Normalized defining polynomial
\( x^{32} - 4 x^{31} - 10 x^{30} + 80 x^{29} + 33 x^{28} - 260 x^{27} - 2622 x^{26} + 5068 x^{25} + \cdots + 279841 \)
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(847622907049404564614012839370162176000000000000000000000000\)
\(\medspace = 2^{88}\cdot 5^{24}\cdot 11^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(74.60\) | 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: | $2^{11/4}5^{3/4}11^{1/2}\approx 74.60300719070477$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(880=2^{4}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{880}(1,·)$, $\chi_{880}(131,·)$, $\chi_{880}(769,·)$, $\chi_{880}(529,·)$, $\chi_{880}(659,·)$, $\chi_{880}(793,·)$, $\chi_{880}(153,·)$, $\chi_{880}(417,·)$, $\chi_{880}(419,·)$, $\chi_{880}(683,·)$, $\chi_{880}(681,·)$, $\chi_{880}(43,·)$, $\chi_{880}(177,·)$, $\chi_{880}(307,·)$, $\chi_{880}(219,·)$, $\chi_{880}(441,·)$, $\chi_{880}(571,·)$, $\chi_{880}(67,·)$, $\chi_{880}(329,·)$, $\chi_{880}(331,·)$, $\chi_{880}(593,·)$, $\chi_{880}(857,·)$, $\chi_{880}(859,·)$, $\chi_{880}(353,·)$, $\chi_{880}(483,·)$, $\chi_{880}(617,·)$, $\chi_{880}(747,·)$, $\chi_{880}(241,·)$, $\chi_{880}(243,·)$, $\chi_{880}(89,·)$, $\chi_{880}(771,·)$, $\chi_{880}(507,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{32768}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{41}a^{24}-\frac{3}{41}a^{23}+\frac{6}{41}a^{22}-\frac{10}{41}a^{21}-\frac{5}{41}a^{20}+\frac{5}{41}a^{19}-\frac{4}{41}a^{18}-\frac{15}{41}a^{17}+\frac{13}{41}a^{16}+\frac{1}{41}a^{15}-\frac{12}{41}a^{14}+\frac{15}{41}a^{13}+\frac{10}{41}a^{12}-\frac{8}{41}a^{11}+\frac{18}{41}a^{10}-\frac{11}{41}a^{9}+\frac{1}{41}a^{8}+\frac{6}{41}a^{7}+\frac{4}{41}a^{6}-\frac{6}{41}a^{5}-\frac{12}{41}a^{4}+\frac{19}{41}a^{3}-\frac{5}{41}a^{2}-\frac{1}{41}a+\frac{10}{41}$, $\frac{1}{943}a^{25}+\frac{8}{943}a^{24}-\frac{150}{943}a^{23}+\frac{466}{943}a^{22}+\frac{418}{943}a^{21}-\frac{296}{943}a^{20}-\frac{113}{943}a^{19}-\frac{305}{943}a^{18}+\frac{176}{943}a^{17}+\frac{390}{943}a^{16}-\frac{247}{943}a^{15}-\frac{404}{943}a^{14}+\frac{93}{943}a^{13}+\frac{143}{943}a^{12}-\frac{193}{943}a^{11}+\frac{1}{41}a^{10}-\frac{243}{943}a^{9}+\frac{222}{943}a^{8}+\frac{439}{943}a^{7}-\frac{249}{943}a^{6}+\frac{86}{943}a^{5}-\frac{441}{943}a^{4}+\frac{409}{943}a^{3}-\frac{15}{943}a^{2}-\frac{247}{943}a+\frac{3}{41}$, $\frac{1}{80544459}a^{26}+\frac{8955}{26848153}a^{25}-\frac{487231}{80544459}a^{24}-\frac{1342664}{26848153}a^{23}-\frac{39211726}{80544459}a^{22}-\frac{34018589}{80544459}a^{21}+\frac{10676494}{26848153}a^{20}-\frac{34201135}{80544459}a^{19}-\frac{4909132}{80544459}a^{18}+\frac{10376252}{80544459}a^{17}+\frac{22662488}{80544459}a^{16}+\frac{7116680}{26848153}a^{15}-\frac{159047}{1167311}a^{14}+\frac{12660095}{80544459}a^{13}-\frac{23411077}{80544459}a^{12}+\frac{11966116}{26848153}a^{11}-\frac{27835280}{80544459}a^{10}+\frac{25996178}{80544459}a^{9}-\frac{29872595}{80544459}a^{8}-\frac{14776406}{80544459}a^{7}+\frac{2721502}{26848153}a^{6}+\frac{5512816}{80544459}a^{5}+\frac{1267883}{3501933}a^{4}+\frac{6803905}{26848153}a^{3}-\frac{38225107}{80544459}a^{2}+\frac{115200}{378143}a+\frac{1610869}{3501933}$, $\frac{1}{80544459}a^{27}+\frac{36872}{80544459}a^{25}+\frac{110683}{26848153}a^{24}-\frac{37363411}{80544459}a^{23}-\frac{10895207}{80544459}a^{22}+\frac{165516}{654833}a^{21}+\frac{7105490}{80544459}a^{20}+\frac{16927232}{80544459}a^{19}+\frac{27382883}{80544459}a^{18}-\frac{40159423}{80544459}a^{17}+\frac{130440}{26848153}a^{16}-\frac{9590209}{26848153}a^{15}+\frac{12571322}{80544459}a^{14}+\frac{3668300}{80544459}a^{13}+\frac{10272362}{26848153}a^{12}-\frac{5203991}{80544459}a^{11}+\frac{24426533}{80544459}a^{10}-\frac{5343125}{80544459}a^{9}+\frac{362009}{3501933}a^{8}+\frac{1567720}{26848153}a^{7}+\frac{21248749}{80544459}a^{6}-\frac{4251077}{80544459}a^{5}-\frac{11909704}{26848153}a^{4}+\frac{15973277}{80544459}a^{3}+\frac{12645817}{26848153}a^{2}+\frac{35646104}{80544459}a+\frac{41204}{1167311}$, $\frac{1}{80544459}a^{28}-\frac{194}{378143}a^{25}+\frac{483451}{80544459}a^{24}-\frac{37391024}{80544459}a^{23}+\frac{7586996}{80544459}a^{22}+\frac{8643247}{26848153}a^{21}+\frac{23986997}{80544459}a^{20}+\frac{20960359}{80544459}a^{19}-\frac{4733675}{80544459}a^{18}-\frac{38707462}{80544459}a^{17}-\frac{30641431}{80544459}a^{16}-\frac{81434}{1964499}a^{15}+\frac{26809292}{80544459}a^{14}+\frac{16416047}{80544459}a^{13}-\frac{2707458}{26848153}a^{12}+\frac{25800077}{80544459}a^{11}-\frac{35046505}{80544459}a^{10}+\frac{7331896}{26848153}a^{9}-\frac{880600}{1964499}a^{8}+\frac{5228456}{80544459}a^{7}-\frac{39144380}{80544459}a^{6}-\frac{24330119}{80544459}a^{5}+\frac{4992448}{26848153}a^{4}-\frac{4381893}{26848153}a^{3}-\frac{1646839}{3501933}a^{2}+\frac{12781262}{26848153}a+\frac{323845}{3501933}$, $\frac{1}{81\!\cdots\!09}a^{29}+\frac{98\!\cdots\!11}{27\!\cdots\!03}a^{28}-\frac{65\!\cdots\!28}{81\!\cdots\!09}a^{27}-\frac{31\!\cdots\!53}{81\!\cdots\!09}a^{26}-\frac{21\!\cdots\!46}{27\!\cdots\!03}a^{25}-\frac{14\!\cdots\!82}{35\!\cdots\!83}a^{24}-\frac{19\!\cdots\!85}{81\!\cdots\!09}a^{23}+\frac{40\!\cdots\!41}{81\!\cdots\!09}a^{22}-\frac{29\!\cdots\!34}{81\!\cdots\!09}a^{21}-\frac{27\!\cdots\!07}{27\!\cdots\!03}a^{20}+\frac{11\!\cdots\!66}{81\!\cdots\!09}a^{19}+\frac{19\!\cdots\!14}{35\!\cdots\!83}a^{18}+\frac{21\!\cdots\!78}{81\!\cdots\!09}a^{17}-\frac{46\!\cdots\!02}{27\!\cdots\!03}a^{16}+\frac{33\!\cdots\!72}{81\!\cdots\!09}a^{15}-\frac{12\!\cdots\!08}{81\!\cdots\!09}a^{14}+\frac{93\!\cdots\!80}{27\!\cdots\!03}a^{13}+\frac{63\!\cdots\!42}{27\!\cdots\!03}a^{12}+\frac{60\!\cdots\!07}{27\!\cdots\!03}a^{11}-\frac{27\!\cdots\!56}{81\!\cdots\!09}a^{10}+\frac{10\!\cdots\!31}{27\!\cdots\!03}a^{9}-\frac{47\!\cdots\!00}{81\!\cdots\!09}a^{8}-\frac{12\!\cdots\!09}{27\!\cdots\!03}a^{7}-\frac{14\!\cdots\!79}{81\!\cdots\!09}a^{6}+\frac{10\!\cdots\!69}{27\!\cdots\!03}a^{5}-\frac{98\!\cdots\!65}{81\!\cdots\!09}a^{4}-\frac{42\!\cdots\!60}{27\!\cdots\!03}a^{3}+\frac{21\!\cdots\!82}{81\!\cdots\!09}a^{2}+\frac{51\!\cdots\!80}{81\!\cdots\!09}a+\frac{10\!\cdots\!11}{35\!\cdots\!83}$, $\frac{1}{18\!\cdots\!07}a^{30}-\frac{4}{18\!\cdots\!07}a^{29}-\frac{40\!\cdots\!62}{18\!\cdots\!07}a^{28}+\frac{38\!\cdots\!49}{62\!\cdots\!69}a^{27}-\frac{10\!\cdots\!02}{18\!\cdots\!07}a^{26}-\frac{26\!\cdots\!59}{62\!\cdots\!69}a^{25}-\frac{35\!\cdots\!80}{81\!\cdots\!09}a^{24}+\frac{65\!\cdots\!12}{18\!\cdots\!07}a^{23}+\frac{31\!\cdots\!66}{18\!\cdots\!07}a^{22}-\frac{35\!\cdots\!30}{86\!\cdots\!63}a^{21}-\frac{38\!\cdots\!47}{18\!\cdots\!07}a^{20}-\frac{74\!\cdots\!63}{18\!\cdots\!07}a^{19}-\frac{40\!\cdots\!30}{81\!\cdots\!09}a^{18}-\frac{23\!\cdots\!53}{18\!\cdots\!07}a^{17}+\frac{15\!\cdots\!11}{18\!\cdots\!07}a^{16}-\frac{53\!\cdots\!84}{18\!\cdots\!07}a^{15}+\frac{83\!\cdots\!17}{27\!\cdots\!03}a^{14}+\frac{63\!\cdots\!92}{18\!\cdots\!07}a^{13}-\frac{35\!\cdots\!81}{87\!\cdots\!39}a^{12}+\frac{20\!\cdots\!84}{62\!\cdots\!69}a^{11}-\frac{27\!\cdots\!17}{62\!\cdots\!69}a^{10}+\frac{31\!\cdots\!08}{18\!\cdots\!07}a^{9}-\frac{20\!\cdots\!09}{62\!\cdots\!69}a^{8}+\frac{26\!\cdots\!41}{18\!\cdots\!07}a^{7}+\frac{14\!\cdots\!26}{62\!\cdots\!69}a^{6}+\frac{88\!\cdots\!97}{18\!\cdots\!07}a^{5}-\frac{14\!\cdots\!23}{18\!\cdots\!07}a^{4}+\frac{30\!\cdots\!01}{18\!\cdots\!07}a^{3}-\frac{22\!\cdots\!27}{62\!\cdots\!69}a^{2}-\frac{32\!\cdots\!45}{81\!\cdots\!09}a+\frac{15\!\cdots\!47}{35\!\cdots\!83}$, $\frac{1}{43\!\cdots\!61}a^{31}-\frac{4}{43\!\cdots\!61}a^{30}-\frac{10}{43\!\cdots\!61}a^{29}-\frac{20\!\cdots\!98}{43\!\cdots\!61}a^{28}-\frac{93\!\cdots\!93}{43\!\cdots\!61}a^{27}+\frac{16\!\cdots\!63}{43\!\cdots\!61}a^{26}-\frac{45\!\cdots\!36}{18\!\cdots\!07}a^{25}+\frac{47\!\cdots\!09}{43\!\cdots\!61}a^{24}+\frac{16\!\cdots\!85}{43\!\cdots\!61}a^{23}+\frac{25\!\cdots\!39}{62\!\cdots\!69}a^{22}+\frac{26\!\cdots\!46}{43\!\cdots\!61}a^{21}-\frac{69\!\cdots\!02}{14\!\cdots\!87}a^{20}-\frac{85\!\cdots\!23}{18\!\cdots\!07}a^{19}-\frac{15\!\cdots\!68}{43\!\cdots\!61}a^{18}-\frac{65\!\cdots\!81}{14\!\cdots\!87}a^{17}+\frac{10\!\cdots\!61}{43\!\cdots\!61}a^{16}-\frac{10\!\cdots\!84}{62\!\cdots\!69}a^{15}+\frac{26\!\cdots\!58}{14\!\cdots\!87}a^{14}+\frac{19\!\cdots\!69}{43\!\cdots\!61}a^{13}+\frac{50\!\cdots\!22}{43\!\cdots\!61}a^{12}-\frac{39\!\cdots\!74}{43\!\cdots\!61}a^{11}+\frac{20\!\cdots\!63}{43\!\cdots\!61}a^{10}+\frac{18\!\cdots\!67}{43\!\cdots\!61}a^{9}+\frac{10\!\cdots\!42}{43\!\cdots\!61}a^{8}-\frac{48\!\cdots\!59}{14\!\cdots\!87}a^{7}-\frac{56\!\cdots\!42}{43\!\cdots\!61}a^{6}-\frac{58\!\cdots\!57}{43\!\cdots\!61}a^{5}-\frac{60\!\cdots\!67}{14\!\cdots\!87}a^{4}-\frac{65\!\cdots\!04}{14\!\cdots\!87}a^{3}+\frac{70\!\cdots\!64}{18\!\cdots\!07}a^{2}+\frac{47\!\cdots\!20}{11\!\cdots\!61}a+\frac{52\!\cdots\!15}{11\!\cdots\!61}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{3480}$, which has order $3480$ (assuming GRH)
Unit group
| Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{2106666720628878475868526987876034520}{1050065174731829290643807548944867305968821} a^{31} + \frac{2728491587602468632036463321901350932}{350021724910609763547935849648289101989607} a^{30} + \frac{21699768149597426199704412020109739328}{1050065174731829290643807548944867305968821} a^{29} - \frac{165106337961696915216373650716506089712}{1050065174731829290643807548944867305968821} a^{28} - \frac{28139228121644107852443327911698258328}{350021724910609763547935849648289101989607} a^{27} + \frac{516970254303805548062550225898175275480}{1050065174731829290643807548944867305968821} a^{26} + \frac{241405532808813770954955245517598437040}{45655007597036056114948154301950752433427} a^{25} - \frac{9980890024985761138135567878264266387789}{1050065174731829290643807548944867305968821} a^{24} - \frac{30815720715125886676714701479916636681360}{350021724910609763547935849648289101989607} a^{23} + \frac{10460024242214767581837776403303372699644}{45655007597036056114948154301950752433427} a^{22} + \frac{178782504619034375802645965202328469748368}{1050065174731829290643807548944867305968821} a^{21} - \frac{290312047141602922699950585536914419752328}{350021724910609763547935849648289101989607} a^{20} - \frac{27008809480489140300617052242410515192400}{45655007597036056114948154301950752433427} a^{19} - \frac{1156798941805965512289534080528004100704332}{350021724910609763547935849648289101989607} a^{18} + \frac{23033442410434154571577217402859027609130112}{1050065174731829290643807548944867305968821} a^{17} - \frac{18098009503076946141524941094205931845453512}{1050065174731829290643807548944867305968821} a^{16} - \frac{3604307379456181258826677676131528982028200}{45655007597036056114948154301950752433427} a^{15} + \frac{172357619804392149018444747209778908070608000}{1050065174731829290643807548944867305968821} a^{14} - \frac{149866137489322176584999170291430541568638704}{1050065174731829290643807548944867305968821} a^{13} + \frac{58118598814395862144847973826031728979576}{2924972631564984096500856682297680518019} a^{12} + \frac{120614667542462230718797212777194691509660936}{1050065174731829290643807548944867305968821} a^{11} - \frac{228992779910557696826949681604820936885661368}{1050065174731829290643807548944867305968821} a^{10} + \frac{186152442826307463125116089722434743591202952}{1050065174731829290643807548944867305968821} a^{9} - \frac{7902426289288953868575919521826951029922456}{350021724910609763547935849648289101989607} a^{8} - \frac{37903610840383315830554824062268654453177952}{350021724910609763547935849648289101989607} a^{7} + \frac{39481430523476047520635625026804655106072268}{350021724910609763547935849648289101989607} a^{6} - \frac{83946447164556606944696023638176602775557216}{1050065174731829290643807548944867305968821} a^{5} + \frac{15232033821296197304215626091374949453112872}{350021724910609763547935849648289101989607} a^{4} - \frac{28400872819925371569882096583727082010933816}{1050065174731829290643807548944867305968821} a^{3} + \frac{679590599948461212499255810187637742381732}{45655007597036056114948154301950752433427} a^{2} - \frac{13842206655799060063440779136860535408736}{1985000330305915483258615404432641410149} a + \frac{218177140528305415525697667068883271208}{86304362187213716663418061062288756963} \)
(order $10$)
| 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: |
$\frac{19\!\cdots\!40}{19\!\cdots\!49}a^{31}-\frac{58\!\cdots\!02}{19\!\cdots\!49}a^{30}-\frac{25\!\cdots\!04}{19\!\cdots\!49}a^{29}+\frac{43\!\cdots\!59}{66\!\cdots\!83}a^{28}+\frac{19\!\cdots\!40}{19\!\cdots\!49}a^{27}-\frac{36\!\cdots\!42}{19\!\cdots\!49}a^{26}-\frac{54\!\cdots\!68}{19\!\cdots\!49}a^{25}+\frac{47\!\cdots\!37}{19\!\cdots\!49}a^{24}+\frac{39\!\cdots\!16}{86\!\cdots\!63}a^{23}-\frac{14\!\cdots\!34}{19\!\cdots\!49}a^{22}-\frac{10\!\cdots\!88}{66\!\cdots\!83}a^{21}+\frac{58\!\cdots\!70}{19\!\cdots\!49}a^{20}+\frac{12\!\cdots\!32}{19\!\cdots\!49}a^{19}+\frac{38\!\cdots\!90}{19\!\cdots\!49}a^{18}-\frac{18\!\cdots\!92}{19\!\cdots\!49}a^{17}-\frac{26\!\cdots\!54}{19\!\cdots\!49}a^{16}+\frac{83\!\cdots\!44}{19\!\cdots\!49}a^{15}-\frac{87\!\cdots\!74}{19\!\cdots\!49}a^{14}+\frac{85\!\cdots\!76}{66\!\cdots\!83}a^{13}+\frac{13\!\cdots\!71}{66\!\cdots\!83}a^{12}-\frac{25\!\cdots\!80}{66\!\cdots\!83}a^{11}+\frac{11\!\cdots\!14}{19\!\cdots\!49}a^{10}-\frac{12\!\cdots\!88}{66\!\cdots\!83}a^{9}-\frac{53\!\cdots\!71}{19\!\cdots\!49}a^{8}+\frac{91\!\cdots\!16}{28\!\cdots\!21}a^{7}-\frac{54\!\cdots\!24}{66\!\cdots\!83}a^{6}+\frac{28\!\cdots\!44}{19\!\cdots\!49}a^{5}-\frac{15\!\cdots\!10}{19\!\cdots\!49}a^{4}+\frac{13\!\cdots\!20}{28\!\cdots\!21}a^{3}-\frac{19\!\cdots\!20}{86\!\cdots\!63}a^{2}+\frac{16\!\cdots\!36}{66\!\cdots\!83}a-\frac{10\!\cdots\!10}{86\!\cdots\!63}$, $\frac{58\!\cdots\!16}{27\!\cdots\!03}a^{31}-\frac{54\!\cdots\!64}{81\!\cdots\!09}a^{30}-\frac{23\!\cdots\!28}{81\!\cdots\!09}a^{29}+\frac{40\!\cdots\!88}{27\!\cdots\!03}a^{28}+\frac{18\!\cdots\!80}{81\!\cdots\!09}a^{27}-\frac{33\!\cdots\!83}{81\!\cdots\!09}a^{26}-\frac{50\!\cdots\!76}{81\!\cdots\!09}a^{25}+\frac{43\!\cdots\!84}{81\!\cdots\!09}a^{24}+\frac{36\!\cdots\!12}{35\!\cdots\!83}a^{23}-\frac{13\!\cdots\!88}{81\!\cdots\!09}a^{22}-\frac{29\!\cdots\!22}{81\!\cdots\!09}a^{21}+\frac{54\!\cdots\!40}{81\!\cdots\!09}a^{20}+\frac{11\!\cdots\!24}{81\!\cdots\!09}a^{19}+\frac{35\!\cdots\!80}{81\!\cdots\!09}a^{18}-\frac{16\!\cdots\!44}{81\!\cdots\!09}a^{17}-\frac{80\!\cdots\!23}{27\!\cdots\!03}a^{16}+\frac{76\!\cdots\!08}{81\!\cdots\!09}a^{15}-\frac{19\!\cdots\!48}{19\!\cdots\!49}a^{14}+\frac{78\!\cdots\!32}{27\!\cdots\!03}a^{13}+\frac{12\!\cdots\!72}{27\!\cdots\!03}a^{12}-\frac{23\!\cdots\!48}{27\!\cdots\!03}a^{11}+\frac{10\!\cdots\!48}{81\!\cdots\!09}a^{10}-\frac{11\!\cdots\!16}{27\!\cdots\!03}a^{9}-\frac{49\!\cdots\!72}{81\!\cdots\!09}a^{8}+\frac{85\!\cdots\!12}{11\!\cdots\!61}a^{7}-\frac{13\!\cdots\!24}{81\!\cdots\!09}a^{6}+\frac{26\!\cdots\!08}{81\!\cdots\!09}a^{5}-\frac{14\!\cdots\!20}{81\!\cdots\!09}a^{4}+\frac{12\!\cdots\!40}{11\!\cdots\!61}a^{3}-\frac{17\!\cdots\!40}{35\!\cdots\!83}a^{2}+\frac{47\!\cdots\!58}{81\!\cdots\!09}a-\frac{15\!\cdots\!04}{35\!\cdots\!83}$, $\frac{38\!\cdots\!00}{43\!\cdots\!61}a^{31}-\frac{19\!\cdots\!55}{43\!\cdots\!61}a^{30}-\frac{25\!\cdots\!72}{43\!\cdots\!61}a^{29}+\frac{35\!\cdots\!79}{43\!\cdots\!61}a^{28}-\frac{15\!\cdots\!16}{43\!\cdots\!61}a^{27}-\frac{41\!\cdots\!14}{14\!\cdots\!87}a^{26}-\frac{98\!\cdots\!48}{45\!\cdots\!27}a^{25}+\frac{29\!\cdots\!56}{43\!\cdots\!61}a^{24}+\frac{15\!\cdots\!08}{43\!\cdots\!61}a^{23}-\frac{27\!\cdots\!02}{18\!\cdots\!07}a^{22}+\frac{18\!\cdots\!48}{10\!\cdots\!61}a^{21}+\frac{21\!\cdots\!37}{43\!\cdots\!61}a^{20}-\frac{68\!\cdots\!52}{62\!\cdots\!69}a^{19}+\frac{43\!\cdots\!06}{43\!\cdots\!61}a^{18}-\frac{16\!\cdots\!28}{14\!\cdots\!87}a^{17}+\frac{25\!\cdots\!66}{14\!\cdots\!87}a^{16}+\frac{59\!\cdots\!40}{18\!\cdots\!07}a^{15}-\frac{50\!\cdots\!74}{43\!\cdots\!61}a^{14}+\frac{55\!\cdots\!32}{43\!\cdots\!61}a^{13}-\frac{17\!\cdots\!61}{43\!\cdots\!61}a^{12}-\frac{31\!\cdots\!36}{43\!\cdots\!61}a^{11}+\frac{22\!\cdots\!09}{14\!\cdots\!87}a^{10}-\frac{67\!\cdots\!56}{43\!\cdots\!61}a^{9}+\frac{22\!\cdots\!67}{43\!\cdots\!61}a^{8}+\frac{10\!\cdots\!00}{14\!\cdots\!87}a^{7}-\frac{14\!\cdots\!32}{14\!\cdots\!87}a^{6}+\frac{92\!\cdots\!82}{14\!\cdots\!87}a^{5}-\frac{15\!\cdots\!06}{43\!\cdots\!61}a^{4}+\frac{95\!\cdots\!76}{43\!\cdots\!61}a^{3}-\frac{23\!\cdots\!52}{18\!\cdots\!07}a^{2}+\frac{38\!\cdots\!56}{86\!\cdots\!63}a+\frac{34\!\cdots\!68}{35\!\cdots\!83}$, $\frac{52\!\cdots\!48}{14\!\cdots\!87}a^{31}-\frac{12\!\cdots\!50}{43\!\cdots\!61}a^{30}+\frac{19\!\cdots\!08}{14\!\cdots\!87}a^{29}+\frac{19\!\cdots\!93}{43\!\cdots\!61}a^{28}-\frac{41\!\cdots\!79}{43\!\cdots\!61}a^{27}-\frac{80\!\cdots\!02}{43\!\cdots\!61}a^{26}-\frac{37\!\cdots\!16}{62\!\cdots\!69}a^{25}+\frac{81\!\cdots\!91}{14\!\cdots\!87}a^{24}+\frac{44\!\cdots\!96}{43\!\cdots\!61}a^{23}-\frac{67\!\cdots\!99}{62\!\cdots\!69}a^{22}+\frac{49\!\cdots\!64}{43\!\cdots\!61}a^{21}+\frac{14\!\cdots\!45}{43\!\cdots\!61}a^{20}-\frac{90\!\cdots\!32}{18\!\cdots\!07}a^{19}-\frac{34\!\cdots\!52}{43\!\cdots\!61}a^{18}-\frac{27\!\cdots\!47}{43\!\cdots\!61}a^{17}+\frac{79\!\cdots\!27}{43\!\cdots\!61}a^{16}+\frac{47\!\cdots\!64}{62\!\cdots\!69}a^{15}-\frac{13\!\cdots\!92}{14\!\cdots\!87}a^{14}+\frac{54\!\cdots\!12}{43\!\cdots\!61}a^{13}-\frac{22\!\cdots\!30}{43\!\cdots\!61}a^{12}-\frac{23\!\cdots\!32}{43\!\cdots\!61}a^{11}+\frac{54\!\cdots\!48}{43\!\cdots\!61}a^{10}-\frac{65\!\cdots\!08}{43\!\cdots\!61}a^{9}+\frac{30\!\cdots\!56}{43\!\cdots\!61}a^{8}+\frac{88\!\cdots\!04}{14\!\cdots\!87}a^{7}-\frac{14\!\cdots\!30}{14\!\cdots\!87}a^{6}+\frac{24\!\cdots\!64}{43\!\cdots\!61}a^{5}-\frac{13\!\cdots\!47}{43\!\cdots\!61}a^{4}+\frac{84\!\cdots\!40}{43\!\cdots\!61}a^{3}-\frac{43\!\cdots\!69}{62\!\cdots\!69}a^{2}+\frac{19\!\cdots\!04}{81\!\cdots\!09}a-\frac{14\!\cdots\!58}{11\!\cdots\!61}$, $\frac{23\!\cdots\!32}{43\!\cdots\!61}a^{31}-\frac{31\!\cdots\!44}{14\!\cdots\!87}a^{30}-\frac{78\!\cdots\!08}{14\!\cdots\!87}a^{29}+\frac{18\!\cdots\!71}{43\!\cdots\!61}a^{28}+\frac{26\!\cdots\!64}{14\!\cdots\!87}a^{27}-\frac{22\!\cdots\!98}{14\!\cdots\!87}a^{26}-\frac{88\!\cdots\!01}{62\!\cdots\!69}a^{25}+\frac{40\!\cdots\!64}{14\!\cdots\!87}a^{24}+\frac{10\!\cdots\!76}{43\!\cdots\!61}a^{23}-\frac{12\!\cdots\!90}{18\!\cdots\!07}a^{22}-\frac{70\!\cdots\!28}{14\!\cdots\!87}a^{21}+\frac{10\!\cdots\!45}{43\!\cdots\!61}a^{20}+\frac{10\!\cdots\!56}{62\!\cdots\!69}a^{19}+\frac{10\!\cdots\!34}{14\!\cdots\!87}a^{18}-\frac{26\!\cdots\!48}{43\!\cdots\!61}a^{17}+\frac{21\!\cdots\!02}{43\!\cdots\!61}a^{16}+\frac{43\!\cdots\!44}{18\!\cdots\!07}a^{15}-\frac{20\!\cdots\!06}{43\!\cdots\!61}a^{14}+\frac{13\!\cdots\!24}{43\!\cdots\!61}a^{13}+\frac{27\!\cdots\!25}{43\!\cdots\!61}a^{12}-\frac{15\!\cdots\!04}{43\!\cdots\!61}a^{11}+\frac{80\!\cdots\!68}{14\!\cdots\!87}a^{10}-\frac{17\!\cdots\!68}{43\!\cdots\!61}a^{9}-\frac{32\!\cdots\!33}{43\!\cdots\!61}a^{8}+\frac{54\!\cdots\!88}{14\!\cdots\!87}a^{7}-\frac{11\!\cdots\!80}{43\!\cdots\!61}a^{6}+\frac{12\!\cdots\!08}{10\!\cdots\!21}a^{5}-\frac{38\!\cdots\!66}{43\!\cdots\!61}a^{4}+\frac{77\!\cdots\!84}{14\!\cdots\!87}a^{3}-\frac{15\!\cdots\!12}{62\!\cdots\!69}a^{2}+\frac{95\!\cdots\!20}{86\!\cdots\!63}a-\frac{26\!\cdots\!91}{98\!\cdots\!37}$, $\frac{38\!\cdots\!48}{43\!\cdots\!61}a^{31}-\frac{10\!\cdots\!98}{43\!\cdots\!61}a^{30}-\frac{60\!\cdots\!60}{43\!\cdots\!61}a^{29}+\frac{25\!\cdots\!03}{43\!\cdots\!61}a^{28}+\frac{56\!\cdots\!43}{43\!\cdots\!61}a^{27}-\frac{27\!\cdots\!06}{14\!\cdots\!87}a^{26}-\frac{50\!\cdots\!76}{18\!\cdots\!07}a^{25}+\frac{52\!\cdots\!29}{43\!\cdots\!61}a^{24}+\frac{19\!\cdots\!56}{43\!\cdots\!61}a^{23}-\frac{96\!\cdots\!77}{18\!\cdots\!07}a^{22}-\frac{96\!\cdots\!92}{43\!\cdots\!61}a^{21}+\frac{39\!\cdots\!31}{14\!\cdots\!87}a^{20}+\frac{51\!\cdots\!60}{62\!\cdots\!69}a^{19}+\frac{77\!\cdots\!12}{43\!\cdots\!61}a^{18}-\frac{35\!\cdots\!41}{43\!\cdots\!61}a^{17}-\frac{24\!\cdots\!73}{43\!\cdots\!61}a^{16}+\frac{29\!\cdots\!64}{62\!\cdots\!69}a^{15}-\frac{34\!\cdots\!44}{14\!\cdots\!87}a^{14}-\frac{62\!\cdots\!12}{14\!\cdots\!87}a^{13}+\frac{30\!\cdots\!05}{43\!\cdots\!61}a^{12}-\frac{20\!\cdots\!20}{43\!\cdots\!61}a^{11}+\frac{74\!\cdots\!48}{43\!\cdots\!61}a^{10}+\frac{21\!\cdots\!96}{43\!\cdots\!61}a^{9}-\frac{39\!\cdots\!54}{43\!\cdots\!61}a^{8}+\frac{18\!\cdots\!32}{43\!\cdots\!61}a^{7}+\frac{13\!\cdots\!58}{43\!\cdots\!61}a^{6}-\frac{12\!\cdots\!16}{43\!\cdots\!61}a^{5}+\frac{63\!\cdots\!21}{43\!\cdots\!61}a^{4}-\frac{14\!\cdots\!52}{14\!\cdots\!87}a^{3}+\frac{26\!\cdots\!99}{62\!\cdots\!69}a^{2}-\frac{95\!\cdots\!20}{27\!\cdots\!03}a+\frac{94\!\cdots\!16}{35\!\cdots\!83}$, $\frac{21\!\cdots\!32}{43\!\cdots\!61}a^{31}-\frac{69\!\cdots\!48}{43\!\cdots\!61}a^{30}-\frac{90\!\cdots\!56}{14\!\cdots\!87}a^{29}+\frac{15\!\cdots\!17}{43\!\cdots\!61}a^{28}+\frac{19\!\cdots\!64}{43\!\cdots\!61}a^{27}-\frac{39\!\cdots\!58}{43\!\cdots\!61}a^{26}-\frac{87\!\cdots\!07}{62\!\cdots\!69}a^{25}+\frac{61\!\cdots\!82}{43\!\cdots\!61}a^{24}+\frac{99\!\cdots\!96}{43\!\cdots\!61}a^{23}-\frac{77\!\cdots\!14}{18\!\cdots\!07}a^{22}-\frac{30\!\cdots\!32}{43\!\cdots\!61}a^{21}+\frac{66\!\cdots\!39}{43\!\cdots\!61}a^{20}+\frac{49\!\cdots\!32}{18\!\cdots\!07}a^{19}+\frac{44\!\cdots\!30}{43\!\cdots\!61}a^{18}-\frac{69\!\cdots\!88}{14\!\cdots\!87}a^{17}+\frac{13\!\cdots\!22}{14\!\cdots\!87}a^{16}+\frac{12\!\cdots\!64}{62\!\cdots\!69}a^{15}-\frac{37\!\cdots\!18}{14\!\cdots\!87}a^{14}+\frac{25\!\cdots\!08}{14\!\cdots\!87}a^{13}+\frac{21\!\cdots\!55}{14\!\cdots\!87}a^{12}-\frac{92\!\cdots\!92}{43\!\cdots\!61}a^{11}+\frac{55\!\cdots\!16}{14\!\cdots\!87}a^{10}-\frac{95\!\cdots\!04}{43\!\cdots\!61}a^{9}-\frac{46\!\cdots\!21}{14\!\cdots\!87}a^{8}+\frac{75\!\cdots\!40}{43\!\cdots\!61}a^{7}-\frac{89\!\cdots\!64}{60\!\cdots\!91}a^{6}+\frac{16\!\cdots\!84}{10\!\cdots\!21}a^{5}-\frac{86\!\cdots\!86}{14\!\cdots\!87}a^{4}+\frac{15\!\cdots\!08}{43\!\cdots\!61}a^{3}-\frac{36\!\cdots\!28}{18\!\cdots\!07}a^{2}+\frac{68\!\cdots\!04}{66\!\cdots\!83}a-\frac{11\!\cdots\!60}{35\!\cdots\!83}$, $\frac{17\!\cdots\!12}{43\!\cdots\!61}a^{31}-\frac{16\!\cdots\!16}{14\!\cdots\!87}a^{30}-\frac{24\!\cdots\!56}{43\!\cdots\!61}a^{29}+\frac{11\!\cdots\!25}{43\!\cdots\!61}a^{28}+\frac{68\!\cdots\!56}{14\!\cdots\!87}a^{27}-\frac{26\!\cdots\!54}{43\!\cdots\!61}a^{26}-\frac{21\!\cdots\!44}{18\!\cdots\!07}a^{25}+\frac{30\!\cdots\!41}{43\!\cdots\!61}a^{24}+\frac{82\!\cdots\!12}{43\!\cdots\!61}a^{23}-\frac{48\!\cdots\!12}{18\!\cdots\!07}a^{22}-\frac{10\!\cdots\!32}{14\!\cdots\!87}a^{21}+\frac{44\!\cdots\!21}{43\!\cdots\!61}a^{20}+\frac{50\!\cdots\!48}{18\!\cdots\!07}a^{19}+\frac{38\!\cdots\!62}{43\!\cdots\!61}a^{18}-\frac{15\!\cdots\!88}{43\!\cdots\!61}a^{17}-\frac{37\!\cdots\!44}{43\!\cdots\!61}a^{16}+\frac{10\!\cdots\!52}{62\!\cdots\!69}a^{15}-\frac{20\!\cdots\!86}{14\!\cdots\!87}a^{14}+\frac{16\!\cdots\!96}{43\!\cdots\!61}a^{13}+\frac{12\!\cdots\!30}{14\!\cdots\!87}a^{12}-\frac{21\!\cdots\!40}{14\!\cdots\!87}a^{11}+\frac{31\!\cdots\!00}{14\!\cdots\!87}a^{10}-\frac{16\!\cdots\!68}{43\!\cdots\!61}a^{9}-\frac{14\!\cdots\!76}{14\!\cdots\!87}a^{8}+\frac{49\!\cdots\!80}{43\!\cdots\!61}a^{7}-\frac{12\!\cdots\!60}{43\!\cdots\!61}a^{6}+\frac{27\!\cdots\!12}{43\!\cdots\!61}a^{5}-\frac{67\!\cdots\!64}{43\!\cdots\!61}a^{4}+\frac{88\!\cdots\!92}{43\!\cdots\!61}a^{3}-\frac{13\!\cdots\!22}{62\!\cdots\!69}a^{2}+\frac{25\!\cdots\!04}{81\!\cdots\!09}a-\frac{34\!\cdots\!94}{35\!\cdots\!83}$, $\frac{90\!\cdots\!48}{43\!\cdots\!61}a^{31}-\frac{32\!\cdots\!66}{43\!\cdots\!61}a^{30}-\frac{10\!\cdots\!84}{43\!\cdots\!61}a^{29}+\frac{68\!\cdots\!72}{43\!\cdots\!61}a^{28}+\frac{59\!\cdots\!20}{43\!\cdots\!61}a^{27}-\frac{22\!\cdots\!84}{43\!\cdots\!61}a^{26}-\frac{46\!\cdots\!36}{81\!\cdots\!09}a^{25}+\frac{12\!\cdots\!57}{14\!\cdots\!87}a^{24}+\frac{13\!\cdots\!16}{14\!\cdots\!87}a^{23}-\frac{39\!\cdots\!94}{18\!\cdots\!07}a^{22}-\frac{11\!\cdots\!20}{43\!\cdots\!61}a^{21}+\frac{35\!\cdots\!01}{43\!\cdots\!61}a^{20}+\frac{18\!\cdots\!16}{18\!\cdots\!07}a^{19}+\frac{51\!\cdots\!40}{14\!\cdots\!87}a^{18}-\frac{96\!\cdots\!12}{43\!\cdots\!61}a^{17}+\frac{44\!\cdots\!39}{43\!\cdots\!61}a^{16}+\frac{16\!\cdots\!32}{18\!\cdots\!07}a^{15}-\frac{61\!\cdots\!88}{43\!\cdots\!61}a^{14}+\frac{11\!\cdots\!04}{14\!\cdots\!87}a^{13}+\frac{60\!\cdots\!89}{43\!\cdots\!61}a^{12}-\frac{41\!\cdots\!64}{43\!\cdots\!61}a^{11}+\frac{23\!\cdots\!78}{14\!\cdots\!87}a^{10}-\frac{48\!\cdots\!64}{43\!\cdots\!61}a^{9}-\frac{91\!\cdots\!58}{43\!\cdots\!61}a^{8}+\frac{34\!\cdots\!28}{43\!\cdots\!61}a^{7}-\frac{21\!\cdots\!68}{43\!\cdots\!61}a^{6}+\frac{20\!\cdots\!88}{43\!\cdots\!61}a^{5}-\frac{67\!\cdots\!34}{14\!\cdots\!87}a^{4}+\frac{40\!\cdots\!68}{14\!\cdots\!87}a^{3}-\frac{32\!\cdots\!98}{18\!\cdots\!07}a^{2}+\frac{16\!\cdots\!04}{27\!\cdots\!03}a-\frac{56\!\cdots\!52}{35\!\cdots\!83}$, $\frac{16\!\cdots\!44}{14\!\cdots\!87}a^{31}-\frac{51\!\cdots\!08}{10\!\cdots\!21}a^{30}-\frac{44\!\cdots\!32}{43\!\cdots\!61}a^{29}+\frac{13\!\cdots\!27}{14\!\cdots\!87}a^{28}+\frac{49\!\cdots\!08}{43\!\cdots\!61}a^{27}-\frac{14\!\cdots\!25}{43\!\cdots\!61}a^{26}-\frac{54\!\cdots\!62}{18\!\cdots\!07}a^{25}+\frac{99\!\cdots\!54}{14\!\cdots\!87}a^{24}+\frac{70\!\cdots\!96}{14\!\cdots\!87}a^{23}-\frac{29\!\cdots\!98}{18\!\cdots\!07}a^{22}-\frac{93\!\cdots\!70}{14\!\cdots\!87}a^{21}+\frac{24\!\cdots\!53}{43\!\cdots\!61}a^{20}+\frac{37\!\cdots\!96}{18\!\cdots\!07}a^{19}+\frac{62\!\cdots\!52}{43\!\cdots\!61}a^{18}-\frac{58\!\cdots\!06}{43\!\cdots\!61}a^{17}+\frac{62\!\cdots\!84}{43\!\cdots\!61}a^{16}+\frac{29\!\cdots\!92}{62\!\cdots\!69}a^{15}-\frac{16\!\cdots\!30}{14\!\cdots\!87}a^{14}+\frac{13\!\cdots\!36}{14\!\cdots\!87}a^{13}-\frac{23\!\cdots\!52}{43\!\cdots\!61}a^{12}-\frac{35\!\cdots\!76}{43\!\cdots\!61}a^{11}+\frac{61\!\cdots\!99}{43\!\cdots\!61}a^{10}-\frac{52\!\cdots\!60}{43\!\cdots\!61}a^{9}+\frac{30\!\cdots\!96}{43\!\cdots\!61}a^{8}+\frac{36\!\cdots\!84}{43\!\cdots\!61}a^{7}-\frac{11\!\cdots\!02}{14\!\cdots\!87}a^{6}+\frac{18\!\cdots\!44}{43\!\cdots\!61}a^{5}-\frac{16\!\cdots\!61}{60\!\cdots\!91}a^{4}+\frac{64\!\cdots\!52}{43\!\cdots\!61}a^{3}-\frac{13\!\cdots\!27}{18\!\cdots\!07}a^{2}+\frac{10\!\cdots\!62}{27\!\cdots\!03}a-\frac{38\!\cdots\!96}{35\!\cdots\!83}$, $\frac{85\!\cdots\!96}{43\!\cdots\!61}a^{31}-\frac{28\!\cdots\!54}{43\!\cdots\!61}a^{30}-\frac{10\!\cdots\!65}{43\!\cdots\!61}a^{29}+\frac{20\!\cdots\!43}{14\!\cdots\!87}a^{28}+\frac{69\!\cdots\!49}{43\!\cdots\!61}a^{27}-\frac{16\!\cdots\!33}{43\!\cdots\!61}a^{26}-\frac{34\!\cdots\!95}{62\!\cdots\!69}a^{25}+\frac{27\!\cdots\!19}{43\!\cdots\!61}a^{24}+\frac{13\!\cdots\!09}{14\!\cdots\!87}a^{23}-\frac{32\!\cdots\!76}{18\!\cdots\!07}a^{22}-\frac{11\!\cdots\!52}{43\!\cdots\!61}a^{21}+\frac{95\!\cdots\!60}{14\!\cdots\!87}a^{20}+\frac{18\!\cdots\!44}{18\!\cdots\!07}a^{19}+\frac{55\!\cdots\!87}{14\!\cdots\!87}a^{18}-\frac{28\!\cdots\!17}{14\!\cdots\!87}a^{17}+\frac{83\!\cdots\!90}{14\!\cdots\!87}a^{16}+\frac{21\!\cdots\!22}{26\!\cdots\!17}a^{15}-\frac{16\!\cdots\!43}{14\!\cdots\!87}a^{14}+\frac{32\!\cdots\!40}{43\!\cdots\!61}a^{13}+\frac{76\!\cdots\!75}{43\!\cdots\!61}a^{12}-\frac{41\!\cdots\!54}{43\!\cdots\!61}a^{11}+\frac{69\!\cdots\!56}{43\!\cdots\!61}a^{10}-\frac{12\!\cdots\!83}{14\!\cdots\!87}a^{9}-\frac{95\!\cdots\!82}{43\!\cdots\!61}a^{8}+\frac{37\!\cdots\!49}{43\!\cdots\!61}a^{7}-\frac{26\!\cdots\!56}{43\!\cdots\!61}a^{6}+\frac{21\!\cdots\!67}{43\!\cdots\!61}a^{5}-\frac{32\!\cdots\!37}{14\!\cdots\!87}a^{4}+\frac{60\!\cdots\!47}{43\!\cdots\!61}a^{3}-\frac{42\!\cdots\!60}{62\!\cdots\!69}a^{2}+\frac{26\!\cdots\!14}{81\!\cdots\!09}a-\frac{47\!\cdots\!34}{49\!\cdots\!73}$, $\frac{30\!\cdots\!81}{43\!\cdots\!61}a^{31}-\frac{10\!\cdots\!59}{14\!\cdots\!87}a^{30}-\frac{61\!\cdots\!86}{43\!\cdots\!61}a^{29}+\frac{13\!\cdots\!97}{43\!\cdots\!61}a^{28}+\frac{76\!\cdots\!71}{43\!\cdots\!61}a^{27}-\frac{25\!\cdots\!95}{14\!\cdots\!87}a^{26}-\frac{43\!\cdots\!14}{18\!\cdots\!07}a^{25}-\frac{32\!\cdots\!99}{14\!\cdots\!87}a^{24}+\frac{16\!\cdots\!87}{43\!\cdots\!61}a^{23}+\frac{22\!\cdots\!97}{18\!\cdots\!07}a^{22}-\frac{10\!\cdots\!31}{43\!\cdots\!61}a^{21}+\frac{74\!\cdots\!37}{43\!\cdots\!61}a^{20}+\frac{17\!\cdots\!32}{18\!\cdots\!07}a^{19}+\frac{95\!\cdots\!22}{43\!\cdots\!61}a^{18}-\frac{58\!\cdots\!12}{14\!\cdots\!87}a^{17}-\frac{62\!\cdots\!60}{43\!\cdots\!61}a^{16}+\frac{65\!\cdots\!02}{18\!\cdots\!07}a^{15}+\frac{11\!\cdots\!40}{43\!\cdots\!61}a^{14}-\frac{30\!\cdots\!19}{43\!\cdots\!61}a^{13}+\frac{93\!\cdots\!97}{14\!\cdots\!87}a^{12}-\frac{36\!\cdots\!23}{14\!\cdots\!87}a^{11}-\frac{10\!\cdots\!03}{43\!\cdots\!61}a^{10}+\frac{44\!\cdots\!13}{43\!\cdots\!61}a^{9}-\frac{38\!\cdots\!84}{43\!\cdots\!61}a^{8}+\frac{30\!\cdots\!52}{43\!\cdots\!61}a^{7}+\frac{69\!\cdots\!35}{14\!\cdots\!87}a^{6}-\frac{14\!\cdots\!92}{43\!\cdots\!61}a^{5}+\frac{16\!\cdots\!12}{43\!\cdots\!61}a^{4}-\frac{62\!\cdots\!64}{43\!\cdots\!61}a^{3}+\frac{60\!\cdots\!59}{18\!\cdots\!07}a^{2}-\frac{52\!\cdots\!27}{81\!\cdots\!09}a+\frac{33\!\cdots\!01}{11\!\cdots\!61}$, $\frac{24\!\cdots\!82}{43\!\cdots\!61}a^{31}-\frac{54\!\cdots\!13}{43\!\cdots\!61}a^{30}-\frac{39\!\cdots\!66}{43\!\cdots\!61}a^{29}+\frac{14\!\cdots\!97}{43\!\cdots\!61}a^{28}+\frac{13\!\cdots\!78}{14\!\cdots\!87}a^{27}-\frac{33\!\cdots\!46}{43\!\cdots\!61}a^{26}-\frac{14\!\cdots\!66}{81\!\cdots\!09}a^{25}+\frac{18\!\cdots\!66}{14\!\cdots\!87}a^{24}+\frac{12\!\cdots\!80}{43\!\cdots\!61}a^{23}-\frac{40\!\cdots\!69}{18\!\cdots\!07}a^{22}-\frac{88\!\cdots\!46}{60\!\cdots\!91}a^{21}+\frac{16\!\cdots\!12}{14\!\cdots\!87}a^{20}+\frac{34\!\cdots\!00}{62\!\cdots\!69}a^{19}+\frac{59\!\cdots\!34}{43\!\cdots\!61}a^{18}-\frac{67\!\cdots\!84}{14\!\cdots\!87}a^{17}-\frac{22\!\cdots\!95}{43\!\cdots\!61}a^{16}+\frac{50\!\cdots\!09}{18\!\cdots\!07}a^{15}-\frac{66\!\cdots\!25}{14\!\cdots\!87}a^{14}-\frac{11\!\cdots\!55}{43\!\cdots\!61}a^{13}+\frac{11\!\cdots\!38}{43\!\cdots\!61}a^{12}-\frac{28\!\cdots\!38}{20\!\cdots\!97}a^{11}+\frac{35\!\cdots\!99}{43\!\cdots\!61}a^{10}+\frac{11\!\cdots\!14}{43\!\cdots\!61}a^{9}-\frac{16\!\cdots\!23}{43\!\cdots\!61}a^{8}+\frac{30\!\cdots\!28}{43\!\cdots\!61}a^{7}+\frac{81\!\cdots\!19}{43\!\cdots\!61}a^{6}-\frac{20\!\cdots\!94}{43\!\cdots\!61}a^{5}+\frac{55\!\cdots\!29}{14\!\cdots\!87}a^{4}-\frac{16\!\cdots\!73}{43\!\cdots\!61}a^{3}+\frac{16\!\cdots\!08}{18\!\cdots\!07}a^{2}-\frac{83\!\cdots\!66}{81\!\cdots\!09}a+\frac{20\!\cdots\!51}{35\!\cdots\!83}$, $\frac{79\!\cdots\!22}{14\!\cdots\!87}a^{31}-\frac{25\!\cdots\!23}{14\!\cdots\!87}a^{30}-\frac{30\!\cdots\!37}{43\!\cdots\!61}a^{29}+\frac{56\!\cdots\!49}{14\!\cdots\!87}a^{28}+\frac{74\!\cdots\!17}{14\!\cdots\!87}a^{27}-\frac{17\!\cdots\!70}{14\!\cdots\!87}a^{26}-\frac{29\!\cdots\!31}{18\!\cdots\!07}a^{25}+\frac{71\!\cdots\!85}{43\!\cdots\!61}a^{24}+\frac{31\!\cdots\!97}{11\!\cdots\!79}a^{23}-\frac{29\!\cdots\!72}{62\!\cdots\!69}a^{22}-\frac{38\!\cdots\!41}{43\!\cdots\!61}a^{21}+\frac{81\!\cdots\!18}{43\!\cdots\!61}a^{20}+\frac{61\!\cdots\!96}{18\!\cdots\!07}a^{19}+\frac{45\!\cdots\!61}{43\!\cdots\!61}a^{18}-\frac{23\!\cdots\!91}{43\!\cdots\!61}a^{17}+\frac{29\!\cdots\!13}{43\!\cdots\!61}a^{16}+\frac{20\!\cdots\!37}{81\!\cdots\!09}a^{15}-\frac{12\!\cdots\!42}{43\!\cdots\!61}a^{14}+\frac{14\!\cdots\!19}{14\!\cdots\!87}a^{13}+\frac{53\!\cdots\!56}{43\!\cdots\!61}a^{12}-\frac{11\!\cdots\!48}{43\!\cdots\!61}a^{11}+\frac{52\!\cdots\!42}{14\!\cdots\!87}a^{10}-\frac{64\!\cdots\!44}{43\!\cdots\!61}a^{9}-\frac{74\!\cdots\!13}{43\!\cdots\!61}a^{8}+\frac{23\!\cdots\!49}{10\!\cdots\!21}a^{7}-\frac{37\!\cdots\!44}{43\!\cdots\!61}a^{6}+\frac{10\!\cdots\!75}{14\!\cdots\!87}a^{5}-\frac{72\!\cdots\!44}{14\!\cdots\!87}a^{4}+\frac{82\!\cdots\!85}{43\!\cdots\!61}a^{3}-\frac{73\!\cdots\!84}{62\!\cdots\!69}a^{2}+\frac{32\!\cdots\!32}{81\!\cdots\!09}a+\frac{33\!\cdots\!13}{11\!\cdots\!61}$, $\frac{27\!\cdots\!69}{43\!\cdots\!61}a^{31}-\frac{10\!\cdots\!27}{43\!\cdots\!61}a^{30}-\frac{29\!\cdots\!64}{43\!\cdots\!61}a^{29}+\frac{22\!\cdots\!72}{43\!\cdots\!61}a^{28}+\frac{12\!\cdots\!23}{43\!\cdots\!61}a^{27}-\frac{78\!\cdots\!43}{43\!\cdots\!61}a^{26}-\frac{32\!\cdots\!94}{18\!\cdots\!07}a^{25}+\frac{13\!\cdots\!91}{43\!\cdots\!61}a^{24}+\frac{41\!\cdots\!34}{14\!\cdots\!87}a^{23}-\frac{47\!\cdots\!55}{62\!\cdots\!69}a^{22}-\frac{96\!\cdots\!20}{14\!\cdots\!87}a^{21}+\frac{43\!\cdots\!00}{14\!\cdots\!87}a^{20}+\frac{13\!\cdots\!98}{62\!\cdots\!69}a^{19}+\frac{39\!\cdots\!14}{43\!\cdots\!61}a^{18}-\frac{10\!\cdots\!13}{14\!\cdots\!87}a^{17}+\frac{22\!\cdots\!85}{43\!\cdots\!61}a^{16}+\frac{53\!\cdots\!98}{18\!\cdots\!07}a^{15}-\frac{23\!\cdots\!21}{43\!\cdots\!61}a^{14}+\frac{14\!\cdots\!90}{43\!\cdots\!61}a^{13}+\frac{21\!\cdots\!72}{14\!\cdots\!87}a^{12}-\frac{20\!\cdots\!51}{43\!\cdots\!61}a^{11}+\frac{27\!\cdots\!55}{43\!\cdots\!61}a^{10}-\frac{55\!\cdots\!59}{14\!\cdots\!87}a^{9}-\frac{77\!\cdots\!81}{43\!\cdots\!61}a^{8}+\frac{21\!\cdots\!38}{43\!\cdots\!61}a^{7}-\frac{12\!\cdots\!22}{43\!\cdots\!61}a^{6}+\frac{10\!\cdots\!47}{14\!\cdots\!87}a^{5}-\frac{18\!\cdots\!09}{43\!\cdots\!61}a^{4}+\frac{71\!\cdots\!41}{14\!\cdots\!87}a^{3}-\frac{17\!\cdots\!91}{62\!\cdots\!69}a^{2}+\frac{16\!\cdots\!47}{27\!\cdots\!03}a+\frac{67\!\cdots\!92}{35\!\cdots\!83}$
| 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: | \( 93528182489607.3 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 93528182489607.3 \cdot 3480}{10\cdot\sqrt{847622907049404564614012839370162176000000000000000000000000}}\cr\approx \mathstrut & 0.208592308290717 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_4^2$ (as 32T36):
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2\times C_4^2$ |
| Character table for $C_2\times C_4^2$ |
Intermediate fields
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 | R | ${\href{/padicField/3.4.0.1}{4} }^{8}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $4$ | $4$ | $44$ | |||
| Deg $16$ | $4$ | $4$ | $44$ | ||||
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ | |
|
\(11\)
| 11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |