Number field 32.0.170615961014359235699341036323165930807590484619140625.1

• Label: 32.0.170...625.1
• Degree: $32$
• Signature: $[0, 16]$
• Discriminant: $1.706\times 10^{53}$
• Root discriminant: \(46.08\)
• Ramified primes: $5,127$
• Class number: $3$ (GRH)
• Class group: [3] (GRH)
• Galois group: $C_8.A_4$ (as 32T402)

Normalized defining polynomial

x^32 - 9*x^31 + 200*x^29 - 221*x^28 - 2631*x^27 + 4402*x^26 + 19766*x^25 - 27449*x^24 - 173128*x^23 + 338537*x^22 + 473336*x^21 - 1004401*x^20 - 3368948*x^19 + 11453717*x^18 - 8139429*x^17 - 16834230*x^16 - 597015*x^15 + 195460786*x^14 - 382818671*x^13 - 27697040*x^12 + 420160443*x^11 + 1322753872*x^10 - 3917503795*x^9 + 2111159122*x^8 + 2050419221*x^7 + 1497324960*x^6 - 10922257252*x^5 + 11658561303*x^4 - 2839464368*x^3 - 2153549419*x^2 + 711827200*x + 313809121

Invariants

Degree:		$32$
Signature:		$[0, 16]$
Discriminant:		\(170615961014359235699341036323165930807590484619140625\) \(\medspace = 5^{28}\cdot 127^{16}\)
Root discriminant:		\(46.08\)
Galois root discriminant:		$5^{7/8}127^{2/3}\approx 103.3081460316515$
Ramified primes:		\(5\), \(127\)
Discriminant root field:		\(\Q\)
$\card{ \Aut(K/\Q) }$:		$8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{23}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{23}-\frac{1}{2}a^{22}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{21}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{22}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{23}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{38}a^{29}+\frac{3}{38}a^{28}+\frac{1}{38}a^{27}+\frac{3}{19}a^{26}+\frac{7}{38}a^{25}-\frac{4}{19}a^{24}-\frac{3}{38}a^{23}-\frac{1}{38}a^{22}-\frac{5}{38}a^{21}-\frac{7}{38}a^{20}-\frac{3}{38}a^{19}-\frac{1}{2}a^{18}+\frac{5}{19}a^{17}-\frac{15}{38}a^{16}+\frac{3}{19}a^{15}-\frac{7}{19}a^{14}+\frac{7}{19}a^{13}-\frac{1}{19}a^{12}-\frac{8}{19}a^{11}+\frac{1}{38}a^{10}-\frac{2}{19}a^{9}-\frac{15}{38}a^{8}-\frac{11}{38}a^{7}+\frac{13}{38}a^{6}+\frac{2}{19}a^{5}-\frac{8}{19}a^{4}+\frac{5}{38}a^{3}+\frac{5}{19}a^{2}-\frac{2}{19}a+\frac{4}{19}$, $\frac{1}{418}a^{30}+\frac{3}{418}a^{29}+\frac{10}{209}a^{28}-\frac{35}{209}a^{27}-\frac{25}{209}a^{26}-\frac{65}{418}a^{25}-\frac{1}{19}a^{24}+\frac{85}{209}a^{23}+\frac{83}{209}a^{22}+\frac{82}{209}a^{21}+\frac{149}{418}a^{20}-\frac{1}{11}a^{19}+\frac{29}{418}a^{18}+\frac{175}{418}a^{17}+\frac{63}{418}a^{16}+\frac{50}{209}a^{15}+\frac{109}{418}a^{14}+\frac{169}{418}a^{13}+\frac{79}{418}a^{12}+\frac{7}{38}a^{11}+\frac{15}{418}a^{10}+\frac{40}{209}a^{9}+\frac{27}{418}a^{8}+\frac{203}{418}a^{7}+\frac{59}{209}a^{6}-\frac{35}{418}a^{5}+\frac{50}{209}a^{4}-\frac{52}{209}a^{3}-\frac{61}{418}a^{2}-\frac{53}{209}a+\frac{3}{22}$, $\frac{1}{70\!\cdots\!58}a^{31}-\frac{50\!\cdots\!01}{70\!\cdots\!58}a^{30}+\frac{41\!\cdots\!70}{35\!\cdots\!79}a^{29}+\frac{72\!\cdots\!70}{35\!\cdots\!79}a^{28}+\frac{23\!\cdots\!52}{11\!\cdots\!09}a^{27}-\frac{63\!\cdots\!21}{35\!\cdots\!79}a^{26}-\frac{12\!\cdots\!03}{70\!\cdots\!58}a^{25}+\frac{60\!\cdots\!45}{35\!\cdots\!79}a^{24}-\frac{79\!\cdots\!71}{37\!\cdots\!82}a^{23}+\frac{78\!\cdots\!35}{70\!\cdots\!58}a^{22}+\frac{86\!\cdots\!66}{35\!\cdots\!79}a^{21}+\frac{16\!\cdots\!85}{35\!\cdots\!79}a^{20}+\frac{16\!\cdots\!27}{70\!\cdots\!58}a^{19}+\frac{12\!\cdots\!54}{35\!\cdots\!79}a^{18}-\frac{12\!\cdots\!45}{35\!\cdots\!79}a^{17}+\frac{53\!\cdots\!75}{35\!\cdots\!79}a^{16}+\frac{30\!\cdots\!39}{70\!\cdots\!58}a^{15}-\frac{25\!\cdots\!83}{70\!\cdots\!58}a^{14}+\frac{22\!\cdots\!43}{31\!\cdots\!89}a^{13}-\frac{87\!\cdots\!33}{70\!\cdots\!58}a^{12}+\frac{54\!\cdots\!87}{35\!\cdots\!79}a^{11}+\frac{76\!\cdots\!20}{35\!\cdots\!79}a^{10}-\frac{91\!\cdots\!84}{35\!\cdots\!79}a^{9}-\frac{17\!\cdots\!79}{70\!\cdots\!58}a^{8}+\frac{40\!\cdots\!37}{31\!\cdots\!89}a^{7}-\frac{55\!\cdots\!91}{70\!\cdots\!58}a^{6}-\frac{29\!\cdots\!91}{70\!\cdots\!58}a^{5}-\frac{13\!\cdots\!45}{63\!\cdots\!78}a^{4}+\frac{36\!\cdots\!89}{70\!\cdots\!58}a^{3}+\frac{11\!\cdots\!83}{70\!\cdots\!58}a^{2}+\frac{87\!\cdots\!39}{63\!\cdots\!78}a+\frac{10\!\cdots\!93}{18\!\cdots\!41}$

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

Unit group

Rank:		$15$
Torsion generator:		\( \frac{2251233301023753864290121373880450245147972155516126734819246581451868446821382038261525463262190275902944857340481}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{31} - \frac{16835458649938533614106570967064377242975920578647151241149123628557657031707751528018003338135924819882433810229809}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{30} - \frac{25636892373650060057736403579186663646059872128402564326161056237188586398361502310508926541978782337989079623537779}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{29} + \frac{411348222381656123189828499719676040394000560913680153209057029289699094376252465627268958071915264325129208154696970}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{28} + \frac{4154689629891382659459474002283323958298956626047513759192006035828691627491854518734616166686534001139502591011033}{32149374026425062715409208228376207384171074320307444206347384291618387326239214977364542718757566472757497694757900029} a^{27} - \frac{5729852957340056528004095201572254248795721113537082172752115172413651068170505470227824908263865447663662798513914899}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{26} + \frac{5672938350678071402026379075639741823904477516092253812255992879504827909805011021268206301261784749048355762074056}{4768567439326205474534380167845274779470350736504931915774013937991244053174237628221535044409017036629102528887535411} a^{25} + \frac{46341307209608985297240138152544386310966468903098329097833803625556347921015239194426798788463840721001218046473594797}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{24} + \frac{8771176401782551765340761894641006741922301108883483159760168754260194045587454977884694790196822116624721165192373821}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{23} - \frac{376709060720830953521040561120479198924483869528599616932132445560805811489126960828918212239027671632047137202488682147}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{22} + \frac{188366973240082804968981161622946393665897766847327522341247598622257132036319111743045815899586358365992292096300634628}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{21} + \frac{1354493869336536344289724588451015090368110278110408007165006419821999182301156149529888564310611649988961660975749569862}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{20} - \frac{198260853335725962685529206207380165108061303507490856015351506475737701185416953100951518696758944916730525385584308510}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{19} - \frac{7893965327105933746980337059842570259758632950597247924090774036492486083760426704357216484811773174314691543718600676097}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{18} + \frac{13762675699941323442524675349948435396012782354856823757021941840014668207079654004422677371187858959707621803645462869269}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{17} + \frac{2666812507278624337999709929964969393842352531400387992376160860165115398590815005251688471246818840980726462102530338208}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{16} - \frac{33884329902622160651807032983467231665367067609459267186603171398798146616043033853337278252922277173173770179543605937738}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{15} - \frac{52980545116736754474704970528341621991596339654793072738818087139388109109308599437089599147725274766889180357094968784918}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{14} + \frac{359562841782018965426963305132127151740712297820276767828422317080662697221134900670026402611616322998908200034662496496182}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{13} - \frac{28546466905352062524755309750282558591015487034568163899089071263459702870795242052623537288894154575193020065285876458962}{90602781347197904016153223189060220809936663993593706399706264821833637010310514936209165843771323695952948048863172809} a^{12} - \frac{542144925796005455505092022204476101303194543635157108139166173101576570001716483810062545301645647011374579524024013935901}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{11} + \frac{120738629273373348254800480170498357550430457316046690845801899412946495846449988657483371840360033385446914559351600207833}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{10} + \frac{3165202330116868036570234090141594486774450051424064051860554956408700286516825433173272133443866446110023334130538949311074}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{9} - \frac{3998113549028306643642103335473462707786701624485925637297912435044138140947228291310523221298036476893003340487934680491552}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{8} - \frac{1349530479843105243775649375215173506649856865238021395200072450676522472829458521973503674660794377968045144311602122367949}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{7} + \frac{2569301802897729828113871552631769692158270375725780767964504860528836358969336871964662118706731061774045127798960934954657}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{6} + \frac{7295790454282644771660072320839886468704473582041051643989635256682140570700964394254810244627396029989037927920198616566459}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{5} - \frac{13481025598585115306440225562049721562627364177235931126685025384903833826281255764352443268261253018146113364457023381734280}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{4} + \frac{5684416368387016013052656777984778653344947414515965604606989011872754072066092786896081478253719917523304177604572532984711}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{3} + \frac{2298331428341924843942439847276976592165013318736886029986276284937317921370809277818076401964586135784877438108393545763604}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a^{2} - \frac{1353956921343978693232465822880583045862985245936820698948338029528100094929279745598193980443911963033371322160506082642512}{996630594819176944177685455079662428909303303929530770396768913040170007113415664298300824281484560655482428537494900899} a - \frac{416687890933487919487196092734471869326479372701430947081006869794606671425936527358733966822277683603035250151071831854}{897057241061365386298546764248120998118184792015779271284220443780531059508024900358506592512587363326266812364981909} \)  (order $10$)
Fundamental units:		$\frac{24\!\cdots\!59}{99\!\cdots\!99}a^{31}-\frac{18\!\cdots\!44}{99\!\cdots\!99}a^{30}-\frac{28\!\cdots\!25}{99\!\cdots\!99}a^{29}+\frac{44\!\cdots\!94}{99\!\cdots\!99}a^{28}+\frac{51\!\cdots\!57}{32\!\cdots\!29}a^{27}-\frac{62\!\cdots\!73}{99\!\cdots\!99}a^{26}+\frac{55\!\cdots\!94}{52\!\cdots\!21}a^{25}+\frac{50\!\cdots\!31}{99\!\cdots\!99}a^{24}+\frac{11\!\cdots\!61}{99\!\cdots\!99}a^{23}-\frac{41\!\cdots\!85}{99\!\cdots\!99}a^{22}+\frac{17\!\cdots\!56}{90\!\cdots\!09}a^{21}+\frac{14\!\cdots\!20}{99\!\cdots\!99}a^{20}-\frac{16\!\cdots\!14}{99\!\cdots\!99}a^{19}-\frac{86\!\cdots\!65}{99\!\cdots\!99}a^{18}+\frac{14\!\cdots\!57}{99\!\cdots\!99}a^{17}+\frac{31\!\cdots\!71}{99\!\cdots\!99}a^{16}-\frac{36\!\cdots\!10}{99\!\cdots\!99}a^{15}-\frac{59\!\cdots\!73}{99\!\cdots\!99}a^{14}+\frac{39\!\cdots\!88}{99\!\cdots\!99}a^{13}-\frac{33\!\cdots\!76}{99\!\cdots\!99}a^{12}-\frac{59\!\cdots\!31}{99\!\cdots\!99}a^{11}+\frac{10\!\cdots\!16}{99\!\cdots\!99}a^{10}+\frac{34\!\cdots\!80}{99\!\cdots\!99}a^{9}-\frac{42\!\cdots\!64}{99\!\cdots\!99}a^{8}-\frac{15\!\cdots\!47}{99\!\cdots\!99}a^{7}+\frac{27\!\cdots\!73}{99\!\cdots\!99}a^{6}+\frac{72\!\cdots\!81}{90\!\cdots\!09}a^{5}-\frac{14\!\cdots\!68}{99\!\cdots\!99}a^{4}+\frac{60\!\cdots\!63}{99\!\cdots\!99}a^{3}+\frac{24\!\cdots\!17}{99\!\cdots\!99}a^{2}-\frac{14\!\cdots\!28}{99\!\cdots\!99}a-\frac{48\!\cdots\!31}{98\!\cdots\!99}$, $\frac{81\!\cdots\!84}{58\!\cdots\!69}a^{31}-\frac{55\!\cdots\!85}{52\!\cdots\!79}a^{30}-\frac{86\!\cdots\!08}{52\!\cdots\!79}a^{29}+\frac{14\!\cdots\!88}{58\!\cdots\!69}a^{28}+\frac{16\!\cdots\!23}{18\!\cdots\!99}a^{27}-\frac{20\!\cdots\!83}{58\!\cdots\!69}a^{26}+\frac{19\!\cdots\!97}{30\!\cdots\!51}a^{25}+\frac{16\!\cdots\!56}{58\!\cdots\!69}a^{24}+\frac{36\!\cdots\!96}{58\!\cdots\!69}a^{23}-\frac{13\!\cdots\!61}{58\!\cdots\!69}a^{22}+\frac{64\!\cdots\!61}{58\!\cdots\!69}a^{21}+\frac{48\!\cdots\!99}{58\!\cdots\!69}a^{20}-\frac{53\!\cdots\!22}{52\!\cdots\!79}a^{19}-\frac{25\!\cdots\!07}{52\!\cdots\!79}a^{18}+\frac{44\!\cdots\!43}{52\!\cdots\!79}a^{17}+\frac{10\!\cdots\!59}{58\!\cdots\!69}a^{16}-\frac{12\!\cdots\!61}{58\!\cdots\!69}a^{15}-\frac{19\!\cdots\!20}{58\!\cdots\!69}a^{14}+\frac{12\!\cdots\!70}{58\!\cdots\!69}a^{13}-\frac{11\!\cdots\!97}{58\!\cdots\!69}a^{12}-\frac{19\!\cdots\!39}{58\!\cdots\!69}a^{11}+\frac{37\!\cdots\!35}{58\!\cdots\!69}a^{10}+\frac{11\!\cdots\!05}{58\!\cdots\!69}a^{9}-\frac{14\!\cdots\!68}{58\!\cdots\!69}a^{8}-\frac{49\!\cdots\!08}{58\!\cdots\!69}a^{7}+\frac{90\!\cdots\!57}{58\!\cdots\!69}a^{6}+\frac{26\!\cdots\!92}{58\!\cdots\!69}a^{5}-\frac{48\!\cdots\!17}{58\!\cdots\!69}a^{4}+\frac{20\!\cdots\!85}{58\!\cdots\!69}a^{3}+\frac{74\!\cdots\!32}{52\!\cdots\!79}a^{2}-\frac{47\!\cdots\!41}{58\!\cdots\!69}a-\frac{16\!\cdots\!79}{57\!\cdots\!69}$, $\frac{70\!\cdots\!27}{58\!\cdots\!69}a^{31}-\frac{53\!\cdots\!69}{58\!\cdots\!69}a^{30}-\frac{73\!\cdots\!91}{58\!\cdots\!69}a^{29}+\frac{13\!\cdots\!29}{58\!\cdots\!69}a^{28}+\frac{77\!\cdots\!33}{18\!\cdots\!99}a^{27}-\frac{18\!\cdots\!19}{58\!\cdots\!69}a^{26}+\frac{28\!\cdots\!54}{27\!\cdots\!41}a^{25}+\frac{14\!\cdots\!00}{58\!\cdots\!69}a^{24}+\frac{97\!\cdots\!09}{58\!\cdots\!69}a^{23}-\frac{12\!\cdots\!15}{58\!\cdots\!69}a^{22}+\frac{71\!\cdots\!27}{58\!\cdots\!69}a^{21}+\frac{43\!\cdots\!10}{58\!\cdots\!69}a^{20}-\frac{10\!\cdots\!74}{58\!\cdots\!69}a^{19}-\frac{25\!\cdots\!90}{58\!\cdots\!69}a^{18}+\frac{45\!\cdots\!71}{58\!\cdots\!69}a^{17}+\frac{67\!\cdots\!55}{58\!\cdots\!69}a^{16}-\frac{11\!\cdots\!43}{58\!\cdots\!69}a^{15}-\frac{15\!\cdots\!97}{58\!\cdots\!69}a^{14}+\frac{11\!\cdots\!24}{58\!\cdots\!69}a^{13}-\frac{98\!\cdots\!94}{52\!\cdots\!79}a^{12}-\frac{17\!\cdots\!08}{58\!\cdots\!69}a^{11}+\frac{60\!\cdots\!55}{58\!\cdots\!69}a^{10}+\frac{10\!\cdots\!31}{58\!\cdots\!69}a^{9}-\frac{13\!\cdots\!96}{58\!\cdots\!69}a^{8}-\frac{42\!\cdots\!47}{58\!\cdots\!69}a^{7}+\frac{91\!\cdots\!91}{58\!\cdots\!69}a^{6}+\frac{23\!\cdots\!08}{58\!\cdots\!69}a^{5}-\frac{44\!\cdots\!81}{58\!\cdots\!69}a^{4}+\frac{19\!\cdots\!66}{58\!\cdots\!69}a^{3}+\frac{76\!\cdots\!70}{58\!\cdots\!69}a^{2}-\frac{45\!\cdots\!89}{58\!\cdots\!69}a-\frac{14\!\cdots\!84}{52\!\cdots\!79}$, $\frac{82\!\cdots\!77}{22\!\cdots\!18}a^{31}-\frac{61\!\cdots\!13}{22\!\cdots\!18}a^{30}-\frac{92\!\cdots\!41}{22\!\cdots\!18}a^{29}+\frac{15\!\cdots\!51}{22\!\cdots\!18}a^{28}+\frac{20\!\cdots\!87}{10\!\cdots\!19}a^{27}-\frac{10\!\cdots\!23}{11\!\cdots\!09}a^{26}+\frac{22\!\cdots\!93}{11\!\cdots\!09}a^{25}+\frac{16\!\cdots\!95}{22\!\cdots\!18}a^{24}+\frac{29\!\cdots\!19}{22\!\cdots\!18}a^{23}-\frac{13\!\cdots\!43}{22\!\cdots\!18}a^{22}+\frac{35\!\cdots\!55}{11\!\cdots\!09}a^{21}+\frac{49\!\cdots\!51}{22\!\cdots\!18}a^{20}-\frac{39\!\cdots\!05}{11\!\cdots\!09}a^{19}-\frac{15\!\cdots\!41}{11\!\cdots\!22}a^{18}+\frac{25\!\cdots\!61}{11\!\cdots\!09}a^{17}+\frac{84\!\cdots\!87}{20\!\cdots\!38}a^{16}-\frac{61\!\cdots\!76}{11\!\cdots\!09}a^{15}-\frac{87\!\cdots\!83}{10\!\cdots\!19}a^{14}+\frac{13\!\cdots\!19}{22\!\cdots\!18}a^{13}-\frac{11\!\cdots\!15}{22\!\cdots\!18}a^{12}-\frac{98\!\cdots\!73}{11\!\cdots\!09}a^{11}+\frac{46\!\cdots\!11}{22\!\cdots\!18}a^{10}+\frac{57\!\cdots\!50}{11\!\cdots\!09}a^{9}-\frac{14\!\cdots\!03}{22\!\cdots\!18}a^{8}-\frac{24\!\cdots\!53}{11\!\cdots\!09}a^{7}+\frac{94\!\cdots\!53}{22\!\cdots\!18}a^{6}+\frac{26\!\cdots\!33}{22\!\cdots\!18}a^{5}-\frac{49\!\cdots\!57}{22\!\cdots\!18}a^{4}+\frac{21\!\cdots\!55}{22\!\cdots\!18}a^{3}+\frac{83\!\cdots\!95}{22\!\cdots\!18}a^{2}-\frac{25\!\cdots\!63}{11\!\cdots\!09}a-\frac{16\!\cdots\!67}{22\!\cdots\!18}$, $\frac{17\!\cdots\!37}{70\!\cdots\!58}a^{31}-\frac{63\!\cdots\!65}{35\!\cdots\!79}a^{30}-\frac{20\!\cdots\!93}{70\!\cdots\!58}a^{29}+\frac{30\!\cdots\!43}{70\!\cdots\!58}a^{28}+\frac{18\!\cdots\!21}{11\!\cdots\!09}a^{27}-\frac{21\!\cdots\!12}{35\!\cdots\!79}a^{26}+\frac{67\!\cdots\!93}{70\!\cdots\!58}a^{25}+\frac{34\!\cdots\!71}{70\!\cdots\!58}a^{24}+\frac{41\!\cdots\!77}{35\!\cdots\!79}a^{23}-\frac{14\!\cdots\!32}{35\!\cdots\!79}a^{22}+\frac{64\!\cdots\!34}{35\!\cdots\!79}a^{21}+\frac{10\!\cdots\!79}{70\!\cdots\!58}a^{20}-\frac{10\!\cdots\!93}{70\!\cdots\!58}a^{19}-\frac{59\!\cdots\!09}{70\!\cdots\!58}a^{18}+\frac{10\!\cdots\!05}{70\!\cdots\!58}a^{17}+\frac{58\!\cdots\!44}{18\!\cdots\!41}a^{16}-\frac{12\!\cdots\!20}{35\!\cdots\!79}a^{15}-\frac{40\!\cdots\!53}{70\!\cdots\!58}a^{14}+\frac{26\!\cdots\!97}{70\!\cdots\!58}a^{13}-\frac{11\!\cdots\!59}{35\!\cdots\!79}a^{12}-\frac{40\!\cdots\!63}{70\!\cdots\!58}a^{11}+\frac{69\!\cdots\!17}{70\!\cdots\!58}a^{10}+\frac{23\!\cdots\!29}{70\!\cdots\!58}a^{9}-\frac{15\!\cdots\!87}{37\!\cdots\!82}a^{8}-\frac{52\!\cdots\!68}{35\!\cdots\!79}a^{7}+\frac{18\!\cdots\!37}{70\!\cdots\!58}a^{6}+\frac{27\!\cdots\!33}{35\!\cdots\!79}a^{5}-\frac{49\!\cdots\!62}{35\!\cdots\!79}a^{4}+\frac{41\!\cdots\!97}{70\!\cdots\!58}a^{3}+\frac{16\!\cdots\!49}{70\!\cdots\!58}a^{2}-\frac{48\!\cdots\!50}{35\!\cdots\!79}a-\frac{17\!\cdots\!95}{36\!\cdots\!82}$, $\frac{11\!\cdots\!29}{70\!\cdots\!58}a^{31}-\frac{54\!\cdots\!63}{70\!\cdots\!58}a^{30}-\frac{14\!\cdots\!22}{35\!\cdots\!79}a^{29}+\frac{13\!\cdots\!49}{70\!\cdots\!58}a^{28}+\frac{14\!\cdots\!31}{22\!\cdots\!18}a^{27}-\frac{87\!\cdots\!48}{35\!\cdots\!79}a^{26}-\frac{23\!\cdots\!15}{35\!\cdots\!79}a^{25}+\frac{59\!\cdots\!30}{35\!\cdots\!79}a^{24}+\frac{41\!\cdots\!03}{70\!\cdots\!58}a^{23}-\frac{80\!\cdots\!95}{70\!\cdots\!58}a^{22}-\frac{16\!\cdots\!85}{70\!\cdots\!58}a^{21}+\frac{10\!\cdots\!73}{31\!\cdots\!89}a^{20}+\frac{74\!\cdots\!47}{70\!\cdots\!58}a^{19}-\frac{50\!\cdots\!91}{18\!\cdots\!41}a^{18}+\frac{54\!\cdots\!04}{35\!\cdots\!79}a^{17}+\frac{47\!\cdots\!57}{70\!\cdots\!58}a^{16}-\frac{37\!\cdots\!21}{70\!\cdots\!58}a^{15}-\frac{40\!\cdots\!13}{70\!\cdots\!58}a^{14}+\frac{69\!\cdots\!33}{70\!\cdots\!58}a^{13}+\frac{61\!\cdots\!69}{70\!\cdots\!58}a^{12}-\frac{55\!\cdots\!76}{35\!\cdots\!79}a^{11}-\frac{17\!\cdots\!19}{35\!\cdots\!79}a^{10}+\frac{65\!\cdots\!03}{70\!\cdots\!58}a^{9}+\frac{28\!\cdots\!03}{33\!\cdots\!62}a^{8}-\frac{27\!\cdots\!03}{35\!\cdots\!79}a^{7}-\frac{33\!\cdots\!22}{35\!\cdots\!79}a^{6}+\frac{18\!\cdots\!49}{70\!\cdots\!58}a^{5}-\frac{52\!\cdots\!64}{35\!\cdots\!79}a^{4}-\frac{67\!\cdots\!06}{35\!\cdots\!79}a^{3}+\frac{11\!\cdots\!83}{35\!\cdots\!79}a^{2}+\frac{13\!\cdots\!83}{70\!\cdots\!58}a-\frac{14\!\cdots\!83}{69\!\cdots\!58}$, $\frac{15\!\cdots\!33}{70\!\cdots\!58}a^{31}-\frac{11\!\cdots\!71}{70\!\cdots\!58}a^{30}-\frac{46\!\cdots\!01}{18\!\cdots\!41}a^{29}+\frac{74\!\cdots\!70}{18\!\cdots\!41}a^{28}+\frac{74\!\cdots\!06}{59\!\cdots\!11}a^{27}-\frac{39\!\cdots\!47}{70\!\cdots\!58}a^{26}+\frac{41\!\cdots\!28}{35\!\cdots\!79}a^{25}+\frac{15\!\cdots\!05}{35\!\cdots\!79}a^{24}+\frac{28\!\cdots\!44}{35\!\cdots\!79}a^{23}-\frac{12\!\cdots\!91}{35\!\cdots\!79}a^{22}+\frac{13\!\cdots\!45}{70\!\cdots\!58}a^{21}+\frac{46\!\cdots\!13}{35\!\cdots\!79}a^{20}-\frac{14\!\cdots\!71}{70\!\cdots\!58}a^{19}-\frac{54\!\cdots\!89}{70\!\cdots\!58}a^{18}+\frac{95\!\cdots\!41}{70\!\cdots\!58}a^{17}+\frac{90\!\cdots\!16}{35\!\cdots\!79}a^{16}-\frac{23\!\cdots\!01}{70\!\cdots\!58}a^{15}-\frac{36\!\cdots\!39}{70\!\cdots\!58}a^{14}+\frac{22\!\cdots\!93}{63\!\cdots\!78}a^{13}-\frac{21\!\cdots\!57}{70\!\cdots\!58}a^{12}-\frac{37\!\cdots\!99}{70\!\cdots\!58}a^{11}+\frac{44\!\cdots\!05}{35\!\cdots\!79}a^{10}+\frac{21\!\cdots\!05}{70\!\cdots\!58}a^{9}-\frac{27\!\cdots\!91}{70\!\cdots\!58}a^{8}-\frac{41\!\cdots\!82}{31\!\cdots\!89}a^{7}+\frac{18\!\cdots\!45}{70\!\cdots\!58}a^{6}+\frac{13\!\cdots\!23}{18\!\cdots\!41}a^{5}-\frac{42\!\cdots\!06}{31\!\cdots\!89}a^{4}+\frac{39\!\cdots\!41}{70\!\cdots\!58}a^{3}+\frac{81\!\cdots\!48}{35\!\cdots\!79}a^{2}-\frac{86\!\cdots\!29}{63\!\cdots\!78}a-\frac{16\!\cdots\!74}{34\!\cdots\!79}$, $\frac{48\!\cdots\!03}{70\!\cdots\!58}a^{31}-\frac{35\!\cdots\!09}{70\!\cdots\!58}a^{30}-\frac{56\!\cdots\!03}{70\!\cdots\!58}a^{29}+\frac{44\!\cdots\!51}{35\!\cdots\!79}a^{28}+\frac{49\!\cdots\!45}{11\!\cdots\!09}a^{27}-\frac{12\!\cdots\!75}{70\!\cdots\!58}a^{26}+\frac{10\!\cdots\!95}{35\!\cdots\!79}a^{25}+\frac{49\!\cdots\!06}{35\!\cdots\!79}a^{24}+\frac{22\!\cdots\!47}{70\!\cdots\!58}a^{23}-\frac{80\!\cdots\!39}{70\!\cdots\!58}a^{22}+\frac{37\!\cdots\!25}{70\!\cdots\!58}a^{21}+\frac{28\!\cdots\!19}{70\!\cdots\!58}a^{20}-\frac{16\!\cdots\!31}{35\!\cdots\!79}a^{19}-\frac{84\!\cdots\!43}{35\!\cdots\!79}a^{18}+\frac{29\!\cdots\!51}{70\!\cdots\!58}a^{17}+\frac{31\!\cdots\!02}{35\!\cdots\!79}a^{16}-\frac{72\!\cdots\!99}{70\!\cdots\!58}a^{15}-\frac{11\!\cdots\!81}{70\!\cdots\!58}a^{14}+\frac{38\!\cdots\!45}{35\!\cdots\!79}a^{13}-\frac{65\!\cdots\!21}{70\!\cdots\!58}a^{12}-\frac{11\!\cdots\!65}{70\!\cdots\!58}a^{11}+\frac{21\!\cdots\!23}{70\!\cdots\!58}a^{10}+\frac{33\!\cdots\!48}{35\!\cdots\!79}a^{9}-\frac{41\!\cdots\!22}{35\!\cdots\!79}a^{8}-\frac{29\!\cdots\!07}{70\!\cdots\!58}a^{7}+\frac{53\!\cdots\!77}{70\!\cdots\!58}a^{6}+\frac{78\!\cdots\!44}{35\!\cdots\!79}a^{5}-\frac{28\!\cdots\!95}{70\!\cdots\!58}a^{4}+\frac{11\!\cdots\!09}{70\!\cdots\!58}a^{3}+\frac{48\!\cdots\!35}{70\!\cdots\!58}a^{2}-\frac{27\!\cdots\!87}{70\!\cdots\!58}a-\frac{95\!\cdots\!85}{69\!\cdots\!58}$, $\frac{44\!\cdots\!77}{70\!\cdots\!58}a^{31}-\frac{14\!\cdots\!33}{31\!\cdots\!89}a^{30}-\frac{46\!\cdots\!15}{63\!\cdots\!78}a^{29}+\frac{80\!\cdots\!81}{70\!\cdots\!58}a^{28}+\frac{86\!\cdots\!51}{22\!\cdots\!18}a^{27}-\frac{55\!\cdots\!37}{35\!\cdots\!79}a^{26}+\frac{10\!\cdots\!62}{35\!\cdots\!79}a^{25}+\frac{90\!\cdots\!77}{70\!\cdots\!58}a^{24}+\frac{18\!\cdots\!05}{70\!\cdots\!58}a^{23}-\frac{36\!\cdots\!12}{35\!\cdots\!79}a^{22}+\frac{94\!\cdots\!51}{18\!\cdots\!41}a^{21}+\frac{13\!\cdots\!32}{35\!\cdots\!79}a^{20}-\frac{32\!\cdots\!85}{63\!\cdots\!78}a^{19}-\frac{13\!\cdots\!47}{63\!\cdots\!78}a^{18}+\frac{12\!\cdots\!58}{31\!\cdots\!89}a^{17}+\frac{52\!\cdots\!35}{70\!\cdots\!58}a^{16}-\frac{65\!\cdots\!97}{70\!\cdots\!58}a^{15}-\frac{52\!\cdots\!57}{35\!\cdots\!79}a^{14}+\frac{34\!\cdots\!35}{35\!\cdots\!79}a^{13}-\frac{60\!\cdots\!85}{70\!\cdots\!58}a^{12}-\frac{10\!\cdots\!09}{70\!\cdots\!58}a^{11}+\frac{11\!\cdots\!25}{35\!\cdots\!79}a^{10}+\frac{61\!\cdots\!57}{70\!\cdots\!58}a^{9}-\frac{77\!\cdots\!15}{70\!\cdots\!58}a^{8}-\frac{26\!\cdots\!99}{70\!\cdots\!58}a^{7}+\frac{24\!\cdots\!46}{35\!\cdots\!79}a^{6}+\frac{70\!\cdots\!34}{35\!\cdots\!79}a^{5}-\frac{26\!\cdots\!97}{70\!\cdots\!58}a^{4}+\frac{11\!\cdots\!81}{70\!\cdots\!58}a^{3}+\frac{40\!\cdots\!69}{63\!\cdots\!78}a^{2}-\frac{13\!\cdots\!51}{35\!\cdots\!79}a-\frac{44\!\cdots\!29}{34\!\cdots\!79}$, $\frac{15\!\cdots\!65}{35\!\cdots\!79}a^{31}-\frac{22\!\cdots\!39}{70\!\cdots\!58}a^{30}-\frac{33\!\cdots\!71}{70\!\cdots\!58}a^{29}+\frac{27\!\cdots\!69}{35\!\cdots\!79}a^{28}+\frac{23\!\cdots\!91}{11\!\cdots\!09}a^{27}-\frac{77\!\cdots\!05}{70\!\cdots\!58}a^{26}+\frac{19\!\cdots\!89}{70\!\cdots\!58}a^{25}+\frac{62\!\cdots\!23}{70\!\cdots\!58}a^{24}+\frac{87\!\cdots\!11}{70\!\cdots\!58}a^{23}-\frac{25\!\cdots\!88}{35\!\cdots\!79}a^{22}+\frac{13\!\cdots\!66}{35\!\cdots\!79}a^{21}+\frac{91\!\cdots\!31}{35\!\cdots\!79}a^{20}-\frac{18\!\cdots\!92}{35\!\cdots\!79}a^{19}-\frac{53\!\cdots\!36}{35\!\cdots\!79}a^{18}+\frac{19\!\cdots\!49}{70\!\cdots\!58}a^{17}+\frac{15\!\cdots\!61}{35\!\cdots\!79}a^{16}-\frac{23\!\cdots\!93}{35\!\cdots\!79}a^{15}-\frac{69\!\cdots\!79}{70\!\cdots\!58}a^{14}+\frac{24\!\cdots\!93}{35\!\cdots\!79}a^{13}-\frac{11\!\cdots\!07}{18\!\cdots\!41}a^{12}-\frac{36\!\cdots\!34}{35\!\cdots\!79}a^{11}+\frac{10\!\cdots\!36}{35\!\cdots\!79}a^{10}+\frac{22\!\cdots\!19}{37\!\cdots\!82}a^{9}-\frac{14\!\cdots\!85}{18\!\cdots\!41}a^{8}-\frac{17\!\cdots\!41}{70\!\cdots\!58}a^{7}+\frac{16\!\cdots\!11}{31\!\cdots\!89}a^{6}+\frac{49\!\cdots\!84}{35\!\cdots\!79}a^{5}-\frac{93\!\cdots\!15}{35\!\cdots\!79}a^{4}+\frac{80\!\cdots\!85}{70\!\cdots\!58}a^{3}+\frac{16\!\cdots\!52}{35\!\cdots\!79}a^{2}-\frac{19\!\cdots\!41}{70\!\cdots\!58}a-\frac{64\!\cdots\!29}{69\!\cdots\!58}$, $\frac{50\!\cdots\!75}{70\!\cdots\!58}a^{31}-\frac{18\!\cdots\!72}{35\!\cdots\!79}a^{30}-\frac{31\!\cdots\!03}{37\!\cdots\!82}a^{29}+\frac{46\!\cdots\!06}{35\!\cdots\!79}a^{28}+\frac{52\!\cdots\!26}{11\!\cdots\!09}a^{27}-\frac{12\!\cdots\!47}{70\!\cdots\!58}a^{26}+\frac{22\!\cdots\!29}{70\!\cdots\!58}a^{25}+\frac{51\!\cdots\!72}{35\!\cdots\!79}a^{24}+\frac{11\!\cdots\!55}{35\!\cdots\!79}a^{23}-\frac{42\!\cdots\!63}{35\!\cdots\!79}a^{22}+\frac{40\!\cdots\!91}{70\!\cdots\!58}a^{21}+\frac{30\!\cdots\!85}{70\!\cdots\!58}a^{20}-\frac{37\!\cdots\!51}{70\!\cdots\!58}a^{19}-\frac{88\!\cdots\!68}{35\!\cdots\!79}a^{18}+\frac{15\!\cdots\!43}{35\!\cdots\!79}a^{17}+\frac{64\!\cdots\!41}{70\!\cdots\!58}a^{16}-\frac{75\!\cdots\!27}{70\!\cdots\!58}a^{15}-\frac{60\!\cdots\!10}{35\!\cdots\!79}a^{14}+\frac{40\!\cdots\!65}{35\!\cdots\!79}a^{13}-\frac{34\!\cdots\!20}{35\!\cdots\!79}a^{12}-\frac{60\!\cdots\!11}{35\!\cdots\!79}a^{11}+\frac{23\!\cdots\!49}{70\!\cdots\!58}a^{10}+\frac{70\!\cdots\!97}{70\!\cdots\!58}a^{9}-\frac{44\!\cdots\!26}{35\!\cdots\!79}a^{8}-\frac{16\!\cdots\!55}{37\!\cdots\!82}a^{7}+\frac{51\!\cdots\!33}{63\!\cdots\!78}a^{6}+\frac{16\!\cdots\!19}{70\!\cdots\!58}a^{5}-\frac{14\!\cdots\!79}{35\!\cdots\!79}a^{4}+\frac{12\!\cdots\!59}{70\!\cdots\!58}a^{3}+\frac{27\!\cdots\!65}{37\!\cdots\!82}a^{2}-\frac{29\!\cdots\!45}{70\!\cdots\!58}a-\frac{10\!\cdots\!59}{69\!\cdots\!58}$, $\frac{82\!\cdots\!49}{70\!\cdots\!58}a^{31}-\frac{24\!\cdots\!75}{37\!\cdots\!82}a^{30}-\frac{92\!\cdots\!86}{35\!\cdots\!79}a^{29}+\frac{59\!\cdots\!77}{35\!\cdots\!79}a^{28}+\frac{45\!\cdots\!58}{11\!\cdots\!09}a^{27}-\frac{82\!\cdots\!03}{35\!\cdots\!79}a^{26}-\frac{28\!\cdots\!33}{70\!\cdots\!58}a^{25}+\frac{12\!\cdots\!45}{70\!\cdots\!58}a^{24}+\frac{14\!\cdots\!04}{35\!\cdots\!79}a^{23}-\frac{90\!\cdots\!59}{70\!\cdots\!58}a^{22}-\frac{12\!\cdots\!41}{70\!\cdots\!58}a^{21}+\frac{31\!\cdots\!81}{70\!\cdots\!58}a^{20}+\frac{31\!\cdots\!10}{35\!\cdots\!79}a^{19}-\frac{93\!\cdots\!85}{35\!\cdots\!79}a^{18}+\frac{50\!\cdots\!57}{35\!\cdots\!79}a^{17}+\frac{38\!\cdots\!25}{70\!\cdots\!58}a^{16}-\frac{26\!\cdots\!70}{35\!\cdots\!79}a^{15}-\frac{32\!\cdots\!21}{70\!\cdots\!58}a^{14}+\frac{36\!\cdots\!47}{35\!\cdots\!79}a^{13}+\frac{46\!\cdots\!71}{70\!\cdots\!58}a^{12}-\frac{14\!\cdots\!73}{70\!\cdots\!58}a^{11}-\frac{14\!\cdots\!88}{35\!\cdots\!79}a^{10}+\frac{69\!\cdots\!57}{70\!\cdots\!58}a^{9}+\frac{73\!\cdots\!65}{70\!\cdots\!58}a^{8}-\frac{36\!\cdots\!31}{37\!\cdots\!82}a^{7}-\frac{59\!\cdots\!37}{63\!\cdots\!78}a^{6}+\frac{18\!\cdots\!51}{70\!\cdots\!58}a^{5}-\frac{79\!\cdots\!59}{70\!\cdots\!58}a^{4}-\frac{18\!\cdots\!03}{35\!\cdots\!79}a^{3}+\frac{91\!\cdots\!18}{35\!\cdots\!79}a^{2}+\frac{16\!\cdots\!87}{70\!\cdots\!58}a+\frac{36\!\cdots\!81}{69\!\cdots\!58}$, $\frac{78\!\cdots\!25}{35\!\cdots\!79}a^{31}-\frac{11\!\cdots\!81}{70\!\cdots\!58}a^{30}-\frac{17\!\cdots\!77}{70\!\cdots\!58}a^{29}+\frac{26\!\cdots\!43}{63\!\cdots\!78}a^{28}+\frac{12\!\cdots\!76}{11\!\cdots\!09}a^{27}-\frac{20\!\cdots\!14}{35\!\cdots\!79}a^{26}+\frac{98\!\cdots\!05}{70\!\cdots\!58}a^{25}+\frac{32\!\cdots\!15}{70\!\cdots\!58}a^{24}+\frac{24\!\cdots\!08}{35\!\cdots\!79}a^{23}-\frac{13\!\cdots\!80}{35\!\cdots\!79}a^{22}+\frac{36\!\cdots\!41}{18\!\cdots\!41}a^{21}+\frac{96\!\cdots\!15}{70\!\cdots\!58}a^{20}-\frac{17\!\cdots\!61}{70\!\cdots\!58}a^{19}-\frac{55\!\cdots\!79}{70\!\cdots\!58}a^{18}+\frac{49\!\cdots\!52}{35\!\cdots\!79}a^{17}+\frac{92\!\cdots\!85}{35\!\cdots\!79}a^{16}-\frac{12\!\cdots\!38}{35\!\cdots\!79}a^{15}-\frac{18\!\cdots\!87}{35\!\cdots\!79}a^{14}+\frac{25\!\cdots\!61}{70\!\cdots\!58}a^{13}-\frac{11\!\cdots\!18}{35\!\cdots\!79}a^{12}-\frac{19\!\cdots\!21}{35\!\cdots\!79}a^{11}+\frac{51\!\cdots\!89}{35\!\cdots\!79}a^{10}+\frac{11\!\cdots\!51}{35\!\cdots\!79}a^{9}-\frac{28\!\cdots\!89}{70\!\cdots\!58}a^{8}-\frac{98\!\cdots\!43}{70\!\cdots\!58}a^{7}+\frac{96\!\cdots\!52}{35\!\cdots\!79}a^{6}+\frac{51\!\cdots\!01}{70\!\cdots\!58}a^{5}-\frac{48\!\cdots\!72}{35\!\cdots\!79}a^{4}+\frac{19\!\cdots\!99}{35\!\cdots\!79}a^{3}+\frac{18\!\cdots\!23}{70\!\cdots\!58}a^{2}-\frac{50\!\cdots\!16}{35\!\cdots\!79}a-\frac{17\!\cdots\!34}{34\!\cdots\!79}$, $\frac{35\!\cdots\!27}{70\!\cdots\!58}a^{31}-\frac{26\!\cdots\!73}{70\!\cdots\!58}a^{30}-\frac{20\!\cdots\!54}{35\!\cdots\!79}a^{29}+\frac{32\!\cdots\!15}{35\!\cdots\!79}a^{28}+\frac{32\!\cdots\!35}{11\!\cdots\!09}a^{27}-\frac{45\!\cdots\!70}{35\!\cdots\!79}a^{26}+\frac{18\!\cdots\!05}{70\!\cdots\!58}a^{25}+\frac{36\!\cdots\!88}{35\!\cdots\!79}a^{24}+\frac{13\!\cdots\!67}{70\!\cdots\!58}a^{23}-\frac{59\!\cdots\!09}{70\!\cdots\!58}a^{22}+\frac{14\!\cdots\!94}{35\!\cdots\!79}a^{21}+\frac{10\!\cdots\!01}{35\!\cdots\!79}a^{20}-\frac{30\!\cdots\!41}{70\!\cdots\!58}a^{19}-\frac{32\!\cdots\!68}{18\!\cdots\!41}a^{18}+\frac{10\!\cdots\!61}{35\!\cdots\!79}a^{17}+\frac{21\!\cdots\!59}{35\!\cdots\!79}a^{16}-\frac{53\!\cdots\!13}{70\!\cdots\!58}a^{15}-\frac{83\!\cdots\!27}{70\!\cdots\!58}a^{14}+\frac{28\!\cdots\!08}{35\!\cdots\!79}a^{13}-\frac{49\!\cdots\!93}{70\!\cdots\!58}a^{12}-\frac{42\!\cdots\!89}{35\!\cdots\!79}a^{11}+\frac{94\!\cdots\!73}{35\!\cdots\!79}a^{10}+\frac{24\!\cdots\!91}{35\!\cdots\!79}a^{9}-\frac{62\!\cdots\!17}{70\!\cdots\!58}a^{8}-\frac{10\!\cdots\!00}{35\!\cdots\!79}a^{7}+\frac{21\!\cdots\!31}{37\!\cdots\!82}a^{6}+\frac{11\!\cdots\!07}{70\!\cdots\!58}a^{5}-\frac{21\!\cdots\!29}{70\!\cdots\!58}a^{4}+\frac{89\!\cdots\!61}{70\!\cdots\!58}a^{3}+\frac{36\!\cdots\!71}{70\!\cdots\!58}a^{2}-\frac{21\!\cdots\!57}{70\!\cdots\!58}a-\frac{36\!\cdots\!11}{34\!\cdots\!79}$, $\frac{96\!\cdots\!68}{35\!\cdots\!79}a^{31}-\frac{13\!\cdots\!87}{70\!\cdots\!58}a^{30}-\frac{13\!\cdots\!37}{35\!\cdots\!79}a^{29}+\frac{16\!\cdots\!58}{35\!\cdots\!79}a^{28}+\frac{74\!\cdots\!85}{22\!\cdots\!18}a^{27}-\frac{23\!\cdots\!10}{35\!\cdots\!79}a^{26}-\frac{64\!\cdots\!43}{70\!\cdots\!58}a^{25}+\frac{37\!\cdots\!55}{70\!\cdots\!58}a^{24}+\frac{10\!\cdots\!21}{35\!\cdots\!79}a^{23}-\frac{78\!\cdots\!96}{18\!\cdots\!41}a^{22}+\frac{66\!\cdots\!85}{70\!\cdots\!58}a^{21}+\frac{53\!\cdots\!11}{35\!\cdots\!79}a^{20}+\frac{16\!\cdots\!17}{70\!\cdots\!58}a^{19}-\frac{31\!\cdots\!40}{35\!\cdots\!79}a^{18}+\frac{98\!\cdots\!61}{70\!\cdots\!58}a^{17}+\frac{38\!\cdots\!43}{70\!\cdots\!58}a^{16}-\frac{25\!\cdots\!41}{70\!\cdots\!58}a^{15}-\frac{25\!\cdots\!73}{35\!\cdots\!79}a^{14}+\frac{28\!\cdots\!57}{70\!\cdots\!58}a^{13}-\frac{18\!\cdots\!89}{70\!\cdots\!58}a^{12}-\frac{44\!\cdots\!97}{70\!\cdots\!58}a^{11}-\frac{15\!\cdots\!85}{18\!\cdots\!41}a^{10}+\frac{24\!\cdots\!55}{70\!\cdots\!58}a^{9}-\frac{26\!\cdots\!59}{70\!\cdots\!58}a^{8}-\frac{13\!\cdots\!45}{70\!\cdots\!58}a^{7}+\frac{14\!\cdots\!39}{70\!\cdots\!58}a^{6}+\frac{59\!\cdots\!97}{70\!\cdots\!58}a^{5}-\frac{95\!\cdots\!57}{70\!\cdots\!58}a^{4}+\frac{35\!\cdots\!51}{70\!\cdots\!58}a^{3}+\frac{84\!\cdots\!76}{35\!\cdots\!79}a^{2}-\frac{82\!\cdots\!29}{70\!\cdots\!58}a-\frac{29\!\cdots\!45}{69\!\cdots\!58}$ (assuming GRH)
Regulator:		\( 103154016774940.02 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 103154016774940.02 \cdot 3}{10\cdot\sqrt{170615961014359235699341036323165930807590484619140625}}\cr\approx \mathstrut & 0.442054141762677 \end{aligned}\] (assuming GRH)

Galois group

$C_8.A_4$ (as 32T402):

A solvable group of order 96

The 28 conjugacy class representatives for $C_8.A_4$

Character table for $C_8.A_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.403225.1, 8.0.162590400625.1, 16.0.16522273984623750244140625.1

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	${\href{/padicField/2.8.0.1}{8} }^{4}$	$24{,}\,{\href{/padicField/3.8.0.1}{8} }$	R	$24{,}\,{\href{/padicField/7.8.0.1}{8} }$	${\href{/padicField/11.3.0.1}{3} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{8}$	$24{,}\,{\href{/padicField/13.8.0.1}{8} }$	$24{,}\,{\href{/padicField/17.8.0.1}{8} }$	${\href{/padicField/19.2.0.1}{2} }^{16}$	$24{,}\,{\href{/padicField/23.8.0.1}{8} }$	${\href{/padicField/29.12.0.1}{12} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$	${\href{/padicField/31.3.0.1}{3} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$	$24{,}\,{\href{/padicField/37.8.0.1}{8} }$	${\href{/padicField/41.3.0.1}{3} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$	$24{,}\,{\href{/padicField/43.8.0.1}{8} }$	${\href{/padicField/47.8.0.1}{8} }^{4}$	$24{,}\,{\href{/padicField/53.8.0.1}{8} }$	${\href{/padicField/59.12.0.1}{12} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$

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
\(5\)	5.16.14.1	$x^{16} - 20 x^{8} - 100$	$8$	$2$	$14$	$C_8: C_2$	$[\ ]_{8}^{2}$
	5.16.14.1	$x^{16} - 20 x^{8} - 100$	$8$	$2$	$14$	$C_8: C_2$	$[\ ]_{8}^{2}$
\(127\)	127.8.0.1	$x^{8} + 3 x^{4} + 104 x^{3} + 55 x^{2} + 8 x + 3$	$1$	$8$	$0$	$C_8$	$[\ ]^{8}$
		Deg $24$	$3$	$8$	$16$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
*	1.5.2t1.a.a	$1$	$ 5 $	\(\Q(\sqrt{5}) \)	$C_2$ (as 2T1)	$1$	$1$
	1.635.6t1.a.a	$1$	$ 5 \cdot 127 $	6.6.32518080125.1	$C_6$ (as 6T1)	$0$	$1$
	1.635.6t1.a.b	$1$	$ 5 \cdot 127 $	6.6.32518080125.1	$C_6$ (as 6T1)	$0$	$1$
	1.127.3t1.a.a	$1$	$ 127 $	3.3.16129.1	$C_3$ (as 3T1)	$0$	$1$
	1.127.3t1.a.b	$1$	$ 127 $	3.3.16129.1	$C_3$ (as 3T1)	$0$	$1$
*	1.5.4t1.a.a	$1$	$ 5 $	\(\Q(\zeta_{5})\)	$C_4$ (as 4T1)	$0$	$-1$
*	1.5.4t1.a.b	$1$	$ 5 $	\(\Q(\zeta_{5})\)	$C_4$ (as 4T1)	$0$	$-1$
	1.635.12t1.a.a	$1$	$ 5 \cdot 127 $	12.0.132178191876990001953125.1	$C_{12}$ (as 12T1)	$0$	$-1$
	1.635.12t1.a.b	$1$	$ 5 \cdot 127 $	12.0.132178191876990001953125.1	$C_{12}$ (as 12T1)	$0$	$-1$
	1.635.12t1.a.c	$1$	$ 5 \cdot 127 $	12.0.132178191876990001953125.1	$C_{12}$ (as 12T1)	$0$	$-1$
	1.635.12t1.a.d	$1$	$ 5 \cdot 127 $	12.0.132178191876990001953125.1	$C_{12}$ (as 12T1)	$0$	$-1$
	2.403225.48.a.a	$2$	$ 5^{2} \cdot 127^{2}$	32.0.170615961014359235699341036323165930807590484619140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
	2.403225.48.a.b	$2$	$ 5^{2} \cdot 127^{2}$	32.0.170615961014359235699341036323165930807590484619140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
	2.403225.48.a.c	$2$	$ 5^{2} \cdot 127^{2}$	32.0.170615961014359235699341036323165930807590484619140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
	2.403225.48.a.d	$2$	$ 5^{2} \cdot 127^{2}$	32.0.170615961014359235699341036323165930807590484619140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.3175.32t402.a.a	$2$	$ 5^{2} \cdot 127 $	32.0.170615961014359235699341036323165930807590484619140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.3175.32t402.a.b	$2$	$ 5^{2} \cdot 127 $	32.0.170615961014359235699341036323165930807590484619140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.3175.32t402.a.c	$2$	$ 5^{2} \cdot 127 $	32.0.170615961014359235699341036323165930807590484619140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.3175.32t402.a.d	$2$	$ 5^{2} \cdot 127 $	32.0.170615961014359235699341036323165930807590484619140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.3175.32t402.a.e	$2$	$ 5^{2} \cdot 127 $	32.0.170615961014359235699341036323165930807590484619140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.3175.32t402.a.f	$2$	$ 5^{2} \cdot 127 $	32.0.170615961014359235699341036323165930807590484619140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.3175.32t402.a.g	$2$	$ 5^{2} \cdot 127 $	32.0.170615961014359235699341036323165930807590484619140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.3175.32t402.a.h	$2$	$ 5^{2} \cdot 127 $	32.0.170615961014359235699341036323165930807590484619140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	3.403225.4t4.a.a	$3$	$ 5^{2} \cdot 127^{2}$	4.0.403225.1	$A_4$ (as 4T4)	$1$	$-1$
*	3.80645.6t6.a.a	$3$	$ 5 \cdot 127^{2}$	6.2.1300723205.1	$A_4\times C_2$ (as 6T6)	$1$	$-1$
*	3.2016125.12t29.a.a	$3$	$ 5^{3} \cdot 127^{2}$	12.8.132178191876990001953125.1	$C_4\times A_4$ (as 12T29)	$0$	$1$
*	3.2016125.12t29.a.b	$3$	$ 5^{3} \cdot 127^{2}$	12.8.132178191876990001953125.1	$C_4\times A_4$ (as 12T29)	$0$	$1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.