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Properties

Label 32.0.236...625.1
Degree $32$
Signature $(0, 16)$
Discriminant $2.363\times 10^{57}$
Root discriminant \(62.08\)
Ramified primes $3,5,7,89,229$
Class number $1590$ (GRH)
Class group [1590] (GRH)
Galois group $D_4^2:C_2^3$ (as 32T12882)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content 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)
 
Copy content 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)
 
Copy content 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);
 
Copy content 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)
 

\( x^{32} - 4 x^{31} - 18 x^{30} + 72 x^{29} + 311 x^{28} - 1090 x^{27} - 2862 x^{26} + 8691 x^{25} + \cdots + 215296 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $32$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(0, 16)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(2363147162130074593554078242531588550681346814117431640625\) \(\medspace = 3^{16}\cdot 5^{16}\cdot 7^{16}\cdot 89^{8}\cdot 229^{4}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(62.08\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(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:   \(3\), \(5\), \(7\), \(89\), \(229\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^3$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2}a^{16}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{7}$, $\frac{1}{4}a^{23}-\frac{1}{4}a^{19}-\frac{1}{4}a^{17}-\frac{1}{4}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{12}a^{24}+\frac{1}{6}a^{22}-\frac{1}{6}a^{21}+\frac{1}{12}a^{20}-\frac{1}{6}a^{19}+\frac{1}{12}a^{18}-\frac{1}{4}a^{17}-\frac{1}{6}a^{16}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{1}{6}a^{11}+\frac{1}{6}a^{10}-\frac{1}{12}a^{9}-\frac{1}{6}a^{8}-\frac{1}{3}a^{6}+\frac{1}{12}a^{5}+\frac{1}{12}a^{3}-\frac{1}{4}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{12}a^{25}-\frac{1}{12}a^{23}-\frac{1}{6}a^{22}+\frac{1}{12}a^{21}-\frac{1}{6}a^{20}-\frac{1}{6}a^{19}-\frac{1}{4}a^{18}+\frac{1}{12}a^{17}-\frac{1}{4}a^{16}+\frac{1}{6}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}-\frac{1}{6}a^{12}+\frac{1}{6}a^{11}+\frac{5}{12}a^{10}+\frac{1}{3}a^{9}+\frac{1}{4}a^{8}+\frac{1}{6}a^{7}-\frac{5}{12}a^{6}-\frac{1}{2}a^{5}-\frac{1}{6}a^{4}+\frac{1}{4}a^{3}-\frac{5}{12}a^{2}+\frac{1}{12}a$, $\frac{1}{24}a^{26}-\frac{1}{24}a^{25}-\frac{1}{24}a^{24}+\frac{1}{12}a^{23}-\frac{1}{8}a^{22}+\frac{1}{8}a^{21}+\frac{1}{12}a^{19}-\frac{1}{12}a^{18}+\frac{5}{24}a^{17}+\frac{1}{12}a^{16}+\frac{1}{3}a^{15}+\frac{1}{6}a^{14}+\frac{1}{12}a^{13}-\frac{1}{3}a^{12}-\frac{3}{8}a^{11}-\frac{7}{24}a^{10}+\frac{5}{24}a^{9}-\frac{1}{6}a^{8}+\frac{5}{24}a^{7}-\frac{1}{24}a^{6}+\frac{5}{12}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{8}a^{2}-\frac{1}{6}a$, $\frac{1}{48}a^{27}-\frac{1}{24}a^{25}-\frac{1}{48}a^{24}-\frac{1}{48}a^{23}+\frac{1}{6}a^{22}+\frac{7}{48}a^{21}+\frac{1}{12}a^{19}+\frac{1}{48}a^{18}+\frac{1}{48}a^{17}-\frac{5}{24}a^{16}+\frac{1}{4}a^{15}-\frac{5}{24}a^{14}+\frac{5}{24}a^{13}+\frac{5}{16}a^{12}+\frac{1}{4}a^{11}-\frac{1}{8}a^{10}-\frac{7}{16}a^{9}-\frac{19}{48}a^{8}-\frac{1}{6}a^{7}-\frac{7}{48}a^{6}-\frac{1}{6}a^{5}-\frac{1}{2}a^{4}+\frac{17}{48}a^{3}-\frac{7}{48}a^{2}-\frac{1}{4}a+\frac{1}{3}$, $\frac{1}{426096}a^{28}+\frac{2209}{426096}a^{27}+\frac{1109}{106524}a^{26}-\frac{16525}{426096}a^{25}+\frac{274}{8877}a^{24}-\frac{495}{4304}a^{23}+\frac{10909}{47344}a^{22}+\frac{55573}{426096}a^{21}-\frac{13637}{106524}a^{20}-\frac{30539}{426096}a^{19}+\frac{9379}{71016}a^{18}-\frac{40507}{426096}a^{17}-\frac{12325}{213048}a^{16}-\frac{25639}{213048}a^{15}-\frac{14911}{53262}a^{14}+\frac{773}{12912}a^{13}+\frac{68083}{426096}a^{12}+\frac{893}{17754}a^{11}+\frac{125903}{426096}a^{10}+\frac{62171}{213048}a^{9}+\frac{144725}{426096}a^{8}-\frac{128449}{426096}a^{7}-\frac{116233}{426096}a^{6}-\frac{3323}{9684}a^{5}+\frac{159949}{426096}a^{4}+\frac{1457}{19368}a^{3}-\frac{191881}{426096}a^{2}+\frac{5817}{11836}a-\frac{12479}{26631}$, $\frac{1}{426096}a^{29}-\frac{443}{106524}a^{27}+\frac{257}{47344}a^{26}+\frac{14653}{426096}a^{25}-\frac{329}{11836}a^{24}-\frac{9689}{142032}a^{23}+\frac{1673}{213048}a^{22}+\frac{1205}{5918}a^{21}-\frac{4217}{142032}a^{20}+\frac{16217}{426096}a^{19}+\frac{1015}{53262}a^{18}-\frac{4327}{26631}a^{17}-\frac{8491}{71016}a^{16}+\frac{10139}{71016}a^{15}+\frac{116603}{426096}a^{14}+\frac{39667}{106524}a^{13}-\frac{19357}{106524}a^{12}+\frac{26009}{426096}a^{11}-\frac{38761}{426096}a^{10}+\frac{12970}{26631}a^{9}-\frac{3561}{47344}a^{8}-\frac{321}{2152}a^{7}-\frac{2009}{17754}a^{6}+\frac{19457}{426096}a^{5}-\frac{26195}{426096}a^{4}-\frac{1633}{71016}a^{3}+\frac{331}{4842}a^{2}+\frac{24755}{53262}a+\frac{3026}{26631}$, $\frac{1}{11\cdots 44}a^{30}+\frac{948710339766505}{36\cdots 48}a^{29}-\frac{56\cdots 47}{55\cdots 72}a^{28}+\frac{10\cdots 43}{11\cdots 44}a^{27}+\frac{32\cdots 89}{18\cdots 24}a^{26}-\frac{30\cdots 13}{11\cdots 44}a^{25}+\frac{11\cdots 71}{33\cdots 68}a^{24}+\frac{90\cdots 99}{11\cdots 44}a^{23}-\frac{24\cdots 75}{61\cdots 08}a^{22}+\frac{50\cdots 61}{11\cdots 44}a^{21}-\frac{31\cdots 87}{30\cdots 04}a^{20}+\frac{70\cdots 91}{36\cdots 48}a^{19}-\frac{99\cdots 10}{62\cdots 69}a^{18}+\frac{34\cdots 19}{55\cdots 72}a^{17}+\frac{17\cdots 54}{69\cdots 59}a^{16}+\frac{13\cdots 79}{36\cdots 48}a^{15}+\frac{47\cdots 53}{11\cdots 44}a^{14}+\frac{89\cdots 17}{55\cdots 72}a^{13}+\frac{12\cdots 03}{12\cdots 16}a^{12}-\frac{58\cdots 49}{16\cdots 08}a^{11}+\frac{14\cdots 19}{36\cdots 48}a^{10}-\frac{52\cdots 23}{11\cdots 44}a^{9}-\frac{34\cdots 47}{11\cdots 44}a^{8}-\frac{13\cdots 41}{55\cdots 72}a^{7}-\frac{23\cdots 19}{11\cdots 44}a^{6}-\frac{23\cdots 24}{23\cdots 53}a^{5}+\frac{45\cdots 33}{10\cdots 04}a^{4}-\frac{26\cdots 51}{61\cdots 08}a^{3}-\frac{51\cdots 03}{15\cdots 02}a^{2}+\frac{21\cdots 51}{27\cdots 36}a-\frac{72\cdots 12}{69\cdots 59}$, $\frac{1}{37\cdots 64}a^{31}-\frac{14\cdots 05}{52\cdots 12}a^{30}-\frac{56\cdots 49}{18\cdots 32}a^{29}+\frac{32\cdots 63}{15\cdots 36}a^{28}+\frac{25\cdots 43}{37\cdots 64}a^{27}+\frac{24\cdots 37}{57\cdots 04}a^{26}+\frac{73\cdots 43}{37\cdots 32}a^{25}-\frac{27\cdots 33}{37\cdots 64}a^{24}-\frac{18\cdots 71}{31\cdots 72}a^{23}+\frac{17\cdots 85}{31\cdots 72}a^{22}+\frac{23\cdots 41}{37\cdots 64}a^{21}-\frac{23\cdots 91}{18\cdots 32}a^{20}-\frac{36\cdots 63}{37\cdots 64}a^{19}+\frac{53\cdots 67}{42\cdots 96}a^{18}+\frac{96\cdots 05}{10\cdots 68}a^{17}+\frac{85\cdots 89}{34\cdots 24}a^{16}+\frac{75\cdots 89}{18\cdots 32}a^{15}+\frac{40\cdots 71}{94\cdots 16}a^{14}-\frac{30\cdots 87}{18\cdots 32}a^{13}-\frac{14\cdots 27}{37\cdots 64}a^{12}+\frac{36\cdots 53}{17\cdots 12}a^{11}+\frac{13\cdots 22}{98\cdots 71}a^{10}-\frac{26\cdots 65}{12\cdots 88}a^{9}-\frac{32\cdots 93}{15\cdots 36}a^{8}-\frac{46\cdots 65}{18\cdots 32}a^{7}+\frac{93\cdots 77}{37\cdots 64}a^{6}+\frac{15\cdots 33}{31\cdots 72}a^{5}+\frac{43\cdots 49}{22\cdots 92}a^{4}-\frac{13\cdots 87}{37\cdots 64}a^{3}-\frac{22\cdots 53}{47\cdots 08}a^{2}+\frac{33\cdots 41}{13\cdots 28}a+\frac{28\cdots 95}{68\cdots 98}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  $C_{1590}$, which has order $1590$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{1590}$, which has order $1590$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 
Relative class number:   $1590$ (assuming GRH)

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $15$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content 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 $6$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{45\cdots 07}{95\cdots 28}a^{31}-\frac{44\cdots 29}{19\cdots 36}a^{30}-\frac{33\cdots 83}{47\cdots 64}a^{29}+\frac{29\cdots 05}{74\cdots 51}a^{28}+\frac{11\cdots 29}{95\cdots 28}a^{27}-\frac{29\cdots 61}{47\cdots 64}a^{26}-\frac{44\cdots 85}{47\cdots 64}a^{25}+\frac{46\cdots 05}{95\cdots 28}a^{24}+\frac{61\cdots 97}{72\cdots 04}a^{23}-\frac{30\cdots 03}{79\cdots 44}a^{22}-\frac{41\cdots 77}{29\cdots 16}a^{21}+\frac{17\cdots 33}{15\cdots 88}a^{20}+\frac{35\cdots 21}{31\cdots 76}a^{19}-\frac{11\cdots 57}{31\cdots 76}a^{18}-\frac{21\cdots 03}{75\cdots 28}a^{17}+\frac{19\cdots 67}{31\cdots 76}a^{16}+\frac{56\cdots 23}{47\cdots 64}a^{15}-\frac{22\cdots 19}{79\cdots 44}a^{14}+\frac{18\cdots 63}{47\cdots 64}a^{13}-\frac{19\cdots 23}{31\cdots 76}a^{12}+\frac{33\cdots 13}{47\cdots 64}a^{11}-\frac{49\cdots 49}{14\cdots 02}a^{10}-\frac{22\cdots 73}{95\cdots 28}a^{9}+\frac{76\cdots 63}{11\cdots 16}a^{8}-\frac{29\cdots 55}{47\cdots 64}a^{7}+\frac{72\cdots 13}{31\cdots 76}a^{6}+\frac{42\cdots 17}{23\cdots 32}a^{5}-\frac{25\cdots 81}{95\cdots 28}a^{4}+\frac{39\cdots 97}{31\cdots 76}a^{3}+\frac{49\cdots 05}{29\cdots 04}a^{2}-\frac{35\cdots 17}{99\cdots 68}a+\frac{80\cdots 57}{51\cdots 38}$, $\frac{30\cdots 83}{70\cdots 28}a^{31}-\frac{35\cdots 61}{17\cdots 32}a^{30}-\frac{21\cdots 43}{32\cdots 24}a^{29}+\frac{20\cdots 75}{58\cdots 44}a^{28}+\frac{80\cdots 41}{70\cdots 28}a^{27}-\frac{18\cdots 69}{35\cdots 64}a^{26}-\frac{32\cdots 65}{35\cdots 64}a^{25}+\frac{30\cdots 09}{70\cdots 28}a^{24}+\frac{72\cdots 13}{88\cdots 16}a^{23}-\frac{14\cdots 37}{44\cdots 08}a^{22}-\frac{12\cdots 49}{70\cdots 28}a^{21}+\frac{10\cdots 43}{10\cdots 08}a^{20}+\frac{80\cdots 15}{70\cdots 28}a^{19}-\frac{22\cdots 31}{70\cdots 28}a^{18}-\frac{60\cdots 65}{20\cdots 36}a^{17}+\frac{34\cdots 93}{70\cdots 28}a^{16}+\frac{40\cdots 21}{35\cdots 64}a^{15}-\frac{10\cdots 99}{44\cdots 08}a^{14}+\frac{11\cdots 77}{35\cdots 64}a^{13}-\frac{38\cdots 73}{70\cdots 28}a^{12}+\frac{18\cdots 03}{32\cdots 24}a^{11}-\frac{22\cdots 95}{88\cdots 16}a^{10}-\frac{50\cdots 67}{23\cdots 76}a^{9}+\frac{95\cdots 69}{17\cdots 32}a^{8}-\frac{19\cdots 61}{39\cdots 96}a^{7}+\frac{11\cdots 39}{70\cdots 28}a^{6}+\frac{27\cdots 97}{17\cdots 32}a^{5}-\frac{14\cdots 89}{70\cdots 28}a^{4}+\frac{12\cdots 29}{13\cdots 28}a^{3}+\frac{13\cdots 19}{88\cdots 16}a^{2}-\frac{18\cdots 33}{73\cdots 18}a+\frac{43\cdots 74}{38\cdots 13}$, $\frac{75\cdots 53}{37\cdots 64}a^{31}-\frac{42\cdots 43}{47\cdots 08}a^{30}-\frac{58\cdots 41}{18\cdots 32}a^{29}+\frac{82\cdots 95}{52\cdots 12}a^{28}+\frac{20\cdots 39}{37\cdots 64}a^{27}-\frac{41\cdots 73}{17\cdots 12}a^{26}-\frac{82\cdots 03}{18\cdots 32}a^{25}+\frac{71\cdots 27}{37\cdots 64}a^{24}+\frac{36\cdots 83}{94\cdots 16}a^{23}-\frac{82\cdots 45}{55\cdots 48}a^{22}-\frac{32\cdots 83}{37\cdots 64}a^{21}+\frac{51\cdots 81}{12\cdots 44}a^{20}+\frac{20\cdots 77}{37\cdots 64}a^{19}-\frac{50\cdots 61}{37\cdots 64}a^{18}-\frac{49\cdots 23}{36\cdots 56}a^{17}+\frac{64\cdots 53}{34\cdots 24}a^{16}+\frac{96\cdots 33}{18\cdots 32}a^{15}-\frac{98\cdots 11}{94\cdots 16}a^{14}+\frac{29\cdots 13}{18\cdots 32}a^{13}-\frac{97\cdots 83}{37\cdots 64}a^{12}+\frac{51\cdots 79}{18\cdots 32}a^{11}-\frac{67\cdots 51}{47\cdots 08}a^{10}-\frac{20\cdots 13}{38\cdots 36}a^{9}+\frac{10\cdots 25}{47\cdots 08}a^{8}-\frac{14\cdots 43}{63\cdots 44}a^{7}+\frac{35\cdots 81}{37\cdots 64}a^{6}+\frac{37\cdots 07}{86\cdots 56}a^{5}-\frac{29\cdots 35}{37\cdots 64}a^{4}+\frac{18\cdots 25}{42\cdots 96}a^{3}+\frac{27\cdots 73}{23\cdots 04}a^{2}-\frac{28\cdots 20}{32\cdots 57}a+\frac{11\cdots 77}{20\cdots 94}$, $\frac{94\cdots 03}{42\cdots 96}a^{31}-\frac{10\cdots 04}{98\cdots 71}a^{30}-\frac{63\cdots 95}{18\cdots 32}a^{29}+\frac{12\cdots 85}{65\cdots 14}a^{28}+\frac{21\cdots 09}{37\cdots 64}a^{27}-\frac{59\cdots 37}{21\cdots 48}a^{26}-\frac{87\cdots 73}{18\cdots 32}a^{25}+\frac{26\cdots 41}{11\cdots 08}a^{24}+\frac{13\cdots 33}{31\cdots 72}a^{23}-\frac{16\cdots 57}{94\cdots 16}a^{22}-\frac{34\cdots 85}{42\cdots 96}a^{21}+\frac{33\cdots 37}{63\cdots 44}a^{20}+\frac{21\cdots 03}{37\cdots 64}a^{19}-\frac{65\cdots 75}{37\cdots 64}a^{18}-\frac{48\cdots 83}{32\cdots 04}a^{17}+\frac{35\cdots 79}{12\cdots 88}a^{16}+\frac{36\cdots 93}{63\cdots 44}a^{15}-\frac{74\cdots 01}{55\cdots 48}a^{14}+\frac{19\cdots 79}{11\cdots 96}a^{13}-\frac{10\cdots 73}{37\cdots 64}a^{12}+\frac{57\cdots 17}{18\cdots 32}a^{11}-\frac{28\cdots 05}{23\cdots 04}a^{10}-\frac{51\cdots 41}{37\cdots 64}a^{9}+\frac{40\cdots 27}{13\cdots 28}a^{8}-\frac{17\cdots 01}{63\cdots 44}a^{7}+\frac{35\cdots 03}{42\cdots 96}a^{6}+\frac{93\cdots 57}{94\cdots 16}a^{5}-\frac{46\cdots 69}{37\cdots 64}a^{4}+\frac{57\cdots 83}{11\cdots 08}a^{3}+\frac{55\cdots 37}{47\cdots 08}a^{2}-\frac{19\cdots 93}{11\cdots 52}a+\frac{12\cdots 25}{20\cdots 94}$, $\frac{61\cdots 53}{23\cdots 88}a^{31}-\frac{17\cdots 67}{13\cdots 88}a^{30}-\frac{41\cdots 89}{11\cdots 44}a^{29}+\frac{33\cdots 09}{14\cdots 68}a^{28}+\frac{14\cdots 51}{23\cdots 88}a^{27}-\frac{37\cdots 17}{10\cdots 04}a^{26}-\frac{16\cdots 09}{39\cdots 48}a^{25}+\frac{66\cdots 95}{23\cdots 88}a^{24}+\frac{21\cdots 27}{53\cdots 52}a^{23}-\frac{14\cdots 07}{65\cdots 08}a^{22}-\frac{62\cdots 31}{23\cdots 88}a^{21}+\frac{75\cdots 17}{11\cdots 44}a^{20}+\frac{10\cdots 77}{23\cdots 88}a^{19}-\frac{52\cdots 93}{23\cdots 88}a^{18}-\frac{18\cdots 11}{17\cdots 84}a^{17}+\frac{89\cdots 35}{23\cdots 88}a^{16}+\frac{20\cdots 45}{35\cdots 68}a^{15}-\frac{68\cdots 13}{38\cdots 24}a^{14}+\frac{17\cdots 29}{69\cdots 32}a^{13}-\frac{91\cdots 35}{23\cdots 88}a^{12}+\frac{18\cdots 01}{39\cdots 48}a^{11}-\frac{80\cdots 51}{29\cdots 36}a^{10}-\frac{71\cdots 33}{78\cdots 96}a^{9}+\frac{30\cdots 31}{81\cdots 26}a^{8}-\frac{48\cdots 45}{11\cdots 44}a^{7}+\frac{14\cdots 47}{78\cdots 96}a^{6}+\frac{45\cdots 21}{58\cdots 72}a^{5}-\frac{38\cdots 35}{23\cdots 88}a^{4}+\frac{18\cdots 23}{21\cdots 08}a^{3}+\frac{10\cdots 99}{24\cdots 78}a^{2}-\frac{79\cdots 65}{36\cdots 17}a+\frac{11\cdots 40}{12\cdots 39}$, $\frac{28\cdots 15}{39\cdots 68}a^{31}-\frac{30\cdots 17}{73\cdots 44}a^{30}-\frac{41\cdots 63}{53\cdots 32}a^{29}+\frac{52\cdots 71}{73\cdots 44}a^{28}+\frac{15\cdots 47}{11\cdots 04}a^{27}-\frac{66\cdots 93}{59\cdots 48}a^{26}-\frac{12\cdots 49}{19\cdots 84}a^{25}+\frac{35\cdots 33}{39\cdots 68}a^{24}+\frac{19\cdots 63}{29\cdots 76}a^{23}-\frac{21\cdots 59}{29\cdots 76}a^{22}+\frac{33\cdots 73}{11\cdots 04}a^{21}+\frac{41\cdots 75}{19\cdots 84}a^{20}+\frac{20\cdots 81}{11\cdots 04}a^{19}-\frac{92\cdots 01}{11\cdots 04}a^{18}+\frac{47\cdots 49}{10\cdots 44}a^{17}+\frac{17\cdots 51}{11\cdots 04}a^{16}+\frac{70\cdots 89}{65\cdots 28}a^{15}-\frac{11\cdots 91}{17\cdots 28}a^{14}+\frac{10\cdots 87}{11\cdots 52}a^{13}-\frac{15\cdots 31}{11\cdots 04}a^{12}+\frac{10\cdots 79}{59\cdots 52}a^{11}-\frac{91\cdots 61}{73\cdots 44}a^{10}-\frac{21\cdots 39}{11\cdots 04}a^{9}+\frac{20\cdots 15}{14\cdots 88}a^{8}-\frac{86\cdots 95}{53\cdots 32}a^{7}+\frac{10\cdots 37}{11\cdots 04}a^{6}+\frac{61\cdots 07}{29\cdots 76}a^{5}-\frac{71\cdots 45}{10\cdots 64}a^{4}+\frac{15\cdots 15}{39\cdots 68}a^{3}-\frac{99\cdots 35}{73\cdots 44}a^{2}-\frac{11\cdots 27}{12\cdots 24}a+\frac{27\cdots 91}{70\cdots 26}$, $\frac{34\cdots 65}{37\cdots 64}a^{31}-\frac{96\cdots 51}{23\cdots 04}a^{30}-\frac{28\cdots 69}{18\cdots 32}a^{29}+\frac{19\cdots 81}{26\cdots 56}a^{28}+\frac{97\cdots 07}{37\cdots 64}a^{27}-\frac{11\cdots 09}{10\cdots 36}a^{26}-\frac{42\cdots 91}{18\cdots 32}a^{25}+\frac{35\cdots 95}{37\cdots 64}a^{24}+\frac{11\cdots 61}{55\cdots 48}a^{23}-\frac{69\cdots 43}{94\cdots 16}a^{22}-\frac{23\cdots 35}{37\cdots 64}a^{21}+\frac{15\cdots 99}{63\cdots 44}a^{20}+\frac{12\cdots 77}{37\cdots 64}a^{19}-\frac{28\cdots 45}{37\cdots 64}a^{18}-\frac{10\cdots 53}{10\cdots 68}a^{17}+\frac{49\cdots 03}{37\cdots 64}a^{16}+\frac{61\cdots 73}{18\cdots 32}a^{15}-\frac{49\cdots 61}{94\cdots 16}a^{14}+\frac{69\cdots 91}{17\cdots 12}a^{13}-\frac{30\cdots 59}{37\cdots 64}a^{12}+\frac{15\cdots 55}{18\cdots 32}a^{11}+\frac{49\cdots 81}{27\cdots 24}a^{10}-\frac{36\cdots 41}{38\cdots 36}a^{9}+\frac{58\cdots 71}{59\cdots 26}a^{8}-\frac{33\cdots 15}{63\cdots 44}a^{7}-\frac{10\cdots 15}{37\cdots 64}a^{6}+\frac{61\cdots 89}{94\cdots 16}a^{5}-\frac{10\cdots 35}{37\cdots 64}a^{4}-\frac{51\cdots 23}{42\cdots 96}a^{3}+\frac{17\cdots 23}{11\cdots 52}a^{2}-\frac{37\cdots 81}{19\cdots 42}a-\frac{54\cdots 79}{20\cdots 94}$, $\frac{27\cdots 85}{37\cdots 64}a^{31}-\frac{37\cdots 97}{11\cdots 52}a^{30}-\frac{18\cdots 79}{17\cdots 12}a^{29}+\frac{17\cdots 01}{32\cdots 57}a^{28}+\frac{69\cdots 31}{37\cdots 64}a^{27}-\frac{15\cdots 03}{18\cdots 32}a^{26}-\frac{27\cdots 83}{18\cdots 32}a^{25}+\frac{23\cdots 95}{37\cdots 64}a^{24}+\frac{12\cdots 41}{94\cdots 16}a^{23}-\frac{46\cdots 95}{94\cdots 16}a^{22}-\frac{77\cdots 87}{37\cdots 64}a^{21}+\frac{77\cdots 91}{63\cdots 44}a^{20}+\frac{57\cdots 61}{37\cdots 64}a^{19}-\frac{15\cdots 53}{37\cdots 64}a^{18}-\frac{32\cdots 83}{10\cdots 68}a^{17}+\frac{18\cdots 87}{37\cdots 64}a^{16}+\frac{14\cdots 97}{11\cdots 96}a^{15}-\frac{33\cdots 27}{94\cdots 16}a^{14}+\frac{13\cdots 73}{18\cdots 32}a^{13}-\frac{34\cdots 49}{34\cdots 24}a^{12}+\frac{19\cdots 55}{18\cdots 32}a^{11}-\frac{17\cdots 89}{23\cdots 04}a^{10}-\frac{15\cdots 77}{12\cdots 88}a^{9}+\frac{30\cdots 28}{29\cdots 13}a^{8}-\frac{71\cdots 23}{63\cdots 44}a^{7}+\frac{22\cdots 77}{37\cdots 64}a^{6}+\frac{50\cdots 63}{55\cdots 48}a^{5}-\frac{17\cdots 15}{37\cdots 64}a^{4}+\frac{14\cdots 53}{42\cdots 96}a^{3}-\frac{87\cdots 73}{27\cdots 24}a^{2}-\frac{15\cdots 65}{23\cdots 52}a+\frac{11\cdots 13}{20\cdots 94}$, $\frac{24\cdots 83}{37\cdots 64}a^{31}-\frac{14\cdots 15}{47\cdots 08}a^{30}-\frac{57\cdots 93}{63\cdots 44}a^{29}+\frac{16\cdots 95}{32\cdots 57}a^{28}+\frac{59\cdots 97}{37\cdots 64}a^{27}-\frac{14\cdots 89}{18\cdots 32}a^{26}-\frac{24\cdots 81}{21\cdots 48}a^{25}+\frac{21\cdots 45}{37\cdots 64}a^{24}+\frac{96\cdots 69}{94\cdots 16}a^{23}-\frac{41\cdots 97}{94\cdots 16}a^{22}-\frac{25\cdots 85}{37\cdots 64}a^{21}+\frac{61\cdots 13}{63\cdots 44}a^{20}+\frac{38\cdots 55}{37\cdots 64}a^{19}-\frac{41\cdots 61}{12\cdots 88}a^{18}-\frac{40\cdots 29}{32\cdots 04}a^{17}+\frac{12\cdots 37}{37\cdots 64}a^{16}+\frac{14\cdots 19}{18\cdots 32}a^{15}-\frac{10\cdots 63}{31\cdots 72}a^{14}+\frac{48\cdots 45}{63\cdots 44}a^{13}-\frac{41\cdots 29}{37\cdots 64}a^{12}+\frac{21\cdots 25}{18\cdots 32}a^{11}-\frac{39\cdots 61}{39\cdots 84}a^{10}+\frac{27\cdots 87}{22\cdots 92}a^{9}+\frac{47\cdots 77}{47\cdots 08}a^{8}-\frac{26\cdots 91}{21\cdots 48}a^{7}+\frac{31\cdots 71}{37\cdots 64}a^{6}-\frac{19\cdots 45}{28\cdots 52}a^{5}-\frac{19\cdots 53}{37\cdots 64}a^{4}+\frac{59\cdots 97}{12\cdots 88}a^{3}-\frac{11\cdots 07}{13\cdots 28}a^{2}-\frac{49\cdots 07}{59\cdots 26}a+\frac{16\cdots 53}{22\cdots 66}$, $\frac{59\cdots 05}{12\cdots 88}a^{31}-\frac{11\cdots 87}{52\cdots 12}a^{30}-\frac{44\cdots 97}{63\cdots 44}a^{29}+\frac{25\cdots 03}{65\cdots 14}a^{28}+\frac{15\cdots 11}{12\cdots 88}a^{27}-\frac{22\cdots 31}{37\cdots 32}a^{26}-\frac{19\cdots 89}{21\cdots 48}a^{25}+\frac{60\cdots 67}{12\cdots 88}a^{24}+\frac{14\cdots 55}{16\cdots 56}a^{23}-\frac{39\cdots 05}{10\cdots 24}a^{22}-\frac{18\cdots 71}{12\cdots 88}a^{21}+\frac{22\cdots 47}{21\cdots 48}a^{20}+\frac{12\cdots 79}{11\cdots 08}a^{19}-\frac{15\cdots 95}{42\cdots 96}a^{18}-\frac{10\cdots 29}{36\cdots 56}a^{17}+\frac{72\cdots 47}{12\cdots 88}a^{16}+\frac{73\cdots 65}{63\cdots 44}a^{15}-\frac{88\cdots 81}{31\cdots 72}a^{14}+\frac{81\cdots 19}{21\cdots 48}a^{13}-\frac{78\cdots 67}{12\cdots 88}a^{12}+\frac{14\cdots 33}{21\cdots 48}a^{11}-\frac{94\cdots 55}{28\cdots 76}a^{10}-\frac{24\cdots 45}{11\cdots 08}a^{9}+\frac{20\cdots 05}{32\cdots 57}a^{8}-\frac{12\cdots 59}{21\cdots 48}a^{7}+\frac{29\cdots 49}{12\cdots 88}a^{6}+\frac{15\cdots 11}{95\cdots 84}a^{5}-\frac{10\cdots 29}{42\cdots 96}a^{4}+\frac{15\cdots 09}{12\cdots 88}a^{3}+\frac{54\cdots 81}{39\cdots 84}a^{2}-\frac{12\cdots 51}{39\cdots 84}a+\frac{99\cdots 79}{68\cdots 98}$, $\frac{36\cdots 47}{22\cdots 92}a^{31}-\frac{20\cdots 91}{23\cdots 04}a^{30}-\frac{33\cdots 13}{19\cdots 68}a^{29}+\frac{75\cdots 85}{52\cdots 12}a^{28}+\frac{11\cdots 29}{37\cdots 64}a^{27}-\frac{42\cdots 17}{18\cdots 32}a^{26}-\frac{10\cdots 79}{63\cdots 44}a^{25}+\frac{64\cdots 65}{37\cdots 64}a^{24}+\frac{15\cdots 17}{94\cdots 16}a^{23}-\frac{12\cdots 55}{94\cdots 16}a^{22}+\frac{18\cdots 39}{37\cdots 64}a^{21}+\frac{19\cdots 89}{63\cdots 44}a^{20}+\frac{49\cdots 09}{34\cdots 24}a^{19}-\frac{15\cdots 69}{12\cdots 88}a^{18}+\frac{38\cdots 01}{32\cdots 04}a^{17}+\frac{66\cdots 05}{37\cdots 64}a^{16}+\frac{46\cdots 99}{18\cdots 32}a^{15}-\frac{34\cdots 41}{31\cdots 72}a^{14}+\frac{13\cdots 69}{63\cdots 44}a^{13}-\frac{14\cdots 77}{37\cdots 64}a^{12}+\frac{51\cdots 71}{10\cdots 36}a^{11}-\frac{17\cdots 31}{35\cdots 44}a^{10}+\frac{10\cdots 75}{37\cdots 64}a^{9}+\frac{13\cdots 85}{43\cdots 28}a^{8}-\frac{16\cdots 21}{63\cdots 44}a^{7}+\frac{11\cdots 35}{37\cdots 64}a^{6}-\frac{16\cdots 61}{10\cdots 24}a^{5}+\frac{55\cdots 27}{37\cdots 64}a^{4}+\frac{64\cdots 85}{12\cdots 88}a^{3}-\frac{62\cdots 01}{15\cdots 36}a^{2}+\frac{17\cdots 39}{11\cdots 52}a-\frac{86\cdots 19}{68\cdots 98}$, $\frac{30\cdots 45}{37\cdots 64}a^{31}-\frac{96\cdots 69}{23\cdots 04}a^{30}-\frac{68\cdots 19}{63\cdots 44}a^{29}+\frac{55\cdots 39}{79\cdots 68}a^{28}+\frac{64\cdots 17}{34\cdots 24}a^{27}-\frac{20\cdots 35}{18\cdots 32}a^{26}-\frac{83\cdots 53}{63\cdots 44}a^{25}+\frac{33\cdots 11}{37\cdots 64}a^{24}+\frac{11\cdots 61}{94\cdots 16}a^{23}-\frac{65\cdots 03}{94\cdots 16}a^{22}-\frac{29\cdots 91}{37\cdots 64}a^{21}+\frac{12\cdots 75}{63\cdots 44}a^{20}+\frac{54\cdots 45}{37\cdots 64}a^{19}-\frac{87\cdots 51}{12\cdots 88}a^{18}-\frac{10\cdots 57}{32\cdots 04}a^{17}+\frac{16\cdots 87}{14\cdots 56}a^{16}+\frac{20\cdots 85}{11\cdots 96}a^{15}-\frac{57\cdots 35}{10\cdots 24}a^{14}+\frac{49\cdots 99}{63\cdots 44}a^{13}-\frac{47\cdots 35}{37\cdots 64}a^{12}+\frac{28\cdots 67}{18\cdots 32}a^{11}-\frac{93\cdots 97}{98\cdots 71}a^{10}-\frac{56\cdots 75}{37\cdots 64}a^{9}+\frac{24\cdots 83}{23\cdots 04}a^{8}-\frac{76\cdots 75}{63\cdots 44}a^{7}+\frac{23\cdots 77}{37\cdots 64}a^{6}+\frac{31\cdots 57}{18\cdots 16}a^{5}-\frac{16\cdots 67}{37\cdots 64}a^{4}+\frac{10\cdots 57}{42\cdots 96}a^{3}+\frac{22\cdots 33}{84\cdots 28}a^{2}-\frac{35\cdots 71}{69\cdots 56}a+\frac{15\cdots 93}{68\cdots 98}$, $\frac{11\cdots 97}{37\cdots 64}a^{31}-\frac{21\cdots 19}{11\cdots 52}a^{30}-\frac{19\cdots 87}{63\cdots 44}a^{29}+\frac{31\cdots 33}{98\cdots 71}a^{28}+\frac{19\cdots 23}{37\cdots 64}a^{27}-\frac{95\cdots 95}{18\cdots 32}a^{26}-\frac{26\cdots 31}{12\cdots 44}a^{25}+\frac{15\cdots 75}{37\cdots 64}a^{24}+\frac{22\cdots 89}{94\cdots 16}a^{23}-\frac{31\cdots 71}{94\cdots 16}a^{22}+\frac{56\cdots 77}{37\cdots 64}a^{21}+\frac{62\cdots 35}{63\cdots 44}a^{20}+\frac{64\cdots 97}{37\cdots 64}a^{19}-\frac{47\cdots 95}{12\cdots 88}a^{18}+\frac{79\cdots 65}{29\cdots 64}a^{17}+\frac{28\cdots 23}{37\cdots 64}a^{16}+\frac{97\cdots 61}{18\cdots 32}a^{15}-\frac{28\cdots 61}{95\cdots 84}a^{14}+\frac{85\cdots 01}{21\cdots 48}a^{13}-\frac{22\cdots 87}{37\cdots 64}a^{12}+\frac{15\cdots 43}{18\cdots 32}a^{11}-\frac{99\cdots 73}{17\cdots 22}a^{10}-\frac{31\cdots 95}{37\cdots 64}a^{9}+\frac{13\cdots 11}{23\cdots 04}a^{8}-\frac{14\cdots 89}{21\cdots 48}a^{7}+\frac{13\cdots 25}{37\cdots 64}a^{6}+\frac{11\cdots 59}{10\cdots 24}a^{5}-\frac{61\cdots 55}{22\cdots 92}a^{4}+\frac{16\cdots 59}{12\cdots 88}a^{3}+\frac{49\cdots 79}{52\cdots 12}a^{2}-\frac{44\cdots 91}{11\cdots 52}a+\frac{62\cdots 93}{68\cdots 98}$, $\frac{19\cdots 65}{47\cdots 08}a^{31}-\frac{90\cdots 67}{47\cdots 08}a^{30}-\frac{14\cdots 31}{23\cdots 04}a^{29}+\frac{59\cdots 41}{17\cdots 22}a^{28}+\frac{16\cdots 07}{15\cdots 36}a^{27}-\frac{24\cdots 67}{47\cdots 08}a^{26}-\frac{21\cdots 27}{26\cdots 56}a^{25}+\frac{19\cdots 89}{47\cdots 08}a^{24}+\frac{34\cdots 37}{47\cdots 08}a^{23}-\frac{69\cdots 91}{21\cdots 64}a^{22}-\frac{21\cdots 31}{15\cdots 36}a^{21}+\frac{44\cdots 23}{47\cdots 08}a^{20}+\frac{14\cdots 41}{14\cdots 76}a^{19}-\frac{60\cdots 11}{19\cdots 42}a^{18}-\frac{40\cdots 27}{16\cdots 52}a^{17}+\frac{77\cdots 77}{15\cdots 36}a^{16}+\frac{53\cdots 73}{52\cdots 12}a^{15}-\frac{56\cdots 87}{23\cdots 04}a^{14}+\frac{97\cdots 07}{29\cdots 13}a^{13}-\frac{27\cdots 91}{52\cdots 12}a^{12}+\frac{27\cdots 71}{47\cdots 08}a^{11}-\frac{10\cdots 65}{39\cdots 84}a^{10}-\frac{87\cdots 41}{43\cdots 28}a^{9}+\frac{25\cdots 07}{47\cdots 08}a^{8}-\frac{91\cdots 10}{17\cdots 89}a^{7}+\frac{80\cdots 75}{43\cdots 28}a^{6}+\frac{24\cdots 35}{15\cdots 36}a^{5}-\frac{35\cdots 35}{15\cdots 36}a^{4}+\frac{24\cdots 63}{23\cdots 04}a^{3}+\frac{69\cdots 55}{47\cdots 08}a^{2}-\frac{87\cdots 77}{29\cdots 13}a+\frac{44\cdots 40}{34\cdots 99}$, $\frac{13\cdots 11}{37\cdots 64}a^{31}-\frac{69\cdots 83}{47\cdots 08}a^{30}-\frac{12\cdots 19}{18\cdots 32}a^{29}+\frac{14\cdots 47}{52\cdots 12}a^{28}+\frac{13\cdots 35}{12\cdots 88}a^{27}-\frac{77\cdots 81}{18\cdots 32}a^{26}-\frac{62\cdots 47}{63\cdots 44}a^{25}+\frac{12\cdots 37}{37\cdots 64}a^{24}+\frac{81\cdots 15}{94\cdots 16}a^{23}-\frac{24\cdots 15}{94\cdots 16}a^{22}-\frac{13\cdots 07}{47\cdots 52}a^{21}+\frac{16\cdots 63}{18\cdots 32}a^{20}+\frac{53\cdots 67}{42\cdots 96}a^{19}-\frac{10\cdots 47}{42\cdots 96}a^{18}-\frac{12\cdots 27}{32\cdots 04}a^{17}+\frac{17\cdots 33}{42\cdots 96}a^{16}+\frac{23\cdots 79}{21\cdots 48}a^{15}-\frac{98\cdots 81}{55\cdots 48}a^{14}+\frac{16\cdots 71}{11\cdots 96}a^{13}-\frac{25\cdots 51}{12\cdots 88}a^{12}+\frac{12\cdots 89}{18\cdots 32}a^{11}+\frac{55\cdots 67}{15\cdots 36}a^{10}-\frac{27\cdots 37}{37\cdots 64}a^{9}+\frac{17\cdots 09}{23\cdots 04}a^{8}-\frac{74\cdots 59}{18\cdots 32}a^{7}-\frac{19\cdots 09}{14\cdots 56}a^{6}+\frac{13\cdots 25}{31\cdots 72}a^{5}-\frac{14\cdots 65}{42\cdots 96}a^{4}+\frac{31\cdots 31}{37\cdots 64}a^{3}+\frac{31\cdots 39}{47\cdots 08}a^{2}-\frac{67\cdots 27}{11\cdots 52}a+\frac{13\cdots 07}{68\cdots 98}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 63453830199620.586 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content 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)

Copy content comment:Analytic class number formula
 
Copy content sage:# 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))))
 
Copy content gp:\\ 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<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); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula 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); 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):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{105}) \), 4.4.2225.1, 4.0.20025.1, 4.0.109025.1, 4.4.981225.1, \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), 8.8.2721997193125.1, 8.0.220481772643125.1, 8.0.1133693125.1, 8.8.91829143125.1, 8.0.121550625.1, 8.0.401000625.1, 8.0.962802500625.6, 8.0.11886450625.2, 8.0.962802500625.3, 8.8.962802500625.2, 8.0.962802500625.2, 16.0.926988655209753125390625.1, 16.0.48612212067854663648609765625.2, 16.0.8432591527071734765625.1, 16.0.7409268719380378547265625.1, 16.0.48612212067854663648609765625.1, 16.16.48612212067854663648609765625.1, 16.0.48612212067854663648609765625.3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content 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.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(5\) Copy content Toggle raw display 5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(7\) Copy content Toggle raw display 7.8.2.8a1.2$x^{16} + 8 x^{11} + 12 x^{10} + 4 x^{9} + 6 x^{8} + 16 x^{6} + 48 x^{5} + 52 x^{4} + 48 x^{3} + 40 x^{2} + 12 x + 16$$2$$8$$8$$C_8\times C_2$$$[\ ]_{2}^{8}$$
7.8.2.8a1.2$x^{16} + 8 x^{11} + 12 x^{10} + 4 x^{9} + 6 x^{8} + 16 x^{6} + 48 x^{5} + 52 x^{4} + 48 x^{3} + 40 x^{2} + 12 x + 16$$2$$8$$8$$C_8\times C_2$$$[\ ]_{2}^{8}$$
\(89\) Copy content Toggle raw display 89.2.2.2a1.2$x^{4} + 164 x^{3} + 6730 x^{2} + 492 x + 98$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
89.2.2.2a1.2$x^{4} + 164 x^{3} + 6730 x^{2} + 492 x + 98$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
89.4.1.0a1.1$x^{4} + 4 x^{2} + 72 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
89.4.1.0a1.1$x^{4} + 4 x^{2} + 72 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
89.2.2.2a1.2$x^{4} + 164 x^{3} + 6730 x^{2} + 492 x + 98$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
89.4.1.0a1.1$x^{4} + 4 x^{2} + 72 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
89.4.1.0a1.1$x^{4} + 4 x^{2} + 72 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
89.2.2.2a1.2$x^{4} + 164 x^{3} + 6730 x^{2} + 492 x + 98$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(229\) Copy content Toggle raw display 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$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)