Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 11*x^31 + 68*x^30 - 336*x^29 + 1315*x^28 - 4335*x^27 + 12498*x^26 - 30713*x^25 + 67152*x^24 - 127199*x^23 + 211471*x^22 - 307078*x^21 + 387026*x^20 - 419617*x^19 + 359728*x^18 - 149755*x^17 - 229142*x^16 + 581095*x^15 - 441722*x^14 - 315448*x^13 + 1021466*x^12 - 946897*x^11 + 366311*x^10 - 50246*x^9 + 132237*x^8 - 215882*x^7 + 157773*x^6 - 97500*x^5 + 79930*x^4 - 54894*x^3 + 25063*x^2 - 7709*x + 3131)
gp: K = bnfinit(y^32 - 11*y^31 + 68*y^30 - 336*y^29 + 1315*y^28 - 4335*y^27 + 12498*y^26 - 30713*y^25 + 67152*y^24 - 127199*y^23 + 211471*y^22 - 307078*y^21 + 387026*y^20 - 419617*y^19 + 359728*y^18 - 149755*y^17 - 229142*y^16 + 581095*y^15 - 441722*y^14 - 315448*y^13 + 1021466*y^12 - 946897*y^11 + 366311*y^10 - 50246*y^9 + 132237*y^8 - 215882*y^7 + 157773*y^6 - 97500*y^5 + 79930*y^4 - 54894*y^3 + 25063*y^2 - 7709*y + 3131, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 11*x^31 + 68*x^30 - 336*x^29 + 1315*x^28 - 4335*x^27 + 12498*x^26 - 30713*x^25 + 67152*x^24 - 127199*x^23 + 211471*x^22 - 307078*x^21 + 387026*x^20 - 419617*x^19 + 359728*x^18 - 149755*x^17 - 229142*x^16 + 581095*x^15 - 441722*x^14 - 315448*x^13 + 1021466*x^12 - 946897*x^11 + 366311*x^10 - 50246*x^9 + 132237*x^8 - 215882*x^7 + 157773*x^6 - 97500*x^5 + 79930*x^4 - 54894*x^3 + 25063*x^2 - 7709*x + 3131);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 11*x^31 + 68*x^30 - 336*x^29 + 1315*x^28 - 4335*x^27 + 12498*x^26 - 30713*x^25 + 67152*x^24 - 127199*x^23 + 211471*x^22 - 307078*x^21 + 387026*x^20 - 419617*x^19 + 359728*x^18 - 149755*x^17 - 229142*x^16 + 581095*x^15 - 441722*x^14 - 315448*x^13 + 1021466*x^12 - 946897*x^11 + 366311*x^10 - 50246*x^9 + 132237*x^8 - 215882*x^7 + 157773*x^6 - 97500*x^5 + 79930*x^4 - 54894*x^3 + 25063*x^2 - 7709*x + 3131)
\( x^{32} - 11 x^{31} + 68 x^{30} - 336 x^{29} + 1315 x^{28} - 4335 x^{27} + 12498 x^{26} - 30713 x^{25} + \cdots + 3131 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
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: |
\(2294081869728198830572049522437155246734619140625\)
\(\medspace = 3^{32}\cdot 5^{28}\cdot 7^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(32.45\) | 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: | $3^{4/3}5^{7/8}7^{2/3}\approx 64.7379764195286$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{101}a^{28}+\frac{35}{101}a^{27}-\frac{33}{101}a^{26}+\frac{20}{101}a^{25}-\frac{7}{101}a^{24}-\frac{10}{101}a^{23}+\frac{3}{101}a^{22}-\frac{7}{101}a^{21}+\frac{21}{101}a^{20}-\frac{11}{101}a^{19}+\frac{7}{101}a^{18}+\frac{24}{101}a^{17}-\frac{24}{101}a^{16}-\frac{22}{101}a^{15}-\frac{48}{101}a^{14}-\frac{21}{101}a^{13}+\frac{31}{101}a^{12}-\frac{36}{101}a^{11}-\frac{20}{101}a^{10}-\frac{47}{101}a^{9}-\frac{19}{101}a^{8}+\frac{20}{101}a^{7}-\frac{9}{101}a^{6}-\frac{48}{101}a^{5}-\frac{6}{101}a^{4}+\frac{27}{101}a^{3}+\frac{8}{101}a^{2}-\frac{13}{101}a$, $\frac{1}{25351}a^{29}+\frac{32}{25351}a^{28}-\frac{4683}{25351}a^{27}+\frac{2947}{25351}a^{26}-\frac{976}{25351}a^{25}-\frac{5443}{25351}a^{24}+\frac{9729}{25351}a^{23}-\frac{420}{25351}a^{22}-\frac{8038}{25351}a^{21}-\frac{10073}{25351}a^{20}-\frac{1374}{25351}a^{19}-\frac{6057}{25351}a^{18}-\frac{3429}{25351}a^{17}+\frac{1363}{25351}a^{16}-\frac{2002}{25351}a^{15}-\frac{7351}{25351}a^{14}+\frac{6255}{25351}a^{13}+\frac{780}{25351}a^{12}-\frac{7891}{25351}a^{11}-\frac{4330}{25351}a^{10}-\frac{1797}{25351}a^{9}+\frac{8763}{25351}a^{8}+\frac{234}{25351}a^{7}-\frac{4566}{25351}a^{6}-\frac{2993}{25351}a^{5}+\frac{6711}{25351}a^{4}+\frac{634}{25351}a^{3}+\frac{11881}{25351}a^{2}+\frac{6806}{25351}a-\frac{40}{251}$, $\frac{1}{19292111}a^{30}-\frac{347}{19292111}a^{29}-\frac{7524}{19292111}a^{28}+\frac{531087}{19292111}a^{27}-\frac{3477791}{19292111}a^{26}-\frac{5736847}{19292111}a^{25}-\frac{780993}{19292111}a^{24}+\frac{8818866}{19292111}a^{23}-\frac{3902508}{19292111}a^{22}+\frac{4897996}{19292111}a^{21}+\frac{3200088}{19292111}a^{20}-\frac{6178728}{19292111}a^{19}-\frac{8315588}{19292111}a^{18}+\frac{1016822}{19292111}a^{17}+\frac{2300653}{19292111}a^{16}-\frac{1988007}{19292111}a^{15}+\frac{9241980}{19292111}a^{14}-\frac{8877291}{19292111}a^{13}-\frac{8406795}{19292111}a^{12}+\frac{852106}{19292111}a^{11}+\frac{5256183}{19292111}a^{10}+\frac{8999432}{19292111}a^{9}+\frac{7226077}{19292111}a^{8}+\frac{6278132}{19292111}a^{7}-\frac{6797945}{19292111}a^{6}+\frac{8680847}{19292111}a^{5}+\frac{190053}{19292111}a^{4}+\frac{7906505}{19292111}a^{3}-\frac{4269691}{19292111}a^{2}-\frac{6709703}{19292111}a-\frac{41817}{191011}$, $\frac{1}{14\!\cdots\!99}a^{31}+\frac{30\!\cdots\!13}{14\!\cdots\!99}a^{30}+\frac{14\!\cdots\!90}{14\!\cdots\!99}a^{29}-\frac{48\!\cdots\!06}{14\!\cdots\!99}a^{28}-\frac{41\!\cdots\!06}{14\!\cdots\!99}a^{27}+\frac{26\!\cdots\!57}{14\!\cdots\!99}a^{26}+\frac{12\!\cdots\!45}{14\!\cdots\!99}a^{25}-\frac{45\!\cdots\!87}{14\!\cdots\!99}a^{24}+\frac{26\!\cdots\!06}{14\!\cdots\!99}a^{23}-\frac{20\!\cdots\!07}{14\!\cdots\!99}a^{22}+\frac{28\!\cdots\!32}{14\!\cdots\!99}a^{21}-\frac{14\!\cdots\!56}{14\!\cdots\!99}a^{20}-\frac{62\!\cdots\!51}{14\!\cdots\!99}a^{19}+\frac{20\!\cdots\!12}{14\!\cdots\!99}a^{18}-\frac{61\!\cdots\!41}{14\!\cdots\!99}a^{17}-\frac{48\!\cdots\!53}{14\!\cdots\!99}a^{16}-\frac{47\!\cdots\!34}{14\!\cdots\!99}a^{15}-\frac{33\!\cdots\!12}{14\!\cdots\!99}a^{14}-\frac{40\!\cdots\!63}{14\!\cdots\!99}a^{13}-\frac{36\!\cdots\!20}{14\!\cdots\!99}a^{12}-\frac{11\!\cdots\!56}{14\!\cdots\!99}a^{11}+\frac{10\!\cdots\!92}{14\!\cdots\!99}a^{10}+\frac{56\!\cdots\!29}{14\!\cdots\!99}a^{9}-\frac{36\!\cdots\!96}{14\!\cdots\!99}a^{8}-\frac{18\!\cdots\!09}{14\!\cdots\!99}a^{7}-\frac{30\!\cdots\!95}{14\!\cdots\!99}a^{6}+\frac{15\!\cdots\!65}{14\!\cdots\!99}a^{5}-\frac{27\!\cdots\!58}{14\!\cdots\!99}a^{4}+\frac{29\!\cdots\!14}{14\!\cdots\!99}a^{3}-\frac{49\!\cdots\!68}{14\!\cdots\!99}a^{2}-\frac{68\!\cdots\!45}{14\!\cdots\!99}a+\frac{16\!\cdots\!30}{14\!\cdots\!99}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{6638407240373683378630867815201395558285438932852502041753695589293418306971}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{31} - \frac{83360160096908894117006724134204811125762338095037488824399616851135042247073}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{30} + \frac{552586014163738577155325274058813465697021226866645545690642768270585828377127}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{29} - \frac{2804666642179450335549806174053850230883742957875856976185287871731565471456526}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{28} + \frac{11448376057233387831714100426239885927797227045726692325427247350711119768648762}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{27} - \frac{38750301851901452722084707016343656282044360782463035469104270631518054084347649}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{26} + \frac{114170354710479830755093491839812819515607372992189798390448294068544665228544452}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{25} - \frac{289781024804178125867948634924493148332501126369397401545487471366754020067608848}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{24} + \frac{642294565831738938401279843876035384621982698559680210429712279484438072942824730}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{23} - \frac{1254643786551414847989052661021681817905378418949421619315407380772118301660628402}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{22} + \frac{2119308879716823472559849162376310767995881815904308276858058405453685192621204068}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{21} - \frac{3147697461597465605827034390643392983115068424290894060040244414995736709618880289}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{20} + \frac{4044401234852695421880787416418196281248982391636883170032454679841276132159812162}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{19} - \frac{4464045586583621418007736119964451771989326114767631768491293495241784993241203536}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{18} + \frac{4004323544016001996908612230217484383514809114810692254535415898424143975324824488}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{17} - \frac{2004683475728018368574031666377328534245129839642259818000104859960660679573890136}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{16} - \frac{1888906683192466235594012371985261427063627709181935555341632451433730519214407584}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{15} + \frac{6147209007970713554223697174773821893593305027167001966564936272839200752533150253}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{14} - \frac{5839198610682105903044455661450306513902137529572679820326119883996285846661576201}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{13} - \frac{2253866069737533778572774913210273797380936431286312886877130995780313755236445307}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{12} + \frac{11056486303856033150293046022056592504630520843935380664945908986247760761649475200}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{11} - \frac{10905253226140676702017327308033055010532233874258641053349183755102273542016509251}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{10} + \frac{3985653179434570741793226675990179085650249929689165464680093821044730224390711367}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{9} - \frac{418220658061310907142377654321005795757674696030982683732573291628270701602078679}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{8} + \frac{1574132625456407462857469230118220107450785261564550312371024038606391021213825259}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{7} - \frac{2295789050320103733137733529913993719794609059227436965884441595334184141780030029}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{6} + \frac{1488203124856075108444096916970177017747200691082406759687608266401704060065732827}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{5} - \frac{1052713753858510674562695324286078925244442381986209996397154848513096067361582265}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{4} + \frac{921264150530854391006387252236104268354459447580578950407124966545463249105098373}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{3} - \frac{583911646184010750699507249489113873344240992705018065478318298849750464246397581}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{2} + \frac{221824581777955967679290896885505956341350187897596320724267372048361108090422140}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a - \frac{1038474931296181961991593062358694559628567681565259383280398696825689577356089}{521845551400641191489504523316972859261484787115789273473422059501610621935209} \)
(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: |
$\frac{50\!\cdots\!46}{52\!\cdots\!09}a^{31}-\frac{61\!\cdots\!48}{52\!\cdots\!09}a^{30}+\frac{39\!\cdots\!79}{52\!\cdots\!09}a^{29}-\frac{20\!\cdots\!46}{52\!\cdots\!09}a^{28}+\frac{80\!\cdots\!02}{52\!\cdots\!09}a^{27}-\frac{26\!\cdots\!54}{52\!\cdots\!09}a^{26}+\frac{78\!\cdots\!65}{52\!\cdots\!09}a^{25}-\frac{19\!\cdots\!25}{52\!\cdots\!09}a^{24}+\frac{42\!\cdots\!96}{52\!\cdots\!09}a^{23}-\frac{81\!\cdots\!62}{52\!\cdots\!09}a^{22}+\frac{13\!\cdots\!22}{52\!\cdots\!09}a^{21}-\frac{19\!\cdots\!76}{52\!\cdots\!09}a^{20}+\frac{23\!\cdots\!95}{52\!\cdots\!09}a^{19}-\frac{25\!\cdots\!65}{52\!\cdots\!09}a^{18}+\frac{20\!\cdots\!79}{52\!\cdots\!09}a^{17}-\frac{60\!\cdots\!36}{52\!\cdots\!09}a^{16}-\frac{19\!\cdots\!98}{52\!\cdots\!09}a^{15}+\frac{43\!\cdots\!94}{52\!\cdots\!09}a^{14}-\frac{32\!\cdots\!81}{52\!\cdots\!09}a^{13}-\frac{28\!\cdots\!85}{52\!\cdots\!09}a^{12}+\frac{83\!\cdots\!63}{52\!\cdots\!09}a^{11}-\frac{66\!\cdots\!14}{52\!\cdots\!09}a^{10}+\frac{70\!\cdots\!55}{52\!\cdots\!09}a^{9}+\frac{14\!\cdots\!96}{52\!\cdots\!09}a^{8}+\frac{50\!\cdots\!74}{52\!\cdots\!09}a^{7}-\frac{16\!\cdots\!16}{52\!\cdots\!09}a^{6}+\frac{10\!\cdots\!76}{52\!\cdots\!09}a^{5}-\frac{47\!\cdots\!26}{52\!\cdots\!09}a^{4}+\frac{36\!\cdots\!61}{52\!\cdots\!09}a^{3}-\frac{25\!\cdots\!03}{52\!\cdots\!09}a^{2}+\frac{95\!\cdots\!74}{52\!\cdots\!09}a-\frac{51\!\cdots\!43}{52\!\cdots\!09}$, $\frac{50\!\cdots\!78}{14\!\cdots\!99}a^{31}-\frac{47\!\cdots\!80}{14\!\cdots\!99}a^{30}+\frac{26\!\cdots\!76}{14\!\cdots\!99}a^{29}-\frac{12\!\cdots\!41}{14\!\cdots\!99}a^{28}+\frac{44\!\cdots\!74}{14\!\cdots\!99}a^{27}-\frac{13\!\cdots\!41}{14\!\cdots\!99}a^{26}+\frac{37\!\cdots\!42}{14\!\cdots\!99}a^{25}-\frac{85\!\cdots\!69}{14\!\cdots\!99}a^{24}+\frac{17\!\cdots\!35}{14\!\cdots\!99}a^{23}-\frac{30\!\cdots\!60}{14\!\cdots\!99}a^{22}+\frac{47\!\cdots\!83}{14\!\cdots\!99}a^{21}-\frac{62\!\cdots\!06}{14\!\cdots\!99}a^{20}+\frac{72\!\cdots\!07}{14\!\cdots\!99}a^{19}-\frac{69\!\cdots\!94}{14\!\cdots\!99}a^{18}+\frac{42\!\cdots\!23}{14\!\cdots\!99}a^{17}+\frac{14\!\cdots\!10}{14\!\cdots\!99}a^{16}-\frac{95\!\cdots\!73}{14\!\cdots\!99}a^{15}+\frac{11\!\cdots\!37}{14\!\cdots\!99}a^{14}+\frac{19\!\cdots\!28}{14\!\cdots\!99}a^{13}-\frac{17\!\cdots\!94}{14\!\cdots\!99}a^{12}+\frac{18\!\cdots\!39}{14\!\cdots\!99}a^{11}-\frac{69\!\cdots\!62}{14\!\cdots\!99}a^{10}+\frac{18\!\cdots\!38}{14\!\cdots\!99}a^{9}-\frac{26\!\cdots\!08}{14\!\cdots\!99}a^{8}+\frac{34\!\cdots\!26}{14\!\cdots\!99}a^{7}-\frac{25\!\cdots\!51}{14\!\cdots\!99}a^{6}+\frac{18\!\cdots\!60}{14\!\cdots\!99}a^{5}-\frac{18\!\cdots\!33}{14\!\cdots\!99}a^{4}+\frac{10\!\cdots\!51}{14\!\cdots\!99}a^{3}-\frac{97\!\cdots\!53}{14\!\cdots\!99}a^{2}+\frac{14\!\cdots\!22}{14\!\cdots\!99}a-\frac{14\!\cdots\!19}{14\!\cdots\!99}$, $\frac{37\!\cdots\!71}{14\!\cdots\!99}a^{31}-\frac{33\!\cdots\!29}{14\!\cdots\!99}a^{30}+\frac{18\!\cdots\!76}{14\!\cdots\!99}a^{29}-\frac{83\!\cdots\!60}{14\!\cdots\!99}a^{28}+\frac{29\!\cdots\!80}{14\!\cdots\!99}a^{27}-\frac{88\!\cdots\!52}{14\!\cdots\!99}a^{26}+\frac{23\!\cdots\!37}{14\!\cdots\!99}a^{25}-\frac{50\!\cdots\!02}{14\!\cdots\!99}a^{24}+\frac{10\!\cdots\!74}{14\!\cdots\!99}a^{23}-\frac{16\!\cdots\!51}{14\!\cdots\!99}a^{22}+\frac{24\!\cdots\!07}{14\!\cdots\!99}a^{21}-\frac{28\!\cdots\!04}{14\!\cdots\!99}a^{20}+\frac{27\!\cdots\!88}{14\!\cdots\!99}a^{19}-\frac{20\!\cdots\!78}{14\!\cdots\!99}a^{18}-\frac{75\!\cdots\!74}{14\!\cdots\!99}a^{17}+\frac{36\!\cdots\!56}{14\!\cdots\!99}a^{16}-\frac{75\!\cdots\!86}{14\!\cdots\!99}a^{15}+\frac{57\!\cdots\!85}{14\!\cdots\!99}a^{14}+\frac{60\!\cdots\!23}{14\!\cdots\!99}a^{13}-\frac{14\!\cdots\!66}{14\!\cdots\!99}a^{12}+\frac{93\!\cdots\!40}{14\!\cdots\!99}a^{11}+\frac{26\!\cdots\!03}{14\!\cdots\!99}a^{10}-\frac{45\!\cdots\!83}{14\!\cdots\!99}a^{9}+\frac{99\!\cdots\!49}{14\!\cdots\!99}a^{8}+\frac{14\!\cdots\!52}{14\!\cdots\!99}a^{7}-\frac{79\!\cdots\!87}{14\!\cdots\!99}a^{6}+\frac{90\!\cdots\!85}{14\!\cdots\!99}a^{5}-\frac{16\!\cdots\!00}{14\!\cdots\!99}a^{4}-\frac{24\!\cdots\!88}{14\!\cdots\!99}a^{3}+\frac{45\!\cdots\!28}{14\!\cdots\!99}a^{2}-\frac{81\!\cdots\!18}{14\!\cdots\!99}a+\frac{81\!\cdots\!02}{14\!\cdots\!99}$, $\frac{12\!\cdots\!88}{14\!\cdots\!99}a^{31}-\frac{13\!\cdots\!37}{14\!\cdots\!99}a^{30}+\frac{86\!\cdots\!64}{14\!\cdots\!99}a^{29}-\frac{42\!\cdots\!96}{14\!\cdots\!99}a^{28}+\frac{16\!\cdots\!09}{14\!\cdots\!99}a^{27}-\frac{54\!\cdots\!78}{14\!\cdots\!99}a^{26}+\frac{15\!\cdots\!45}{14\!\cdots\!99}a^{25}-\frac{38\!\cdots\!71}{14\!\cdots\!99}a^{24}+\frac{82\!\cdots\!93}{14\!\cdots\!99}a^{23}-\frac{15\!\cdots\!81}{14\!\cdots\!99}a^{22}+\frac{25\!\cdots\!88}{14\!\cdots\!99}a^{21}-\frac{36\!\cdots\!38}{14\!\cdots\!99}a^{20}+\frac{44\!\cdots\!31}{14\!\cdots\!99}a^{19}-\frac{46\!\cdots\!89}{14\!\cdots\!99}a^{18}+\frac{38\!\cdots\!32}{14\!\cdots\!99}a^{17}-\frac{10\!\cdots\!48}{14\!\cdots\!99}a^{16}-\frac{35\!\cdots\!81}{14\!\cdots\!99}a^{15}+\frac{74\!\cdots\!25}{14\!\cdots\!99}a^{14}-\frac{46\!\cdots\!70}{14\!\cdots\!99}a^{13}-\frac{58\!\cdots\!33}{14\!\cdots\!99}a^{12}+\frac{13\!\cdots\!62}{14\!\cdots\!99}a^{11}-\frac{98\!\cdots\!89}{14\!\cdots\!99}a^{10}+\frac{17\!\cdots\!81}{14\!\cdots\!99}a^{9}+\frac{13\!\cdots\!36}{14\!\cdots\!99}a^{8}+\frac{21\!\cdots\!14}{14\!\cdots\!99}a^{7}-\frac{25\!\cdots\!26}{14\!\cdots\!99}a^{6}+\frac{13\!\cdots\!47}{14\!\cdots\!99}a^{5}-\frac{94\!\cdots\!75}{14\!\cdots\!99}a^{4}+\frac{89\!\cdots\!66}{14\!\cdots\!99}a^{3}-\frac{47\!\cdots\!00}{14\!\cdots\!99}a^{2}+\frac{17\!\cdots\!84}{14\!\cdots\!99}a-\frac{93\!\cdots\!61}{14\!\cdots\!99}$, $\frac{20\!\cdots\!48}{14\!\cdots\!99}a^{31}-\frac{21\!\cdots\!33}{14\!\cdots\!99}a^{30}+\frac{12\!\cdots\!43}{14\!\cdots\!99}a^{29}-\frac{60\!\cdots\!57}{14\!\cdots\!99}a^{28}+\frac{22\!\cdots\!30}{14\!\cdots\!99}a^{27}-\frac{72\!\cdots\!84}{14\!\cdots\!99}a^{26}+\frac{20\!\cdots\!21}{14\!\cdots\!99}a^{25}-\frac{47\!\cdots\!84}{14\!\cdots\!99}a^{24}+\frac{98\!\cdots\!24}{14\!\cdots\!99}a^{23}-\frac{17\!\cdots\!81}{14\!\cdots\!99}a^{22}+\frac{27\!\cdots\!51}{14\!\cdots\!99}a^{21}-\frac{37\!\cdots\!95}{14\!\cdots\!99}a^{20}+\frac{42\!\cdots\!52}{14\!\cdots\!99}a^{19}-\frac{41\!\cdots\!71}{14\!\cdots\!99}a^{18}+\frac{27\!\cdots\!09}{14\!\cdots\!99}a^{17}+\frac{56\!\cdots\!51}{14\!\cdots\!99}a^{16}-\frac{55\!\cdots\!14}{14\!\cdots\!99}a^{15}+\frac{78\!\cdots\!96}{14\!\cdots\!99}a^{14}-\frac{96\!\cdots\!57}{14\!\cdots\!99}a^{13}-\frac{10\!\cdots\!10}{14\!\cdots\!99}a^{12}+\frac{13\!\cdots\!66}{14\!\cdots\!99}a^{11}-\frac{43\!\cdots\!48}{14\!\cdots\!99}a^{10}-\frac{17\!\cdots\!55}{14\!\cdots\!99}a^{9}-\frac{12\!\cdots\!01}{14\!\cdots\!99}a^{8}+\frac{39\!\cdots\!93}{14\!\cdots\!99}a^{7}-\frac{22\!\cdots\!08}{14\!\cdots\!99}a^{6}+\frac{33\!\cdots\!73}{14\!\cdots\!99}a^{5}-\frac{52\!\cdots\!36}{14\!\cdots\!99}a^{4}+\frac{64\!\cdots\!07}{14\!\cdots\!99}a^{3}-\frac{49\!\cdots\!83}{14\!\cdots\!99}a^{2}+\frac{15\!\cdots\!78}{14\!\cdots\!99}a-\frac{10\!\cdots\!45}{14\!\cdots\!99}$, $\frac{31\!\cdots\!11}{14\!\cdots\!99}a^{31}-\frac{64\!\cdots\!73}{14\!\cdots\!99}a^{30}+\frac{49\!\cdots\!92}{14\!\cdots\!99}a^{29}-\frac{26\!\cdots\!24}{14\!\cdots\!99}a^{28}+\frac{11\!\cdots\!90}{14\!\cdots\!99}a^{27}-\frac{40\!\cdots\!97}{14\!\cdots\!99}a^{26}+\frac{12\!\cdots\!14}{14\!\cdots\!99}a^{25}-\frac{32\!\cdots\!20}{14\!\cdots\!99}a^{24}+\frac{71\!\cdots\!23}{14\!\cdots\!99}a^{23}-\frac{14\!\cdots\!05}{14\!\cdots\!99}a^{22}+\frac{23\!\cdots\!02}{14\!\cdots\!99}a^{21}-\frac{36\!\cdots\!41}{14\!\cdots\!99}a^{20}+\frac{46\!\cdots\!52}{14\!\cdots\!99}a^{19}-\frac{50\!\cdots\!89}{14\!\cdots\!99}a^{18}+\frac{44\!\cdots\!13}{14\!\cdots\!99}a^{17}-\frac{20\!\cdots\!21}{14\!\cdots\!99}a^{16}-\frac{27\!\cdots\!56}{14\!\cdots\!99}a^{15}+\frac{84\!\cdots\!61}{14\!\cdots\!99}a^{14}-\frac{83\!\cdots\!86}{14\!\cdots\!99}a^{13}-\frac{32\!\cdots\!01}{14\!\cdots\!99}a^{12}+\frac{15\!\cdots\!76}{14\!\cdots\!99}a^{11}-\frac{13\!\cdots\!35}{14\!\cdots\!99}a^{10}+\frac{28\!\cdots\!53}{14\!\cdots\!99}a^{9}+\frac{83\!\cdots\!73}{14\!\cdots\!99}a^{8}+\frac{14\!\cdots\!65}{14\!\cdots\!99}a^{7}-\frac{23\!\cdots\!59}{14\!\cdots\!99}a^{6}+\frac{94\!\cdots\!82}{14\!\cdots\!99}a^{5}-\frac{97\!\cdots\!26}{14\!\cdots\!99}a^{4}+\frac{77\!\cdots\!45}{14\!\cdots\!99}a^{3}-\frac{46\!\cdots\!70}{14\!\cdots\!99}a^{2}+\frac{21\!\cdots\!02}{14\!\cdots\!99}a-\frac{52\!\cdots\!32}{14\!\cdots\!99}$, $\frac{27\!\cdots\!64}{14\!\cdots\!99}a^{31}-\frac{25\!\cdots\!45}{14\!\cdots\!99}a^{30}+\frac{12\!\cdots\!12}{14\!\cdots\!99}a^{29}-\frac{48\!\cdots\!90}{14\!\cdots\!99}a^{28}+\frac{12\!\cdots\!77}{14\!\cdots\!99}a^{27}-\frac{20\!\cdots\!15}{14\!\cdots\!99}a^{26}-\frac{18\!\cdots\!07}{14\!\cdots\!99}a^{25}+\frac{19\!\cdots\!45}{14\!\cdots\!99}a^{24}-\frac{88\!\cdots\!17}{14\!\cdots\!99}a^{23}+\frac{27\!\cdots\!94}{14\!\cdots\!99}a^{22}-\frac{67\!\cdots\!73}{14\!\cdots\!99}a^{21}+\frac{13\!\cdots\!89}{14\!\cdots\!99}a^{20}-\frac{23\!\cdots\!65}{14\!\cdots\!99}a^{19}+\frac{34\!\cdots\!64}{14\!\cdots\!99}a^{18}-\frac{43\!\cdots\!29}{14\!\cdots\!99}a^{17}+\frac{47\!\cdots\!34}{14\!\cdots\!99}a^{16}-\frac{40\!\cdots\!55}{14\!\cdots\!99}a^{15}+\frac{91\!\cdots\!29}{14\!\cdots\!99}a^{14}+\frac{47\!\cdots\!77}{14\!\cdots\!99}a^{13}-\frac{88\!\cdots\!39}{14\!\cdots\!99}a^{12}+\frac{44\!\cdots\!20}{14\!\cdots\!99}a^{11}+\frac{61\!\cdots\!64}{14\!\cdots\!99}a^{10}-\frac{11\!\cdots\!14}{14\!\cdots\!99}a^{9}+\frac{78\!\cdots\!50}{14\!\cdots\!99}a^{8}-\frac{35\!\cdots\!12}{14\!\cdots\!99}a^{7}+\frac{26\!\cdots\!23}{14\!\cdots\!99}a^{6}-\frac{20\!\cdots\!91}{14\!\cdots\!99}a^{5}+\frac{62\!\cdots\!34}{14\!\cdots\!99}a^{4}+\frac{11\!\cdots\!59}{14\!\cdots\!99}a^{3}+\frac{42\!\cdots\!81}{14\!\cdots\!99}a^{2}-\frac{91\!\cdots\!12}{14\!\cdots\!99}a-\frac{35\!\cdots\!30}{14\!\cdots\!99}$, $\frac{21\!\cdots\!69}{14\!\cdots\!99}a^{31}-\frac{21\!\cdots\!84}{14\!\cdots\!99}a^{30}+\frac{11\!\cdots\!22}{14\!\cdots\!99}a^{29}-\frac{54\!\cdots\!88}{14\!\cdots\!99}a^{28}+\frac{19\!\cdots\!22}{14\!\cdots\!99}a^{27}-\frac{59\!\cdots\!37}{14\!\cdots\!99}a^{26}+\frac{15\!\cdots\!28}{14\!\cdots\!99}a^{25}-\frac{34\!\cdots\!85}{14\!\cdots\!99}a^{24}+\frac{67\!\cdots\!42}{14\!\cdots\!99}a^{23}-\frac{10\!\cdots\!16}{14\!\cdots\!99}a^{22}+\frac{14\!\cdots\!00}{14\!\cdots\!99}a^{21}-\frac{16\!\cdots\!98}{14\!\cdots\!99}a^{20}+\frac{12\!\cdots\!46}{14\!\cdots\!99}a^{19}-\frac{42\!\cdots\!31}{14\!\cdots\!99}a^{18}-\frac{11\!\cdots\!76}{14\!\cdots\!99}a^{17}+\frac{38\!\cdots\!14}{14\!\cdots\!99}a^{16}-\frac{63\!\cdots\!99}{14\!\cdots\!99}a^{15}+\frac{47\!\cdots\!26}{14\!\cdots\!99}a^{14}+\frac{60\!\cdots\!70}{14\!\cdots\!99}a^{13}-\frac{15\!\cdots\!19}{14\!\cdots\!99}a^{12}+\frac{80\!\cdots\!59}{14\!\cdots\!99}a^{11}+\frac{76\!\cdots\!79}{14\!\cdots\!99}a^{10}-\frac{10\!\cdots\!47}{14\!\cdots\!99}a^{9}+\frac{27\!\cdots\!06}{14\!\cdots\!99}a^{8}+\frac{40\!\cdots\!93}{14\!\cdots\!99}a^{7}-\frac{73\!\cdots\!34}{14\!\cdots\!99}a^{6}-\frac{12\!\cdots\!44}{14\!\cdots\!99}a^{5}+\frac{61\!\cdots\!76}{14\!\cdots\!99}a^{4}+\frac{57\!\cdots\!18}{14\!\cdots\!99}a^{3}+\frac{26\!\cdots\!17}{14\!\cdots\!99}a^{2}-\frac{36\!\cdots\!14}{14\!\cdots\!99}a-\frac{32\!\cdots\!67}{14\!\cdots\!99}$, $\frac{48\!\cdots\!98}{14\!\cdots\!99}a^{31}-\frac{59\!\cdots\!05}{14\!\cdots\!99}a^{30}+\frac{37\!\cdots\!72}{14\!\cdots\!99}a^{29}-\frac{18\!\cdots\!63}{14\!\cdots\!99}a^{28}+\frac{74\!\cdots\!35}{14\!\cdots\!99}a^{27}-\frac{24\!\cdots\!62}{14\!\cdots\!99}a^{26}+\frac{70\!\cdots\!73}{14\!\cdots\!99}a^{25}-\frac{17\!\cdots\!61}{14\!\cdots\!99}a^{24}+\frac{37\!\cdots\!43}{14\!\cdots\!99}a^{23}-\frac{72\!\cdots\!68}{14\!\cdots\!99}a^{22}+\frac{11\!\cdots\!10}{14\!\cdots\!99}a^{21}-\frac{17\!\cdots\!62}{14\!\cdots\!99}a^{20}+\frac{21\!\cdots\!41}{14\!\cdots\!99}a^{19}-\frac{23\!\cdots\!30}{14\!\cdots\!99}a^{18}+\frac{20\!\cdots\!41}{14\!\cdots\!99}a^{17}-\frac{90\!\cdots\!57}{14\!\cdots\!99}a^{16}-\frac{11\!\cdots\!12}{14\!\cdots\!99}a^{15}+\frac{30\!\cdots\!39}{14\!\cdots\!99}a^{14}-\frac{17\!\cdots\!92}{14\!\cdots\!99}a^{13}-\frac{30\!\cdots\!82}{14\!\cdots\!99}a^{12}+\frac{57\!\cdots\!64}{14\!\cdots\!99}a^{11}-\frac{33\!\cdots\!26}{14\!\cdots\!99}a^{10}+\frac{34\!\cdots\!43}{14\!\cdots\!99}a^{9}-\frac{97\!\cdots\!05}{14\!\cdots\!99}a^{8}+\frac{29\!\cdots\!60}{14\!\cdots\!99}a^{7}-\frac{21\!\cdots\!13}{14\!\cdots\!99}a^{6}+\frac{52\!\cdots\!06}{14\!\cdots\!99}a^{5}-\frac{41\!\cdots\!06}{14\!\cdots\!99}a^{4}+\frac{58\!\cdots\!27}{14\!\cdots\!99}a^{3}-\frac{29\!\cdots\!77}{14\!\cdots\!99}a^{2}+\frac{27\!\cdots\!19}{14\!\cdots\!99}a-\frac{22\!\cdots\!61}{14\!\cdots\!99}$, $\frac{11\!\cdots\!24}{14\!\cdots\!99}a^{31}-\frac{37\!\cdots\!66}{14\!\cdots\!99}a^{30}-\frac{91\!\cdots\!91}{14\!\cdots\!99}a^{29}+\frac{11\!\cdots\!58}{14\!\cdots\!99}a^{28}-\frac{83\!\cdots\!16}{14\!\cdots\!99}a^{27}+\frac{36\!\cdots\!77}{14\!\cdots\!99}a^{26}-\frac{12\!\cdots\!98}{14\!\cdots\!99}a^{25}+\frac{38\!\cdots\!07}{14\!\cdots\!99}a^{24}-\frac{92\!\cdots\!77}{14\!\cdots\!99}a^{23}+\frac{20\!\cdots\!57}{14\!\cdots\!99}a^{22}-\frac{36\!\cdots\!39}{14\!\cdots\!99}a^{21}+\frac{58\!\cdots\!23}{14\!\cdots\!99}a^{20}-\frac{79\!\cdots\!93}{14\!\cdots\!99}a^{19}+\frac{89\!\cdots\!76}{14\!\cdots\!99}a^{18}-\frac{87\!\cdots\!56}{14\!\cdots\!99}a^{17}+\frac{57\!\cdots\!66}{14\!\cdots\!99}a^{16}+\frac{18\!\cdots\!18}{14\!\cdots\!99}a^{15}-\frac{14\!\cdots\!75}{14\!\cdots\!99}a^{14}+\frac{20\!\cdots\!94}{14\!\cdots\!99}a^{13}-\frac{69\!\cdots\!26}{14\!\cdots\!99}a^{12}-\frac{30\!\cdots\!32}{14\!\cdots\!99}a^{11}+\frac{32\!\cdots\!63}{14\!\cdots\!99}a^{10}-\frac{44\!\cdots\!44}{14\!\cdots\!99}a^{9}-\frac{81\!\cdots\!43}{14\!\cdots\!99}a^{8}-\frac{28\!\cdots\!77}{14\!\cdots\!99}a^{7}+\frac{79\!\cdots\!04}{14\!\cdots\!99}a^{6}-\frac{22\!\cdots\!68}{14\!\cdots\!99}a^{5}+\frac{85\!\cdots\!99}{14\!\cdots\!99}a^{4}-\frac{23\!\cdots\!81}{14\!\cdots\!99}a^{3}+\frac{14\!\cdots\!59}{14\!\cdots\!99}a^{2}-\frac{13\!\cdots\!76}{14\!\cdots\!99}a+\frac{28\!\cdots\!88}{14\!\cdots\!99}$, $\frac{22\!\cdots\!65}{14\!\cdots\!99}a^{31}-\frac{26\!\cdots\!87}{14\!\cdots\!99}a^{30}+\frac{17\!\cdots\!62}{14\!\cdots\!99}a^{29}-\frac{84\!\cdots\!83}{14\!\cdots\!99}a^{28}+\frac{33\!\cdots\!91}{14\!\cdots\!99}a^{27}-\frac{11\!\cdots\!29}{14\!\cdots\!99}a^{26}+\frac{31\!\cdots\!77}{14\!\cdots\!99}a^{25}-\frac{78\!\cdots\!95}{14\!\cdots\!99}a^{24}+\frac{16\!\cdots\!30}{14\!\cdots\!99}a^{23}-\frac{32\!\cdots\!90}{14\!\cdots\!99}a^{22}+\frac{52\!\cdots\!94}{14\!\cdots\!99}a^{21}-\frac{74\!\cdots\!45}{14\!\cdots\!99}a^{20}+\frac{91\!\cdots\!93}{14\!\cdots\!99}a^{19}-\frac{94\!\cdots\!64}{14\!\cdots\!99}a^{18}+\frac{77\!\cdots\!62}{14\!\cdots\!99}a^{17}-\frac{21\!\cdots\!86}{14\!\cdots\!99}a^{16}-\frac{75\!\cdots\!05}{14\!\cdots\!99}a^{15}+\frac{15\!\cdots\!59}{14\!\cdots\!99}a^{14}-\frac{95\!\cdots\!31}{14\!\cdots\!99}a^{13}-\frac{14\!\cdots\!95}{14\!\cdots\!99}a^{12}+\frac{29\!\cdots\!23}{14\!\cdots\!99}a^{11}-\frac{17\!\cdots\!88}{14\!\cdots\!99}a^{10}+\frac{44\!\cdots\!56}{14\!\cdots\!99}a^{9}-\frac{54\!\cdots\!30}{14\!\cdots\!99}a^{8}+\frac{71\!\cdots\!65}{14\!\cdots\!99}a^{7}-\frac{55\!\cdots\!00}{14\!\cdots\!99}a^{6}+\frac{22\!\cdots\!54}{14\!\cdots\!99}a^{5}-\frac{17\!\cdots\!07}{14\!\cdots\!99}a^{4}+\frac{20\!\cdots\!13}{14\!\cdots\!99}a^{3}-\frac{10\!\cdots\!13}{14\!\cdots\!99}a^{2}+\frac{38\!\cdots\!78}{14\!\cdots\!99}a-\frac{18\!\cdots\!18}{14\!\cdots\!99}$, $\frac{16\!\cdots\!48}{14\!\cdots\!99}a^{31}-\frac{11\!\cdots\!00}{14\!\cdots\!99}a^{30}+\frac{41\!\cdots\!26}{14\!\cdots\!99}a^{29}-\frac{14\!\cdots\!79}{14\!\cdots\!99}a^{28}+\frac{21\!\cdots\!77}{14\!\cdots\!99}a^{27}+\frac{16\!\cdots\!70}{14\!\cdots\!99}a^{26}-\frac{25\!\cdots\!14}{14\!\cdots\!99}a^{25}+\frac{13\!\cdots\!30}{14\!\cdots\!99}a^{24}-\frac{39\!\cdots\!96}{14\!\cdots\!99}a^{23}+\frac{10\!\cdots\!75}{14\!\cdots\!99}a^{22}-\frac{21\!\cdots\!26}{14\!\cdots\!99}a^{21}+\frac{37\!\cdots\!38}{14\!\cdots\!99}a^{20}-\frac{55\!\cdots\!85}{14\!\cdots\!99}a^{19}+\frac{68\!\cdots\!44}{14\!\cdots\!99}a^{18}-\frac{75\!\cdots\!62}{14\!\cdots\!99}a^{17}+\frac{65\!\cdots\!52}{14\!\cdots\!99}a^{16}-\frac{21\!\cdots\!51}{14\!\cdots\!99}a^{15}-\frac{76\!\cdots\!30}{14\!\cdots\!99}a^{14}+\frac{17\!\cdots\!62}{14\!\cdots\!99}a^{13}-\frac{90\!\cdots\!72}{14\!\cdots\!99}a^{12}-\frac{16\!\cdots\!94}{14\!\cdots\!99}a^{11}+\frac{28\!\cdots\!49}{14\!\cdots\!99}a^{10}-\frac{11\!\cdots\!19}{14\!\cdots\!99}a^{9}-\frac{46\!\cdots\!50}{14\!\cdots\!99}a^{8}+\frac{98\!\cdots\!37}{14\!\cdots\!99}a^{7}+\frac{57\!\cdots\!60}{14\!\cdots\!99}a^{6}-\frac{40\!\cdots\!10}{14\!\cdots\!99}a^{5}+\frac{14\!\cdots\!21}{14\!\cdots\!99}a^{4}-\frac{14\!\cdots\!99}{14\!\cdots\!99}a^{3}+\frac{12\!\cdots\!61}{14\!\cdots\!99}a^{2}-\frac{41\!\cdots\!33}{14\!\cdots\!99}a+\frac{32\!\cdots\!89}{14\!\cdots\!99}$, $\frac{14\!\cdots\!26}{14\!\cdots\!99}a^{31}-\frac{14\!\cdots\!81}{14\!\cdots\!99}a^{30}+\frac{83\!\cdots\!70}{14\!\cdots\!99}a^{29}-\frac{39\!\cdots\!45}{14\!\cdots\!99}a^{28}+\frac{14\!\cdots\!65}{14\!\cdots\!99}a^{27}-\frac{45\!\cdots\!21}{14\!\cdots\!99}a^{26}+\frac{12\!\cdots\!46}{14\!\cdots\!99}a^{25}-\frac{28\!\cdots\!48}{14\!\cdots\!99}a^{24}+\frac{59\!\cdots\!10}{14\!\cdots\!99}a^{23}-\frac{10\!\cdots\!84}{14\!\cdots\!99}a^{22}+\frac{15\!\cdots\!10}{14\!\cdots\!99}a^{21}-\frac{20\!\cdots\!94}{14\!\cdots\!99}a^{20}+\frac{22\!\cdots\!03}{14\!\cdots\!99}a^{19}-\frac{20\!\cdots\!03}{14\!\cdots\!99}a^{18}+\frac{11\!\cdots\!33}{14\!\cdots\!99}a^{17}+\frac{93\!\cdots\!86}{14\!\cdots\!99}a^{16}-\frac{37\!\cdots\!96}{14\!\cdots\!99}a^{15}+\frac{45\!\cdots\!79}{14\!\cdots\!99}a^{14}+\frac{59\!\cdots\!76}{14\!\cdots\!99}a^{13}-\frac{77\!\cdots\!51}{14\!\cdots\!99}a^{12}+\frac{80\!\cdots\!52}{14\!\cdots\!99}a^{11}-\frac{14\!\cdots\!49}{14\!\cdots\!99}a^{10}-\frac{24\!\cdots\!66}{14\!\cdots\!99}a^{9}+\frac{22\!\cdots\!46}{14\!\cdots\!99}a^{8}+\frac{20\!\cdots\!57}{14\!\cdots\!99}a^{7}-\frac{14\!\cdots\!76}{14\!\cdots\!99}a^{6}+\frac{36\!\cdots\!39}{14\!\cdots\!99}a^{5}-\frac{24\!\cdots\!56}{18\!\cdots\!59}a^{4}+\frac{24\!\cdots\!96}{14\!\cdots\!99}a^{3}-\frac{89\!\cdots\!88}{14\!\cdots\!99}a^{2}+\frac{31\!\cdots\!01}{14\!\cdots\!99}a-\frac{18\!\cdots\!02}{14\!\cdots\!99}$, $\frac{16\!\cdots\!80}{14\!\cdots\!99}a^{31}-\frac{16\!\cdots\!48}{14\!\cdots\!99}a^{30}+\frac{96\!\cdots\!67}{14\!\cdots\!99}a^{29}-\frac{46\!\cdots\!01}{14\!\cdots\!99}a^{28}+\frac{17\!\cdots\!01}{14\!\cdots\!99}a^{27}-\frac{55\!\cdots\!54}{14\!\cdots\!99}a^{26}+\frac{15\!\cdots\!68}{14\!\cdots\!99}a^{25}-\frac{37\!\cdots\!12}{14\!\cdots\!99}a^{24}+\frac{79\!\cdots\!41}{14\!\cdots\!99}a^{23}-\frac{14\!\cdots\!02}{14\!\cdots\!99}a^{22}+\frac{23\!\cdots\!58}{14\!\cdots\!99}a^{21}-\frac{33\!\cdots\!59}{14\!\cdots\!99}a^{20}+\frac{41\!\cdots\!61}{14\!\cdots\!99}a^{19}-\frac{44\!\cdots\!79}{14\!\cdots\!99}a^{18}+\frac{37\!\cdots\!56}{14\!\cdots\!99}a^{17}-\frac{13\!\cdots\!24}{14\!\cdots\!99}a^{16}-\frac{26\!\cdots\!86}{14\!\cdots\!99}a^{15}+\frac{54\!\cdots\!89}{14\!\cdots\!99}a^{14}-\frac{21\!\cdots\!65}{14\!\cdots\!99}a^{13}-\frac{45\!\cdots\!69}{14\!\cdots\!99}a^{12}+\frac{74\!\cdots\!40}{14\!\cdots\!99}a^{11}-\frac{63\!\cdots\!87}{14\!\cdots\!99}a^{10}+\frac{55\!\cdots\!52}{14\!\cdots\!99}a^{9}-\frac{49\!\cdots\!52}{14\!\cdots\!99}a^{8}+\frac{15\!\cdots\!26}{14\!\cdots\!99}a^{7}-\frac{29\!\cdots\!82}{14\!\cdots\!99}a^{6}+\frac{12\!\cdots\!20}{14\!\cdots\!99}a^{5}-\frac{24\!\cdots\!72}{14\!\cdots\!99}a^{4}+\frac{92\!\cdots\!43}{14\!\cdots\!99}a^{3}-\frac{34\!\cdots\!43}{14\!\cdots\!99}a^{2}+\frac{20\!\cdots\!52}{14\!\cdots\!99}a-\frac{32\!\cdots\!46}{14\!\cdots\!99}$, $\frac{11\!\cdots\!87}{14\!\cdots\!99}a^{31}-\frac{13\!\cdots\!18}{14\!\cdots\!99}a^{30}+\frac{82\!\cdots\!71}{14\!\cdots\!99}a^{29}-\frac{41\!\cdots\!30}{14\!\cdots\!99}a^{28}+\frac{16\!\cdots\!60}{14\!\cdots\!99}a^{27}-\frac{53\!\cdots\!62}{14\!\cdots\!99}a^{26}+\frac{15\!\cdots\!18}{14\!\cdots\!99}a^{25}-\frac{38\!\cdots\!28}{14\!\cdots\!99}a^{24}+\frac{83\!\cdots\!21}{14\!\cdots\!99}a^{23}-\frac{15\!\cdots\!00}{14\!\cdots\!99}a^{22}+\frac{26\!\cdots\!31}{14\!\cdots\!99}a^{21}-\frac{37\!\cdots\!19}{14\!\cdots\!99}a^{20}+\frac{47\!\cdots\!41}{14\!\cdots\!99}a^{19}-\frac{50\!\cdots\!92}{14\!\cdots\!99}a^{18}+\frac{42\!\cdots\!72}{14\!\cdots\!99}a^{17}-\frac{15\!\cdots\!02}{14\!\cdots\!99}a^{16}-\frac{32\!\cdots\!33}{14\!\cdots\!99}a^{15}+\frac{77\!\cdots\!47}{14\!\cdots\!99}a^{14}-\frac{58\!\cdots\!24}{14\!\cdots\!99}a^{13}-\frac{43\!\cdots\!96}{14\!\cdots\!99}a^{12}+\frac{13\!\cdots\!29}{14\!\cdots\!99}a^{11}-\frac{12\!\cdots\!26}{14\!\cdots\!99}a^{10}+\frac{37\!\cdots\!91}{14\!\cdots\!99}a^{9}+\frac{10\!\cdots\!82}{14\!\cdots\!99}a^{8}+\frac{82\!\cdots\!14}{14\!\cdots\!99}a^{7}-\frac{29\!\cdots\!99}{14\!\cdots\!99}a^{6}+\frac{21\!\cdots\!18}{14\!\cdots\!99}a^{5}-\frac{98\!\cdots\!81}{14\!\cdots\!99}a^{4}+\frac{72\!\cdots\!77}{14\!\cdots\!99}a^{3}-\frac{58\!\cdots\!37}{14\!\cdots\!99}a^{2}+\frac{24\!\cdots\!34}{14\!\cdots\!99}a-\frac{41\!\cdots\!45}{14\!\cdots\!99}$
| 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: | \( 584489557946.5273 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 584489557946.5273 \cdot 1}{10\cdot\sqrt{2294081869728198830572049522437155246734619140625}}\cr\approx \mathstrut & 0.227693269976021 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 - 11*x^31 + 68*x^30 - 336*x^29 + 1315*x^28 - 4335*x^27 + 12498*x^26 - 30713*x^25 + 67152*x^24 - 127199*x^23 + 211471*x^22 - 307078*x^21 + 387026*x^20 - 419617*x^19 + 359728*x^18 - 149755*x^17 - 229142*x^16 + 581095*x^15 - 441722*x^14 - 315448*x^13 + 1021466*x^12 - 946897*x^11 + 366311*x^10 - 50246*x^9 + 132237*x^8 - 215882*x^7 + 157773*x^6 - 97500*x^5 + 79930*x^4 - 54894*x^3 + 25063*x^2 - 7709*x + 3131)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^32 - 11*x^31 + 68*x^30 - 336*x^29 + 1315*x^28 - 4335*x^27 + 12498*x^26 - 30713*x^25 + 67152*x^24 - 127199*x^23 + 211471*x^22 - 307078*x^21 + 387026*x^20 - 419617*x^19 + 359728*x^18 - 149755*x^17 - 229142*x^16 + 581095*x^15 - 441722*x^14 - 315448*x^13 + 1021466*x^12 - 946897*x^11 + 366311*x^10 - 50246*x^9 + 132237*x^8 - 215882*x^7 + 157773*x^6 - 97500*x^5 + 79930*x^4 - 54894*x^3 + 25063*x^2 - 7709*x + 3131, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 - 11*x^31 + 68*x^30 - 336*x^29 + 1315*x^28 - 4335*x^27 + 12498*x^26 - 30713*x^25 + 67152*x^24 - 127199*x^23 + 211471*x^22 - 307078*x^21 + 387026*x^20 - 419617*x^19 + 359728*x^18 - 149755*x^17 - 229142*x^16 + 581095*x^15 - 441722*x^14 - 315448*x^13 + 1021466*x^12 - 946897*x^11 + 366311*x^10 - 50246*x^9 + 132237*x^8 - 215882*x^7 + 157773*x^6 - 97500*x^5 + 79930*x^4 - 54894*x^3 + 25063*x^2 - 7709*x + 3131);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 11*x^31 + 68*x^30 - 336*x^29 + 1315*x^28 - 4335*x^27 + 12498*x^26 - 30713*x^25 + 67152*x^24 - 127199*x^23 + 211471*x^22 - 307078*x^21 + 387026*x^20 - 419617*x^19 + 359728*x^18 - 149755*x^17 - 229142*x^16 + 581095*x^15 - 441722*x^14 - 315448*x^13 + 1021466*x^12 - 946897*x^11 + 366311*x^10 - 50246*x^9 + 132237*x^8 - 215882*x^7 + 157773*x^6 - 97500*x^5 + 79930*x^4 - 54894*x^3 + 25063*x^2 - 7709*x + 3131);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 96 |
| The 28 conjugacy class representatives for $C_8.A_4$ |
| Character table for $C_8.A_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.99225.1, 8.0.9845600625.1, 16.0.60584907291875244140625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $24{,}\,{\href{/padicField/2.8.0.1}{8} }$ | R | R | R | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | $24{,}\,{\href{/padicField/13.8.0.1}{8} }$ | $24{,}\,{\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.12.0.1}{12} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{4}$ | ${\href{/padicField/29.12.0.1}{12} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ | $24{,}\,{\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.6.0.1}{6} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | $24{,}\,{\href{/padicField/43.8.0.1}{8} }$ | $24{,}\,{\href{/padicField/47.8.0.1}{8} }$ | $24{,}\,{\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.12.0.1}{12} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.8.0.1 | $x^{8} + 2 x^{5} + x^{4} + 2 x^{2} + 2 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
| Deg $24$ | $3$ | $8$ | $32$ | ||||
|
\(5\)
| 5.16.14.1 | $x^{16} - 20 x^{8} - 100$ | $8$ | $2$ | $14$ | $C_8: C_2$ | $[\ ]_{8}^{2}$ |
| 5.16.14.1 | $x^{16} - 20 x^{8} - 100$ | $8$ | $2$ | $14$ | $C_8: C_2$ | $[\ ]_{8}^{2}$ | |
|
\(7\)
| 7.8.0.1 | $x^{8} + 4 x^{3} + 6 x^{2} + 2 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
| 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) | $1$ | $1$ |
| 1.315.6t1.b.a | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.6.1969120125.2 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.315.6t1.b.b | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.6.1969120125.2 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.63.3t1.a.a | $1$ | $ 3^{2} \cdot 7 $ | 3.3.3969.2 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.63.3t1.a.b | $1$ | $ 3^{2} \cdot 7 $ | 3.3.3969.2 | $C_3$ (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) | $0$ | $-1$ |
| 1.315.12t1.b.a | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 12.0.484679258335001953125.2 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.315.12t1.b.b | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 12.0.484679258335001953125.2 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.315.12t1.b.c | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 12.0.484679258335001953125.2 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.315.12t1.b.d | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 12.0.484679258335001953125.2 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 2.99225.48.a.a | $2$ | $ 3^{4} \cdot 5^{2} \cdot 7^{2}$ | 32.0.2294081869728198830572049522437155246734619140625.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ | |
| 2.99225.48.a.b | $2$ | $ 3^{4} \cdot 5^{2} \cdot 7^{2}$ | 32.0.2294081869728198830572049522437155246734619140625.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ | |
| 2.99225.48.a.c | $2$ | $ 3^{4} \cdot 5^{2} \cdot 7^{2}$ | 32.0.2294081869728198830572049522437155246734619140625.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ | |
| 2.99225.48.a.d | $2$ | $ 3^{4} \cdot 5^{2} \cdot 7^{2}$ | 32.0.2294081869728198830572049522437155246734619140625.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ | |
| * | 2.1575.32t402.a.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7 $ | 32.0.2294081869728198830572049522437155246734619140625.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 2.1575.32t402.a.b | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7 $ | 32.0.2294081869728198830572049522437155246734619140625.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 2.1575.32t402.a.c | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7 $ | 32.0.2294081869728198830572049522437155246734619140625.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 2.1575.32t402.a.d | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7 $ | 32.0.2294081869728198830572049522437155246734619140625.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 2.1575.32t402.a.e | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7 $ | 32.0.2294081869728198830572049522437155246734619140625.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 2.1575.32t402.a.f | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7 $ | 32.0.2294081869728198830572049522437155246734619140625.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 2.1575.32t402.a.g | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7 $ | 32.0.2294081869728198830572049522437155246734619140625.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 2.1575.32t402.a.h | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7 $ | 32.0.2294081869728198830572049522437155246734619140625.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 3.99225.4t4.a.a | $3$ | $ 3^{4} \cdot 5^{2} \cdot 7^{2}$ | 4.0.99225.1 | $A_4$ (as 4T4) | $1$ | $-1$ |
| * | 3.19845.6t6.a.a | $3$ | $ 3^{4} \cdot 5 \cdot 7^{2}$ | 6.2.78764805.1 | $A_4\times C_2$ (as 6T6) | $1$ | $-1$ |
| * | 3.496125.12t29.a.a | $3$ | $ 3^{4} \cdot 5^{3} \cdot 7^{2}$ | 12.8.484679258335001953125.1 | $C_4\times A_4$ (as 12T29) | $0$ | $1$ |
| * | 3.496125.12t29.a.b | $3$ | $ 3^{4} \cdot 5^{3} \cdot 7^{2}$ | 12.8.484679258335001953125.1 | $C_4\times A_4$ (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 *.