67}a^{30}+\frac{171691647294723}{22\!\cdots\!67}a^{29}-\frac{949241372413865}{22\!\cdots\!67}a^{28}+\frac{332178509014470}{22\!\cdots\!67}a^{27}+\frac{10\!\cdots\!65}{22\!\cdots\!67}a^{26}-\frac{383217560010110}{22\!\cdots\!67}a^{25}+\frac{372801926219074}{22\!\cdots\!67}a^{24}-\frac{10\!\cdots\!21}{22\!\cdots\!67}a^{23}-\frac{767510511419400}{22\!\cdots\!67}a^{22}+\frac{264482103288142}{22\!\cdots\!67}a^{21}-\frac{313924335370548}{22\!\cdots\!67}a^{20}-\frac{300213545255435}{22\!\cdots\!67}a^{19}+\frac{926800850902677}{22\!\cdots\!67}a^{18}-\frac{98630737827108}{22\!\cdots\!67}a^{17}-\frac{716077003733672}{22\!\cdots\!67}a^{16}-\frac{702051294682896}{22\!\cdots\!67}a^{15}+\frac{918077618927187}{22\!\cdots\!67}a^{14}+\frac{297837528679717}{22\!\cdots\!67}a^{13}+\frac{10\!\cdots\!47}{22\!\cdots\!67}a^{12}+\frac{10\!\cdots\!35}{22\!\cdots\!67}a^{11}+\frac{450629628010674}{22\!\cdots\!67}a^{10}-\frac{11\!\cdots\!63}{22\!\cdots\!67}a^{9}+\frac{526039468515532}{22\!\cdots\!67}a^{8}+\frac{207630882839486}{22\!\cdots\!67}a^{7}-\frac{423333591896739}{22\!\cdots\!67}a^{6}+\frac{10\!\cdots\!97}{22\!\cdots\!67}a^{5}+\frac{881399663085203}{22\!\cdots\!67}a^{4}-\frac{119407666427453}{22\!\cdots\!67}a^{3}+\frac{86064239039028}{22\!\cdots\!67}a^{2}-\frac{877126224056394}{22\!\cdots\!67}a-\frac{887143913933421}{22\!\cdots\!67}$, $\frac{1}{22\!\cdots\!67}a^{35}+\frac{8244229067945}{22\!\cdots\!67}a^{33}+\frac{691278913657533}{22\!\cdots\!67}a^{32}-\frac{893678332992650}{22\!\cdots\!67}a^{31}-\frac{287395601800894}{22\!\cdots\!67}a^{30}-\frac{459141341721954}{22\!\cdots\!67}a^{29}-\frac{749596435148456}{22\!\cdots\!67}a^{28}-\frac{454894747251211}{22\!\cdots\!67}a^{27}+\frac{10\!\cdots\!71}{22\!\cdots\!67}a^{26}-\frac{240615557372136}{22\!\cdots\!67}a^{25}-\frac{11\!\cdots\!67}{22\!\cdots\!67}a^{24}-\frac{736323173198752}{22\!\cdots\!67}a^{23}-\frac{10\!\cdots\!24}{22\!\cdots\!67}a^{22}+\frac{680389533684884}{22\!\cdots\!67}a^{21}+\frac{775512867090856}{22\!\cdots\!67}a^{20}-\frac{393143216352802}{22\!\cdots\!67}a^{19}+\frac{442820084856026}{22\!\cdots\!67}a^{18}-\frac{890310037283172}{22\!\cdots\!67}a^{17}+\frac{631090058588183}{22\!\cdots\!67}a^{16}-\frac{199829105870762}{22\!\cdots\!67}a^{15}-\frac{10\!\cdots\!91}{22\!\cdots\!67}a^{14}-\frac{873280563606007}{22\!\cdots\!67}a^{13}-\frac{235056339464359}{22\!\cdots\!67}a^{12}-\frac{480093408015428}{22\!\cdots\!67}a^{11}+\frac{928382007959791}{22\!\cdots\!67}a^{10}+\frac{810697453107927}{22\!\cdots\!67}a^{9}-\frac{303512844144325}{22\!\cdots\!67}a^{8}-\frac{598097370704873}{22\!\cdots\!67}a^{7}-\frac{185624750550091}{22\!\cdots\!67}a^{6}-\frac{835676318696354}{22\!\cdots\!67}a^{5}+\frac{367480232647917}{22\!\cdots\!67}a^{4}-\frac{931513413969189}{22\!\cdots\!67}a^{3}+\frac{235917931439519}{22\!\cdots\!67}a^{2}+\frac{941805560185985}{22\!\cdots\!67}a+\frac{577800473940566}{22\!\cdots\!67}$, $\frac{1}{22\!\cdots\!67}a^{36}+\frac{522961692316}{22\!\cdots\!67}a^{33}-\frac{502321057734001}{22\!\cdots\!67}a^{32}+\frac{236883119666154}{22\!\cdots\!67}a^{31}+\frac{832722093807129}{22\!\cdots\!67}a^{30}+\frac{270533099127026}{22\!\cdots\!67}a^{29}-\frac{959358845985506}{22\!\cdots\!67}a^{28}+\frac{953718111579022}{22\!\cdots\!67}a^{27}+\frac{981296213714708}{22\!\cdots\!67}a^{26}-\frac{506744675558648}{22\!\cdots\!67}a^{25}-\frac{304128459662635}{22\!\cdots\!67}a^{24}-\frac{941009815376074}{22\!\cdots\!67}a^{23}-\frac{186283304075852}{22\!\cdots\!67}a^{22}+\frac{453634164093674}{22\!\cdots\!67}a^{21}-\frac{737660827747590}{22\!\cdots\!67}a^{20}-\frac{270782892405096}{22\!\cdots\!67}a^{19}+\frac{917466001634140}{22\!\cdots\!67}a^{18}+\frac{259292214813248}{22\!\cdots\!67}a^{17}+\frac{10\!\cdots\!27}{22\!\cdots\!67}a^{16}-\frac{993280932561415}{22\!\cdots\!67}a^{15}-\frac{585108031648633}{22\!\cdots\!67}a^{14}+\frac{196390648786019}{22\!\cdots\!67}a^{13}+\frac{112783649148577}{22\!\cdots\!67}a^{12}+\frac{329265381179598}{22\!\cdots\!67}a^{11}+\frac{11\!\cdots\!10}{22\!\cdots\!67}a^{10}+\frac{11\!\cdots\!64}{22\!\cdots\!67}a^{9}+\frac{747612441440723}{22\!\cdots\!67}a^{8}+\frac{198571356853698}{22\!\cdots\!67}a^{7}-\frac{107758574260928}{22\!\cdots\!67}a^{6}-\frac{897417748728642}{22\!\cdots\!67}a^{5}-\frac{207992004369570}{22\!\cdots\!67}a^{4}-\frac{102773537852359}{22\!\cdots\!67}a^{3}+\frac{199628168311764}{22\!\cdots\!67}a^{2}+\frac{188981402743915}{22\!\cdots\!67}a-\frac{991816909031218}{22\!\cdots\!67}$, $\frac{1}{22\!\cdots\!67}a^{37}-\frac{2010104925250}{22\!\cdots\!67}a^{33}-\frac{689357063079322}{22\!\cdots\!67}a^{32}-\frac{364194323588789}{22\!\cdots\!67}a^{31}-\frac{163822788769697}{22\!\cdots\!67}a^{30}-\frac{10\!\cdots\!76}{22\!\cdots\!67}a^{29}+\frac{78853063352512}{22\!\cdots\!67}a^{28}-\frac{197443924308262}{22\!\cdots\!67}a^{27}-\frac{958904502651046}{22\!\cdots\!67}a^{26}+\frac{259279165542973}{22\!\cdots\!67}a^{25}+\frac{117583809662097}{22\!\cdots\!67}a^{24}-\frac{428966753800203}{22\!\cdots\!67}a^{23}-\frac{205056516226964}{22\!\cdots\!67}a^{22}+\frac{971218989785806}{22\!\cdots\!67}a^{21}-\frac{869830792263963}{22\!\cdots\!67}a^{20}-\frac{92655849762553}{22\!\cdots\!67}a^{19}-\frac{626673330060151}{22\!\cdots\!67}a^{18}-\frac{178764127572864}{22\!\cdots\!67}a^{17}-\frac{10\!\cdots\!46}{22\!\cdots\!67}a^{16}-\frac{616642772074987}{22\!\cdots\!67}a^{15}-\frac{75661703631167}{22\!\cdots\!67}a^{14}-\frac{100665602322676}{22\!\cdots\!67}a^{13}-\frac{405470258298178}{22\!\cdots\!67}a^{12}+\frac{461703826484588}{22\!\cdots\!67}a^{11}-\frac{106901923574250}{22\!\cdots\!67}a^{10}-\frac{155483813167640}{22\!\cdots\!67}a^{9}+\frac{47844327376861}{22\!\cdots\!67}a^{8}-\frac{562699865623436}{22\!\cdots\!67}a^{7}+\frac{10\!\cdots\!32}{22\!\cdots\!67}a^{6}-\frac{126524980634099}{22\!\cdots\!67}a^{5}+\frac{300511201954757}{22\!\cdots\!67}a^{4}+\frac{923849263802092}{22\!\cdots\!67}a^{3}-\frac{98797429517386}{22\!\cdots\!67}a^{2}-\frac{594161601301260}{22\!\cdots\!67}a+\frac{414443916002075}{22\!\cdots\!67}$, $\frac{1}{22\!\cdots\!67}a^{38}+\frac{5291369847994}{22\!\cdots\!67}a^{33}+\frac{896012877280495}{22\!\cdots\!67}a^{32}-\frac{564596814684050}{22\!\cdots\!67}a^{31}+\frac{449993160468202}{22\!\cdots\!67}a^{30}+\frac{952205702416732}{22\!\cdots\!67}a^{29}+\frac{646613604640463}{22\!\cdots\!67}a^{28}+\frac{10\!\cdots\!63}{22\!\cdots\!67}a^{27}-\frac{870506408225841}{22\!\cdots\!67}a^{26}-\frac{311807175803011}{22\!\cdots\!67}a^{25}-\frac{890383898513765}{22\!\cdots\!67}a^{24}+\frac{747304342208907}{22\!\cdots\!67}a^{23}+\frac{544491092533848}{22\!\cdots\!67}a^{22}-\frac{838314797613240}{22\!\cdots\!67}a^{21}+\frac{122705784078782}{22\!\cdots\!67}a^{20}-\frac{137261780183995}{22\!\cdots\!67}a^{19}+\frac{88892337897347}{22\!\cdots\!67}a^{18}-\frac{10\!\cdots\!17}{22\!\cdots\!67}a^{17}+\frac{806927495122775}{22\!\cdots\!67}a^{16}+\frac{726858463416968}{22\!\cdots\!67}a^{15}+\frac{991738763563153}{22\!\cdots\!67}a^{14}+\frac{465805173763051}{22\!\cdots\!67}a^{13}+\frac{329844932858149}{22\!\cdots\!67}a^{12}+\frac{455089107688238}{22\!\cdots\!67}a^{11}-\frac{45702789751654}{22\!\cdots\!67}a^{10}+\frac{688239317260221}{22\!\cdots\!67}a^{9}-\frac{4502565756941}{16530532223453}a^{8}+\frac{207404908523724}{22\!\cdots\!67}a^{7}+\frac{10\!\cdots\!13}{22\!\cdots\!67}a^{6}-\frac{11\!\cdots\!73}{22\!\cdots\!67}a^{5}+\frac{358956833837074}{22\!\cdots\!67}a^{4}+\frac{995324929295495}{22\!\cdots\!67}a^{3}-\frac{251379395713364}{22\!\cdots\!67}a^{2}+\frac{969580334955717}{22\!\cdots\!67}a-\frac{375487486147798}{22\!\cdots\!67}$, $\frac{1}{22\!\cdots\!67}a^{39}-\frac{7199811372735}{22\!\cdots\!67}a^{33}+\frac{420570877637476}{22\!\cdots\!67}a^{32}-\frac{10\!\cdots\!62}{22\!\cdots\!67}a^{31}+\frac{615402472242010}{22\!\cdots\!67}a^{30}-\frac{270619663252872}{22\!\cdots\!67}a^{29}+\frac{956639617613120}{22\!\cdots\!67}a^{28}-\frac{511720129686397}{22\!\cdots\!67}a^{27}-\frac{960968381836401}{22\!\cdots\!67}a^{26}-\frac{2708653715529}{16530532223453}a^{25}+\frac{631845537587556}{22\!\cdots\!67}a^{24}-\frac{796547622739457}{22\!\cdots\!67}a^{23}-\frac{304476421228804}{22\!\cdots\!67}a^{22}-\frac{684206643405430}{22\!\cdots\!67}a^{21}-\frac{300948968854860}{22\!\cdots\!67}a^{20}-\frac{602639386240790}{22\!\cdots\!67}a^{19}-\frac{10\!\cdots\!33}{22\!\cdots\!67}a^{18}+\frac{265775616330019}{22\!\cdots\!67}a^{17}+\frac{879482160086036}{22\!\cdots\!67}a^{16}+\frac{134709656723506}{22\!\cdots\!67}a^{15}+\frac{481811969759663}{22\!\cdots\!67}a^{14}-\frac{86741164086447}{22\!\cdots\!67}a^{13}+\frac{601662064983059}{22\!\cdots\!67}a^{12}-\frac{395535285582116}{22\!\cdots\!67}a^{11}+\frac{10\!\cdots\!94}{22\!\cdots\!67}a^{10}+\frac{885008442120184}{22\!\cdots\!67}a^{9}-\frac{179086292063370}{22\!\cdots\!67}a^{8}+\frac{11\!\cdots\!82}{22\!\cdots\!67}a^{7}-\frac{511740307930557}{22\!\cdots\!67}a^{6}-\frac{10\!\cdots\!84}{22\!\cdots\!67}a^{5}-\frac{992365920544978}{22\!\cdots\!67}a^{4}-\frac{511472734302692}{22\!\cdots\!67}a^{3}+\frac{419348538883149}{22\!\cdots\!67}a^{2}-\frac{312995324776061}{22\!\cdots\!67}a+\frac{664618652459166}{22\!\cdots\!67}$, $\frac{1}{22\!\cdots\!67}a^{40}-\frac{795558285358}{22\!\cdots\!67}a^{33}-\frac{385610578092827}{22\!\cdots\!67}a^{32}-\frac{272994587469920}{22\!\cdots\!67}a^{31}-\frac{161119692659477}{22\!\cdots\!67}a^{30}-\frac{718150667615346}{22\!\cdots\!67}a^{29}-\frac{749615176090614}{22\!\cdots\!67}a^{28}+\frac{432798750539435}{22\!\cdots\!67}a^{27}-\frac{533278522404671}{22\!\cdots\!67}a^{26}+\frac{387736873210801}{22\!\cdots\!67}a^{25}-\frac{91986089118230}{22\!\cdots\!67}a^{24}-\frac{796135492622356}{22\!\cdots\!67}a^{23}-\frac{3779258825861}{16530532223453}a^{22}-\frac{420487878971722}{22\!\cdots\!67}a^{21}+\frac{51555092867009}{22\!\cdots\!67}a^{20}+\frac{422660528220450}{22\!\cdots\!67}a^{19}-\frac{364660979589156}{22\!\cdots\!67}a^{18}+\frac{610667950661309}{22\!\cdots\!67}a^{17}-\frac{304273250640754}{22\!\cdots\!67}a^{16}-\frac{590741848696212}{22\!\cdots\!67}a^{15}-\frac{532005554961418}{22\!\cdots\!67}a^{14}-\frac{105423623079580}{22\!\cdots\!67}a^{13}-\frac{893278437859077}{22\!\cdots\!67}a^{12}+\frac{622249281305680}{22\!\cdots\!67}a^{11}+\frac{348984297038900}{22\!\cdots\!67}a^{10}+\frac{468070679744141}{22\!\cdots\!67}a^{9}-\frac{61511213701345}{22\!\cdots\!67}a^{8}+\frac{691497906550133}{22\!\cdots\!67}a^{7}+\frac{940809069086109}{22\!\cdots\!67}a^{6}+\frac{11\!\cdots\!84}{22\!\cdots\!67}a^{5}-\frac{81501044550700}{22\!\cdots\!67}a^{4}-\frac{3826529850810}{22\!\cdots\!67}a^{3}+\frac{556138380868247}{22\!\cdots\!67}a^{2}-\frac{39579659174631}{22\!\cdots\!67}a+\frac{750002656230122}{22\!\cdots\!67}$, $\frac{1}{63\!\cdots\!59}a^{41}+\frac{18}{63\!\cdots\!59}a^{40}+\frac{19}{63\!\cdots\!59}a^{39}-\frac{25}{63\!\cdots\!59}a^{38}-\frac{64}{63\!\cdots\!59}a^{37}+\frac{5}{63\!\cdots\!59}a^{36}+\frac{79}{63\!\cdots\!59}a^{35}-\frac{127}{63\!\cdots\!59}a^{34}+\frac{562973793894962}{63\!\cdots\!59}a^{33}-\frac{26\!\cdots\!71}{63\!\cdots\!59}a^{32}-\frac{19\!\cdots\!17}{63\!\cdots\!59}a^{31}+\frac{25\!\cdots\!35}{63\!\cdots\!59}a^{30}-\frac{26\!\cdots\!59}{63\!\cdots\!59}a^{29}+\frac{19\!\cdots\!21}{63\!\cdots\!59}a^{28}+\frac{38\!\cdots\!76}{63\!\cdots\!59}a^{27}-\frac{16\!\cdots\!66}{63\!\cdots\!59}a^{26}+\frac{12\!\cdots\!05}{63\!\cdots\!59}a^{25}+\frac{22\!\cdots\!70}{63\!\cdots\!59}a^{24}-\frac{90\!\cdots\!09}{63\!\cdots\!59}a^{23}-\frac{12\!\cdots\!92}{63\!\cdots\!59}a^{22}-\frac{19\!\cdots\!49}{63\!\cdots\!59}a^{21}+\frac{14\!\cdots\!19}{63\!\cdots\!59}a^{20}+\frac{21\!\cdots\!40}{63\!\cdots\!59}a^{19}+\frac{18\!\cdots\!69}{63\!\cdots\!59}a^{18}-\frac{11\!\cdots\!52}{63\!\cdots\!59}a^{17}-\frac{17\!\cdots\!40}{63\!\cdots\!59}a^{16}-\frac{18\!\cdots\!06}{63\!\cdots\!59}a^{15}+\frac{26\!\cdots\!21}{63\!\cdots\!59}a^{14}-\frac{20\!\cdots\!00}{63\!\cdots\!59}a^{13}-\frac{64\!\cdots\!15}{63\!\cdots\!59}a^{12}+\frac{30\!\cdots\!88}{63\!\cdots\!59}a^{11}-\frac{12\!\cdots\!98}{63\!\cdots\!59}a^{10}+\frac{31\!\cdots\!67}{63\!\cdots\!59}a^{9}+\frac{19\!\cdots\!08}{63\!\cdots\!59}a^{8}+\frac{49\!\cdots\!76}{63\!\cdots\!59}a^{7}+\frac{18\!\cdots\!43}{63\!\cdots\!59}a^{6}-\frac{25\!\cdots\!83}{63\!\cdots\!59}a^{5}-\frac{875455801563379}{22\!\cdots\!67}a^{4}-\frac{12\!\cdots\!65}{63\!\cdots\!59}a^{3}+\frac{23\!\cdots\!20}{63\!\cdots\!59}a^{2}+\frac{14\!\cdots\!21}{63\!\cdots\!59}a-\frac{89\!\cdots\!36}{63\!\cdots\!59}$, $\frac{1}{63\!\cdots\!59}a^{42}-\frac{28}{63\!\cdots\!59}a^{40}-\frac{90}{63\!\cdots\!59}a^{39}+\frac{109}{63\!\cdots\!59}a^{38}+\frac{49}{63\!\cdots\!59}a^{37}-\frac{11}{63\!\cdots\!59}a^{36}+\frac{113}{63\!\cdots\!59}a^{35}+\frac{28}{63\!\cdots\!59}a^{34}-\frac{19\!\cdots\!75}{63\!\cdots\!59}a^{33}+\frac{17\!\cdots\!50}{63\!\cdots\!59}a^{32}-\frac{19\!\cdots\!36}{63\!\cdots\!59}a^{31}-\frac{23\!\cdots\!35}{63\!\cdots\!59}a^{30}-\frac{19\!\cdots\!85}{63\!\cdots\!59}a^{29}-\frac{17\!\cdots\!01}{63\!\cdots\!59}a^{28}+\frac{27\!\cdots\!35}{63\!\cdots\!59}a^{27}-\frac{10\!\cdots\!48}{63\!\cdots\!59}a^{26}+\frac{38\!\cdots\!51}{63\!\cdots\!59}a^{25}+\frac{13\!\cdots\!12}{63\!\cdots\!59}a^{24}+\frac{24\!\cdots\!61}{63\!\cdots\!59}a^{23}-\frac{64\!\cdots\!33}{63\!\cdots\!59}a^{22}-\frac{26\!\cdots\!53}{63\!\cdots\!59}a^{21}-\frac{91\!\cdots\!65}{63\!\cdots\!59}a^{20}+\frac{63\!\cdots\!54}{63\!\cdots\!59}a^{19}-\frac{94\!\cdots\!85}{63\!\cdots\!59}a^{18}+\frac{15\!\cdots\!96}{63\!\cdots\!59}a^{17}-\frac{60\!\cdots\!20}{63\!\cdots\!59}a^{16}+\frac{62\!\cdots\!83}{63\!\cdots\!59}a^{15}+\frac{18\!\cdots\!45}{63\!\cdots\!59}a^{14}-\frac{22\!\cdots\!78}{63\!\cdots\!59}a^{13}+\frac{10\!\cdots\!75}{63\!\cdots\!59}a^{12}-\frac{43\!\cdots\!65}{63\!\cdots\!59}a^{11}+\frac{12\!\cdots\!75}{63\!\cdots\!59}a^{10}+\frac{62\!\cdots\!07}{63\!\cdots\!59}a^{9}+\frac{10\!\cdots\!27}{63\!\cdots\!59}a^{8}+\frac{15\!\cdots\!81}{63\!\cdots\!59}a^{7}+\frac{30\!\cdots\!84}{63\!\cdots\!59}a^{6}+\frac{31\!\cdots\!32}{63\!\cdots\!59}a^{5}-\frac{16\!\cdots\!48}{63\!\cdots\!59}a^{4}-\frac{28\!\cdots\!58}{63\!\cdots\!59}a^{3}-\frac{14\!\cdots\!19}{63\!\cdots\!59}a^{2}-\frac{75\!\cdots\!92}{63\!\cdots\!59}a-\frac{12\!\cdots\!12}{63\!\cdots\!59}$, $\frac{1}{10\!\cdots\!63}a^{43}-\frac{48\!\cdots\!62}{10\!\cdots\!63}a^{42}-\frac{56\!\cdots\!36}{10\!\cdots\!63}a^{41}+\frac{17\!\cdots\!33}{10\!\cdots\!63}a^{40}+\frac{17\!\cdots\!02}{10\!\cdots\!63}a^{39}-\frac{15\!\cdots\!39}{10\!\cdots\!63}a^{38}+\frac{41\!\cdots\!84}{10\!\cdots\!63}a^{37}-\frac{11\!\cdots\!47}{10\!\cdots\!63}a^{36}-\frac{56\!\cdots\!04}{10\!\cdots\!63}a^{35}-\frac{10\!\cdots\!66}{10\!\cdots\!63}a^{34}+\frac{20\!\cdots\!50}{10\!\cdots\!63}a^{33}+\frac{40\!\cdots\!53}{10\!\cdots\!63}a^{32}-\frac{18\!\cdots\!24}{10\!\cdots\!63}a^{31}-\frac{10\!\cdots\!83}{10\!\cdots\!63}a^{30}+\frac{30\!\cdots\!66}{10\!\cdots\!63}a^{29}-\frac{13\!\cdots\!93}{10\!\cdots\!63}a^{28}+\frac{19\!\cdots\!43}{10\!\cdots\!63}a^{27}+\frac{26\!\cdots\!63}{10\!\cdots\!63}a^{26}-\frac{48\!\cdots\!61}{10\!\cdots\!63}a^{25}+\frac{47\!\cdots\!81}{10\!\cdots\!63}a^{24}+\frac{48\!\cdots\!71}{10\!\cdots\!63}a^{23}-\frac{23\!\cdots\!53}{10\!\cdots\!63}a^{22}-\frac{26\!\cdots\!44}{10\!\cdots\!63}a^{21}-\frac{20\!\cdots\!47}{10\!\cdots\!63}a^{20}+\frac{21\!\cdots\!76}{10\!\cdots\!63}a^{19}-\frac{14\!\cdots\!11}{10\!\cdots\!63}a^{18}-\frac{31\!\cdots\!38}{10\!\cdots\!63}a^{17}-\frac{35\!\cdots\!18}{10\!\cdots\!63}a^{16}-\frac{21\!\cdots\!46}{10\!\cdots\!63}a^{15}+\frac{43\!\cdots\!20}{10\!\cdots\!63}a^{14}+\frac{13\!\cdots\!20}{10\!\cdots\!63}a^{13}+\frac{24\!\cdots\!02}{10\!\cdots\!63}a^{12}-\frac{30\!\cdots\!59}{10\!\cdots\!63}a^{11}-\frac{26\!\cdots\!15}{10\!\cdots\!63}a^{10}+\frac{42\!\cdots\!13}{10\!\cdots\!63}a^{9}+\frac{47\!\cdots\!77}{10\!\cdots\!63}a^{8}-\frac{20\!\cdots\!47}{10\!\cdots\!63}a^{7}+\frac{25\!\cdots\!92}{10\!\cdots\!63}a^{6}-\frac{29\!\cdots\!63}{10\!\cdots\!63}a^{5}-\frac{23\!\cdots\!94}{10\!\cdots\!63}a^{4}+\frac{29\!\cdots\!36}{10\!\cdots\!63}a^{3}-\frac{27\!\cdots\!84}{10\!\cdots\!63}a^{2}-\frac{18\!\cdots\!42}{10\!\cdots\!63}a-\frac{38\!\cdots\!10}{10\!\cdots\!63}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $43$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(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 $
Galois group
| A cyclic group of order 44 |
| The 44 conjugacy class representatives for $C_{44}$ |
| Character table for $C_{44}$ |
Intermediate fields
| \(\Q(\sqrt{29}) \), 4.4.12901781.1, \(\Q(\zeta_{23})^+\), 22.22.20937975979670626213353681795476767790826629.1 |
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 | $44$ | $44$ | ${\href{/padicField/5.11.0.1}{11} }^{4}$ | $22^{2}$ | $44$ | $22^{2}$ | $44$ | $44$ | R | R | $44$ | $44$ | $44$ | $44$ | ${\href{/padicField/47.4.0.1}{4} }^{11}$ | $22^{2}$ | ${\href{/padicField/59.11.0.1}{11} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.22.21.17 | $x^{22} + 23$ | $22$ | $1$ | $21$ | 22T1 | $[\ ]_{22}$ |
| 23.22.21.17 | $x^{22} + 23$ | $22$ | $1$ | $21$ | 22T1 | $[\ ]_{22}$ | |
|
\(29\)
| Deg $44$ | $4$ | $11$ | $33$ |