Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 - 18*x^30 + 72*x^29 + 311*x^28 - 1090*x^27 - 2862*x^26 + 8691*x^25 + 24872*x^24 - 66988*x^23 - 86403*x^22 + 207758*x^21 + 395325*x^20 - 595053*x^19 - 1119168*x^18 + 799235*x^17 + 3298870*x^16 - 4221332*x^15 + 4111698*x^14 - 7461943*x^13 + 5469898*x^12 + 2888264*x^11 - 9443335*x^10 + 9992412*x^9 - 3874794*x^8 - 4061403*x^7 + 6823432*x^6 - 2937855*x^5 - 1141251*x^4 + 2061220*x^3 - 477856*x^2 - 165184*x + 215296)
gp: K = bnfinit(y^32 - 4*y^31 - 18*y^30 + 72*y^29 + 311*y^28 - 1090*y^27 - 2862*y^26 + 8691*y^25 + 24872*y^24 - 66988*y^23 - 86403*y^22 + 207758*y^21 + 395325*y^20 - 595053*y^19 - 1119168*y^18 + 799235*y^17 + 3298870*y^16 - 4221332*y^15 + 4111698*y^14 - 7461943*y^13 + 5469898*y^12 + 2888264*y^11 - 9443335*y^10 + 9992412*y^9 - 3874794*y^8 - 4061403*y^7 + 6823432*y^6 - 2937855*y^5 - 1141251*y^4 + 2061220*y^3 - 477856*y^2 - 165184*y + 215296, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 4*x^31 - 18*x^30 + 72*x^29 + 311*x^28 - 1090*x^27 - 2862*x^26 + 8691*x^25 + 24872*x^24 - 66988*x^23 - 86403*x^22 + 207758*x^21 + 395325*x^20 - 595053*x^19 - 1119168*x^18 + 799235*x^17 + 3298870*x^16 - 4221332*x^15 + 4111698*x^14 - 7461943*x^13 + 5469898*x^12 + 2888264*x^11 - 9443335*x^10 + 9992412*x^9 - 3874794*x^8 - 4061403*x^7 + 6823432*x^6 - 2937855*x^5 - 1141251*x^4 + 2061220*x^3 - 477856*x^2 - 165184*x + 215296);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 4*x^31 - 18*x^30 + 72*x^29 + 311*x^28 - 1090*x^27 - 2862*x^26 + 8691*x^25 + 24872*x^24 - 66988*x^23 - 86403*x^22 + 207758*x^21 + 395325*x^20 - 595053*x^19 - 1119168*x^18 + 799235*x^17 + 3298870*x^16 - 4221332*x^15 + 4111698*x^14 - 7461943*x^13 + 5469898*x^12 + 2888264*x^11 - 9443335*x^10 + 9992412*x^9 - 3874794*x^8 - 4061403*x^7 + 6823432*x^6 - 2937855*x^5 - 1141251*x^4 + 2061220*x^3 - 477856*x^2 - 165184*x + 215296)
\( x^{32} - 4 x^{31} - 18 x^{30} + 72 x^{29} + 311 x^{28} - 1090 x^{27} - 2862 x^{26} + 8691 x^{25} + \cdots + 215296 \)
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: |
\(2363147162130074593554078242531588550681346814117431640625\)
\(\medspace = 3^{16}\cdot 5^{16}\cdot 7^{16}\cdot 89^{8}\cdot 229^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(62.08\) | 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: | $3^{1/2}5^{1/2}7^{1/2}89^{1/2}229^{1/2}\approx 1462.8755927965988$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\), \(89\), \(229\)
| 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 a CM field. | |||
| Reflex fields: | unavailable$^{32768}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2}a^{16}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{7}$, $\frac{1}{4}a^{23}-\frac{1}{4}a^{19}-\frac{1}{4}a^{17}-\frac{1}{4}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{12}a^{24}+\frac{1}{6}a^{22}-\frac{1}{6}a^{21}+\frac{1}{12}a^{20}-\frac{1}{6}a^{19}+\frac{1}{12}a^{18}-\frac{1}{4}a^{17}-\frac{1}{6}a^{16}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{1}{6}a^{11}+\frac{1}{6}a^{10}-\frac{1}{12}a^{9}-\frac{1}{6}a^{8}-\frac{1}{3}a^{6}+\frac{1}{12}a^{5}+\frac{1}{12}a^{3}-\frac{1}{4}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{12}a^{25}-\frac{1}{12}a^{23}-\frac{1}{6}a^{22}+\frac{1}{12}a^{21}-\frac{1}{6}a^{20}-\frac{1}{6}a^{19}-\frac{1}{4}a^{18}+\frac{1}{12}a^{17}-\frac{1}{4}a^{16}+\frac{1}{6}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}-\frac{1}{6}a^{12}+\frac{1}{6}a^{11}+\frac{5}{12}a^{10}+\frac{1}{3}a^{9}+\frac{1}{4}a^{8}+\frac{1}{6}a^{7}-\frac{5}{12}a^{6}-\frac{1}{2}a^{5}-\frac{1}{6}a^{4}+\frac{1}{4}a^{3}-\frac{5}{12}a^{2}+\frac{1}{12}a$, $\frac{1}{24}a^{26}-\frac{1}{24}a^{25}-\frac{1}{24}a^{24}+\frac{1}{12}a^{23}-\frac{1}{8}a^{22}+\frac{1}{8}a^{21}+\frac{1}{12}a^{19}-\frac{1}{12}a^{18}+\frac{5}{24}a^{17}+\frac{1}{12}a^{16}+\frac{1}{3}a^{15}+\frac{1}{6}a^{14}+\frac{1}{12}a^{13}-\frac{1}{3}a^{12}-\frac{3}{8}a^{11}-\frac{7}{24}a^{10}+\frac{5}{24}a^{9}-\frac{1}{6}a^{8}+\frac{5}{24}a^{7}-\frac{1}{24}a^{6}+\frac{5}{12}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{8}a^{2}-\frac{1}{6}a$, $\frac{1}{48}a^{27}-\frac{1}{24}a^{25}-\frac{1}{48}a^{24}-\frac{1}{48}a^{23}+\frac{1}{6}a^{22}+\frac{7}{48}a^{21}+\frac{1}{12}a^{19}+\frac{1}{48}a^{18}+\frac{1}{48}a^{17}-\frac{5}{24}a^{16}+\frac{1}{4}a^{15}-\frac{5}{24}a^{14}+\frac{5}{24}a^{13}+\frac{5}{16}a^{12}+\frac{1}{4}a^{11}-\frac{1}{8}a^{10}-\frac{7}{16}a^{9}-\frac{19}{48}a^{8}-\frac{1}{6}a^{7}-\frac{7}{48}a^{6}-\frac{1}{6}a^{5}-\frac{1}{2}a^{4}+\frac{17}{48}a^{3}-\frac{7}{48}a^{2}-\frac{1}{4}a+\frac{1}{3}$, $\frac{1}{426096}a^{28}+\frac{2209}{426096}a^{27}+\frac{1109}{106524}a^{26}-\frac{16525}{426096}a^{25}+\frac{274}{8877}a^{24}-\frac{495}{4304}a^{23}+\frac{10909}{47344}a^{22}+\frac{55573}{426096}a^{21}-\frac{13637}{106524}a^{20}-\frac{30539}{426096}a^{19}+\frac{9379}{71016}a^{18}-\frac{40507}{426096}a^{17}-\frac{12325}{213048}a^{16}-\frac{25639}{213048}a^{15}-\frac{14911}{53262}a^{14}+\frac{773}{12912}a^{13}+\frac{68083}{426096}a^{12}+\frac{893}{17754}a^{11}+\frac{125903}{426096}a^{10}+\frac{62171}{213048}a^{9}+\frac{144725}{426096}a^{8}-\frac{128449}{426096}a^{7}-\frac{116233}{426096}a^{6}-\frac{3323}{9684}a^{5}+\frac{159949}{426096}a^{4}+\frac{1457}{19368}a^{3}-\frac{191881}{426096}a^{2}+\frac{5817}{11836}a-\frac{12479}{26631}$, $\frac{1}{426096}a^{29}-\frac{443}{106524}a^{27}+\frac{257}{47344}a^{26}+\frac{14653}{426096}a^{25}-\frac{329}{11836}a^{24}-\frac{9689}{142032}a^{23}+\frac{1673}{213048}a^{22}+\frac{1205}{5918}a^{21}-\frac{4217}{142032}a^{20}+\frac{16217}{426096}a^{19}+\frac{1015}{53262}a^{18}-\frac{4327}{26631}a^{17}-\frac{8491}{71016}a^{16}+\frac{10139}{71016}a^{15}+\frac{116603}{426096}a^{14}+\frac{39667}{106524}a^{13}-\frac{19357}{106524}a^{12}+\frac{26009}{426096}a^{11}-\frac{38761}{426096}a^{10}+\frac{12970}{26631}a^{9}-\frac{3561}{47344}a^{8}-\frac{321}{2152}a^{7}-\frac{2009}{17754}a^{6}+\frac{19457}{426096}a^{5}-\frac{26195}{426096}a^{4}-\frac{1633}{71016}a^{3}+\frac{331}{4842}a^{2}+\frac{24755}{53262}a+\frac{3026}{26631}$, $\frac{1}{11\!\cdots\!44}a^{30}+\frac{948710339766505}{36\!\cdots\!48}a^{29}-\frac{56\!\cdots\!47}{55\!\cdots\!72}a^{28}+\frac{10\!\cdots\!43}{11\!\cdots\!44}a^{27}+\frac{32\!\cdots\!89}{18\!\cdots\!24}a^{26}-\frac{30\!\cdots\!13}{11\!\cdots\!44}a^{25}+\frac{11\!\cdots\!71}{33\!\cdots\!68}a^{24}+\frac{90\!\cdots\!99}{11\!\cdots\!44}a^{23}-\frac{24\!\cdots\!75}{61\!\cdots\!08}a^{22}+\frac{50\!\cdots\!61}{11\!\cdots\!44}a^{21}-\frac{31\!\cdots\!87}{30\!\cdots\!04}a^{20}+\frac{70\!\cdots\!91}{36\!\cdots\!48}a^{19}-\frac{99\!\cdots\!10}{62\!\cdots\!69}a^{18}+\frac{34\!\cdots\!19}{55\!\cdots\!72}a^{17}+\frac{17\!\cdots\!54}{69\!\cdots\!59}a^{16}+\frac{13\!\cdots\!79}{36\!\cdots\!48}a^{15}+\frac{47\!\cdots\!53}{11\!\cdots\!44}a^{14}+\frac{89\!\cdots\!17}{55\!\cdots\!72}a^{13}+\frac{12\!\cdots\!03}{12\!\cdots\!16}a^{12}-\frac{58\!\cdots\!49}{16\!\cdots\!08}a^{11}+\frac{14\!\cdots\!19}{36\!\cdots\!48}a^{10}-\frac{52\!\cdots\!23}{11\!\cdots\!44}a^{9}-\frac{34\!\cdots\!47}{11\!\cdots\!44}a^{8}-\frac{13\!\cdots\!41}{55\!\cdots\!72}a^{7}-\frac{23\!\cdots\!19}{11\!\cdots\!44}a^{6}-\frac{23\!\cdots\!24}{23\!\cdots\!53}a^{5}+\frac{45\!\cdots\!33}{10\!\cdots\!04}a^{4}-\frac{26\!\cdots\!51}{61\!\cdots\!08}a^{3}-\frac{51\!\cdots\!03}{15\!\cdots\!02}a^{2}+\frac{21\!\cdots\!51}{27\!\cdots\!36}a-\frac{72\!\cdots\!12}{69\!\cdots\!59}$, $\frac{1}{37\!\cdots\!64}a^{31}-\frac{14\!\cdots\!05}{52\!\cdots\!12}a^{30}-\frac{56\!\cdots\!49}{18\!\cdots\!32}a^{29}+\frac{32\!\cdots\!63}{15\!\cdots\!36}a^{28}+\frac{25\!\cdots\!43}{37\!\cdots\!64}a^{27}+\frac{24\!\cdots\!37}{57\!\cdots\!04}a^{26}+\frac{73\!\cdots\!43}{37\!\cdots\!32}a^{25}-\frac{27\!\cdots\!33}{37\!\cdots\!64}a^{24}-\frac{18\!\cdots\!71}{31\!\cdots\!72}a^{23}+\frac{17\!\cdots\!85}{31\!\cdots\!72}a^{22}+\frac{23\!\cdots\!41}{37\!\cdots\!64}a^{21}-\frac{23\!\cdots\!91}{18\!\cdots\!32}a^{20}-\frac{36\!\cdots\!63}{37\!\cdots\!64}a^{19}+\frac{53\!\cdots\!67}{42\!\cdots\!96}a^{18}+\frac{96\!\cdots\!05}{10\!\cdots\!68}a^{17}+\frac{85\!\cdots\!89}{34\!\cdots\!24}a^{16}+\frac{75\!\cdots\!89}{18\!\cdots\!32}a^{15}+\frac{40\!\cdots\!71}{94\!\cdots\!16}a^{14}-\frac{30\!\cdots\!87}{18\!\cdots\!32}a^{13}-\frac{14\!\cdots\!27}{37\!\cdots\!64}a^{12}+\frac{36\!\cdots\!53}{17\!\cdots\!12}a^{11}+\frac{13\!\cdots\!22}{98\!\cdots\!71}a^{10}-\frac{26\!\cdots\!65}{12\!\cdots\!88}a^{9}-\frac{32\!\cdots\!93}{15\!\cdots\!36}a^{8}-\frac{46\!\cdots\!65}{18\!\cdots\!32}a^{7}+\frac{93\!\cdots\!77}{37\!\cdots\!64}a^{6}+\frac{15\!\cdots\!33}{31\!\cdots\!72}a^{5}+\frac{43\!\cdots\!49}{22\!\cdots\!92}a^{4}-\frac{13\!\cdots\!87}{37\!\cdots\!64}a^{3}-\frac{22\!\cdots\!53}{47\!\cdots\!08}a^{2}+\frac{33\!\cdots\!41}{13\!\cdots\!28}a+\frac{28\!\cdots\!95}{68\!\cdots\!98}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{1590}$, which has order $1590$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $1590$ (assuming GRH)
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{46490688600500137277704913805153691790529148290111965811742321277}{604565565706998913088635993669365979176838043334452770474853299363456} a^{31} + \frac{40961587498400958920282900505473238336356832212250327520963506829}{113356043570062296204119248813006121095657133125209894464034993630648} a^{30} + \frac{1027320267532457048724801496255122756573848237473801874922216573743}{906848348560498369632953990504048968765257065001679155712279949045184} a^{29} - \frac{10881382628665478298980266833761433078612473829038743200230645389}{1717515811667610548547261345651607895388744441291059007030833236828} a^{28} - \frac{11790794426717774707773343421566612384619568139315988376210643631707}{604565565706998913088635993669365979176838043334452770474853299363456} a^{27} + \frac{88489020141622612092308608373970645618449564691283274324437839475957}{906848348560498369632953990504048968765257065001679155712279949045184} a^{26} + \frac{137978463141508099783249375026661028921845049856431101022312847378973}{906848348560498369632953990504048968765257065001679155712279949045184} a^{25} - \frac{469539429552014992008290801083921045773542604136214674576856792102367}{604565565706998913088635993669365979176838043334452770474853299363456} a^{24} - \frac{621932323461347305349416772092881713249390131766634369179165653817591}{453424174280249184816476995252024484382628532500839577856139974522592} a^{23} + \frac{2777946327752054129903079878009646694272752827863860168548600496802181}{453424174280249184816476995252024484382628532500839577856139974522592} a^{22} + \frac{392459044740538267727773950825804946874176433210897823063621384429487}{164881517920090612660537089182554357957319466363941664674959990735488} a^{21} - \frac{16115910488488492320601198423088918728114209799415107515862604406246847}{906848348560498369632953990504048968765257065001679155712279949045184} a^{20} - \frac{10876154469129216394284382321972039807996251907509525747643758931587633}{604565565706998913088635993669365979176838043334452770474853299363456} a^{19} + \frac{106570862945997287190796196502285916071208818167900803375703268187364047}{1813696697120996739265907981008097937530514130003358311424559898090368} a^{18} + \frac{235763111207489661223754606120183218692086599627929670654157261879305}{5211772118163783733522724083356603268765845201159075607541838787616} a^{17} - \frac{171458058149313701721011683949566864066117022321216103742415598245198389}{1813696697120996739265907981008097937530514130003358311424559898090368} a^{16} - \frac{15529745885244195597827876312216811136245926706850973987489587924229885}{82440758960045306330268544591277178978659733181970832337479995367744} a^{15} + \frac{208158306306045823034226541857023747068264019375255831414302997040031325}{453424174280249184816476995252024484382628532500839577856139974522592} a^{14} - \frac{191416731597290399150116969632872859215276431427703418971373757119917373}{302282782853499456544317996834682989588419021667226385237426649681728} a^{13} + \frac{608427673425722745716055163605663748050887090711974741742423731121111059}{604565565706998913088635993669365979176838043334452770474853299363456} a^{12} - \frac{1009642491523817429083364284659998633030543736093735103527441290100596545}{906848348560498369632953990504048968765257065001679155712279949045184} a^{11} + \frac{60927717153856303483891460190350990798244811198466457241022958056684485}{113356043570062296204119248813006121095657133125209894464034993630648} a^{10} + \frac{20410483213415441312954695003852022522986335953924437426108063498138193}{54960505973363537553512363060851452652439822121313888224986663578496} a^{9} - \frac{117255665817276556177738058388383546403103125617897990379290438924626505}{113356043570062296204119248813006121095657133125209894464034993630648} a^{8} + \frac{916279785465334608575977624980334915263212182327584041279138483045378891}{906848348560498369632953990504048968765257065001679155712279949045184} a^{7} - \frac{223975118706813177243433828962879478864276703578843681830475004848007609}{604565565706998913088635993669365979176838043334452770474853299363456} a^{6} - \frac{11804096325784304340051606658201726465207199504340027662786993457288991}{41220379480022653165134272295638589489329866590985416168739997683872} a^{5} + \frac{259416200077417218992350585738676206405202243537726609654864726522487187}{604565565706998913088635993669365979176838043334452770474853299363456} a^{4} - \frac{368451981026373455964775129906921309416869560366304331311526418400029351}{1813696697120996739265907981008097937530514130003358311424559898090368} a^{3} - \frac{1986680370355795566241415371684354050918743372564085865858351799833129}{75570695713374864136079499208670747397104755416806596309356662420432} a^{2} + \frac{1095435858418094593916116348284349365760180713178823880298507413453133}{18892673928343716034019874802167686849276188854201649077339165605108} a - \frac{24960553132748417358091142682485962633355138791398306609677292090277}{977207272155709450035510765629363112893595975217326676414094772678} \)
(order $6$)
| 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: |
$\frac{45\!\cdots\!07}{95\!\cdots\!28}a^{31}-\frac{44\!\cdots\!29}{19\!\cdots\!36}a^{30}-\frac{33\!\cdots\!83}{47\!\cdots\!64}a^{29}+\frac{29\!\cdots\!05}{74\!\cdots\!51}a^{28}+\frac{11\!\cdots\!29}{95\!\cdots\!28}a^{27}-\frac{29\!\cdots\!61}{47\!\cdots\!64}a^{26}-\frac{44\!\cdots\!85}{47\!\cdots\!64}a^{25}+\frac{46\!\cdots\!05}{95\!\cdots\!28}a^{24}+\frac{61\!\cdots\!97}{72\!\cdots\!04}a^{23}-\frac{30\!\cdots\!03}{79\!\cdots\!44}a^{22}-\frac{41\!\cdots\!77}{29\!\cdots\!16}a^{21}+\frac{17\!\cdots\!33}{15\!\cdots\!88}a^{20}+\frac{35\!\cdots\!21}{31\!\cdots\!76}a^{19}-\frac{11\!\cdots\!57}{31\!\cdots\!76}a^{18}-\frac{21\!\cdots\!03}{75\!\cdots\!28}a^{17}+\frac{19\!\cdots\!67}{31\!\cdots\!76}a^{16}+\frac{56\!\cdots\!23}{47\!\cdots\!64}a^{15}-\frac{22\!\cdots\!19}{79\!\cdots\!44}a^{14}+\frac{18\!\cdots\!63}{47\!\cdots\!64}a^{13}-\frac{19\!\cdots\!23}{31\!\cdots\!76}a^{12}+\frac{33\!\cdots\!13}{47\!\cdots\!64}a^{11}-\frac{49\!\cdots\!49}{14\!\cdots\!02}a^{10}-\frac{22\!\cdots\!73}{95\!\cdots\!28}a^{9}+\frac{76\!\cdots\!63}{11\!\cdots\!16}a^{8}-\frac{29\!\cdots\!55}{47\!\cdots\!64}a^{7}+\frac{72\!\cdots\!13}{31\!\cdots\!76}a^{6}+\frac{42\!\cdots\!17}{23\!\cdots\!32}a^{5}-\frac{25\!\cdots\!81}{95\!\cdots\!28}a^{4}+\frac{39\!\cdots\!97}{31\!\cdots\!76}a^{3}+\frac{49\!\cdots\!05}{29\!\cdots\!04}a^{2}-\frac{35\!\cdots\!17}{99\!\cdots\!68}a+\frac{80\!\cdots\!57}{51\!\cdots\!38}$, $\frac{30\!\cdots\!83}{70\!\cdots\!28}a^{31}-\frac{35\!\cdots\!61}{17\!\cdots\!32}a^{30}-\frac{21\!\cdots\!43}{32\!\cdots\!24}a^{29}+\frac{20\!\cdots\!75}{58\!\cdots\!44}a^{28}+\frac{80\!\cdots\!41}{70\!\cdots\!28}a^{27}-\frac{18\!\cdots\!69}{35\!\cdots\!64}a^{26}-\frac{32\!\cdots\!65}{35\!\cdots\!64}a^{25}+\frac{30\!\cdots\!09}{70\!\cdots\!28}a^{24}+\frac{72\!\cdots\!13}{88\!\cdots\!16}a^{23}-\frac{14\!\cdots\!37}{44\!\cdots\!08}a^{22}-\frac{12\!\cdots\!49}{70\!\cdots\!28}a^{21}+\frac{10\!\cdots\!43}{10\!\cdots\!08}a^{20}+\frac{80\!\cdots\!15}{70\!\cdots\!28}a^{19}-\frac{22\!\cdots\!31}{70\!\cdots\!28}a^{18}-\frac{60\!\cdots\!65}{20\!\cdots\!36}a^{17}+\frac{34\!\cdots\!93}{70\!\cdots\!28}a^{16}+\frac{40\!\cdots\!21}{35\!\cdots\!64}a^{15}-\frac{10\!\cdots\!99}{44\!\cdots\!08}a^{14}+\frac{11\!\cdots\!77}{35\!\cdots\!64}a^{13}-\frac{38\!\cdots\!73}{70\!\cdots\!28}a^{12}+\frac{18\!\cdots\!03}{32\!\cdots\!24}a^{11}-\frac{22\!\cdots\!95}{88\!\cdots\!16}a^{10}-\frac{50\!\cdots\!67}{23\!\cdots\!76}a^{9}+\frac{95\!\cdots\!69}{17\!\cdots\!32}a^{8}-\frac{19\!\cdots\!61}{39\!\cdots\!96}a^{7}+\frac{11\!\cdots\!39}{70\!\cdots\!28}a^{6}+\frac{27\!\cdots\!97}{17\!\cdots\!32}a^{5}-\frac{14\!\cdots\!89}{70\!\cdots\!28}a^{4}+\frac{12\!\cdots\!29}{13\!\cdots\!28}a^{3}+\frac{13\!\cdots\!19}{88\!\cdots\!16}a^{2}-\frac{18\!\cdots\!33}{73\!\cdots\!18}a+\frac{43\!\cdots\!74}{38\!\cdots\!13}$, $\frac{75\!\cdots\!53}{37\!\cdots\!64}a^{31}-\frac{42\!\cdots\!43}{47\!\cdots\!08}a^{30}-\frac{58\!\cdots\!41}{18\!\cdots\!32}a^{29}+\frac{82\!\cdots\!95}{52\!\cdots\!12}a^{28}+\frac{20\!\cdots\!39}{37\!\cdots\!64}a^{27}-\frac{41\!\cdots\!73}{17\!\cdots\!12}a^{26}-\frac{82\!\cdots\!03}{18\!\cdots\!32}a^{25}+\frac{71\!\cdots\!27}{37\!\cdots\!64}a^{24}+\frac{36\!\cdots\!83}{94\!\cdots\!16}a^{23}-\frac{82\!\cdots\!45}{55\!\cdots\!48}a^{22}-\frac{32\!\cdots\!83}{37\!\cdots\!64}a^{21}+\frac{51\!\cdots\!81}{12\!\cdots\!44}a^{20}+\frac{20\!\cdots\!77}{37\!\cdots\!64}a^{19}-\frac{50\!\cdots\!61}{37\!\cdots\!64}a^{18}-\frac{49\!\cdots\!23}{36\!\cdots\!56}a^{17}+\frac{64\!\cdots\!53}{34\!\cdots\!24}a^{16}+\frac{96\!\cdots\!33}{18\!\cdots\!32}a^{15}-\frac{98\!\cdots\!11}{94\!\cdots\!16}a^{14}+\frac{29\!\cdots\!13}{18\!\cdots\!32}a^{13}-\frac{97\!\cdots\!83}{37\!\cdots\!64}a^{12}+\frac{51\!\cdots\!79}{18\!\cdots\!32}a^{11}-\frac{67\!\cdots\!51}{47\!\cdots\!08}a^{10}-\frac{20\!\cdots\!13}{38\!\cdots\!36}a^{9}+\frac{10\!\cdots\!25}{47\!\cdots\!08}a^{8}-\frac{14\!\cdots\!43}{63\!\cdots\!44}a^{7}+\frac{35\!\cdots\!81}{37\!\cdots\!64}a^{6}+\frac{37\!\cdots\!07}{86\!\cdots\!56}a^{5}-\frac{29\!\cdots\!35}{37\!\cdots\!64}a^{4}+\frac{18\!\cdots\!25}{42\!\cdots\!96}a^{3}+\frac{27\!\cdots\!73}{23\!\cdots\!04}a^{2}-\frac{28\!\cdots\!20}{32\!\cdots\!57}a+\frac{11\!\cdots\!77}{20\!\cdots\!94}$, $\frac{94\!\cdots\!03}{42\!\cdots\!96}a^{31}-\frac{10\!\cdots\!04}{98\!\cdots\!71}a^{30}-\frac{63\!\cdots\!95}{18\!\cdots\!32}a^{29}+\frac{12\!\cdots\!85}{65\!\cdots\!14}a^{28}+\frac{21\!\cdots\!09}{37\!\cdots\!64}a^{27}-\frac{59\!\cdots\!37}{21\!\cdots\!48}a^{26}-\frac{87\!\cdots\!73}{18\!\cdots\!32}a^{25}+\frac{26\!\cdots\!41}{11\!\cdots\!08}a^{24}+\frac{13\!\cdots\!33}{31\!\cdots\!72}a^{23}-\frac{16\!\cdots\!57}{94\!\cdots\!16}a^{22}-\frac{34\!\cdots\!85}{42\!\cdots\!96}a^{21}+\frac{33\!\cdots\!37}{63\!\cdots\!44}a^{20}+\frac{21\!\cdots\!03}{37\!\cdots\!64}a^{19}-\frac{65\!\cdots\!75}{37\!\cdots\!64}a^{18}-\frac{48\!\cdots\!83}{32\!\cdots\!04}a^{17}+\frac{35\!\cdots\!79}{12\!\cdots\!88}a^{16}+\frac{36\!\cdots\!93}{63\!\cdots\!44}a^{15}-\frac{74\!\cdots\!01}{55\!\cdots\!48}a^{14}+\frac{19\!\cdots\!79}{11\!\cdots\!96}a^{13}-\frac{10\!\cdots\!73}{37\!\cdots\!64}a^{12}+\frac{57\!\cdots\!17}{18\!\cdots\!32}a^{11}-\frac{28\!\cdots\!05}{23\!\cdots\!04}a^{10}-\frac{51\!\cdots\!41}{37\!\cdots\!64}a^{9}+\frac{40\!\cdots\!27}{13\!\cdots\!28}a^{8}-\frac{17\!\cdots\!01}{63\!\cdots\!44}a^{7}+\frac{35\!\cdots\!03}{42\!\cdots\!96}a^{6}+\frac{93\!\cdots\!57}{94\!\cdots\!16}a^{5}-\frac{46\!\cdots\!69}{37\!\cdots\!64}a^{4}+\frac{57\!\cdots\!83}{11\!\cdots\!08}a^{3}+\frac{55\!\cdots\!37}{47\!\cdots\!08}a^{2}-\frac{19\!\cdots\!93}{11\!\cdots\!52}a+\frac{12\!\cdots\!25}{20\!\cdots\!94}$, $\frac{61\!\cdots\!53}{23\!\cdots\!88}a^{31}-\frac{17\!\cdots\!67}{13\!\cdots\!88}a^{30}-\frac{41\!\cdots\!89}{11\!\cdots\!44}a^{29}+\frac{33\!\cdots\!09}{14\!\cdots\!68}a^{28}+\frac{14\!\cdots\!51}{23\!\cdots\!88}a^{27}-\frac{37\!\cdots\!17}{10\!\cdots\!04}a^{26}-\frac{16\!\cdots\!09}{39\!\cdots\!48}a^{25}+\frac{66\!\cdots\!95}{23\!\cdots\!88}a^{24}+\frac{21\!\cdots\!27}{53\!\cdots\!52}a^{23}-\frac{14\!\cdots\!07}{65\!\cdots\!08}a^{22}-\frac{62\!\cdots\!31}{23\!\cdots\!88}a^{21}+\frac{75\!\cdots\!17}{11\!\cdots\!44}a^{20}+\frac{10\!\cdots\!77}{23\!\cdots\!88}a^{19}-\frac{52\!\cdots\!93}{23\!\cdots\!88}a^{18}-\frac{18\!\cdots\!11}{17\!\cdots\!84}a^{17}+\frac{89\!\cdots\!35}{23\!\cdots\!88}a^{16}+\frac{20\!\cdots\!45}{35\!\cdots\!68}a^{15}-\frac{68\!\cdots\!13}{38\!\cdots\!24}a^{14}+\frac{17\!\cdots\!29}{69\!\cdots\!32}a^{13}-\frac{91\!\cdots\!35}{23\!\cdots\!88}a^{12}+\frac{18\!\cdots\!01}{39\!\cdots\!48}a^{11}-\frac{80\!\cdots\!51}{29\!\cdots\!36}a^{10}-\frac{71\!\cdots\!33}{78\!\cdots\!96}a^{9}+\frac{30\!\cdots\!31}{81\!\cdots\!26}a^{8}-\frac{48\!\cdots\!45}{11\!\cdots\!44}a^{7}+\frac{14\!\cdots\!47}{78\!\cdots\!96}a^{6}+\frac{45\!\cdots\!21}{58\!\cdots\!72}a^{5}-\frac{38\!\cdots\!35}{23\!\cdots\!88}a^{4}+\frac{18\!\cdots\!23}{21\!\cdots\!08}a^{3}+\frac{10\!\cdots\!99}{24\!\cdots\!78}a^{2}-\frac{79\!\cdots\!65}{36\!\cdots\!17}a+\frac{11\!\cdots\!40}{12\!\cdots\!39}$, $\frac{28\!\cdots\!15}{39\!\cdots\!68}a^{31}-\frac{30\!\cdots\!17}{73\!\cdots\!44}a^{30}-\frac{41\!\cdots\!63}{53\!\cdots\!32}a^{29}+\frac{52\!\cdots\!71}{73\!\cdots\!44}a^{28}+\frac{15\!\cdots\!47}{11\!\cdots\!04}a^{27}-\frac{66\!\cdots\!93}{59\!\cdots\!48}a^{26}-\frac{12\!\cdots\!49}{19\!\cdots\!84}a^{25}+\frac{35\!\cdots\!33}{39\!\cdots\!68}a^{24}+\frac{19\!\cdots\!63}{29\!\cdots\!76}a^{23}-\frac{21\!\cdots\!59}{29\!\cdots\!76}a^{22}+\frac{33\!\cdots\!73}{11\!\cdots\!04}a^{21}+\frac{41\!\cdots\!75}{19\!\cdots\!84}a^{20}+\frac{20\!\cdots\!81}{11\!\cdots\!04}a^{19}-\frac{92\!\cdots\!01}{11\!\cdots\!04}a^{18}+\frac{47\!\cdots\!49}{10\!\cdots\!44}a^{17}+\frac{17\!\cdots\!51}{11\!\cdots\!04}a^{16}+\frac{70\!\cdots\!89}{65\!\cdots\!28}a^{15}-\frac{11\!\cdots\!91}{17\!\cdots\!28}a^{14}+\frac{10\!\cdots\!87}{11\!\cdots\!52}a^{13}-\frac{15\!\cdots\!31}{11\!\cdots\!04}a^{12}+\frac{10\!\cdots\!79}{59\!\cdots\!52}a^{11}-\frac{91\!\cdots\!61}{73\!\cdots\!44}a^{10}-\frac{21\!\cdots\!39}{11\!\cdots\!04}a^{9}+\frac{20\!\cdots\!15}{14\!\cdots\!88}a^{8}-\frac{86\!\cdots\!95}{53\!\cdots\!32}a^{7}+\frac{10\!\cdots\!37}{11\!\cdots\!04}a^{6}+\frac{61\!\cdots\!07}{29\!\cdots\!76}a^{5}-\frac{71\!\cdots\!45}{10\!\cdots\!64}a^{4}+\frac{15\!\cdots\!15}{39\!\cdots\!68}a^{3}-\frac{99\!\cdots\!35}{73\!\cdots\!44}a^{2}-\frac{11\!\cdots\!27}{12\!\cdots\!24}a+\frac{27\!\cdots\!91}{70\!\cdots\!26}$, $\frac{34\!\cdots\!65}{37\!\cdots\!64}a^{31}-\frac{96\!\cdots\!51}{23\!\cdots\!04}a^{30}-\frac{28\!\cdots\!69}{18\!\cdots\!32}a^{29}+\frac{19\!\cdots\!81}{26\!\cdots\!56}a^{28}+\frac{97\!\cdots\!07}{37\!\cdots\!64}a^{27}-\frac{11\!\cdots\!09}{10\!\cdots\!36}a^{26}-\frac{42\!\cdots\!91}{18\!\cdots\!32}a^{25}+\frac{35\!\cdots\!95}{37\!\cdots\!64}a^{24}+\frac{11\!\cdots\!61}{55\!\cdots\!48}a^{23}-\frac{69\!\cdots\!43}{94\!\cdots\!16}a^{22}-\frac{23\!\cdots\!35}{37\!\cdots\!64}a^{21}+\frac{15\!\cdots\!99}{63\!\cdots\!44}a^{20}+\frac{12\!\cdots\!77}{37\!\cdots\!64}a^{19}-\frac{28\!\cdots\!45}{37\!\cdots\!64}a^{18}-\frac{10\!\cdots\!53}{10\!\cdots\!68}a^{17}+\frac{49\!\cdots\!03}{37\!\cdots\!64}a^{16}+\frac{61\!\cdots\!73}{18\!\cdots\!32}a^{15}-\frac{49\!\cdots\!61}{94\!\cdots\!16}a^{14}+\frac{69\!\cdots\!91}{17\!\cdots\!12}a^{13}-\frac{30\!\cdots\!59}{37\!\cdots\!64}a^{12}+\frac{15\!\cdots\!55}{18\!\cdots\!32}a^{11}+\frac{49\!\cdots\!81}{27\!\cdots\!24}a^{10}-\frac{36\!\cdots\!41}{38\!\cdots\!36}a^{9}+\frac{58\!\cdots\!71}{59\!\cdots\!26}a^{8}-\frac{33\!\cdots\!15}{63\!\cdots\!44}a^{7}-\frac{10\!\cdots\!15}{37\!\cdots\!64}a^{6}+\frac{61\!\cdots\!89}{94\!\cdots\!16}a^{5}-\frac{10\!\cdots\!35}{37\!\cdots\!64}a^{4}-\frac{51\!\cdots\!23}{42\!\cdots\!96}a^{3}+\frac{17\!\cdots\!23}{11\!\cdots\!52}a^{2}-\frac{37\!\cdots\!81}{19\!\cdots\!42}a-\frac{54\!\cdots\!79}{20\!\cdots\!94}$, $\frac{27\!\cdots\!85}{37\!\cdots\!64}a^{31}-\frac{37\!\cdots\!97}{11\!\cdots\!52}a^{30}-\frac{18\!\cdots\!79}{17\!\cdots\!12}a^{29}+\frac{17\!\cdots\!01}{32\!\cdots\!57}a^{28}+\frac{69\!\cdots\!31}{37\!\cdots\!64}a^{27}-\frac{15\!\cdots\!03}{18\!\cdots\!32}a^{26}-\frac{27\!\cdots\!83}{18\!\cdots\!32}a^{25}+\frac{23\!\cdots\!95}{37\!\cdots\!64}a^{24}+\frac{12\!\cdots\!41}{94\!\cdots\!16}a^{23}-\frac{46\!\cdots\!95}{94\!\cdots\!16}a^{22}-\frac{77\!\cdots\!87}{37\!\cdots\!64}a^{21}+\frac{77\!\cdots\!91}{63\!\cdots\!44}a^{20}+\frac{57\!\cdots\!61}{37\!\cdots\!64}a^{19}-\frac{15\!\cdots\!53}{37\!\cdots\!64}a^{18}-\frac{32\!\cdots\!83}{10\!\cdots\!68}a^{17}+\frac{18\!\cdots\!87}{37\!\cdots\!64}a^{16}+\frac{14\!\cdots\!97}{11\!\cdots\!96}a^{15}-\frac{33\!\cdots\!27}{94\!\cdots\!16}a^{14}+\frac{13\!\cdots\!73}{18\!\cdots\!32}a^{13}-\frac{34\!\cdots\!49}{34\!\cdots\!24}a^{12}+\frac{19\!\cdots\!55}{18\!\cdots\!32}a^{11}-\frac{17\!\cdots\!89}{23\!\cdots\!04}a^{10}-\frac{15\!\cdots\!77}{12\!\cdots\!88}a^{9}+\frac{30\!\cdots\!28}{29\!\cdots\!13}a^{8}-\frac{71\!\cdots\!23}{63\!\cdots\!44}a^{7}+\frac{22\!\cdots\!77}{37\!\cdots\!64}a^{6}+\frac{50\!\cdots\!63}{55\!\cdots\!48}a^{5}-\frac{17\!\cdots\!15}{37\!\cdots\!64}a^{4}+\frac{14\!\cdots\!53}{42\!\cdots\!96}a^{3}-\frac{87\!\cdots\!73}{27\!\cdots\!24}a^{2}-\frac{15\!\cdots\!65}{23\!\cdots\!52}a+\frac{11\!\cdots\!13}{20\!\cdots\!94}$, $\frac{24\!\cdots\!83}{37\!\cdots\!64}a^{31}-\frac{14\!\cdots\!15}{47\!\cdots\!08}a^{30}-\frac{57\!\cdots\!93}{63\!\cdots\!44}a^{29}+\frac{16\!\cdots\!95}{32\!\cdots\!57}a^{28}+\frac{59\!\cdots\!97}{37\!\cdots\!64}a^{27}-\frac{14\!\cdots\!89}{18\!\cdots\!32}a^{26}-\frac{24\!\cdots\!81}{21\!\cdots\!48}a^{25}+\frac{21\!\cdots\!45}{37\!\cdots\!64}a^{24}+\frac{96\!\cdots\!69}{94\!\cdots\!16}a^{23}-\frac{41\!\cdots\!97}{94\!\cdots\!16}a^{22}-\frac{25\!\cdots\!85}{37\!\cdots\!64}a^{21}+\frac{61\!\cdots\!13}{63\!\cdots\!44}a^{20}+\frac{38\!\cdots\!55}{37\!\cdots\!64}a^{19}-\frac{41\!\cdots\!61}{12\!\cdots\!88}a^{18}-\frac{40\!\cdots\!29}{32\!\cdots\!04}a^{17}+\frac{12\!\cdots\!37}{37\!\cdots\!64}a^{16}+\frac{14\!\cdots\!19}{18\!\cdots\!32}a^{15}-\frac{10\!\cdots\!63}{31\!\cdots\!72}a^{14}+\frac{48\!\cdots\!45}{63\!\cdots\!44}a^{13}-\frac{41\!\cdots\!29}{37\!\cdots\!64}a^{12}+\frac{21\!\cdots\!25}{18\!\cdots\!32}a^{11}-\frac{39\!\cdots\!61}{39\!\cdots\!84}a^{10}+\frac{27\!\cdots\!87}{22\!\cdots\!92}a^{9}+\frac{47\!\cdots\!77}{47\!\cdots\!08}a^{8}-\frac{26\!\cdots\!91}{21\!\cdots\!48}a^{7}+\frac{31\!\cdots\!71}{37\!\cdots\!64}a^{6}-\frac{19\!\cdots\!45}{28\!\cdots\!52}a^{5}-\frac{19\!\cdots\!53}{37\!\cdots\!64}a^{4}+\frac{59\!\cdots\!97}{12\!\cdots\!88}a^{3}-\frac{11\!\cdots\!07}{13\!\cdots\!28}a^{2}-\frac{49\!\cdots\!07}{59\!\cdots\!26}a+\frac{16\!\cdots\!53}{22\!\cdots\!66}$, $\frac{59\!\cdots\!05}{12\!\cdots\!88}a^{31}-\frac{11\!\cdots\!87}{52\!\cdots\!12}a^{30}-\frac{44\!\cdots\!97}{63\!\cdots\!44}a^{29}+\frac{25\!\cdots\!03}{65\!\cdots\!14}a^{28}+\frac{15\!\cdots\!11}{12\!\cdots\!88}a^{27}-\frac{22\!\cdots\!31}{37\!\cdots\!32}a^{26}-\frac{19\!\cdots\!89}{21\!\cdots\!48}a^{25}+\frac{60\!\cdots\!67}{12\!\cdots\!88}a^{24}+\frac{14\!\cdots\!55}{16\!\cdots\!56}a^{23}-\frac{39\!\cdots\!05}{10\!\cdots\!24}a^{22}-\frac{18\!\cdots\!71}{12\!\cdots\!88}a^{21}+\frac{22\!\cdots\!47}{21\!\cdots\!48}a^{20}+\frac{12\!\cdots\!79}{11\!\cdots\!08}a^{19}-\frac{15\!\cdots\!95}{42\!\cdots\!96}a^{18}-\frac{10\!\cdots\!29}{36\!\cdots\!56}a^{17}+\frac{72\!\cdots\!47}{12\!\cdots\!88}a^{16}+\frac{73\!\cdots\!65}{63\!\cdots\!44}a^{15}-\frac{88\!\cdots\!81}{31\!\cdots\!72}a^{14}+\frac{81\!\cdots\!19}{21\!\cdots\!48}a^{13}-\frac{78\!\cdots\!67}{12\!\cdots\!88}a^{12}+\frac{14\!\cdots\!33}{21\!\cdots\!48}a^{11}-\frac{94\!\cdots\!55}{28\!\cdots\!76}a^{10}-\frac{24\!\cdots\!45}{11\!\cdots\!08}a^{9}+\frac{20\!\cdots\!05}{32\!\cdots\!57}a^{8}-\frac{12\!\cdots\!59}{21\!\cdots\!48}a^{7}+\frac{29\!\cdots\!49}{12\!\cdots\!88}a^{6}+\frac{15\!\cdots\!11}{95\!\cdots\!84}a^{5}-\frac{10\!\cdots\!29}{42\!\cdots\!96}a^{4}+\frac{15\!\cdots\!09}{12\!\cdots\!88}a^{3}+\frac{54\!\cdots\!81}{39\!\cdots\!84}a^{2}-\frac{12\!\cdots\!51}{39\!\cdots\!84}a+\frac{99\!\cdots\!79}{68\!\cdots\!98}$, $\frac{36\!\cdots\!47}{22\!\cdots\!92}a^{31}-\frac{20\!\cdots\!91}{23\!\cdots\!04}a^{30}-\frac{33\!\cdots\!13}{19\!\cdots\!68}a^{29}+\frac{75\!\cdots\!85}{52\!\cdots\!12}a^{28}+\frac{11\!\cdots\!29}{37\!\cdots\!64}a^{27}-\frac{42\!\cdots\!17}{18\!\cdots\!32}a^{26}-\frac{10\!\cdots\!79}{63\!\cdots\!44}a^{25}+\frac{64\!\cdots\!65}{37\!\cdots\!64}a^{24}+\frac{15\!\cdots\!17}{94\!\cdots\!16}a^{23}-\frac{12\!\cdots\!55}{94\!\cdots\!16}a^{22}+\frac{18\!\cdots\!39}{37\!\cdots\!64}a^{21}+\frac{19\!\cdots\!89}{63\!\cdots\!44}a^{20}+\frac{49\!\cdots\!09}{34\!\cdots\!24}a^{19}-\frac{15\!\cdots\!69}{12\!\cdots\!88}a^{18}+\frac{38\!\cdots\!01}{32\!\cdots\!04}a^{17}+\frac{66\!\cdots\!05}{37\!\cdots\!64}a^{16}+\frac{46\!\cdots\!99}{18\!\cdots\!32}a^{15}-\frac{34\!\cdots\!41}{31\!\cdots\!72}a^{14}+\frac{13\!\cdots\!69}{63\!\cdots\!44}a^{13}-\frac{14\!\cdots\!77}{37\!\cdots\!64}a^{12}+\frac{51\!\cdots\!71}{10\!\cdots\!36}a^{11}-\frac{17\!\cdots\!31}{35\!\cdots\!44}a^{10}+\frac{10\!\cdots\!75}{37\!\cdots\!64}a^{9}+\frac{13\!\cdots\!85}{43\!\cdots\!28}a^{8}-\frac{16\!\cdots\!21}{63\!\cdots\!44}a^{7}+\frac{11\!\cdots\!35}{37\!\cdots\!64}a^{6}-\frac{16\!\cdots\!61}{10\!\cdots\!24}a^{5}+\frac{55\!\cdots\!27}{37\!\cdots\!64}a^{4}+\frac{64\!\cdots\!85}{12\!\cdots\!88}a^{3}-\frac{62\!\cdots\!01}{15\!\cdots\!36}a^{2}+\frac{17\!\cdots\!39}{11\!\cdots\!52}a-\frac{86\!\cdots\!19}{68\!\cdots\!98}$, $\frac{30\!\cdots\!45}{37\!\cdots\!64}a^{31}-\frac{96\!\cdots\!69}{23\!\cdots\!04}a^{30}-\frac{68\!\cdots\!19}{63\!\cdots\!44}a^{29}+\frac{55\!\cdots\!39}{79\!\cdots\!68}a^{28}+\frac{64\!\cdots\!17}{34\!\cdots\!24}a^{27}-\frac{20\!\cdots\!35}{18\!\cdots\!32}a^{26}-\frac{83\!\cdots\!53}{63\!\cdots\!44}a^{25}+\frac{33\!\cdots\!11}{37\!\cdots\!64}a^{24}+\frac{11\!\cdots\!61}{94\!\cdots\!16}a^{23}-\frac{65\!\cdots\!03}{94\!\cdots\!16}a^{22}-\frac{29\!\cdots\!91}{37\!\cdots\!64}a^{21}+\frac{12\!\cdots\!75}{63\!\cdots\!44}a^{20}+\frac{54\!\cdots\!45}{37\!\cdots\!64}a^{19}-\frac{87\!\cdots\!51}{12\!\cdots\!88}a^{18}-\frac{10\!\cdots\!57}{32\!\cdots\!04}a^{17}+\frac{16\!\cdots\!87}{14\!\cdots\!56}a^{16}+\frac{20\!\cdots\!85}{11\!\cdots\!96}a^{15}-\frac{57\!\cdots\!35}{10\!\cdots\!24}a^{14}+\frac{49\!\cdots\!99}{63\!\cdots\!44}a^{13}-\frac{47\!\cdots\!35}{37\!\cdots\!64}a^{12}+\frac{28\!\cdots\!67}{18\!\cdots\!32}a^{11}-\frac{93\!\cdots\!97}{98\!\cdots\!71}a^{10}-\frac{56\!\cdots\!75}{37\!\cdots\!64}a^{9}+\frac{24\!\cdots\!83}{23\!\cdots\!04}a^{8}-\frac{76\!\cdots\!75}{63\!\cdots\!44}a^{7}+\frac{23\!\cdots\!77}{37\!\cdots\!64}a^{6}+\frac{31\!\cdots\!57}{18\!\cdots\!16}a^{5}-\frac{16\!\cdots\!67}{37\!\cdots\!64}a^{4}+\frac{10\!\cdots\!57}{42\!\cdots\!96}a^{3}+\frac{22\!\cdots\!33}{84\!\cdots\!28}a^{2}-\frac{35\!\cdots\!71}{69\!\cdots\!56}a+\frac{15\!\cdots\!93}{68\!\cdots\!98}$, $\frac{11\!\cdots\!97}{37\!\cdots\!64}a^{31}-\frac{21\!\cdots\!19}{11\!\cdots\!52}a^{30}-\frac{19\!\cdots\!87}{63\!\cdots\!44}a^{29}+\frac{31\!\cdots\!33}{98\!\cdots\!71}a^{28}+\frac{19\!\cdots\!23}{37\!\cdots\!64}a^{27}-\frac{95\!\cdots\!95}{18\!\cdots\!32}a^{26}-\frac{26\!\cdots\!31}{12\!\cdots\!44}a^{25}+\frac{15\!\cdots\!75}{37\!\cdots\!64}a^{24}+\frac{22\!\cdots\!89}{94\!\cdots\!16}a^{23}-\frac{31\!\cdots\!71}{94\!\cdots\!16}a^{22}+\frac{56\!\cdots\!77}{37\!\cdots\!64}a^{21}+\frac{62\!\cdots\!35}{63\!\cdots\!44}a^{20}+\frac{64\!\cdots\!97}{37\!\cdots\!64}a^{19}-\frac{47\!\cdots\!95}{12\!\cdots\!88}a^{18}+\frac{79\!\cdots\!65}{29\!\cdots\!64}a^{17}+\frac{28\!\cdots\!23}{37\!\cdots\!64}a^{16}+\frac{97\!\cdots\!61}{18\!\cdots\!32}a^{15}-\frac{28\!\cdots\!61}{95\!\cdots\!84}a^{14}+\frac{85\!\cdots\!01}{21\!\cdots\!48}a^{13}-\frac{22\!\cdots\!87}{37\!\cdots\!64}a^{12}+\frac{15\!\cdots\!43}{18\!\cdots\!32}a^{11}-\frac{99\!\cdots\!73}{17\!\cdots\!22}a^{10}-\frac{31\!\cdots\!95}{37\!\cdots\!64}a^{9}+\frac{13\!\cdots\!11}{23\!\cdots\!04}a^{8}-\frac{14\!\cdots\!89}{21\!\cdots\!48}a^{7}+\frac{13\!\cdots\!25}{37\!\cdots\!64}a^{6}+\frac{11\!\cdots\!59}{10\!\cdots\!24}a^{5}-\frac{61\!\cdots\!55}{22\!\cdots\!92}a^{4}+\frac{16\!\cdots\!59}{12\!\cdots\!88}a^{3}+\frac{49\!\cdots\!79}{52\!\cdots\!12}a^{2}-\frac{44\!\cdots\!91}{11\!\cdots\!52}a+\frac{62\!\cdots\!93}{68\!\cdots\!98}$, $\frac{19\!\cdots\!65}{47\!\cdots\!08}a^{31}-\frac{90\!\cdots\!67}{47\!\cdots\!08}a^{30}-\frac{14\!\cdots\!31}{23\!\cdots\!04}a^{29}+\frac{59\!\cdots\!41}{17\!\cdots\!22}a^{28}+\frac{16\!\cdots\!07}{15\!\cdots\!36}a^{27}-\frac{24\!\cdots\!67}{47\!\cdots\!08}a^{26}-\frac{21\!\cdots\!27}{26\!\cdots\!56}a^{25}+\frac{19\!\cdots\!89}{47\!\cdots\!08}a^{24}+\frac{34\!\cdots\!37}{47\!\cdots\!08}a^{23}-\frac{69\!\cdots\!91}{21\!\cdots\!64}a^{22}-\frac{21\!\cdots\!31}{15\!\cdots\!36}a^{21}+\frac{44\!\cdots\!23}{47\!\cdots\!08}a^{20}+\frac{14\!\cdots\!41}{14\!\cdots\!76}a^{19}-\frac{60\!\cdots\!11}{19\!\cdots\!42}a^{18}-\frac{40\!\cdots\!27}{16\!\cdots\!52}a^{17}+\frac{77\!\cdots\!77}{15\!\cdots\!36}a^{16}+\frac{53\!\cdots\!73}{52\!\cdots\!12}a^{15}-\frac{56\!\cdots\!87}{23\!\cdots\!04}a^{14}+\frac{97\!\cdots\!07}{29\!\cdots\!13}a^{13}-\frac{27\!\cdots\!91}{52\!\cdots\!12}a^{12}+\frac{27\!\cdots\!71}{47\!\cdots\!08}a^{11}-\frac{10\!\cdots\!65}{39\!\cdots\!84}a^{10}-\frac{87\!\cdots\!41}{43\!\cdots\!28}a^{9}+\frac{25\!\cdots\!07}{47\!\cdots\!08}a^{8}-\frac{91\!\cdots\!10}{17\!\cdots\!89}a^{7}+\frac{80\!\cdots\!75}{43\!\cdots\!28}a^{6}+\frac{24\!\cdots\!35}{15\!\cdots\!36}a^{5}-\frac{35\!\cdots\!35}{15\!\cdots\!36}a^{4}+\frac{24\!\cdots\!63}{23\!\cdots\!04}a^{3}+\frac{69\!\cdots\!55}{47\!\cdots\!08}a^{2}-\frac{87\!\cdots\!77}{29\!\cdots\!13}a+\frac{44\!\cdots\!40}{34\!\cdots\!99}$, $\frac{13\!\cdots\!11}{37\!\cdots\!64}a^{31}-\frac{69\!\cdots\!83}{47\!\cdots\!08}a^{30}-\frac{12\!\cdots\!19}{18\!\cdots\!32}a^{29}+\frac{14\!\cdots\!47}{52\!\cdots\!12}a^{28}+\frac{13\!\cdots\!35}{12\!\cdots\!88}a^{27}-\frac{77\!\cdots\!81}{18\!\cdots\!32}a^{26}-\frac{62\!\cdots\!47}{63\!\cdots\!44}a^{25}+\frac{12\!\cdots\!37}{37\!\cdots\!64}a^{24}+\frac{81\!\cdots\!15}{94\!\cdots\!16}a^{23}-\frac{24\!\cdots\!15}{94\!\cdots\!16}a^{22}-\frac{13\!\cdots\!07}{47\!\cdots\!52}a^{21}+\frac{16\!\cdots\!63}{18\!\cdots\!32}a^{20}+\frac{53\!\cdots\!67}{42\!\cdots\!96}a^{19}-\frac{10\!\cdots\!47}{42\!\cdots\!96}a^{18}-\frac{12\!\cdots\!27}{32\!\cdots\!04}a^{17}+\frac{17\!\cdots\!33}{42\!\cdots\!96}a^{16}+\frac{23\!\cdots\!79}{21\!\cdots\!48}a^{15}-\frac{98\!\cdots\!81}{55\!\cdots\!48}a^{14}+\frac{16\!\cdots\!71}{11\!\cdots\!96}a^{13}-\frac{25\!\cdots\!51}{12\!\cdots\!88}a^{12}+\frac{12\!\cdots\!89}{18\!\cdots\!32}a^{11}+\frac{55\!\cdots\!67}{15\!\cdots\!36}a^{10}-\frac{27\!\cdots\!37}{37\!\cdots\!64}a^{9}+\frac{17\!\cdots\!09}{23\!\cdots\!04}a^{8}-\frac{74\!\cdots\!59}{18\!\cdots\!32}a^{7}-\frac{19\!\cdots\!09}{14\!\cdots\!56}a^{6}+\frac{13\!\cdots\!25}{31\!\cdots\!72}a^{5}-\frac{14\!\cdots\!65}{42\!\cdots\!96}a^{4}+\frac{31\!\cdots\!31}{37\!\cdots\!64}a^{3}+\frac{31\!\cdots\!39}{47\!\cdots\!08}a^{2}-\frac{67\!\cdots\!27}{11\!\cdots\!52}a+\frac{13\!\cdots\!07}{68\!\cdots\!98}$
| 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: | \( 63453830199620.586 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 63453830199620.586 \cdot 1590}{6\cdot\sqrt{2363147162130074593554078242531588550681346814117431640625}}\cr\approx \mathstrut & 2.04096814292068 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 - 18*x^30 + 72*x^29 + 311*x^28 - 1090*x^27 - 2862*x^26 + 8691*x^25 + 24872*x^24 - 66988*x^23 - 86403*x^22 + 207758*x^21 + 395325*x^20 - 595053*x^19 - 1119168*x^18 + 799235*x^17 + 3298870*x^16 - 4221332*x^15 + 4111698*x^14 - 7461943*x^13 + 5469898*x^12 + 2888264*x^11 - 9443335*x^10 + 9992412*x^9 - 3874794*x^8 - 4061403*x^7 + 6823432*x^6 - 2937855*x^5 - 1141251*x^4 + 2061220*x^3 - 477856*x^2 - 165184*x + 215296)
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 - 4*x^31 - 18*x^30 + 72*x^29 + 311*x^28 - 1090*x^27 - 2862*x^26 + 8691*x^25 + 24872*x^24 - 66988*x^23 - 86403*x^22 + 207758*x^21 + 395325*x^20 - 595053*x^19 - 1119168*x^18 + 799235*x^17 + 3298870*x^16 - 4221332*x^15 + 4111698*x^14 - 7461943*x^13 + 5469898*x^12 + 2888264*x^11 - 9443335*x^10 + 9992412*x^9 - 3874794*x^8 - 4061403*x^7 + 6823432*x^6 - 2937855*x^5 - 1141251*x^4 + 2061220*x^3 - 477856*x^2 - 165184*x + 215296, 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 - 4*x^31 - 18*x^30 + 72*x^29 + 311*x^28 - 1090*x^27 - 2862*x^26 + 8691*x^25 + 24872*x^24 - 66988*x^23 - 86403*x^22 + 207758*x^21 + 395325*x^20 - 595053*x^19 - 1119168*x^18 + 799235*x^17 + 3298870*x^16 - 4221332*x^15 + 4111698*x^14 - 7461943*x^13 + 5469898*x^12 + 2888264*x^11 - 9443335*x^10 + 9992412*x^9 - 3874794*x^8 - 4061403*x^7 + 6823432*x^6 - 2937855*x^5 - 1141251*x^4 + 2061220*x^3 - 477856*x^2 - 165184*x + 215296);
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 - 4*x^31 - 18*x^30 + 72*x^29 + 311*x^28 - 1090*x^27 - 2862*x^26 + 8691*x^25 + 24872*x^24 - 66988*x^23 - 86403*x^22 + 207758*x^21 + 395325*x^20 - 595053*x^19 - 1119168*x^18 + 799235*x^17 + 3298870*x^16 - 4221332*x^15 + 4111698*x^14 - 7461943*x^13 + 5469898*x^12 + 2888264*x^11 - 9443335*x^10 + 9992412*x^9 - 3874794*x^8 - 4061403*x^7 + 6823432*x^6 - 2937855*x^5 - 1141251*x^4 + 2061220*x^3 - 477856*x^2 - 165184*x + 215296);
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
$D_4^2:C_2^3$ (as 32T12882):
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 512 |
| The 80 conjugacy class representatives for $D_4^2:C_2^3$ |
| Character table for $D_4^2:C_2^3$ |
Intermediate fields
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]
Sibling fields
| Degree 32 siblings: | data not computed |
| Minimal sibling: | not computed |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}{,}\,{\href{/padicField/2.2.0.1}{2} }^{8}$ | R | R | R | ${\href{/padicField/11.2.0.1}{2} }^{16}$ | ${\href{/padicField/13.8.0.1}{8} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{16}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{16}$ | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(7\)
| 7.16.8.1 | $x^{16} + 56 x^{14} + 1372 x^{12} + 8 x^{11} + 19220 x^{10} - 388 x^{9} + 166984 x^{8} - 9184 x^{7} + 931800 x^{6} - 35624 x^{5} + 3372764 x^{4} + 135176 x^{3} + 6908172 x^{2} + 607080 x + 5583776$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
| 7.16.8.1 | $x^{16} + 56 x^{14} + 1372 x^{12} + 8 x^{11} + 19220 x^{10} - 388 x^{9} + 166984 x^{8} - 9184 x^{7} + 931800 x^{6} - 35624 x^{5} + 3372764 x^{4} + 135176 x^{3} + 6908172 x^{2} + 607080 x + 5583776$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ | |
|
\(89\)
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(229\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |