Properties

Label 32.0.480...776.1
Degree $32$
Signature $[0, 16]$
Discriminant $4.810\times 10^{50}$
Root discriminant \(38.35\)
Ramified primes $2,3,13,17$
Class number $16$ (GRH)
Class group [2, 2, 4] (GRH)
Galois group $C_2^4:D_4$ (as 32T1369)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 20*x^30 + 236*x^28 + 1948*x^26 + 12276*x^24 + 58148*x^22 + 212422*x^20 + 567784*x^18 + 1013620*x^16 + 772588*x^14 + 788932*x^12 + 1345276*x^10 + 364625*x^8 - 228564*x^6 + 145744*x^4 - 25856*x^2 + 4096)
 
gp: K = bnfinit(y^32 + 20*y^30 + 236*y^28 + 1948*y^26 + 12276*y^24 + 58148*y^22 + 212422*y^20 + 567784*y^18 + 1013620*y^16 + 772588*y^14 + 788932*y^12 + 1345276*y^10 + 364625*y^8 - 228564*y^6 + 145744*y^4 - 25856*y^2 + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 20*x^30 + 236*x^28 + 1948*x^26 + 12276*x^24 + 58148*x^22 + 212422*x^20 + 567784*x^18 + 1013620*x^16 + 772588*x^14 + 788932*x^12 + 1345276*x^10 + 364625*x^8 - 228564*x^6 + 145744*x^4 - 25856*x^2 + 4096);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 20*x^30 + 236*x^28 + 1948*x^26 + 12276*x^24 + 58148*x^22 + 212422*x^20 + 567784*x^18 + 1013620*x^16 + 772588*x^14 + 788932*x^12 + 1345276*x^10 + 364625*x^8 - 228564*x^6 + 145744*x^4 - 25856*x^2 + 4096)
 

\( x^{32} + 20 x^{30} + 236 x^{28} + 1948 x^{26} + 12276 x^{24} + 58148 x^{22} + 212422 x^{20} + 567784 x^{18} + 1013620 x^{16} + 772588 x^{14} + 788932 x^{12} + 1345276 x^{10} + \cdots + 4096 \) 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:   \(480960519379403029833827263813614000556122650443776\) \(\medspace = 2^{48}\cdot 3^{16}\cdot 13^{8}\cdot 17^{16}\) 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:  \(38.35\)
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:  $2^{2}3^{1/2}13^{1/2}17^{1/2}\approx 102.99514551666987$
Ramified primes:   \(2\), \(3\), \(13\), \(17\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $16$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{5}$, $\frac{1}{8}a^{18}-\frac{1}{8}a^{15}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{8}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{3}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{19}-\frac{1}{8}a^{16}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{8}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{3}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{136}a^{20}+\frac{3}{68}a^{18}-\frac{1}{8}a^{17}+\frac{5}{68}a^{16}+\frac{3}{34}a^{14}-\frac{1}{4}a^{13}+\frac{2}{17}a^{12}-\frac{1}{4}a^{11}+\frac{11}{68}a^{10}+\frac{9}{136}a^{8}-\frac{1}{4}a^{7}-\frac{21}{68}a^{6}+\frac{3}{8}a^{5}+\frac{3}{17}a^{4}-\frac{1}{2}a^{3}-\frac{27}{68}a^{2}-\frac{1}{2}a-\frac{8}{17}$, $\frac{1}{136}a^{21}+\frac{3}{68}a^{19}+\frac{5}{68}a^{17}-\frac{5}{136}a^{15}+\frac{2}{17}a^{13}-\frac{3}{34}a^{11}-\frac{25}{136}a^{9}+\frac{13}{68}a^{7}+\frac{29}{68}a^{5}-\frac{3}{136}a^{3}-\frac{8}{17}a$, $\frac{1}{136}a^{22}+\frac{1}{17}a^{18}+\frac{3}{136}a^{16}+\frac{3}{34}a^{14}+\frac{7}{34}a^{12}-\frac{21}{136}a^{10}-\frac{7}{34}a^{8}+\frac{1}{34}a^{6}+\frac{57}{136}a^{4}+\frac{7}{17}a^{2}-\frac{3}{17}$, $\frac{1}{272}a^{23}-\frac{1}{272}a^{22}+\frac{1}{34}a^{19}-\frac{1}{34}a^{18}+\frac{3}{272}a^{17}-\frac{3}{272}a^{16}+\frac{3}{68}a^{15}-\frac{3}{68}a^{14}+\frac{7}{68}a^{13}-\frac{7}{68}a^{12}+\frac{47}{272}a^{11}-\frac{47}{272}a^{10}+\frac{5}{34}a^{9}-\frac{5}{34}a^{8}+\frac{1}{68}a^{7}+\frac{33}{68}a^{6}-\frac{11}{272}a^{5}+\frac{11}{272}a^{4}-\frac{3}{68}a^{3}+\frac{3}{68}a^{2}-\frac{3}{34}a-\frac{7}{17}$, $\frac{1}{272}a^{24}-\frac{1}{272}a^{22}+\frac{15}{272}a^{18}-\frac{3}{272}a^{16}-\frac{3}{68}a^{14}+\frac{27}{272}a^{12}-\frac{47}{272}a^{10}+\frac{7}{68}a^{8}-\frac{19}{272}a^{6}-\frac{125}{272}a^{4}-\frac{7}{34}a^{2}+\frac{8}{17}$, $\frac{1}{1088}a^{25}+\frac{1}{1088}a^{23}-\frac{1}{272}a^{22}+\frac{31}{1088}a^{19}-\frac{1}{34}a^{18}+\frac{71}{1088}a^{17}-\frac{3}{272}a^{16}-\frac{7}{136}a^{15}-\frac{3}{68}a^{14}-\frac{189}{1088}a^{13}-\frac{7}{68}a^{12}+\frac{183}{1088}a^{11}-\frac{47}{272}a^{10}-\frac{7}{272}a^{9}-\frac{5}{34}a^{8}+\frac{261}{1088}a^{7}+\frac{33}{68}a^{6}-\frac{351}{1088}a^{5}+\frac{11}{272}a^{4}-\frac{37}{272}a^{3}+\frac{3}{68}a^{2}-\frac{29}{68}a-\frac{7}{17}$, $\frac{1}{1088}a^{26}+\frac{1}{1088}a^{24}-\frac{1}{272}a^{22}-\frac{1}{1088}a^{20}-\frac{1}{64}a^{18}-\frac{29}{272}a^{16}-\frac{1}{8}a^{15}-\frac{77}{1088}a^{14}-\frac{169}{1088}a^{12}-\frac{1}{4}a^{11}+\frac{21}{136}a^{10}-\frac{1}{4}a^{9}+\frac{5}{64}a^{8}+\frac{25}{1088}a^{6}+\frac{1}{4}a^{5}+\frac{61}{136}a^{4}+\frac{3}{8}a^{3}+\frac{31}{68}a^{2}-\frac{1}{2}a+\frac{8}{17}$, $\frac{1}{1088}a^{27}-\frac{1}{1088}a^{23}-\frac{1}{1088}a^{21}-\frac{1}{68}a^{19}+\frac{97}{1088}a^{17}-\frac{1}{8}a^{16}+\frac{27}{1088}a^{15}+\frac{33}{272}a^{13}-\frac{1}{4}a^{12}+\frac{173}{1088}a^{11}-\frac{1}{4}a^{10}-\frac{271}{1088}a^{9}-\frac{55}{272}a^{7}+\frac{1}{4}a^{6}+\frac{523}{1088}a^{5}+\frac{3}{8}a^{4}+\frac{13}{272}a^{3}-\frac{1}{2}a^{2}-\frac{13}{68}a$, $\frac{1}{239206592}a^{28}+\frac{12879}{239206592}a^{26}-\frac{193411}{119603296}a^{24}+\frac{181391}{239206592}a^{22}-\frac{725967}{239206592}a^{20}+\frac{173463}{119603296}a^{18}-\frac{1}{8}a^{17}-\frac{28584061}{239206592}a^{16}+\frac{21811377}{239206592}a^{14}-\frac{1}{4}a^{13}+\frac{23518525}{119603296}a^{12}-\frac{1}{4}a^{11}+\frac{2659581}{239206592}a^{10}+\frac{46079727}{239206592}a^{8}-\frac{1}{4}a^{7}-\frac{28592741}{119603296}a^{6}+\frac{3}{8}a^{5}-\frac{1159657}{29900824}a^{4}-\frac{1}{2}a^{3}-\frac{326555}{14950412}a^{2}-\frac{1}{2}a-\frac{958939}{3737603}$, $\frac{1}{239206592}a^{29}+\frac{12879}{239206592}a^{27}+\frac{1653}{7475206}a^{25}-\frac{258327}{239206592}a^{23}-\frac{1}{272}a^{22}-\frac{725967}{239206592}a^{21}+\frac{867837}{29900824}a^{19}-\frac{1}{34}a^{18}-\frac{2391}{239206592}a^{17}-\frac{3}{272}a^{16}+\frac{972633}{14070976}a^{15}-\frac{3}{68}a^{14}+\frac{14727359}{59801648}a^{13}-\frac{7}{68}a^{12}-\frac{1059245}{14070976}a^{11}-\frac{47}{272}a^{10}+\frac{58391831}{239206592}a^{9}-\frac{5}{34}a^{8}+\frac{13515793}{59801648}a^{7}+\frac{33}{68}a^{6}-\frac{47071415}{119603296}a^{5}+\frac{11}{272}a^{4}+\frac{2804819}{7475206}a^{3}+\frac{3}{68}a^{2}-\frac{79503}{3737603}a-\frac{7}{17}$, $\frac{1}{84\!\cdots\!12}a^{30}+\frac{41\!\cdots\!65}{21\!\cdots\!28}a^{28}-\frac{54\!\cdots\!99}{15\!\cdots\!44}a^{26}-\frac{11\!\cdots\!27}{12\!\cdots\!84}a^{24}-\frac{31\!\cdots\!71}{21\!\cdots\!28}a^{22}+\frac{77\!\cdots\!23}{21\!\cdots\!28}a^{20}-\frac{25\!\cdots\!13}{42\!\cdots\!56}a^{18}-\frac{1}{8}a^{17}+\frac{97\!\cdots\!71}{10\!\cdots\!64}a^{16}+\frac{18\!\cdots\!19}{21\!\cdots\!28}a^{14}-\frac{1}{4}a^{13}-\frac{32\!\cdots\!03}{21\!\cdots\!28}a^{12}-\frac{1}{4}a^{11}-\frac{42\!\cdots\!79}{12\!\cdots\!84}a^{10}-\frac{36\!\cdots\!75}{21\!\cdots\!28}a^{8}-\frac{1}{4}a^{7}-\frac{41\!\cdots\!63}{84\!\cdots\!12}a^{6}+\frac{3}{8}a^{5}+\frac{58\!\cdots\!31}{21\!\cdots\!28}a^{4}-\frac{1}{2}a^{3}+\frac{18\!\cdots\!65}{52\!\cdots\!32}a^{2}-\frac{1}{2}a+\frac{73\!\cdots\!68}{33\!\cdots\!77}$, $\frac{1}{33\!\cdots\!48}a^{31}+\frac{41\!\cdots\!65}{84\!\cdots\!12}a^{29}+\frac{22\!\cdots\!27}{61\!\cdots\!76}a^{27}+\frac{20\!\cdots\!03}{84\!\cdots\!12}a^{25}-\frac{31\!\cdots\!71}{84\!\cdots\!12}a^{23}-\frac{1}{272}a^{22}-\frac{11\!\cdots\!87}{84\!\cdots\!12}a^{21}+\frac{73\!\cdots\!35}{16\!\cdots\!24}a^{19}-\frac{1}{34}a^{18}-\frac{18\!\cdots\!93}{42\!\cdots\!56}a^{17}+\frac{31}{272}a^{16}+\frac{41\!\cdots\!77}{84\!\cdots\!12}a^{15}-\frac{3}{68}a^{14}-\frac{79\!\cdots\!05}{84\!\cdots\!12}a^{13}+\frac{5}{34}a^{12}+\frac{96\!\cdots\!73}{84\!\cdots\!12}a^{11}+\frac{21}{272}a^{10}-\frac{61\!\cdots\!81}{84\!\cdots\!12}a^{9}-\frac{5}{34}a^{8}+\frac{54\!\cdots\!53}{33\!\cdots\!48}a^{7}+\frac{4}{17}a^{6}+\frac{40\!\cdots\!35}{84\!\cdots\!12}a^{5}-\frac{91}{272}a^{4}+\frac{26\!\cdots\!33}{21\!\cdots\!28}a^{3}-\frac{31}{68}a^{2}-\frac{24\!\cdots\!85}{13\!\cdots\!08}a-\frac{7}{17}$ 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:  $2$

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{31733846347289050006707609339}{338268464032869881049690213045248} a^{31} + \frac{159066371668502704786194320775}{84567116008217470262422553261312} a^{29} + \frac{13722982620419582052588929353}{617278219038083724543230315776} a^{27} + \frac{15544719819385212136664947501213}{84567116008217470262422553261312} a^{25} + \frac{98127949168048442982113604227447}{84567116008217470262422553261312} a^{23} + \frac{465912278340159774396442411472379}{84567116008217470262422553261312} a^{21} + \frac{3413485261243466901479279119063505}{169134232016434940524845106522624} a^{19} + \frac{2291173245929342645683680194076411}{42283558004108735131211276630656} a^{17} + \frac{8247282890832293527114323360526327}{84567116008217470262422553261312} a^{15} + \frac{151071612981110777014047569238667}{1966677116470173727033082633984} a^{13} + \frac{6552031141224874223212789373396867}{84567116008217470262422553261312} a^{11} + \frac{11119375391339031974950856247776221}{84567116008217470262422553261312} a^{9} + \frac{13520292253964077881824453407256171}{338268464032869881049690213045248} a^{7} - \frac{1617629871718382246042141050801871}{84567116008217470262422553261312} a^{5} + \frac{18716742129237302188519841608487}{1243634058944374562682684606784} a^{3} - \frac{3516851695558134935161606145415}{1321361187628397972850352394708} a \)  (order $12$) 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:   $\frac{82\!\cdots\!07}{82\!\cdots\!24}a^{31}-\frac{52\!\cdots\!73}{16\!\cdots\!24}a^{30}+\frac{41\!\cdots\!15}{20\!\cdots\!56}a^{29}-\frac{61\!\cdots\!95}{96\!\cdots\!92}a^{28}+\frac{49\!\cdots\!13}{20\!\cdots\!56}a^{27}-\frac{22\!\cdots\!63}{30\!\cdots\!88}a^{26}+\frac{40\!\cdots\!45}{20\!\cdots\!56}a^{25}-\frac{25\!\cdots\!63}{41\!\cdots\!56}a^{24}+\frac{25\!\cdots\!31}{20\!\cdots\!56}a^{23}-\frac{16\!\cdots\!13}{41\!\cdots\!56}a^{22}+\frac{12\!\cdots\!67}{20\!\cdots\!56}a^{21}-\frac{78\!\cdots\!29}{41\!\cdots\!56}a^{20}+\frac{90\!\cdots\!65}{41\!\cdots\!12}a^{19}-\frac{34\!\cdots\!11}{48\!\cdots\!36}a^{18}+\frac{61\!\cdots\!83}{10\!\cdots\!28}a^{17}-\frac{39\!\cdots\!69}{20\!\cdots\!28}a^{16}+\frac{22\!\cdots\!31}{20\!\cdots\!56}a^{15}-\frac{14\!\cdots\!85}{41\!\cdots\!56}a^{14}+\frac{42\!\cdots\!71}{47\!\cdots\!92}a^{13}-\frac{71\!\cdots\!71}{24\!\cdots\!68}a^{12}+\frac{18\!\cdots\!51}{20\!\cdots\!56}a^{11}-\frac{11\!\cdots\!81}{41\!\cdots\!56}a^{10}+\frac{30\!\cdots\!73}{20\!\cdots\!56}a^{9}-\frac{20\!\cdots\!15}{41\!\cdots\!56}a^{8}+\frac{43\!\cdots\!03}{82\!\cdots\!24}a^{7}-\frac{29\!\cdots\!21}{16\!\cdots\!24}a^{6}-\frac{31\!\cdots\!43}{20\!\cdots\!56}a^{5}+\frac{22\!\cdots\!69}{41\!\cdots\!56}a^{4}+\frac{74\!\cdots\!87}{51\!\cdots\!64}a^{3}-\frac{63\!\cdots\!17}{10\!\cdots\!64}a^{2}-\frac{19\!\cdots\!19}{80\!\cdots\!01}a+\frac{70\!\cdots\!73}{64\!\cdots\!29}$, $\frac{57\!\cdots\!99}{17\!\cdots\!96}a^{31}-\frac{11\!\cdots\!37}{29\!\cdots\!08}a^{30}+\frac{29\!\cdots\!63}{44\!\cdots\!24}a^{29}-\frac{47\!\cdots\!11}{58\!\cdots\!16}a^{28}+\frac{25\!\cdots\!03}{32\!\cdots\!52}a^{27}-\frac{10\!\cdots\!81}{10\!\cdots\!92}a^{26}+\frac{28\!\cdots\!59}{44\!\cdots\!24}a^{25}-\frac{31\!\cdots\!39}{36\!\cdots\!01}a^{24}+\frac{18\!\cdots\!99}{44\!\cdots\!24}a^{23}-\frac{33\!\cdots\!45}{58\!\cdots\!16}a^{22}+\frac{87\!\cdots\!37}{44\!\cdots\!24}a^{21}-\frac{41\!\cdots\!13}{14\!\cdots\!04}a^{20}+\frac{64\!\cdots\!13}{89\!\cdots\!48}a^{19}-\frac{18\!\cdots\!79}{17\!\cdots\!24}a^{18}+\frac{44\!\cdots\!15}{22\!\cdots\!12}a^{17}-\frac{18\!\cdots\!61}{58\!\cdots\!16}a^{16}+\frac{16\!\cdots\!05}{44\!\cdots\!24}a^{15}-\frac{49\!\cdots\!91}{73\!\cdots\!02}a^{14}+\frac{14\!\cdots\!87}{44\!\cdots\!24}a^{13}-\frac{61\!\cdots\!59}{73\!\cdots\!02}a^{12}+\frac{13\!\cdots\!99}{44\!\cdots\!24}a^{11}-\frac{45\!\cdots\!47}{58\!\cdots\!16}a^{10}+\frac{21\!\cdots\!55}{44\!\cdots\!24}a^{9}-\frac{69\!\cdots\!81}{73\!\cdots\!02}a^{8}+\frac{35\!\cdots\!51}{17\!\cdots\!96}a^{7}-\frac{31\!\cdots\!62}{36\!\cdots\!01}a^{6}-\frac{26\!\cdots\!43}{44\!\cdots\!24}a^{5}-\frac{18\!\cdots\!12}{85\!\cdots\!07}a^{4}+\frac{18\!\cdots\!91}{11\!\cdots\!56}a^{3}+\frac{13\!\cdots\!32}{36\!\cdots\!01}a^{2}-\frac{68\!\cdots\!80}{70\!\cdots\!91}a-\frac{73\!\cdots\!11}{21\!\cdots\!53}$, $\frac{31\!\cdots\!53}{24\!\cdots\!36}a^{30}+\frac{15\!\cdots\!53}{61\!\cdots\!84}a^{28}+\frac{18\!\cdots\!75}{61\!\cdots\!84}a^{26}+\frac{15\!\cdots\!35}{61\!\cdots\!84}a^{24}+\frac{97\!\cdots\!93}{61\!\cdots\!84}a^{22}+\frac{46\!\cdots\!41}{61\!\cdots\!84}a^{20}+\frac{34\!\cdots\!63}{12\!\cdots\!68}a^{18}+\frac{22\!\cdots\!77}{30\!\cdots\!92}a^{16}+\frac{82\!\cdots\!41}{61\!\cdots\!84}a^{14}+\frac{65\!\cdots\!03}{61\!\cdots\!84}a^{12}+\frac{14\!\cdots\!95}{14\!\cdots\!88}a^{10}+\frac{10\!\cdots\!07}{61\!\cdots\!84}a^{8}+\frac{32\!\cdots\!19}{57\!\cdots\!52}a^{6}-\frac{20\!\cdots\!93}{61\!\cdots\!84}a^{4}+\frac{17\!\cdots\!53}{15\!\cdots\!96}a^{2}-\frac{11\!\cdots\!07}{96\!\cdots\!31}$, $\frac{50\!\cdots\!91}{84\!\cdots\!12}a^{31}-\frac{58\!\cdots\!17}{84\!\cdots\!12}a^{30}+\frac{19\!\cdots\!87}{21\!\cdots\!28}a^{29}-\frac{29\!\cdots\!53}{21\!\cdots\!28}a^{28}+\frac{18\!\cdots\!05}{21\!\cdots\!24}a^{27}-\frac{25\!\cdots\!31}{15\!\cdots\!44}a^{26}+\frac{10\!\cdots\!05}{21\!\cdots\!28}a^{25}-\frac{29\!\cdots\!11}{21\!\cdots\!28}a^{24}+\frac{39\!\cdots\!97}{21\!\cdots\!28}a^{23}-\frac{18\!\cdots\!29}{21\!\cdots\!28}a^{22}-\frac{34\!\cdots\!67}{21\!\cdots\!28}a^{21}-\frac{87\!\cdots\!65}{21\!\cdots\!28}a^{20}-\frac{10\!\cdots\!51}{24\!\cdots\!68}a^{19}-\frac{64\!\cdots\!75}{42\!\cdots\!56}a^{18}-\frac{94\!\cdots\!94}{33\!\cdots\!77}a^{17}-\frac{43\!\cdots\!99}{10\!\cdots\!64}a^{16}-\frac{23\!\cdots\!55}{21\!\cdots\!28}a^{15}-\frac{37\!\cdots\!55}{49\!\cdots\!96}a^{14}-\frac{58\!\cdots\!91}{21\!\cdots\!28}a^{13}-\frac{13\!\cdots\!71}{21\!\cdots\!28}a^{12}-\frac{51\!\cdots\!55}{21\!\cdots\!28}a^{11}-\frac{12\!\cdots\!73}{21\!\cdots\!28}a^{10}-\frac{41\!\cdots\!01}{21\!\cdots\!28}a^{9}-\frac{12\!\cdots\!43}{12\!\cdots\!84}a^{8}-\frac{34\!\cdots\!69}{84\!\cdots\!12}a^{7}-\frac{30\!\cdots\!01}{84\!\cdots\!12}a^{6}-\frac{45\!\cdots\!57}{21\!\cdots\!28}a^{5}+\frac{32\!\cdots\!57}{21\!\cdots\!28}a^{4}+\frac{53\!\cdots\!35}{12\!\cdots\!24}a^{3}-\frac{31\!\cdots\!69}{52\!\cdots\!32}a^{2}-\frac{11\!\cdots\!97}{66\!\cdots\!54}a+\frac{18\!\cdots\!75}{33\!\cdots\!77}$, $\frac{12\!\cdots\!53}{16\!\cdots\!24}a^{31}-\frac{24\!\cdots\!91}{42\!\cdots\!56}a^{30}+\frac{62\!\cdots\!79}{42\!\cdots\!56}a^{29}-\frac{12\!\cdots\!41}{10\!\cdots\!64}a^{28}+\frac{53\!\cdots\!87}{30\!\cdots\!88}a^{27}-\frac{10\!\cdots\!85}{77\!\cdots\!72}a^{26}+\frac{59\!\cdots\!15}{42\!\cdots\!56}a^{25}-\frac{12\!\cdots\!21}{10\!\cdots\!64}a^{24}+\frac{37\!\cdots\!49}{42\!\cdots\!56}a^{23}-\frac{75\!\cdots\!29}{10\!\cdots\!64}a^{22}+\frac{17\!\cdots\!77}{42\!\cdots\!56}a^{21}-\frac{35\!\cdots\!79}{10\!\cdots\!64}a^{20}+\frac{73\!\cdots\!23}{49\!\cdots\!36}a^{19}-\frac{26\!\cdots\!05}{21\!\cdots\!28}a^{18}+\frac{81\!\cdots\!87}{21\!\cdots\!28}a^{17}-\frac{87\!\cdots\!07}{26\!\cdots\!16}a^{16}+\frac{27\!\cdots\!37}{42\!\cdots\!56}a^{15}-\frac{62\!\cdots\!51}{10\!\cdots\!64}a^{14}+\frac{15\!\cdots\!27}{42\!\cdots\!56}a^{13}-\frac{46\!\cdots\!13}{10\!\cdots\!64}a^{12}+\frac{16\!\cdots\!77}{42\!\cdots\!56}a^{11}-\frac{45\!\cdots\!25}{10\!\cdots\!64}a^{10}+\frac{34\!\cdots\!67}{42\!\cdots\!56}a^{9}-\frac{80\!\cdots\!77}{10\!\cdots\!64}a^{8}-\frac{10\!\cdots\!95}{16\!\cdots\!24}a^{7}-\frac{83\!\cdots\!91}{42\!\cdots\!56}a^{6}-\frac{14\!\cdots\!95}{42\!\cdots\!56}a^{5}+\frac{11\!\cdots\!67}{62\!\cdots\!92}a^{4}+\frac{10\!\cdots\!63}{10\!\cdots\!64}a^{3}-\frac{14\!\cdots\!87}{26\!\cdots\!16}a^{2}-\frac{31\!\cdots\!11}{13\!\cdots\!08}a+\frac{28\!\cdots\!74}{33\!\cdots\!77}$, $\frac{11\!\cdots\!71}{42\!\cdots\!56}a^{31}+\frac{37\!\cdots\!19}{84\!\cdots\!12}a^{30}+\frac{12\!\cdots\!89}{21\!\cdots\!28}a^{29}+\frac{27\!\cdots\!59}{31\!\cdots\!96}a^{28}+\frac{55\!\cdots\!43}{77\!\cdots\!72}a^{27}+\frac{82\!\cdots\!45}{77\!\cdots\!72}a^{26}+\frac{13\!\cdots\!69}{21\!\cdots\!28}a^{25}+\frac{18\!\cdots\!65}{21\!\cdots\!28}a^{24}+\frac{86\!\cdots\!53}{21\!\cdots\!28}a^{23}+\frac{60\!\cdots\!01}{10\!\cdots\!64}a^{22}+\frac{21\!\cdots\!61}{10\!\cdots\!64}a^{21}+\frac{28\!\cdots\!97}{10\!\cdots\!64}a^{20}+\frac{85\!\cdots\!45}{10\!\cdots\!64}a^{19}+\frac{42\!\cdots\!21}{42\!\cdots\!56}a^{18}+\frac{50\!\cdots\!35}{21\!\cdots\!28}a^{17}+\frac{58\!\cdots\!09}{21\!\cdots\!28}a^{16}+\frac{53\!\cdots\!23}{10\!\cdots\!64}a^{15}+\frac{54\!\cdots\!63}{10\!\cdots\!64}a^{14}+\frac{13\!\cdots\!07}{21\!\cdots\!28}a^{13}+\frac{96\!\cdots\!93}{21\!\cdots\!28}a^{12}+\frac{12\!\cdots\!83}{21\!\cdots\!28}a^{11}+\frac{12\!\cdots\!57}{31\!\cdots\!96}a^{10}+\frac{75\!\cdots\!97}{10\!\cdots\!64}a^{9}+\frac{77\!\cdots\!07}{10\!\cdots\!64}a^{8}+\frac{27\!\cdots\!33}{42\!\cdots\!56}a^{7}+\frac{26\!\cdots\!99}{84\!\cdots\!12}a^{6}+\frac{10\!\cdots\!15}{66\!\cdots\!54}a^{5}-\frac{19\!\cdots\!23}{21\!\cdots\!28}a^{4}-\frac{93\!\cdots\!20}{33\!\cdots\!77}a^{3}+\frac{52\!\cdots\!75}{52\!\cdots\!32}a^{2}+\frac{64\!\cdots\!39}{33\!\cdots\!77}a-\frac{57\!\cdots\!67}{33\!\cdots\!77}$, $\frac{30\!\cdots\!33}{71\!\cdots\!84}a^{31}+\frac{11\!\cdots\!37}{29\!\cdots\!08}a^{30}+\frac{15\!\cdots\!17}{17\!\cdots\!96}a^{29}+\frac{47\!\cdots\!11}{58\!\cdots\!16}a^{28}+\frac{78\!\cdots\!63}{77\!\cdots\!24}a^{27}+\frac{10\!\cdots\!81}{10\!\cdots\!92}a^{26}+\frac{35\!\cdots\!17}{41\!\cdots\!72}a^{25}+\frac{31\!\cdots\!39}{36\!\cdots\!01}a^{24}+\frac{95\!\cdots\!17}{17\!\cdots\!96}a^{23}+\frac{33\!\cdots\!45}{58\!\cdots\!16}a^{22}+\frac{45\!\cdots\!45}{17\!\cdots\!96}a^{21}+\frac{41\!\cdots\!13}{14\!\cdots\!04}a^{20}+\frac{33\!\cdots\!67}{35\!\cdots\!92}a^{19}+\frac{18\!\cdots\!79}{17\!\cdots\!24}a^{18}+\frac{22\!\cdots\!13}{89\!\cdots\!48}a^{17}+\frac{18\!\cdots\!61}{58\!\cdots\!16}a^{16}+\frac{49\!\cdots\!73}{10\!\cdots\!88}a^{15}+\frac{49\!\cdots\!91}{73\!\cdots\!02}a^{14}+\frac{72\!\cdots\!71}{17\!\cdots\!96}a^{13}+\frac{61\!\cdots\!59}{73\!\cdots\!02}a^{12}+\frac{69\!\cdots\!37}{17\!\cdots\!96}a^{11}+\frac{45\!\cdots\!47}{58\!\cdots\!16}a^{10}+\frac{11\!\cdots\!63}{17\!\cdots\!96}a^{9}+\frac{69\!\cdots\!81}{73\!\cdots\!02}a^{8}+\frac{18\!\cdots\!77}{71\!\cdots\!84}a^{7}+\frac{31\!\cdots\!62}{36\!\cdots\!01}a^{6}-\frac{13\!\cdots\!85}{17\!\cdots\!96}a^{5}+\frac{18\!\cdots\!12}{85\!\cdots\!07}a^{4}+\frac{14\!\cdots\!81}{44\!\cdots\!24}a^{3}-\frac{13\!\cdots\!32}{36\!\cdots\!01}a^{2}-\frac{85\!\cdots\!09}{70\!\cdots\!91}a+\frac{51\!\cdots\!58}{21\!\cdots\!53}$, $\frac{27\!\cdots\!19}{33\!\cdots\!48}a^{31}+\frac{13\!\cdots\!89}{84\!\cdots\!12}a^{30}+\frac{14\!\cdots\!03}{84\!\cdots\!12}a^{29}+\frac{69\!\cdots\!39}{21\!\cdots\!28}a^{28}+\frac{13\!\cdots\!65}{61\!\cdots\!76}a^{27}+\frac{60\!\cdots\!21}{15\!\cdots\!44}a^{26}+\frac{15\!\cdots\!77}{84\!\cdots\!12}a^{25}+\frac{69\!\cdots\!27}{21\!\cdots\!28}a^{24}+\frac{10\!\cdots\!71}{84\!\cdots\!12}a^{23}+\frac{44\!\cdots\!99}{21\!\cdots\!28}a^{22}+\frac{52\!\cdots\!63}{84\!\cdots\!12}a^{21}+\frac{21\!\cdots\!19}{21\!\cdots\!28}a^{20}+\frac{40\!\cdots\!09}{16\!\cdots\!24}a^{19}+\frac{15\!\cdots\!75}{42\!\cdots\!56}a^{18}+\frac{30\!\cdots\!29}{42\!\cdots\!56}a^{17}+\frac{27\!\cdots\!93}{26\!\cdots\!16}a^{16}+\frac{12\!\cdots\!79}{84\!\cdots\!12}a^{15}+\frac{41\!\cdots\!87}{21\!\cdots\!28}a^{14}+\frac{16\!\cdots\!41}{84\!\cdots\!12}a^{13}+\frac{23\!\cdots\!27}{12\!\cdots\!84}a^{12}+\frac{14\!\cdots\!31}{84\!\cdots\!12}a^{11}+\frac{37\!\cdots\!99}{21\!\cdots\!28}a^{10}+\frac{18\!\cdots\!13}{84\!\cdots\!12}a^{9}+\frac{55\!\cdots\!13}{21\!\cdots\!28}a^{8}+\frac{67\!\cdots\!75}{33\!\cdots\!48}a^{7}+\frac{12\!\cdots\!41}{84\!\cdots\!12}a^{6}+\frac{10\!\cdots\!15}{19\!\cdots\!84}a^{5}-\frac{20\!\cdots\!93}{21\!\cdots\!28}a^{4}-\frac{19\!\cdots\!25}{21\!\cdots\!28}a^{3}+\frac{37\!\cdots\!85}{52\!\cdots\!32}a^{2}+\frac{20\!\cdots\!33}{33\!\cdots\!77}a+\frac{45\!\cdots\!10}{33\!\cdots\!77}$, $\frac{14\!\cdots\!45}{33\!\cdots\!48}a^{31}+\frac{27\!\cdots\!01}{84\!\cdots\!12}a^{30}+\frac{73\!\cdots\!09}{84\!\cdots\!12}a^{29}+\frac{69\!\cdots\!51}{10\!\cdots\!64}a^{28}+\frac{64\!\cdots\!67}{61\!\cdots\!76}a^{27}+\frac{29\!\cdots\!15}{38\!\cdots\!36}a^{26}+\frac{73\!\cdots\!11}{84\!\cdots\!12}a^{25}+\frac{13\!\cdots\!47}{21\!\cdots\!28}a^{24}+\frac{47\!\cdots\!17}{84\!\cdots\!12}a^{23}+\frac{21\!\cdots\!23}{52\!\cdots\!32}a^{22}+\frac{22\!\cdots\!41}{84\!\cdots\!12}a^{21}+\frac{25\!\cdots\!43}{13\!\cdots\!08}a^{20}+\frac{99\!\cdots\!11}{99\!\cdots\!72}a^{19}+\frac{30\!\cdots\!95}{42\!\cdots\!56}a^{18}+\frac{11\!\cdots\!15}{42\!\cdots\!56}a^{17}+\frac{41\!\cdots\!61}{21\!\cdots\!28}a^{16}+\frac{44\!\cdots\!85}{84\!\cdots\!12}a^{15}+\frac{19\!\cdots\!77}{52\!\cdots\!32}a^{14}+\frac{24\!\cdots\!03}{49\!\cdots\!36}a^{13}+\frac{66\!\cdots\!07}{21\!\cdots\!28}a^{12}+\frac{39\!\cdots\!81}{84\!\cdots\!12}a^{11}+\frac{31\!\cdots\!87}{10\!\cdots\!64}a^{10}+\frac{59\!\cdots\!11}{84\!\cdots\!12}a^{9}+\frac{31\!\cdots\!19}{66\!\cdots\!54}a^{8}+\frac{12\!\cdots\!29}{33\!\cdots\!48}a^{7}+\frac{16\!\cdots\!97}{84\!\cdots\!12}a^{6}-\frac{25\!\cdots\!13}{84\!\cdots\!12}a^{5}-\frac{73\!\cdots\!53}{12\!\cdots\!84}a^{4}+\frac{46\!\cdots\!61}{21\!\cdots\!28}a^{3}+\frac{12\!\cdots\!29}{52\!\cdots\!32}a^{2}+\frac{34\!\cdots\!75}{13\!\cdots\!08}a-\frac{45\!\cdots\!75}{33\!\cdots\!77}$, $\frac{39\!\cdots\!63}{33\!\cdots\!48}a^{31}+\frac{81\!\cdots\!77}{42\!\cdots\!56}a^{30}+\frac{21\!\cdots\!07}{84\!\cdots\!12}a^{29}+\frac{83\!\cdots\!99}{21\!\cdots\!28}a^{28}+\frac{19\!\cdots\!97}{61\!\cdots\!76}a^{27}+\frac{73\!\cdots\!13}{15\!\cdots\!44}a^{26}+\frac{23\!\cdots\!61}{84\!\cdots\!12}a^{25}+\frac{21\!\cdots\!01}{52\!\cdots\!32}a^{24}+\frac{94\!\cdots\!99}{49\!\cdots\!36}a^{23}+\frac{55\!\cdots\!53}{21\!\cdots\!28}a^{22}+\frac{81\!\cdots\!95}{84\!\cdots\!12}a^{21}+\frac{26\!\cdots\!43}{21\!\cdots\!28}a^{20}+\frac{65\!\cdots\!53}{16\!\cdots\!24}a^{19}+\frac{10\!\cdots\!89}{21\!\cdots\!28}a^{18}+\frac{49\!\cdots\!63}{42\!\cdots\!56}a^{17}+\frac{28\!\cdots\!51}{21\!\cdots\!28}a^{16}+\frac{21\!\cdots\!11}{84\!\cdots\!12}a^{15}+\frac{56\!\cdots\!51}{21\!\cdots\!28}a^{14}+\frac{29\!\cdots\!93}{84\!\cdots\!12}a^{13}+\frac{15\!\cdots\!29}{52\!\cdots\!32}a^{12}+\frac{27\!\cdots\!31}{84\!\cdots\!12}a^{11}+\frac{56\!\cdots\!15}{21\!\cdots\!28}a^{10}+\frac{31\!\cdots\!73}{84\!\cdots\!12}a^{9}+\frac{77\!\cdots\!05}{21\!\cdots\!28}a^{8}+\frac{12\!\cdots\!71}{33\!\cdots\!48}a^{7}+\frac{10\!\cdots\!53}{42\!\cdots\!56}a^{6}+\frac{10\!\cdots\!77}{84\!\cdots\!12}a^{5}+\frac{28\!\cdots\!63}{10\!\cdots\!64}a^{4}-\frac{34\!\cdots\!49}{21\!\cdots\!28}a^{3}+\frac{95\!\cdots\!49}{26\!\cdots\!16}a^{2}+\frac{16\!\cdots\!59}{13\!\cdots\!08}a+\frac{15\!\cdots\!20}{33\!\cdots\!77}$, $\frac{21\!\cdots\!45}{33\!\cdots\!48}a^{31}+\frac{42\!\cdots\!63}{42\!\cdots\!56}a^{30}+\frac{11\!\cdots\!53}{84\!\cdots\!12}a^{29}+\frac{44\!\cdots\!11}{21\!\cdots\!28}a^{28}+\frac{62\!\cdots\!83}{36\!\cdots\!28}a^{27}+\frac{38\!\cdots\!91}{15\!\cdots\!44}a^{26}+\frac{12\!\cdots\!83}{84\!\cdots\!12}a^{25}+\frac{22\!\cdots\!75}{10\!\cdots\!64}a^{24}+\frac{85\!\cdots\!09}{84\!\cdots\!12}a^{23}+\frac{29\!\cdots\!61}{21\!\cdots\!28}a^{22}+\frac{43\!\cdots\!81}{84\!\cdots\!12}a^{21}+\frac{83\!\cdots\!37}{12\!\cdots\!84}a^{20}+\frac{34\!\cdots\!95}{16\!\cdots\!24}a^{19}+\frac{12\!\cdots\!63}{49\!\cdots\!96}a^{18}+\frac{26\!\cdots\!29}{42\!\cdots\!56}a^{17}+\frac{15\!\cdots\!67}{21\!\cdots\!28}a^{16}+\frac{11\!\cdots\!49}{84\!\cdots\!12}a^{15}+\frac{30\!\cdots\!41}{21\!\cdots\!28}a^{14}+\frac{15\!\cdots\!99}{84\!\cdots\!12}a^{13}+\frac{15\!\cdots\!57}{10\!\cdots\!64}a^{12}+\frac{14\!\cdots\!41}{84\!\cdots\!12}a^{11}+\frac{29\!\cdots\!15}{21\!\cdots\!28}a^{10}+\frac{16\!\cdots\!07}{84\!\cdots\!12}a^{9}+\frac{41\!\cdots\!87}{21\!\cdots\!28}a^{8}+\frac{68\!\cdots\!61}{33\!\cdots\!48}a^{7}+\frac{55\!\cdots\!91}{42\!\cdots\!56}a^{6}+\frac{55\!\cdots\!07}{84\!\cdots\!12}a^{5}+\frac{15\!\cdots\!09}{10\!\cdots\!64}a^{4}-\frac{18\!\cdots\!59}{21\!\cdots\!28}a^{3}+\frac{51\!\cdots\!67}{26\!\cdots\!16}a^{2}+\frac{85\!\cdots\!49}{13\!\cdots\!08}a+\frac{83\!\cdots\!24}{33\!\cdots\!77}$, $\frac{31\!\cdots\!75}{33\!\cdots\!48}a^{31}-\frac{10\!\cdots\!31}{42\!\cdots\!56}a^{30}+\frac{15\!\cdots\!79}{84\!\cdots\!12}a^{29}-\frac{13\!\cdots\!99}{26\!\cdots\!16}a^{28}+\frac{12\!\cdots\!37}{61\!\cdots\!76}a^{27}-\frac{47\!\cdots\!73}{77\!\cdots\!72}a^{26}+\frac{13\!\cdots\!49}{84\!\cdots\!12}a^{25}-\frac{26\!\cdots\!67}{52\!\cdots\!32}a^{24}+\frac{81\!\cdots\!07}{84\!\cdots\!12}a^{23}-\frac{84\!\cdots\!35}{26\!\cdots\!16}a^{22}+\frac{36\!\cdots\!71}{84\!\cdots\!12}a^{21}-\frac{16\!\cdots\!15}{10\!\cdots\!64}a^{20}+\frac{14\!\cdots\!53}{99\!\cdots\!72}a^{19}-\frac{11\!\cdots\!43}{21\!\cdots\!28}a^{18}+\frac{13\!\cdots\!11}{42\!\cdots\!56}a^{17}-\frac{16\!\cdots\!31}{10\!\cdots\!64}a^{16}+\frac{71\!\cdots\!49}{19\!\cdots\!84}a^{15}-\frac{29\!\cdots\!91}{10\!\cdots\!64}a^{14}-\frac{31\!\cdots\!75}{84\!\cdots\!12}a^{13}-\frac{59\!\cdots\!25}{26\!\cdots\!16}a^{12}-\frac{22\!\cdots\!69}{84\!\cdots\!12}a^{11}-\frac{11\!\cdots\!59}{52\!\cdots\!32}a^{10}+\frac{26\!\cdots\!25}{84\!\cdots\!12}a^{9}-\frac{38\!\cdots\!65}{10\!\cdots\!64}a^{8}-\frac{37\!\cdots\!33}{33\!\cdots\!48}a^{7}-\frac{54\!\cdots\!51}{42\!\cdots\!56}a^{6}-\frac{78\!\cdots\!99}{84\!\cdots\!12}a^{5}+\frac{63\!\cdots\!73}{10\!\cdots\!64}a^{4}+\frac{52\!\cdots\!55}{21\!\cdots\!28}a^{3}-\frac{58\!\cdots\!93}{26\!\cdots\!16}a^{2}-\frac{91\!\cdots\!61}{13\!\cdots\!08}a+\frac{60\!\cdots\!95}{33\!\cdots\!77}$, $\frac{42\!\cdots\!93}{24\!\cdots\!04}a^{31}+\frac{94\!\cdots\!35}{84\!\cdots\!12}a^{30}+\frac{22\!\cdots\!65}{61\!\cdots\!76}a^{29}+\frac{46\!\cdots\!67}{21\!\cdots\!28}a^{28}+\frac{27\!\cdots\!71}{61\!\cdots\!76}a^{27}+\frac{39\!\cdots\!55}{15\!\cdots\!44}a^{26}+\frac{23\!\cdots\!35}{61\!\cdots\!76}a^{25}+\frac{43\!\cdots\!27}{21\!\cdots\!28}a^{24}+\frac{89\!\cdots\!49}{36\!\cdots\!28}a^{23}+\frac{27\!\cdots\!43}{21\!\cdots\!28}a^{22}+\frac{75\!\cdots\!41}{61\!\cdots\!76}a^{21}+\frac{12\!\cdots\!17}{21\!\cdots\!28}a^{20}+\frac{58\!\cdots\!67}{12\!\cdots\!52}a^{19}+\frac{88\!\cdots\!61}{42\!\cdots\!56}a^{18}+\frac{42\!\cdots\!05}{30\!\cdots\!88}a^{17}+\frac{32\!\cdots\!29}{62\!\cdots\!92}a^{16}+\frac{17\!\cdots\!57}{61\!\cdots\!76}a^{15}+\frac{17\!\cdots\!09}{21\!\cdots\!28}a^{14}+\frac{20\!\cdots\!87}{61\!\cdots\!76}a^{13}+\frac{56\!\cdots\!87}{21\!\cdots\!28}a^{12}+\frac{19\!\cdots\!13}{61\!\cdots\!76}a^{11}+\frac{58\!\cdots\!35}{21\!\cdots\!28}a^{10}+\frac{14\!\cdots\!47}{36\!\cdots\!28}a^{9}+\frac{19\!\cdots\!31}{21\!\cdots\!28}a^{8}+\frac{48\!\cdots\!93}{14\!\cdots\!12}a^{7}-\frac{34\!\cdots\!69}{84\!\cdots\!12}a^{6}+\frac{52\!\cdots\!83}{61\!\cdots\!76}a^{5}-\frac{16\!\cdots\!71}{21\!\cdots\!28}a^{4}+\frac{42\!\cdots\!85}{15\!\cdots\!44}a^{3}+\frac{46\!\cdots\!87}{52\!\cdots\!32}a^{2}+\frac{56\!\cdots\!81}{48\!\cdots\!42}a-\frac{15\!\cdots\!96}{33\!\cdots\!77}$, $\frac{73\!\cdots\!49}{21\!\cdots\!28}a^{31}+\frac{21\!\cdots\!27}{42\!\cdots\!56}a^{30}+\frac{74\!\cdots\!91}{10\!\cdots\!64}a^{29}+\frac{10\!\cdots\!71}{10\!\cdots\!64}a^{28}+\frac{12\!\cdots\!69}{15\!\cdots\!44}a^{27}+\frac{46\!\cdots\!53}{38\!\cdots\!36}a^{26}+\frac{14\!\cdots\!87}{21\!\cdots\!28}a^{25}+\frac{26\!\cdots\!51}{26\!\cdots\!16}a^{24}+\frac{47\!\cdots\!39}{10\!\cdots\!64}a^{23}+\frac{67\!\cdots\!03}{10\!\cdots\!64}a^{22}+\frac{45\!\cdots\!23}{21\!\cdots\!28}a^{21}+\frac{16\!\cdots\!87}{52\!\cdots\!32}a^{20}+\frac{16\!\cdots\!27}{21\!\cdots\!28}a^{19}+\frac{24\!\cdots\!15}{21\!\cdots\!28}a^{18}+\frac{23\!\cdots\!33}{10\!\cdots\!64}a^{17}+\frac{16\!\cdots\!05}{52\!\cdots\!32}a^{16}+\frac{86\!\cdots\!39}{21\!\cdots\!28}a^{15}+\frac{30\!\cdots\!01}{52\!\cdots\!32}a^{14}+\frac{78\!\cdots\!05}{21\!\cdots\!28}a^{13}+\frac{26\!\cdots\!25}{52\!\cdots\!32}a^{12}+\frac{38\!\cdots\!19}{10\!\cdots\!64}a^{11}+\frac{49\!\cdots\!79}{10\!\cdots\!64}a^{10}+\frac{11\!\cdots\!73}{21\!\cdots\!28}a^{9}+\frac{39\!\cdots\!39}{52\!\cdots\!32}a^{8}+\frac{14\!\cdots\!47}{52\!\cdots\!32}a^{7}+\frac{13\!\cdots\!31}{42\!\cdots\!56}a^{6}-\frac{13\!\cdots\!41}{52\!\cdots\!32}a^{5}-\frac{10\!\cdots\!91}{10\!\cdots\!64}a^{4}+\frac{13\!\cdots\!83}{66\!\cdots\!54}a^{3}+\frac{10\!\cdots\!35}{26\!\cdots\!16}a^{2}+\frac{57\!\cdots\!87}{66\!\cdots\!54}a-\frac{84\!\cdots\!41}{33\!\cdots\!77}$, $\frac{23\!\cdots\!31}{33\!\cdots\!48}a^{31}-\frac{16\!\cdots\!47}{84\!\cdots\!12}a^{30}+\frac{11\!\cdots\!47}{84\!\cdots\!12}a^{29}-\frac{82\!\cdots\!93}{21\!\cdots\!28}a^{28}+\frac{10\!\cdots\!77}{61\!\cdots\!76}a^{27}-\frac{71\!\cdots\!39}{15\!\cdots\!44}a^{26}+\frac{11\!\cdots\!25}{84\!\cdots\!12}a^{25}-\frac{48\!\cdots\!97}{12\!\cdots\!84}a^{24}+\frac{72\!\cdots\!95}{84\!\cdots\!12}a^{23}-\frac{52\!\cdots\!93}{21\!\cdots\!28}a^{22}+\frac{34\!\cdots\!59}{84\!\cdots\!12}a^{21}-\frac{25\!\cdots\!89}{21\!\cdots\!28}a^{20}+\frac{24\!\cdots\!69}{16\!\cdots\!24}a^{19}-\frac{18\!\cdots\!05}{42\!\cdots\!56}a^{18}+\frac{16\!\cdots\!95}{42\!\cdots\!56}a^{17}-\frac{32\!\cdots\!09}{26\!\cdots\!16}a^{16}+\frac{57\!\cdots\!23}{84\!\cdots\!12}a^{15}-\frac{49\!\cdots\!97}{21\!\cdots\!28}a^{14}+\frac{40\!\cdots\!13}{84\!\cdots\!12}a^{13}-\frac{46\!\cdots\!25}{21\!\cdots\!28}a^{12}+\frac{40\!\cdots\!19}{84\!\cdots\!12}a^{11}-\frac{42\!\cdots\!37}{21\!\cdots\!28}a^{10}+\frac{74\!\cdots\!13}{84\!\cdots\!12}a^{9}-\frac{64\!\cdots\!19}{21\!\cdots\!28}a^{8}+\frac{56\!\cdots\!03}{33\!\cdots\!48}a^{7}-\frac{13\!\cdots\!87}{84\!\cdots\!12}a^{6}-\frac{19\!\cdots\!03}{84\!\cdots\!12}a^{5}+\frac{56\!\cdots\!15}{21\!\cdots\!28}a^{4}+\frac{20\!\cdots\!27}{21\!\cdots\!28}a^{3}-\frac{16\!\cdots\!59}{52\!\cdots\!32}a^{2}-\frac{93\!\cdots\!01}{66\!\cdots\!54}a-\frac{86\!\cdots\!89}{33\!\cdots\!77}$ 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:  \( 2205277237958.931 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 2205277237958.931 \cdot 16}{12\cdot\sqrt{480960519379403029833827263813614000556122650443776}}\cr\approx \mathstrut & 0.791088572881245 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 + 20*x^30 + 236*x^28 + 1948*x^26 + 12276*x^24 + 58148*x^22 + 212422*x^20 + 567784*x^18 + 1013620*x^16 + 772588*x^14 + 788932*x^12 + 1345276*x^10 + 364625*x^8 - 228564*x^6 + 145744*x^4 - 25856*x^2 + 4096)
 
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 + 20*x^30 + 236*x^28 + 1948*x^26 + 12276*x^24 + 58148*x^22 + 212422*x^20 + 567784*x^18 + 1013620*x^16 + 772588*x^14 + 788932*x^12 + 1345276*x^10 + 364625*x^8 - 228564*x^6 + 145744*x^4 - 25856*x^2 + 4096, 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 + 20*x^30 + 236*x^28 + 1948*x^26 + 12276*x^24 + 58148*x^22 + 212422*x^20 + 567784*x^18 + 1013620*x^16 + 772588*x^14 + 788932*x^12 + 1345276*x^10 + 364625*x^8 - 228564*x^6 + 145744*x^4 - 25856*x^2 + 4096);
 
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 + 20*x^30 + 236*x^28 + 1948*x^26 + 12276*x^24 + 58148*x^22 + 212422*x^20 + 567784*x^18 + 1013620*x^16 + 772588*x^14 + 788932*x^12 + 1345276*x^10 + 364625*x^8 - 228564*x^6 + 145744*x^4 - 25856*x^2 + 4096);
 
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

$C_2^4:D_4$ (as 32T1369):

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 128
The 56 conjugacy class representatives for $C_2^4:D_4$ are not computed
Character table for $C_2^4:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{51}) \), \(\Q(\sqrt{-51}) \), 4.0.120224.1, 4.4.20808.1, 4.4.30056.1, 4.0.83232.1, 4.0.2312.1, 4.4.1082016.1, 4.4.9248.1, 4.0.270504.1, \(\Q(i, \sqrt{51})\), 4.0.541008.2, \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{17})\), 4.4.60112.1, \(\Q(\sqrt{-3}, \sqrt{-17})\), \(\Q(\sqrt{3}, \sqrt{17})\), 4.0.3757.1, \(\Q(\sqrt{3}, \sqrt{-17})\), \(\Q(\sqrt{-3}, \sqrt{17})\), 4.4.33813.1, 8.0.292689656064.35, 8.0.292689656064.9, 8.0.3613452544.3, 8.0.292689656064.39, 8.0.1731891456.1, 8.8.292689656064.1, 8.0.1143318969.1, 8.0.4683034497024.8, 8.0.4683034497024.5, 8.0.4683034497024.90, 8.0.4683034497024.31, 8.0.57815240704.2, 8.0.27710263296.14, 8.0.342102016.5, 8.0.4683034497024.9, 8.0.57815240704.21, 8.8.4683034497024.2, 8.8.57815240704.3, 8.0.4683034497024.3, 8.0.4683034497024.66, 8.8.27710263296.2, 8.8.4683034497024.6, 8.0.27710263296.13, 8.0.14453810176.1, 8.0.903363136.1, 8.0.73172414016.5, 8.0.1170758624256.10, 8.0.1170758624256.4, 8.0.432972864.2, 8.0.73172414016.6, 8.0.6927565824.3, 8.0.1170758624256.17, 8.8.73172414016.1, 8.0.73172414016.14, 8.8.1170758624256.2, 16.0.85667234766862611972096.1, 16.0.21930812100316828664856576.4, 16.0.21930812100316828664856576.16, 16.0.21930812100316828664856576.17, 16.0.21930812100316828664856576.8, 16.0.3342602057661458415616.4, 16.0.21930812100316828664856576.19, 16.0.21930812100316828664856576.15, 16.0.21930812100316828664856576.13, 16.0.21930812100316828664856576.20, 16.0.767858691933644783616.8, 16.0.21930812100316828664856576.18, 16.16.21930812100316828664856576.1, 16.0.1370675756269801791553536.1, 16.0.5354202172928913248256.2

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 R ${\href{/padicField/5.4.0.1}{4} }^{8}$ ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{8}$ R R ${\href{/padicField/19.2.0.1}{2} }^{16}$ ${\href{/padicField/23.2.0.1}{2} }^{16}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{16}$ ${\href{/padicField/47.2.0.1}{2} }^{16}$ ${\href{/padicField/53.2.0.1}{2} }^{16}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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 2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
\(3\) Copy content Toggle raw display 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}$
\(13\) Copy content Toggle raw display 13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(17\) Copy content Toggle raw display 17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$