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Properties

Label 32.0.385...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $3.856\times 10^{52}$
Root discriminant \(43.99\)
Ramified primes $2,5,31,89$
Class number $96$ (GRH)
Class group [2, 2, 24] (GRH)
Galois group $D_4^2:C_2^3$ (as 32T12882)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 12*x^30 + 102*x^28 - 452*x^26 + 1409*x^24 - 1772*x^22 + 2946*x^20 - 15768*x^18 + 33540*x^16 - 20668*x^14 + 94478*x^12 - 7132*x^10 + 205785*x^8 + 131700*x^6 + 83794*x^4 + 9720*x^2 + 2025)
 
gp: K = bnfinit(y^32 - 12*y^30 + 102*y^28 - 452*y^26 + 1409*y^24 - 1772*y^22 + 2946*y^20 - 15768*y^18 + 33540*y^16 - 20668*y^14 + 94478*y^12 - 7132*y^10 + 205785*y^8 + 131700*y^6 + 83794*y^4 + 9720*y^2 + 2025, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 12*x^30 + 102*x^28 - 452*x^26 + 1409*x^24 - 1772*x^22 + 2946*x^20 - 15768*x^18 + 33540*x^16 - 20668*x^14 + 94478*x^12 - 7132*x^10 + 205785*x^8 + 131700*x^6 + 83794*x^4 + 9720*x^2 + 2025);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 12*x^30 + 102*x^28 - 452*x^26 + 1409*x^24 - 1772*x^22 + 2946*x^20 - 15768*x^18 + 33540*x^16 - 20668*x^14 + 94478*x^12 - 7132*x^10 + 205785*x^8 + 131700*x^6 + 83794*x^4 + 9720*x^2 + 2025)
 

\( x^{32} - 12 x^{30} + 102 x^{28} - 452 x^{26} + 1409 x^{24} - 1772 x^{22} + 2946 x^{20} - 15768 x^{18} + \cdots + 2025 \) Copy content Toggle raw display

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:   \(38559672304350233766313499377700700160000000000000000\) \(\medspace = 2^{72}\cdot 5^{16}\cdot 31^{8}\cdot 89^{4}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(43.99\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{9/4}5^{1/2}31^{1/2}89^{1/2}\approx 558.699577131565$
Ramified primes:   \(2\), \(5\), \(31\), \(89\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{8}a^{12}+\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{2}+\frac{1}{8}$, $\frac{1}{8}a^{13}+\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{2}a^{3}+\frac{1}{8}a$, $\frac{1}{8}a^{14}-\frac{1}{2}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}+\frac{1}{8}a^{2}-\frac{1}{4}$, $\frac{1}{8}a^{15}-\frac{1}{2}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}+\frac{1}{8}a^{3}-\frac{1}{4}a$, $\frac{1}{8}a^{16}-\frac{1}{4}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}+\frac{1}{8}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{17}-\frac{1}{4}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}+\frac{1}{8}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{40}a^{18}+\frac{1}{20}a^{16}+\frac{1}{40}a^{12}-\frac{1}{20}a^{10}-\frac{1}{10}a^{8}-\frac{1}{8}a^{6}-\frac{1}{5}a^{4}-\frac{1}{10}a^{2}-\frac{1}{8}$, $\frac{1}{40}a^{19}+\frac{1}{20}a^{17}+\frac{1}{40}a^{13}-\frac{1}{20}a^{11}-\frac{1}{10}a^{9}-\frac{1}{8}a^{7}-\frac{1}{5}a^{5}-\frac{1}{10}a^{3}-\frac{1}{8}a$, $\frac{1}{40}a^{20}+\frac{1}{40}a^{16}+\frac{1}{40}a^{14}+\frac{1}{40}a^{12}-\frac{17}{40}a^{8}+\frac{3}{10}a^{6}+\frac{17}{40}a^{4}+\frac{13}{40}a^{2}-\frac{1}{8}$, $\frac{1}{240}a^{21}-\frac{1}{120}a^{19}-\frac{1}{80}a^{18}+\frac{1}{20}a^{17}-\frac{1}{40}a^{16}+\frac{11}{240}a^{15}+\frac{7}{120}a^{13}-\frac{1}{80}a^{12}-\frac{3}{20}a^{11}+\frac{1}{40}a^{10}+\frac{71}{240}a^{9}+\frac{1}{20}a^{8}-\frac{17}{60}a^{7}+\frac{1}{16}a^{6}+\frac{1}{5}a^{5}+\frac{1}{10}a^{4}+\frac{101}{240}a^{3}-\frac{9}{20}a^{2}+\frac{5}{12}a-\frac{7}{16}$, $\frac{1}{240}a^{22}-\frac{1}{120}a^{20}-\frac{1}{80}a^{19}-\frac{1}{40}a^{17}-\frac{13}{240}a^{16}+\frac{7}{120}a^{14}-\frac{1}{80}a^{13}+\frac{1}{20}a^{12}+\frac{1}{40}a^{11}-\frac{5}{48}a^{10}+\frac{1}{20}a^{9}-\frac{1}{12}a^{8}+\frac{1}{16}a^{7}-\frac{1}{20}a^{6}+\frac{1}{10}a^{5}-\frac{43}{240}a^{4}-\frac{9}{20}a^{3}-\frac{23}{60}a^{2}-\frac{7}{16}a-\frac{1}{2}$, $\frac{1}{240}a^{23}-\frac{1}{80}a^{20}+\frac{1}{120}a^{19}-\frac{7}{240}a^{17}+\frac{1}{20}a^{16}+\frac{1}{40}a^{15}-\frac{1}{80}a^{14}-\frac{7}{120}a^{13}+\frac{1}{20}a^{12}-\frac{49}{240}a^{11}+\frac{19}{120}a^{9}-\frac{3}{80}a^{8}-\frac{29}{120}a^{7}-\frac{1}{40}a^{6}+\frac{19}{48}a^{5}+\frac{7}{20}a^{4}+\frac{29}{60}a^{3}+\frac{37}{80}a^{2}-\frac{7}{24}a-\frac{1}{8}$, $\frac{1}{960}a^{24}-\frac{1}{240}a^{20}-\frac{1}{80}a^{19}-\frac{1}{240}a^{18}-\frac{1}{40}a^{17}+\frac{3}{80}a^{16}+\frac{1}{24}a^{14}-\frac{1}{80}a^{13}+\frac{19}{480}a^{12}+\frac{1}{40}a^{11}+\frac{19}{120}a^{10}+\frac{1}{20}a^{9}+\frac{1}{30}a^{8}+\frac{1}{16}a^{7}-\frac{13}{240}a^{6}+\frac{1}{10}a^{5}+\frac{1}{48}a^{4}-\frac{9}{20}a^{3}+\frac{1}{3}a^{2}-\frac{7}{16}a+\frac{31}{64}$, $\frac{1}{960}a^{25}-\frac{1}{80}a^{20}-\frac{1}{80}a^{19}-\frac{1}{80}a^{18}-\frac{3}{80}a^{17}+\frac{1}{40}a^{16}-\frac{3}{80}a^{15}-\frac{1}{80}a^{14}-\frac{13}{480}a^{13}+\frac{3}{80}a^{12}-\frac{59}{120}a^{11}+\frac{1}{40}a^{10}+\frac{19}{240}a^{9}+\frac{1}{80}a^{8}-\frac{7}{80}a^{7}+\frac{3}{80}a^{6}-\frac{97}{240}a^{5}+\frac{9}{20}a^{4}+\frac{91}{240}a^{3}+\frac{1}{80}a^{2}-\frac{91}{192}a+\frac{7}{16}$, $\frac{1}{36480}a^{26}+\frac{1}{2432}a^{24}-\frac{1}{760}a^{22}-\frac{1}{760}a^{20}+\frac{23}{3040}a^{18}-\frac{1}{76}a^{16}-\frac{113}{3648}a^{14}+\frac{871}{18240}a^{12}-\frac{2957}{9120}a^{10}+\frac{1171}{3040}a^{8}-\frac{3109}{9120}a^{6}-\frac{1549}{4560}a^{4}+\frac{6577}{36480}a^{2}+\frac{877}{2432}$, $\frac{1}{36480}a^{27}+\frac{1}{2432}a^{25}-\frac{1}{760}a^{23}-\frac{1}{760}a^{21}+\frac{23}{3040}a^{19}-\frac{1}{76}a^{17}-\frac{113}{3648}a^{15}+\frac{871}{18240}a^{13}-\frac{2957}{9120}a^{11}+\frac{1171}{3040}a^{9}-\frac{3109}{9120}a^{7}-\frac{1549}{4560}a^{5}+\frac{6577}{36480}a^{3}+\frac{877}{2432}a$, $\frac{1}{10068480}a^{28}+\frac{1}{629280}a^{26}+\frac{841}{10068480}a^{24}+\frac{1261}{1258560}a^{22}-\frac{947}{132480}a^{20}-\frac{1}{80}a^{19}+\frac{101}{21888}a^{18}-\frac{1}{40}a^{17}-\frac{271517}{5034240}a^{16}+\frac{13073}{2517120}a^{14}-\frac{1}{80}a^{13}+\frac{84979}{5034240}a^{12}+\frac{1}{40}a^{11}-\frac{375181}{1258560}a^{10}+\frac{1}{20}a^{9}-\frac{52249}{104880}a^{8}+\frac{1}{16}a^{7}-\frac{107629}{2517120}a^{6}+\frac{1}{10}a^{5}+\frac{93247}{437760}a^{4}-\frac{9}{20}a^{3}+\frac{1095989}{2517120}a^{2}-\frac{7}{16}a+\frac{101841}{223744}$, $\frac{1}{100684800}a^{29}-\frac{1}{20136960}a^{28}-\frac{341}{25171200}a^{27}+\frac{13}{1006848}a^{26}+\frac{2321}{20136960}a^{25}-\frac{7189}{20136960}a^{24}+\frac{9541}{12585600}a^{23}-\frac{2917}{2517120}a^{22}+\frac{51007}{25171200}a^{21}-\frac{37759}{5034240}a^{20}+\frac{1381}{1094400}a^{19}-\frac{103}{11520}a^{18}+\frac{751891}{50342400}a^{17}-\frac{72215}{2013696}a^{16}-\frac{1191253}{25171200}a^{15}+\frac{55789}{5034240}a^{14}-\frac{2574833}{50342400}a^{13}+\frac{333713}{10068480}a^{12}-\frac{1135147}{12585600}a^{11}+\frac{774691}{2517120}a^{10}+\frac{19235}{83904}a^{9}-\frac{61177}{139840}a^{8}-\frac{8539153}{25171200}a^{7}+\frac{2091793}{5034240}a^{6}+\frac{1072063}{4377600}a^{5}-\frac{130687}{875520}a^{4}-\frac{1557073}{6292800}a^{3}+\frac{15959}{39330}a^{2}-\frac{2306633}{6712320}a-\frac{101565}{447488}$, $\frac{1}{13\!\cdots\!00}a^{30}+\frac{47\!\cdots\!61}{13\!\cdots\!00}a^{28}-\frac{20\!\cdots\!47}{27\!\cdots\!40}a^{26}-\frac{54\!\cdots\!07}{13\!\cdots\!00}a^{24}-\frac{75\!\cdots\!83}{33\!\cdots\!00}a^{22}-\frac{62\!\cdots\!61}{16\!\cdots\!00}a^{20}-\frac{74\!\cdots\!39}{67\!\cdots\!00}a^{18}+\frac{41\!\cdots\!69}{67\!\cdots\!00}a^{16}+\frac{16\!\cdots\!77}{67\!\cdots\!00}a^{14}-\frac{22\!\cdots\!33}{67\!\cdots\!00}a^{12}-\frac{43\!\cdots\!99}{11\!\cdots\!60}a^{10}-\frac{10\!\cdots\!13}{33\!\cdots\!00}a^{8}+\frac{53\!\cdots\!49}{13\!\cdots\!00}a^{6}+\frac{66\!\cdots\!59}{31\!\cdots\!00}a^{4}+\frac{52\!\cdots\!59}{15\!\cdots\!40}a^{2}+\frac{25\!\cdots\!45}{60\!\cdots\!92}$, $\frac{1}{40\!\cdots\!00}a^{31}-\frac{36\!\cdots\!27}{16\!\cdots\!00}a^{29}-\frac{1}{20136960}a^{28}+\frac{60\!\cdots\!09}{45\!\cdots\!00}a^{27}+\frac{13}{1006848}a^{26}+\frac{31\!\cdots\!17}{10\!\cdots\!00}a^{25}-\frac{7189}{20136960}a^{24}+\frac{20\!\cdots\!39}{10\!\cdots\!00}a^{23}-\frac{2917}{2517120}a^{22}+\frac{36\!\cdots\!39}{20\!\cdots\!80}a^{21}+\frac{25169}{5034240}a^{20}-\frac{57\!\cdots\!51}{67\!\cdots\!00}a^{19}-\frac{103}{11520}a^{18}-\frac{36\!\cdots\!01}{67\!\cdots\!60}a^{17}+\frac{394061}{10068480}a^{16}+\frac{16\!\cdots\!73}{75\!\cdots\!00}a^{15}+\frac{118717}{5034240}a^{14}-\frac{24\!\cdots\!37}{25\!\cdots\!00}a^{13}-\frac{169711}{10068480}a^{12}-\frac{18\!\cdots\!21}{25\!\cdots\!00}a^{11}+\frac{145411}{2517120}a^{10}-\frac{21\!\cdots\!33}{10\!\cdots\!00}a^{9}+\frac{13987}{139840}a^{8}+\frac{32\!\cdots\!79}{13\!\cdots\!00}a^{7}-\frac{299471}{5034240}a^{6}-\frac{35\!\cdots\!89}{78\!\cdots\!00}a^{5}-\frac{327679}{875520}a^{4}-\frac{10\!\cdots\!83}{40\!\cdots\!00}a^{3}-\frac{96509}{314640}a^{2}-\frac{13\!\cdots\!63}{75\!\cdots\!40}a+\frac{178115}{447488}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$

Class group and class number

$C_{2}\times C_{2}\times C_{24}$, which has order $96$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Relative class number: $96$ (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:   \( -\frac{3634050601013182169041}{13520050999672801963000320} a^{31} + \frac{219267863053442750990557}{67600254998364009815001600} a^{29} - \frac{1867607758621843888325573}{67600254998364009815001600} a^{27} + \frac{1666392427169855097857353}{13520050999672801963000320} a^{25} - \frac{6525754570296588639733601}{16900063749591002453750400} a^{23} + \frac{1052004941562606144328333}{2112507968698875306718800} a^{21} - \frac{27396064151061876512527433}{33800127499182004907500800} a^{19} + \frac{144192330748985395189298317}{33800127499182004907500800} a^{17} - \frac{312844531256508390190109117}{33800127499182004907500800} a^{15} + \frac{203409760803353643798310139}{33800127499182004907500800} a^{13} - \frac{23864650103756411064009761}{938892430532833469652800} a^{11} + \frac{10615996324566417618182077}{3380012749918200490750080} a^{9} - \frac{3671677510991242606251849889}{67600254998364009815001600} a^{7} - \frac{51245094508751156027902189}{1572098953450325809651200} a^{5} - \frac{7286818896614725946105151}{395323128645403566169600} a^{3} - \frac{8257764890987172615047401}{4506683666557600654333440} a \)  (order $8$) Copy content Toggle raw display
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:   $\frac{485450893471}{78\!\cdots\!60}a^{30}-\frac{6160434374317}{78\!\cdots\!60}a^{28}+\frac{10715733541147}{15\!\cdots\!52}a^{26}-\frac{28230209629037}{873077796592640}a^{24}+\frac{23313402331647}{218269449148160}a^{22}-\frac{56126788916219}{327404173722240}a^{20}+\frac{205981969444747}{785770016933376}a^{18}-\frac{43\!\cdots\!13}{39\!\cdots\!80}a^{16}+\frac{10\!\cdots\!99}{39\!\cdots\!80}a^{14}-\frac{12\!\cdots\!43}{436538898296320}a^{12}+\frac{67\!\cdots\!61}{982212521166720}a^{10}-\frac{369906117718733}{85409784449280}a^{8}+\frac{20\!\cdots\!67}{15\!\cdots\!52}a^{6}-\frac{10104727857611}{182737213240320}a^{4}+\frac{22\!\cdots\!23}{78\!\cdots\!60}a^{2}-\frac{288636099896213}{174615559318528}$, $\frac{23\!\cdots\!41}{20\!\cdots\!00}a^{31}-\frac{68439044336341}{56\!\cdots\!00}a^{30}-\frac{97\!\cdots\!97}{67\!\cdots\!00}a^{29}+\frac{32\!\cdots\!21}{22\!\cdots\!00}a^{28}+\frac{27\!\cdots\!97}{22\!\cdots\!00}a^{27}-\frac{277186166046383}{22\!\cdots\!56}a^{26}-\frac{11\!\cdots\!47}{20\!\cdots\!00}a^{25}+\frac{13\!\cdots\!67}{25\!\cdots\!00}a^{24}+\frac{89\!\cdots\!93}{50\!\cdots\!00}a^{23}-\frac{62\!\cdots\!37}{37\!\cdots\!00}a^{22}-\frac{60\!\cdots\!53}{25\!\cdots\!00}a^{21}+\frac{37\!\cdots\!11}{18\!\cdots\!00}a^{20}+\frac{12\!\cdots\!43}{33\!\cdots\!00}a^{19}-\frac{37\!\cdots\!69}{11\!\cdots\!00}a^{18}-\frac{64\!\cdots\!41}{33\!\cdots\!00}a^{17}+\frac{10\!\cdots\!37}{56\!\cdots\!00}a^{16}+\frac{16\!\cdots\!27}{37\!\cdots\!00}a^{15}-\frac{24\!\cdots\!22}{61\!\cdots\!75}a^{14}-\frac{31\!\cdots\!37}{10\!\cdots\!00}a^{13}+\frac{98\!\cdots\!98}{47\!\cdots\!75}a^{12}+\frac{28\!\cdots\!39}{25\!\cdots\!00}a^{11}-\frac{12\!\cdots\!29}{11\!\cdots\!40}a^{10}-\frac{14\!\cdots\!73}{50\!\cdots\!00}a^{9}-\frac{44\!\cdots\!63}{56\!\cdots\!00}a^{8}+\frac{15\!\cdots\!91}{67\!\cdots\!00}a^{7}-\frac{26\!\cdots\!93}{11\!\cdots\!00}a^{6}+\frac{16\!\cdots\!61}{15\!\cdots\!00}a^{5}-\frac{92\!\cdots\!51}{52\!\cdots\!00}a^{4}+\frac{69\!\cdots\!01}{20\!\cdots\!00}a^{3}-\frac{23\!\cdots\!13}{28\!\cdots\!45}a^{2}-\frac{30\!\cdots\!61}{15\!\cdots\!80}a-\frac{17\!\cdots\!31}{10\!\cdots\!36}$, $\frac{85\!\cdots\!93}{22\!\cdots\!00}a^{31}+\frac{485450893471}{78\!\cdots\!60}a^{30}-\frac{22\!\cdots\!83}{48\!\cdots\!00}a^{29}-\frac{6160434374317}{78\!\cdots\!60}a^{28}+\frac{29\!\cdots\!17}{73\!\cdots\!00}a^{27}+\frac{10715733541147}{15\!\cdots\!52}a^{26}-\frac{79\!\cdots\!27}{44\!\cdots\!00}a^{25}-\frac{28230209629037}{873077796592640}a^{24}+\frac{19\!\cdots\!43}{34\!\cdots\!50}a^{23}+\frac{23313402331647}{218269449148160}a^{22}-\frac{81\!\cdots\!63}{11\!\cdots\!00}a^{21}-\frac{56126788916219}{327404173722240}a^{20}+\frac{72\!\cdots\!63}{61\!\cdots\!00}a^{19}+\frac{205981969444747}{785770016933376}a^{18}-\frac{45\!\cdots\!23}{73\!\cdots\!00}a^{17}-\frac{43\!\cdots\!13}{39\!\cdots\!80}a^{16}+\frac{49\!\cdots\!33}{36\!\cdots\!40}a^{15}+\frac{10\!\cdots\!99}{39\!\cdots\!80}a^{14}-\frac{20\!\cdots\!49}{22\!\cdots\!00}a^{13}-\frac{12\!\cdots\!43}{436538898296320}a^{12}+\frac{20\!\cdots\!41}{55\!\cdots\!00}a^{11}+\frac{67\!\cdots\!61}{982212521166720}a^{10}-\frac{33\!\cdots\!59}{55\!\cdots\!00}a^{9}-\frac{369906117718733}{85409784449280}a^{8}+\frac{11\!\cdots\!53}{14\!\cdots\!60}a^{7}+\frac{20\!\cdots\!67}{15\!\cdots\!52}a^{6}+\frac{19\!\cdots\!99}{45\!\cdots\!96}a^{5}-\frac{10104727857611}{182737213240320}a^{4}+\frac{48\!\cdots\!17}{22\!\cdots\!00}a^{3}+\frac{22\!\cdots\!23}{78\!\cdots\!60}a^{2}-\frac{19\!\cdots\!67}{97\!\cdots\!40}a-\frac{114020540577685}{174615559318528}$, $\frac{54\!\cdots\!13}{20\!\cdots\!00}a^{31}+\frac{17\!\cdots\!79}{55\!\cdots\!00}a^{30}-\frac{48\!\cdots\!91}{15\!\cdots\!00}a^{29}-\frac{79\!\cdots\!39}{22\!\cdots\!00}a^{28}+\frac{18\!\cdots\!77}{67\!\cdots\!00}a^{27}+\frac{82\!\cdots\!07}{27\!\cdots\!40}a^{26}-\frac{50\!\cdots\!17}{40\!\cdots\!00}a^{25}-\frac{30\!\cdots\!23}{24\!\cdots\!00}a^{24}+\frac{98\!\cdots\!49}{25\!\cdots\!00}a^{23}+\frac{10\!\cdots\!99}{30\!\cdots\!00}a^{22}-\frac{51\!\cdots\!53}{10\!\cdots\!00}a^{21}-\frac{46\!\cdots\!99}{18\!\cdots\!00}a^{20}+\frac{59\!\cdots\!81}{70\!\cdots\!00}a^{19}+\frac{23\!\cdots\!33}{55\!\cdots\!00}a^{18}-\frac{29\!\cdots\!73}{67\!\cdots\!00}a^{17}-\frac{45\!\cdots\!01}{11\!\cdots\!00}a^{16}+\frac{15\!\cdots\!09}{16\!\cdots\!40}a^{15}+\frac{40\!\cdots\!81}{55\!\cdots\!00}a^{14}-\frac{12\!\cdots\!99}{20\!\cdots\!00}a^{13}+\frac{18\!\cdots\!23}{12\!\cdots\!00}a^{12}+\frac{13\!\cdots\!31}{50\!\cdots\!00}a^{11}+\frac{23\!\cdots\!73}{11\!\cdots\!56}a^{10}-\frac{19\!\cdots\!59}{50\!\cdots\!00}a^{9}+\frac{43\!\cdots\!97}{21\!\cdots\!25}a^{8}+\frac{76\!\cdots\!13}{13\!\cdots\!20}a^{7}+\frac{13\!\cdots\!63}{27\!\cdots\!00}a^{6}+\frac{67\!\cdots\!13}{20\!\cdots\!60}a^{5}+\frac{43\!\cdots\!79}{51\!\cdots\!00}a^{4}+\frac{44\!\cdots\!77}{20\!\cdots\!00}a^{3}+\frac{22\!\cdots\!73}{11\!\cdots\!60}a^{2}+\frac{22\!\cdots\!83}{90\!\cdots\!80}a+\frac{15\!\cdots\!61}{98\!\cdots\!72}$, $\frac{11\!\cdots\!69}{17\!\cdots\!00}a^{31}-\frac{36\!\cdots\!47}{24\!\cdots\!00}a^{30}-\frac{15\!\cdots\!49}{33\!\cdots\!00}a^{29}+\frac{27\!\cdots\!43}{15\!\cdots\!00}a^{28}+\frac{22\!\cdots\!47}{90\!\cdots\!80}a^{27}-\frac{22\!\cdots\!17}{14\!\cdots\!80}a^{26}+\frac{41\!\cdots\!37}{50\!\cdots\!00}a^{25}+\frac{18\!\cdots\!47}{26\!\cdots\!00}a^{24}-\frac{91\!\cdots\!91}{10\!\cdots\!00}a^{23}-\frac{39\!\cdots\!47}{18\!\cdots\!00}a^{22}+\frac{50\!\cdots\!01}{10\!\cdots\!00}a^{21}+\frac{49\!\cdots\!37}{18\!\cdots\!00}a^{20}-\frac{23\!\cdots\!47}{29\!\cdots\!00}a^{19}-\frac{51\!\cdots\!37}{12\!\cdots\!00}a^{18}+\frac{14\!\cdots\!43}{33\!\cdots\!00}a^{17}+\frac{42\!\cdots\!03}{18\!\cdots\!00}a^{16}-\frac{29\!\cdots\!53}{75\!\cdots\!00}a^{15}-\frac{62\!\cdots\!69}{12\!\cdots\!00}a^{14}+\frac{73\!\cdots\!71}{50\!\cdots\!00}a^{13}+\frac{72\!\cdots\!89}{24\!\cdots\!00}a^{12}-\frac{50\!\cdots\!61}{50\!\cdots\!20}a^{11}-\frac{41\!\cdots\!99}{30\!\cdots\!60}a^{10}+\frac{37\!\cdots\!09}{10\!\cdots\!00}a^{9}+\frac{13\!\cdots\!93}{18\!\cdots\!00}a^{8}-\frac{16\!\cdots\!79}{13\!\cdots\!00}a^{7}-\frac{20\!\cdots\!89}{73\!\cdots\!00}a^{6}+\frac{16\!\cdots\!13}{21\!\cdots\!00}a^{5}-\frac{27\!\cdots\!97}{14\!\cdots\!00}a^{4}-\frac{58\!\cdots\!73}{81\!\cdots\!20}a^{3}-\frac{93\!\cdots\!09}{14\!\cdots\!80}a^{2}-\frac{54\!\cdots\!49}{37\!\cdots\!12}a-\frac{21\!\cdots\!09}{24\!\cdots\!68}$, $\frac{10\!\cdots\!69}{33\!\cdots\!80}a^{31}+\frac{30\!\cdots\!47}{22\!\cdots\!00}a^{30}-\frac{10\!\cdots\!07}{27\!\cdots\!00}a^{29}-\frac{20\!\cdots\!53}{13\!\cdots\!00}a^{28}+\frac{62\!\cdots\!37}{18\!\cdots\!00}a^{27}+\frac{33\!\cdots\!79}{27\!\cdots\!64}a^{26}-\frac{11\!\cdots\!09}{72\!\cdots\!80}a^{25}-\frac{65\!\cdots\!09}{13\!\cdots\!00}a^{24}+\frac{21\!\cdots\!71}{41\!\cdots\!00}a^{23}+\frac{11\!\cdots\!51}{84\!\cdots\!00}a^{22}-\frac{35\!\cdots\!39}{41\!\cdots\!00}a^{21}-\frac{21\!\cdots\!49}{33\!\cdots\!00}a^{20}+\frac{38\!\cdots\!21}{27\!\cdots\!00}a^{19}+\frac{12\!\cdots\!29}{84\!\cdots\!00}a^{18}-\frac{38\!\cdots\!11}{69\!\cdots\!00}a^{17}-\frac{11\!\cdots\!27}{67\!\cdots\!00}a^{16}+\frac{40\!\cdots\!21}{31\!\cdots\!00}a^{15}+\frac{11\!\cdots\!97}{44\!\cdots\!00}a^{14}-\frac{56\!\cdots\!67}{41\!\cdots\!00}a^{13}+\frac{11\!\cdots\!29}{67\!\cdots\!00}a^{12}+\frac{94\!\cdots\!73}{26\!\cdots\!00}a^{11}+\frac{30\!\cdots\!91}{33\!\cdots\!80}a^{10}-\frac{16\!\cdots\!97}{83\!\cdots\!20}a^{9}+\frac{57\!\cdots\!39}{56\!\cdots\!00}a^{8}+\frac{39\!\cdots\!23}{55\!\cdots\!00}a^{7}+\frac{13\!\cdots\!99}{67\!\cdots\!00}a^{6}+\frac{67\!\cdots\!49}{64\!\cdots\!00}a^{5}+\frac{13\!\cdots\!33}{31\!\cdots\!00}a^{4}+\frac{31\!\cdots\!21}{16\!\cdots\!00}a^{3}+\frac{16\!\cdots\!89}{13\!\cdots\!20}a^{2}+\frac{84\!\cdots\!41}{18\!\cdots\!60}a+\frac{18\!\cdots\!39}{60\!\cdots\!92}$, $\frac{10\!\cdots\!11}{16\!\cdots\!00}a^{31}-\frac{64\!\cdots\!89}{24\!\cdots\!00}a^{30}-\frac{52\!\cdots\!31}{67\!\cdots\!00}a^{29}+\frac{79\!\cdots\!21}{24\!\cdots\!00}a^{28}+\frac{22\!\cdots\!73}{33\!\cdots\!80}a^{27}-\frac{13\!\cdots\!77}{49\!\cdots\!60}a^{26}-\frac{65\!\cdots\!01}{22\!\cdots\!00}a^{25}+\frac{31\!\cdots\!13}{24\!\cdots\!00}a^{24}+\frac{25\!\cdots\!13}{28\!\cdots\!00}a^{23}-\frac{13\!\cdots\!47}{32\!\cdots\!00}a^{22}-\frac{63\!\cdots\!31}{56\!\cdots\!00}a^{21}+\frac{89\!\cdots\!97}{15\!\cdots\!00}a^{20}+\frac{31\!\cdots\!57}{16\!\cdots\!00}a^{19}-\frac{59\!\cdots\!31}{64\!\cdots\!00}a^{18}-\frac{14\!\cdots\!03}{14\!\cdots\!00}a^{17}+\frac{53\!\cdots\!69}{12\!\cdots\!00}a^{16}+\frac{36\!\cdots\!19}{16\!\cdots\!00}a^{15}-\frac{12\!\cdots\!73}{12\!\cdots\!00}a^{14}-\frac{47\!\cdots\!93}{37\!\cdots\!00}a^{13}+\frac{10\!\cdots\!67}{12\!\cdots\!00}a^{12}+\frac{10\!\cdots\!23}{16\!\cdots\!40}a^{11}-\frac{16\!\cdots\!97}{61\!\cdots\!20}a^{10}-\frac{15\!\cdots\!43}{42\!\cdots\!00}a^{9}+\frac{59\!\cdots\!17}{61\!\cdots\!00}a^{8}+\frac{58\!\cdots\!13}{44\!\cdots\!00}a^{7}-\frac{13\!\cdots\!21}{24\!\cdots\!00}a^{6}+\frac{13\!\cdots\!31}{15\!\cdots\!00}a^{5}-\frac{10\!\cdots\!61}{57\!\cdots\!00}a^{4}+\frac{87\!\cdots\!93}{16\!\cdots\!40}a^{3}-\frac{52\!\cdots\!21}{49\!\cdots\!60}a^{2}+\frac{17\!\cdots\!19}{90\!\cdots\!88}a+\frac{18\!\cdots\!73}{98\!\cdots\!72}$, $\frac{21\!\cdots\!77}{40\!\cdots\!00}a^{31}-\frac{93\!\cdots\!21}{36\!\cdots\!00}a^{30}-\frac{87\!\cdots\!17}{13\!\cdots\!00}a^{29}-\frac{20\!\cdots\!91}{36\!\cdots\!00}a^{28}+\frac{24\!\cdots\!93}{45\!\cdots\!00}a^{27}+\frac{58\!\cdots\!87}{73\!\cdots\!40}a^{26}-\frac{10\!\cdots\!99}{40\!\cdots\!00}a^{25}-\frac{96\!\cdots\!01}{12\!\cdots\!00}a^{24}+\frac{35\!\cdots\!91}{44\!\cdots\!00}a^{23}+\frac{11\!\cdots\!61}{30\!\cdots\!00}a^{22}-\frac{11\!\cdots\!77}{10\!\cdots\!40}a^{21}-\frac{94\!\cdots\!19}{76\!\cdots\!00}a^{20}+\frac{12\!\cdots\!03}{67\!\cdots\!00}a^{19}+\frac{29\!\cdots\!19}{18\!\cdots\!00}a^{18}-\frac{11\!\cdots\!13}{13\!\cdots\!20}a^{17}-\frac{41\!\cdots\!59}{18\!\cdots\!00}a^{16}+\frac{44\!\cdots\!13}{22\!\cdots\!00}a^{15}+\frac{24\!\cdots\!03}{18\!\cdots\!00}a^{14}-\frac{32\!\cdots\!97}{20\!\cdots\!00}a^{13}-\frac{18\!\cdots\!99}{61\!\cdots\!00}a^{12}+\frac{27\!\cdots\!51}{50\!\cdots\!00}a^{11}+\frac{16\!\cdots\!01}{92\!\cdots\!80}a^{10}-\frac{17\!\cdots\!41}{10\!\cdots\!00}a^{9}-\frac{74\!\cdots\!27}{92\!\cdots\!00}a^{8}+\frac{15\!\cdots\!43}{13\!\cdots\!00}a^{7}+\frac{26\!\cdots\!71}{36\!\cdots\!00}a^{6}+\frac{12\!\cdots\!13}{31\!\cdots\!00}a^{5}-\frac{15\!\cdots\!29}{85\!\cdots\!00}a^{4}+\frac{12\!\cdots\!89}{40\!\cdots\!00}a^{3}-\frac{69\!\cdots\!81}{73\!\cdots\!40}a^{2}-\frac{11\!\cdots\!69}{30\!\cdots\!60}a-\frac{17\!\cdots\!65}{49\!\cdots\!36}$, $\frac{17\!\cdots\!59}{36\!\cdots\!00}a^{31}-\frac{33\!\cdots\!67}{59\!\cdots\!00}a^{29}+\frac{27\!\cdots\!43}{56\!\cdots\!00}a^{27}-\frac{73\!\cdots\!21}{33\!\cdots\!00}a^{25}+\frac{28\!\cdots\!51}{42\!\cdots\!00}a^{23}-\frac{29\!\cdots\!09}{33\!\cdots\!08}a^{21}+\frac{57\!\cdots\!89}{40\!\cdots\!00}a^{19}-\frac{84\!\cdots\!99}{11\!\cdots\!60}a^{17}+\frac{11\!\cdots\!17}{70\!\cdots\!00}a^{15}-\frac{17\!\cdots\!03}{16\!\cdots\!00}a^{13}+\frac{47\!\cdots\!51}{10\!\cdots\!00}a^{11}-\frac{10\!\cdots\!73}{22\!\cdots\!00}a^{9}+\frac{13\!\cdots\!99}{14\!\cdots\!00}a^{7}+\frac{22\!\cdots\!33}{37\!\cdots\!00}a^{5}+\frac{56\!\cdots\!03}{16\!\cdots\!00}a^{3}+\frac{75\!\cdots\!81}{22\!\cdots\!20}a$, $\frac{20\!\cdots\!29}{22\!\cdots\!00}a^{31}+\frac{47\!\cdots\!27}{13\!\cdots\!00}a^{30}-\frac{39\!\cdots\!33}{76\!\cdots\!00}a^{29}-\frac{60\!\cdots\!23}{13\!\cdots\!00}a^{28}+\frac{58\!\cdots\!93}{25\!\cdots\!00}a^{27}+\frac{21\!\cdots\!71}{54\!\cdots\!28}a^{26}+\frac{40\!\cdots\!57}{22\!\cdots\!00}a^{25}-\frac{27\!\cdots\!11}{15\!\cdots\!00}a^{24}-\frac{82\!\cdots\!23}{57\!\cdots\!00}a^{23}+\frac{68\!\cdots\!93}{11\!\cdots\!00}a^{22}+\frac{20\!\cdots\!53}{28\!\cdots\!00}a^{21}-\frac{33\!\cdots\!99}{35\!\cdots\!00}a^{20}-\frac{68\!\cdots\!49}{67\!\cdots\!00}a^{19}+\frac{89\!\cdots\!67}{67\!\cdots\!00}a^{18}+\frac{19\!\cdots\!91}{38\!\cdots\!00}a^{17}-\frac{40\!\cdots\!67}{67\!\cdots\!00}a^{16}-\frac{22\!\cdots\!33}{38\!\cdots\!00}a^{15}+\frac{10\!\cdots\!79}{67\!\cdots\!00}a^{14}+\frac{22\!\cdots\!27}{11\!\cdots\!00}a^{13}-\frac{32\!\cdots\!87}{22\!\cdots\!00}a^{12}-\frac{13\!\cdots\!23}{12\!\cdots\!00}a^{11}+\frac{11\!\cdots\!83}{33\!\cdots\!80}a^{10}+\frac{29\!\cdots\!03}{57\!\cdots\!00}a^{9}-\frac{67\!\cdots\!71}{33\!\cdots\!00}a^{8}-\frac{44\!\cdots\!41}{76\!\cdots\!00}a^{7}+\frac{48\!\cdots\!77}{71\!\cdots\!00}a^{6}+\frac{21\!\cdots\!49}{17\!\cdots\!00}a^{5}+\frac{49\!\cdots\!43}{31\!\cdots\!00}a^{4}+\frac{10\!\cdots\!69}{22\!\cdots\!00}a^{3}-\frac{37\!\cdots\!21}{27\!\cdots\!40}a^{2}+\frac{19\!\cdots\!19}{15\!\cdots\!20}a-\frac{20\!\cdots\!83}{60\!\cdots\!92}$, $\frac{86\!\cdots\!31}{20\!\cdots\!80}a^{31}+\frac{60\!\cdots\!67}{22\!\cdots\!00}a^{30}-\frac{19\!\cdots\!39}{37\!\cdots\!00}a^{29}-\frac{74\!\cdots\!13}{22\!\cdots\!00}a^{28}+\frac{14\!\cdots\!39}{33\!\cdots\!00}a^{27}+\frac{12\!\cdots\!47}{45\!\cdots\!60}a^{26}-\frac{39\!\cdots\!39}{20\!\cdots\!80}a^{25}-\frac{96\!\cdots\!03}{76\!\cdots\!00}a^{24}+\frac{35\!\cdots\!27}{58\!\cdots\!00}a^{23}+\frac{76\!\cdots\!33}{19\!\cdots\!00}a^{22}-\frac{10\!\cdots\!23}{12\!\cdots\!00}a^{21}-\frac{53\!\cdots\!99}{95\!\cdots\!00}a^{20}+\frac{71\!\cdots\!83}{56\!\cdots\!00}a^{19}+\frac{98\!\cdots\!47}{11\!\cdots\!00}a^{18}-\frac{11\!\cdots\!31}{16\!\cdots\!00}a^{17}-\frac{51\!\cdots\!57}{11\!\cdots\!00}a^{16}+\frac{13\!\cdots\!09}{88\!\cdots\!00}a^{15}+\frac{11\!\cdots\!99}{11\!\cdots\!00}a^{14}-\frac{49\!\cdots\!21}{50\!\cdots\!00}a^{13}-\frac{28\!\cdots\!97}{38\!\cdots\!00}a^{12}+\frac{24\!\cdots\!03}{63\!\cdots\!00}a^{11}+\frac{14\!\cdots\!69}{57\!\cdots\!20}a^{10}-\frac{30\!\cdots\!33}{50\!\cdots\!20}a^{9}-\frac{84\!\cdots\!11}{57\!\cdots\!00}a^{8}+\frac{28\!\cdots\!57}{33\!\cdots\!00}a^{7}+\frac{11\!\cdots\!43}{22\!\cdots\!00}a^{6}+\frac{12\!\cdots\!49}{26\!\cdots\!00}a^{5}+\frac{45\!\cdots\!13}{53\!\cdots\!00}a^{4}+\frac{20\!\cdots\!69}{10\!\cdots\!00}a^{3}+\frac{11\!\cdots\!99}{91\!\cdots\!92}a^{2}+\frac{10\!\cdots\!71}{75\!\cdots\!40}a-\frac{43\!\cdots\!53}{10\!\cdots\!88}$, $\frac{38\!\cdots\!47}{16\!\cdots\!84}a^{31}+\frac{46\!\cdots\!31}{45\!\cdots\!00}a^{30}-\frac{26\!\cdots\!67}{93\!\cdots\!00}a^{29}-\frac{93\!\cdots\!09}{75\!\cdots\!00}a^{28}+\frac{17\!\cdots\!23}{71\!\cdots\!00}a^{27}+\frac{96\!\cdots\!99}{90\!\cdots\!80}a^{26}-\frac{21\!\cdots\!13}{20\!\cdots\!80}a^{25}-\frac{10\!\cdots\!41}{22\!\cdots\!00}a^{24}+\frac{14\!\cdots\!59}{44\!\cdots\!00}a^{23}+\frac{17\!\cdots\!17}{11\!\cdots\!00}a^{22}-\frac{42\!\cdots\!43}{10\!\cdots\!00}a^{21}-\frac{75\!\cdots\!99}{37\!\cdots\!00}a^{20}+\frac{15\!\cdots\!89}{22\!\cdots\!00}a^{19}+\frac{24\!\cdots\!27}{75\!\cdots\!00}a^{18}-\frac{12\!\cdots\!99}{33\!\cdots\!00}a^{17}-\frac{92\!\cdots\!89}{56\!\cdots\!00}a^{16}+\frac{53\!\cdots\!43}{67\!\cdots\!00}a^{15}+\frac{82\!\cdots\!37}{22\!\cdots\!00}a^{14}-\frac{11\!\cdots\!61}{25\!\cdots\!00}a^{13}-\frac{28\!\cdots\!29}{11\!\cdots\!00}a^{12}+\frac{55\!\cdots\!99}{25\!\cdots\!00}a^{11}+\frac{27\!\cdots\!49}{28\!\cdots\!40}a^{10}-\frac{24\!\cdots\!19}{20\!\cdots\!80}a^{9}-\frac{12\!\cdots\!17}{59\!\cdots\!00}a^{8}+\frac{63\!\cdots\!41}{13\!\cdots\!00}a^{7}+\frac{91\!\cdots\!59}{45\!\cdots\!00}a^{6}+\frac{27\!\cdots\!31}{87\!\cdots\!00}a^{5}+\frac{56\!\cdots\!17}{52\!\cdots\!00}a^{4}+\frac{67\!\cdots\!27}{40\!\cdots\!00}a^{3}+\frac{42\!\cdots\!03}{90\!\cdots\!80}a^{2}+\frac{85\!\cdots\!41}{22\!\cdots\!20}a-\frac{11\!\cdots\!15}{30\!\cdots\!96}$, $\frac{82\!\cdots\!91}{40\!\cdots\!00}a^{31}-\frac{16\!\cdots\!61}{45\!\cdots\!00}a^{30}-\frac{12\!\cdots\!47}{47\!\cdots\!20}a^{29}+\frac{51\!\cdots\!07}{13\!\cdots\!00}a^{28}+\frac{30\!\cdots\!73}{13\!\cdots\!00}a^{27}-\frac{80\!\cdots\!59}{27\!\cdots\!40}a^{26}-\frac{42\!\cdots\!37}{40\!\cdots\!00}a^{25}+\frac{13\!\cdots\!91}{13\!\cdots\!00}a^{24}+\frac{70\!\cdots\!27}{20\!\cdots\!80}a^{23}-\frac{68\!\cdots\!71}{33\!\cdots\!00}a^{22}-\frac{28\!\cdots\!47}{50\!\cdots\!00}a^{21}-\frac{16\!\cdots\!23}{42\!\cdots\!00}a^{20}+\frac{12\!\cdots\!37}{15\!\cdots\!80}a^{19}+\frac{44\!\cdots\!77}{67\!\cdots\!00}a^{18}-\frac{24\!\cdots\!59}{67\!\cdots\!00}a^{17}+\frac{20\!\cdots\!43}{67\!\cdots\!00}a^{16}+\frac{60\!\cdots\!61}{67\!\cdots\!00}a^{15}-\frac{10\!\cdots\!51}{67\!\cdots\!00}a^{14}-\frac{36\!\cdots\!51}{40\!\cdots\!60}a^{13}-\frac{13\!\cdots\!31}{67\!\cdots\!00}a^{12}+\frac{11\!\cdots\!57}{50\!\cdots\!00}a^{11}-\frac{30\!\cdots\!67}{67\!\cdots\!16}a^{10}-\frac{13\!\cdots\!63}{10\!\cdots\!00}a^{9}-\frac{24\!\cdots\!09}{37\!\cdots\!00}a^{8}+\frac{58\!\cdots\!57}{13\!\cdots\!00}a^{7}-\frac{37\!\cdots\!67}{13\!\cdots\!00}a^{6}+\frac{14\!\cdots\!89}{10\!\cdots\!00}a^{5}-\frac{55\!\cdots\!07}{31\!\cdots\!00}a^{4}+\frac{85\!\cdots\!43}{40\!\cdots\!00}a^{3}-\frac{42\!\cdots\!87}{27\!\cdots\!40}a^{2}-\frac{13\!\cdots\!89}{90\!\cdots\!80}a-\frac{36\!\cdots\!47}{31\!\cdots\!68}$, $\frac{45\!\cdots\!83}{56\!\cdots\!00}a^{31}-\frac{51\!\cdots\!65}{27\!\cdots\!64}a^{30}-\frac{32\!\cdots\!23}{33\!\cdots\!00}a^{29}+\frac{45\!\cdots\!77}{18\!\cdots\!60}a^{28}+\frac{68\!\cdots\!01}{84\!\cdots\!00}a^{27}-\frac{96\!\cdots\!39}{45\!\cdots\!40}a^{26}-\frac{11\!\cdots\!81}{33\!\cdots\!00}a^{25}+\frac{17\!\cdots\!13}{16\!\cdots\!40}a^{24}+\frac{22\!\cdots\!01}{21\!\cdots\!00}a^{23}-\frac{11\!\cdots\!07}{33\!\cdots\!80}a^{22}-\frac{10\!\cdots\!63}{84\!\cdots\!00}a^{21}+\frac{19\!\cdots\!01}{33\!\cdots\!80}a^{20}+\frac{11\!\cdots\!43}{55\!\cdots\!00}a^{19}-\frac{20\!\cdots\!41}{22\!\cdots\!20}a^{18}-\frac{21\!\cdots\!09}{16\!\cdots\!00}a^{17}+\frac{39\!\cdots\!33}{11\!\cdots\!60}a^{16}+\frac{21\!\cdots\!11}{84\!\cdots\!00}a^{15}-\frac{20\!\cdots\!81}{22\!\cdots\!20}a^{14}-\frac{19\!\cdots\!31}{16\!\cdots\!00}a^{13}+\frac{17\!\cdots\!09}{16\!\cdots\!40}a^{12}+\frac{33\!\cdots\!41}{45\!\cdots\!00}a^{11}-\frac{46\!\cdots\!31}{21\!\cdots\!80}a^{10}+\frac{59\!\cdots\!21}{88\!\cdots\!50}a^{9}+\frac{58\!\cdots\!29}{33\!\cdots\!80}a^{8}+\frac{27\!\cdots\!27}{16\!\cdots\!00}a^{7}-\frac{18\!\cdots\!67}{45\!\cdots\!40}a^{6}+\frac{10\!\cdots\!79}{78\!\cdots\!00}a^{5}+\frac{76\!\cdots\!67}{87\!\cdots\!40}a^{4}+\frac{17\!\cdots\!43}{21\!\cdots\!00}a^{3}+\frac{38\!\cdots\!67}{13\!\cdots\!20}a^{2}+\frac{71\!\cdots\!91}{75\!\cdots\!40}a+\frac{98\!\cdots\!87}{18\!\cdots\!56}$, $\frac{18\!\cdots\!29}{20\!\cdots\!00}a^{31}+\frac{40\!\cdots\!99}{11\!\cdots\!00}a^{30}-\frac{74\!\cdots\!43}{67\!\cdots\!00}a^{29}-\frac{17\!\cdots\!51}{39\!\cdots\!00}a^{28}+\frac{63\!\cdots\!99}{67\!\cdots\!00}a^{27}+\frac{44\!\cdots\!33}{11\!\cdots\!80}a^{26}-\frac{83\!\cdots\!33}{20\!\cdots\!00}a^{25}-\frac{37\!\cdots\!51}{22\!\cdots\!00}a^{24}+\frac{63\!\cdots\!17}{50\!\cdots\!00}a^{23}+\frac{74\!\cdots\!99}{14\!\cdots\!00}a^{22}-\frac{18\!\cdots\!71}{12\!\cdots\!00}a^{21}-\frac{39\!\cdots\!11}{56\!\cdots\!00}a^{20}+\frac{84\!\cdots\!87}{33\!\cdots\!00}a^{19}+\frac{54\!\cdots\!97}{46\!\cdots\!00}a^{18}-\frac{16\!\cdots\!53}{11\!\cdots\!00}a^{17}-\frac{22\!\cdots\!31}{37\!\cdots\!00}a^{16}+\frac{99\!\cdots\!87}{33\!\cdots\!00}a^{15}+\frac{35\!\cdots\!09}{28\!\cdots\!00}a^{14}-\frac{15\!\cdots\!23}{10\!\cdots\!00}a^{13}-\frac{33\!\cdots\!43}{37\!\cdots\!00}a^{12}+\frac{21\!\cdots\!71}{25\!\cdots\!00}a^{11}+\frac{20\!\cdots\!59}{56\!\cdots\!80}a^{10}+\frac{19\!\cdots\!23}{50\!\cdots\!00}a^{9}-\frac{16\!\cdots\!67}{28\!\cdots\!00}a^{8}+\frac{14\!\cdots\!11}{75\!\cdots\!00}a^{7}+\frac{88\!\cdots\!81}{11\!\cdots\!00}a^{6}+\frac{22\!\cdots\!19}{15\!\cdots\!00}a^{5}+\frac{21\!\cdots\!67}{52\!\cdots\!00}a^{4}+\frac{17\!\cdots\!29}{20\!\cdots\!00}a^{3}+\frac{25\!\cdots\!19}{75\!\cdots\!40}a^{2}+\frac{11\!\cdots\!21}{15\!\cdots\!80}a+\frac{86\!\cdots\!23}{30\!\cdots\!96}$ Copy content Toggle raw display (assuming GRH)
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:  \( 2307780676559.752 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 2307780676559.752 \cdot 96}{8\cdot\sqrt{38559672304350233766313499377700700160000000000000000}}\cr\approx \mathstrut & 0.832122001804932 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 12*x^30 + 102*x^28 - 452*x^26 + 1409*x^24 - 1772*x^22 + 2946*x^20 - 15768*x^18 + 33540*x^16 - 20668*x^14 + 94478*x^12 - 7132*x^10 + 205785*x^8 + 131700*x^6 + 83794*x^4 + 9720*x^2 + 2025)
 
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 - 12*x^30 + 102*x^28 - 452*x^26 + 1409*x^24 - 1772*x^22 + 2946*x^20 - 15768*x^18 + 33540*x^16 - 20668*x^14 + 94478*x^12 - 7132*x^10 + 205785*x^8 + 131700*x^6 + 83794*x^4 + 9720*x^2 + 2025, 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 - 12*x^30 + 102*x^28 - 452*x^26 + 1409*x^24 - 1772*x^22 + 2946*x^20 - 15768*x^18 + 33540*x^16 - 20668*x^14 + 94478*x^12 - 7132*x^10 + 205785*x^8 + 131700*x^6 + 83794*x^4 + 9720*x^2 + 2025);
 
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 - 12*x^30 + 102*x^28 - 452*x^26 + 1409*x^24 - 1772*x^22 + 2946*x^20 - 15768*x^18 + 33540*x^16 - 20668*x^14 + 94478*x^12 - 7132*x^10 + 205785*x^8 + 131700*x^6 + 83794*x^4 + 9720*x^2 + 2025);
 
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.49600.1, 4.4.12400.1, \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{-2}, \sqrt{5})\), 4.4.49600.1, 4.0.12400.1, 8.0.39362560000.2, 8.8.39362560000.1, 8.0.40960000.1, 8.0.39362560000.9, 8.0.39362560000.4, 8.0.2460160000.5, 8.0.39362560000.1, 8.8.3503267840000.1, 8.8.218954240000.1, 8.0.13684640000.1, 8.0.3503267840000.1, 16.0.24790578076057600000000.1, 16.16.196366168940452249600000000.1, 16.0.12272885558778265600000000.3, 16.0.12272885558778265600000000.2, 16.0.196366168940452249600000000.1, 16.0.47940959213977600000000.1, 16.0.196366168940452249600000000.2

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: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }^{8}$ R ${\href{/padicField/7.8.0.1}{8} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.8.0.1}{8} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{16}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{16}$ R ${\href{/padicField/37.8.0.1}{8} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $16$$8$$2$$36$
Deg $16$$8$$2$$36$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(31\) Copy content Toggle raw display 31.4.0.1$x^{4} + 3 x^{2} + 16 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.0.1$x^{4} + 3 x^{2} + 16 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} + 3 x^{2} + 16 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.0.1$x^{4} + 3 x^{2} + 16 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
\(89\) Copy content Toggle raw display $\Q_{89}$$x + 86$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 86$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 86$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 86$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 86$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 86$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 86$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 86$$1$$1$$0$Trivial$[\ ]$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{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}$
89.4.0.1$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}$