Properties

Label 29.1.142...409.1
Degree $29$
Signature $[1, 14]$
Discriminant $1.424\times 10^{50}$
Root discriminant \(53.63\)
Ramified prime $3823$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{29}$ (as 29T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^29 - x^28 + 33*x^27 - 64*x^26 + 535*x^25 - 1144*x^24 + 5145*x^23 - 10327*x^22 + 31220*x^21 - 55583*x^20 + 124455*x^19 - 188951*x^18 + 323227*x^17 - 411652*x^16 + 521736*x^15 - 591430*x^14 + 489137*x^13 - 738131*x^12 + 175881*x^11 - 1211717*x^10 + 917449*x^9 - 3034006*x^8 + 2818536*x^7 - 6798367*x^6 + 6818314*x^5 - 8408506*x^4 + 7552950*x^3 - 8000650*x^2 + 5448750*x - 1749375)
 
gp: K = bnfinit(y^29 - y^28 + 33*y^27 - 64*y^26 + 535*y^25 - 1144*y^24 + 5145*y^23 - 10327*y^22 + 31220*y^21 - 55583*y^20 + 124455*y^19 - 188951*y^18 + 323227*y^17 - 411652*y^16 + 521736*y^15 - 591430*y^14 + 489137*y^13 - 738131*y^12 + 175881*y^11 - 1211717*y^10 + 917449*y^9 - 3034006*y^8 + 2818536*y^7 - 6798367*y^6 + 6818314*y^5 - 8408506*y^4 + 7552950*y^3 - 8000650*y^2 + 5448750*y - 1749375, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - x^28 + 33*x^27 - 64*x^26 + 535*x^25 - 1144*x^24 + 5145*x^23 - 10327*x^22 + 31220*x^21 - 55583*x^20 + 124455*x^19 - 188951*x^18 + 323227*x^17 - 411652*x^16 + 521736*x^15 - 591430*x^14 + 489137*x^13 - 738131*x^12 + 175881*x^11 - 1211717*x^10 + 917449*x^9 - 3034006*x^8 + 2818536*x^7 - 6798367*x^6 + 6818314*x^5 - 8408506*x^4 + 7552950*x^3 - 8000650*x^2 + 5448750*x - 1749375);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 - x^28 + 33*x^27 - 64*x^26 + 535*x^25 - 1144*x^24 + 5145*x^23 - 10327*x^22 + 31220*x^21 - 55583*x^20 + 124455*x^19 - 188951*x^18 + 323227*x^17 - 411652*x^16 + 521736*x^15 - 591430*x^14 + 489137*x^13 - 738131*x^12 + 175881*x^11 - 1211717*x^10 + 917449*x^9 - 3034006*x^8 + 2818536*x^7 - 6798367*x^6 + 6818314*x^5 - 8408506*x^4 + 7552950*x^3 - 8000650*x^2 + 5448750*x - 1749375)
 

\( x^{29} - x^{28} + 33 x^{27} - 64 x^{26} + 535 x^{25} - 1144 x^{24} + 5145 x^{23} - 10327 x^{22} + 31220 x^{21} - 55583 x^{20} + 124455 x^{19} - 188951 x^{18} + 323227 x^{17} + \cdots - 1749375 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $29$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(142449251725173555024565249558533387292386826365409\) \(\medspace = 3823^{14}\) 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:  \(53.63\)
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:  $3823^{1/2}\approx 61.83041322844284$
Ramified primes:   \(3823\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{7}$, $\frac{1}{15}a^{16}-\frac{1}{15}a^{15}-\frac{1}{15}a^{14}-\frac{2}{15}a^{13}-\frac{1}{15}a^{12}+\frac{1}{15}a^{11}+\frac{1}{15}a^{10}-\frac{1}{15}a^{9}+\frac{2}{15}a^{8}+\frac{7}{15}a^{7}+\frac{7}{15}a^{6}-\frac{7}{15}a^{5}+\frac{4}{15}a^{4}-\frac{4}{15}a^{3}+\frac{1}{3}a^{2}+\frac{4}{15}a$, $\frac{1}{15}a^{17}-\frac{2}{15}a^{15}+\frac{2}{15}a^{14}+\frac{2}{15}a^{13}+\frac{2}{15}a^{11}+\frac{1}{15}a^{9}-\frac{2}{5}a^{8}-\frac{1}{15}a^{7}-\frac{1}{3}a^{6}+\frac{7}{15}a^{5}+\frac{1}{15}a^{3}-\frac{2}{5}a^{2}+\frac{4}{15}a$, $\frac{1}{45}a^{18}+\frac{2}{15}a^{13}-\frac{1}{15}a^{11}-\frac{2}{45}a^{10}-\frac{1}{15}a^{9}+\frac{2}{5}a^{8}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{7}{15}a^{5}+\frac{1}{5}a^{4}+\frac{7}{15}a^{3}+\frac{19}{45}a^{2}+\frac{1}{15}a$, $\frac{1}{45}a^{19}+\frac{2}{15}a^{14}-\frac{1}{15}a^{12}-\frac{2}{45}a^{11}-\frac{1}{15}a^{10}+\frac{1}{15}a^{9}+\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{7}{15}a^{6}+\frac{1}{5}a^{5}+\frac{7}{15}a^{4}+\frac{19}{45}a^{3}+\frac{1}{15}a^{2}+\frac{1}{3}a$, $\frac{1}{315}a^{20}+\frac{1}{105}a^{19}+\frac{2}{315}a^{18}-\frac{2}{105}a^{17}+\frac{2}{105}a^{16}+\frac{2}{15}a^{15}-\frac{1}{7}a^{14}+\frac{1}{7}a^{13}-\frac{32}{315}a^{12}+\frac{1}{35}a^{11}+\frac{26}{315}a^{10}-\frac{2}{21}a^{9}+\frac{7}{15}a^{8}-\frac{4}{21}a^{7}-\frac{16}{35}a^{6}-\frac{1}{35}a^{5}+\frac{7}{45}a^{4}-\frac{31}{105}a^{3}+\frac{14}{45}a^{2}-\frac{6}{35}a-\frac{3}{7}$, $\frac{1}{315}a^{21}+\frac{2}{315}a^{18}+\frac{1}{105}a^{17}+\frac{1}{105}a^{16}-\frac{1}{105}a^{15}-\frac{1}{35}a^{14}+\frac{22}{315}a^{13}-\frac{1}{21}a^{11}+\frac{32}{315}a^{10}+\frac{2}{105}a^{9}-\frac{34}{105}a^{8}-\frac{44}{105}a^{7}-\frac{2}{35}a^{6}+\frac{13}{315}a^{5}+\frac{6}{35}a^{4}+\frac{44}{105}a^{3}+\frac{13}{63}a^{2}-\frac{11}{35}a+\frac{2}{7}$, $\frac{1}{315}a^{22}+\frac{2}{315}a^{19}+\frac{1}{105}a^{18}+\frac{1}{105}a^{17}-\frac{1}{105}a^{16}-\frac{1}{35}a^{15}+\frac{22}{315}a^{14}-\frac{1}{21}a^{12}+\frac{32}{315}a^{11}+\frac{2}{105}a^{10}+\frac{1}{105}a^{9}-\frac{44}{105}a^{8}-\frac{2}{35}a^{7}+\frac{13}{315}a^{6}+\frac{6}{35}a^{5}+\frac{44}{105}a^{4}+\frac{13}{63}a^{3}-\frac{11}{35}a^{2}-\frac{1}{21}a$, $\frac{1}{1575}a^{23}-\frac{1}{1575}a^{22}-\frac{1}{1575}a^{21}-\frac{1}{315}a^{19}+\frac{1}{105}a^{18}+\frac{8}{525}a^{17}-\frac{2}{75}a^{16}+\frac{34}{1575}a^{15}+\frac{7}{45}a^{14}-\frac{127}{1575}a^{13}-\frac{61}{525}a^{12}-\frac{71}{1575}a^{11}-\frac{1}{35}a^{10}-\frac{34}{525}a^{9}-\frac{166}{525}a^{8}-\frac{116}{1575}a^{7}+\frac{116}{1575}a^{6}-\frac{547}{1575}a^{5}-\frac{248}{525}a^{4}+\frac{436}{1575}a^{3}-\frac{248}{525}a^{2}-\frac{44}{105}a-\frac{2}{7}$, $\frac{1}{4725}a^{24}+\frac{1}{4725}a^{23}+\frac{2}{4725}a^{22}+\frac{1}{1575}a^{21}+\frac{1}{945}a^{20}-\frac{1}{189}a^{19}-\frac{41}{4725}a^{18}-\frac{8}{1575}a^{17}+\frac{2}{945}a^{16}-\frac{377}{4725}a^{15}+\frac{83}{4725}a^{14}-\frac{239}{1575}a^{13}+\frac{428}{4725}a^{12}-\frac{502}{4725}a^{11}-\frac{302}{4725}a^{10}-\frac{74}{1575}a^{9}+\frac{334}{675}a^{8}-\frac{1046}{4725}a^{7}+\frac{442}{945}a^{6}-\frac{146}{1575}a^{5}-\frac{262}{4725}a^{4}+\frac{113}{4725}a^{3}-\frac{8}{4725}a^{2}+\frac{19}{63}a-\frac{1}{21}$, $\frac{1}{70875}a^{25}-\frac{1}{70875}a^{24}+\frac{2}{23625}a^{23}-\frac{16}{10125}a^{22}+\frac{14}{10125}a^{21}-\frac{4}{14175}a^{20}+\frac{8}{23625}a^{19}-\frac{662}{70875}a^{18}-\frac{29}{10125}a^{17}+\frac{1016}{70875}a^{16}+\frac{137}{23625}a^{15}+\frac{2642}{70875}a^{14}+\frac{2266}{14175}a^{13}-\frac{4511}{70875}a^{12}+\frac{3392}{23625}a^{11}-\frac{2858}{70875}a^{10}-\frac{538}{14175}a^{9}-\frac{6908}{14175}a^{8}-\frac{6238}{23625}a^{7}-\frac{6112}{70875}a^{6}+\frac{17777}{70875}a^{5}-\frac{22307}{70875}a^{4}+\frac{167}{3375}a^{3}+\frac{21547}{70875}a^{2}-\frac{29}{945}a+\frac{44}{315}$, $\frac{1}{496125}a^{26}+\frac{2}{496125}a^{25}+\frac{2}{55125}a^{24}+\frac{8}{70875}a^{23}+\frac{107}{496125}a^{22}-\frac{41}{496125}a^{21}-\frac{62}{165375}a^{20}-\frac{913}{99225}a^{19}-\frac{47}{70875}a^{18}-\frac{4138}{496125}a^{17}+\frac{971}{55125}a^{16}-\frac{5783}{99225}a^{15}-\frac{47224}{496125}a^{14}-\frac{8678}{70875}a^{13}-\frac{16514}{165375}a^{12}+\frac{15026}{99225}a^{11}-\frac{32219}{496125}a^{10}+\frac{1693}{99225}a^{9}-\frac{947}{18375}a^{8}+\frac{243896}{496125}a^{7}-\frac{41749}{496125}a^{6}-\frac{21263}{70875}a^{5}-\frac{15503}{165375}a^{4}-\frac{60152}{496125}a^{3}+\frac{54727}{165375}a^{2}+\frac{866}{2205}a-\frac{12}{245}$, $\frac{1}{496125}a^{27}+\frac{34}{496125}a^{24}-\frac{89}{496125}a^{23}-\frac{262}{496125}a^{22}+\frac{11}{55125}a^{21}-\frac{109}{70875}a^{20}+\frac{748}{99225}a^{19}-\frac{512}{496125}a^{18}+\frac{319}{165375}a^{17}+\frac{808}{496125}a^{16}-\frac{51848}{496125}a^{15}+\frac{37664}{496125}a^{14}-\frac{946}{11025}a^{13}-\frac{5182}{496125}a^{12}+\frac{48332}{496125}a^{11}-\frac{1357}{99225}a^{10}+\frac{12562}{165375}a^{9}+\frac{65119}{496125}a^{8}-\frac{47839}{99225}a^{7}-\frac{8704}{19845}a^{6}-\frac{3484}{11025}a^{5}-\frac{152536}{496125}a^{4}+\frac{136162}{496125}a^{3}-\frac{22144}{99225}a^{2}+\frac{997}{6615}a+\frac{109}{441}$, $\frac{1}{10\!\cdots\!25}a^{28}-\frac{88\!\cdots\!66}{10\!\cdots\!25}a^{27}+\frac{39\!\cdots\!43}{47\!\cdots\!25}a^{26}+\frac{41\!\cdots\!36}{10\!\cdots\!25}a^{25}-\frac{14\!\cdots\!92}{40\!\cdots\!25}a^{24}+\frac{25\!\cdots\!96}{10\!\cdots\!25}a^{23}-\frac{34\!\cdots\!47}{67\!\cdots\!75}a^{22}+\frac{12\!\cdots\!23}{10\!\cdots\!25}a^{21}+\frac{99\!\cdots\!49}{20\!\cdots\!25}a^{20}-\frac{21\!\cdots\!18}{10\!\cdots\!25}a^{19}-\frac{54\!\cdots\!36}{67\!\cdots\!75}a^{18}+\frac{18\!\cdots\!79}{10\!\cdots\!25}a^{17}+\frac{13\!\cdots\!97}{10\!\cdots\!25}a^{16}+\frac{37\!\cdots\!28}{10\!\cdots\!25}a^{15}-\frac{40\!\cdots\!78}{33\!\cdots\!75}a^{14}-\frac{87\!\cdots\!39}{80\!\cdots\!05}a^{13}-\frac{75\!\cdots\!33}{10\!\cdots\!25}a^{12}-\frac{13\!\cdots\!26}{10\!\cdots\!25}a^{11}+\frac{46\!\cdots\!57}{33\!\cdots\!75}a^{10}+\frac{13\!\cdots\!68}{10\!\cdots\!25}a^{9}+\frac{67\!\cdots\!41}{20\!\cdots\!25}a^{8}+\frac{33\!\cdots\!74}{10\!\cdots\!25}a^{7}+\frac{16\!\cdots\!87}{33\!\cdots\!75}a^{6}-\frac{65\!\cdots\!81}{14\!\cdots\!75}a^{5}-\frac{11\!\cdots\!82}{33\!\cdots\!75}a^{4}-\frac{75\!\cdots\!02}{33\!\cdots\!75}a^{3}+\frac{42\!\cdots\!43}{95\!\cdots\!25}a^{2}-\frac{84\!\cdots\!16}{63\!\cdots\!75}a+\frac{82\!\cdots\!02}{29\!\cdots\!15}$ 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:  $3$, $5$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{69\!\cdots\!07}{10\!\cdots\!25}a^{28}-\frac{34\!\cdots\!69}{33\!\cdots\!75}a^{27}+\frac{31\!\cdots\!73}{14\!\cdots\!75}a^{26}-\frac{18\!\cdots\!96}{33\!\cdots\!75}a^{25}+\frac{23\!\cdots\!09}{67\!\cdots\!75}a^{24}-\frac{30\!\cdots\!26}{33\!\cdots\!75}a^{23}+\frac{14\!\cdots\!38}{44\!\cdots\!25}a^{22}-\frac{88\!\cdots\!66}{11\!\cdots\!25}a^{21}+\frac{13\!\cdots\!09}{67\!\cdots\!75}a^{20}-\frac{45\!\cdots\!59}{11\!\cdots\!25}a^{19}+\frac{17\!\cdots\!72}{22\!\cdots\!25}a^{18}-\frac{44\!\cdots\!49}{33\!\cdots\!75}a^{17}+\frac{63\!\cdots\!28}{33\!\cdots\!75}a^{16}-\frac{92\!\cdots\!63}{33\!\cdots\!75}a^{15}+\frac{10\!\cdots\!16}{37\!\cdots\!75}a^{14}-\frac{27\!\cdots\!69}{74\!\cdots\!75}a^{13}+\frac{96\!\cdots\!33}{33\!\cdots\!75}a^{12}-\frac{40\!\cdots\!63}{11\!\cdots\!25}a^{11}+\frac{91\!\cdots\!24}{33\!\cdots\!75}a^{10}-\frac{99\!\cdots\!23}{33\!\cdots\!75}a^{9}+\frac{66\!\cdots\!88}{47\!\cdots\!25}a^{8}-\frac{35\!\cdots\!29}{33\!\cdots\!75}a^{7}+\frac{28\!\cdots\!98}{11\!\cdots\!25}a^{6}-\frac{81\!\cdots\!14}{22\!\cdots\!25}a^{5}+\frac{56\!\cdots\!03}{10\!\cdots\!25}a^{4}-\frac{10\!\cdots\!09}{33\!\cdots\!75}a^{3}+\frac{15\!\cdots\!33}{28\!\cdots\!75}a^{2}-\frac{98\!\cdots\!36}{19\!\cdots\!25}a+\frac{16\!\cdots\!47}{89\!\cdots\!45}$, $\frac{24\!\cdots\!38}{47\!\cdots\!25}a^{28}+\frac{77\!\cdots\!83}{20\!\cdots\!25}a^{27}+\frac{24\!\cdots\!17}{14\!\cdots\!75}a^{26}-\frac{90\!\cdots\!91}{14\!\cdots\!75}a^{25}+\frac{74\!\cdots\!23}{28\!\cdots\!75}a^{24}-\frac{33\!\cdots\!21}{14\!\cdots\!75}a^{23}+\frac{74\!\cdots\!38}{31\!\cdots\!75}a^{22}-\frac{39\!\cdots\!18}{14\!\cdots\!75}a^{21}+\frac{40\!\cdots\!43}{28\!\cdots\!75}a^{20}-\frac{26\!\cdots\!82}{14\!\cdots\!75}a^{19}+\frac{56\!\cdots\!08}{95\!\cdots\!25}a^{18}-\frac{11\!\cdots\!74}{14\!\cdots\!75}a^{17}+\frac{35\!\cdots\!79}{20\!\cdots\!25}a^{16}-\frac{29\!\cdots\!93}{14\!\cdots\!75}a^{15}+\frac{23\!\cdots\!86}{76\!\cdots\!75}a^{14}-\frac{20\!\cdots\!19}{57\!\cdots\!75}a^{13}+\frac{34\!\cdots\!48}{14\!\cdots\!75}a^{12}-\frac{61\!\cdots\!24}{14\!\cdots\!75}a^{11}-\frac{42\!\cdots\!94}{15\!\cdots\!75}a^{10}-\frac{12\!\cdots\!93}{14\!\cdots\!75}a^{9}-\frac{26\!\cdots\!79}{14\!\cdots\!75}a^{8}-\frac{23\!\cdots\!84}{14\!\cdots\!75}a^{7}+\frac{58\!\cdots\!72}{52\!\cdots\!25}a^{6}-\frac{65\!\cdots\!28}{14\!\cdots\!75}a^{5}+\frac{35\!\cdots\!91}{14\!\cdots\!75}a^{4}-\frac{78\!\cdots\!71}{15\!\cdots\!75}a^{3}+\frac{23\!\cdots\!57}{41\!\cdots\!25}a^{2}-\frac{87\!\cdots\!59}{19\!\cdots\!25}a+\frac{48\!\cdots\!28}{12\!\cdots\!35}$, $\frac{59\!\cdots\!48}{10\!\cdots\!25}a^{28}-\frac{18\!\cdots\!72}{11\!\cdots\!25}a^{27}+\frac{91\!\cdots\!39}{47\!\cdots\!25}a^{26}-\frac{24\!\cdots\!57}{10\!\cdots\!25}a^{25}+\frac{57\!\cdots\!53}{20\!\cdots\!25}a^{24}-\frac{47\!\cdots\!07}{10\!\cdots\!25}a^{23}+\frac{17\!\cdots\!51}{67\!\cdots\!75}a^{22}-\frac{41\!\cdots\!46}{10\!\cdots\!25}a^{21}+\frac{56\!\cdots\!17}{40\!\cdots\!25}a^{20}-\frac{21\!\cdots\!04}{10\!\cdots\!25}a^{19}+\frac{33\!\cdots\!99}{67\!\cdots\!75}a^{18}-\frac{69\!\cdots\!13}{10\!\cdots\!25}a^{17}+\frac{11\!\cdots\!56}{10\!\cdots\!25}a^{16}-\frac{13\!\cdots\!76}{10\!\cdots\!25}a^{15}+\frac{47\!\cdots\!76}{33\!\cdots\!75}a^{14}-\frac{79\!\cdots\!87}{40\!\cdots\!25}a^{13}+\frac{68\!\cdots\!36}{10\!\cdots\!25}a^{12}-\frac{30\!\cdots\!43}{10\!\cdots\!25}a^{11}-\frac{33\!\cdots\!29}{33\!\cdots\!75}a^{10}-\frac{56\!\cdots\!66}{10\!\cdots\!25}a^{9}+\frac{60\!\cdots\!41}{14\!\cdots\!75}a^{8}-\frac{81\!\cdots\!33}{10\!\cdots\!25}a^{7}+\frac{50\!\cdots\!91}{33\!\cdots\!75}a^{6}-\frac{26\!\cdots\!48}{14\!\cdots\!75}a^{5}+\frac{24\!\cdots\!67}{10\!\cdots\!25}a^{4}-\frac{19\!\cdots\!18}{10\!\cdots\!25}a^{3}+\frac{16\!\cdots\!27}{71\!\cdots\!75}a^{2}-\frac{12\!\cdots\!08}{63\!\cdots\!75}a+\frac{39\!\cdots\!43}{59\!\cdots\!43}$, $\frac{16\!\cdots\!47}{21\!\cdots\!75}a^{28}+\frac{33\!\cdots\!92}{23\!\cdots\!75}a^{27}+\frac{76\!\cdots\!73}{30\!\cdots\!25}a^{26}+\frac{43\!\cdots\!62}{21\!\cdots\!75}a^{25}+\frac{94\!\cdots\!83}{28\!\cdots\!25}a^{24}+\frac{32\!\cdots\!17}{21\!\cdots\!75}a^{23}+\frac{10\!\cdots\!67}{42\!\cdots\!75}a^{22}+\frac{22\!\cdots\!46}{21\!\cdots\!75}a^{21}+\frac{47\!\cdots\!74}{47\!\cdots\!75}a^{20}+\frac{13\!\cdots\!49}{21\!\cdots\!75}a^{19}+\frac{98\!\cdots\!34}{42\!\cdots\!75}a^{18}+\frac{64\!\cdots\!98}{21\!\cdots\!75}a^{17}+\frac{12\!\cdots\!78}{71\!\cdots\!25}a^{16}+\frac{18\!\cdots\!56}{21\!\cdots\!75}a^{15}-\frac{14\!\cdots\!98}{21\!\cdots\!75}a^{14}+\frac{34\!\cdots\!11}{42\!\cdots\!75}a^{13}-\frac{21\!\cdots\!77}{71\!\cdots\!25}a^{12}-\frac{61\!\cdots\!82}{21\!\cdots\!75}a^{11}-\frac{19\!\cdots\!08}{21\!\cdots\!75}a^{10}-\frac{30\!\cdots\!74}{21\!\cdots\!75}a^{9}-\frac{14\!\cdots\!97}{10\!\cdots\!75}a^{8}-\frac{26\!\cdots\!72}{21\!\cdots\!75}a^{7}-\frac{29\!\cdots\!23}{21\!\cdots\!75}a^{6}-\frac{74\!\cdots\!91}{43\!\cdots\!75}a^{5}-\frac{85\!\cdots\!57}{21\!\cdots\!75}a^{4}+\frac{15\!\cdots\!73}{21\!\cdots\!75}a^{3}-\frac{19\!\cdots\!72}{87\!\cdots\!75}a^{2}+\frac{16\!\cdots\!29}{40\!\cdots\!75}a-\frac{39\!\cdots\!41}{19\!\cdots\!35}$, $\frac{14\!\cdots\!21}{10\!\cdots\!25}a^{28}-\frac{12\!\cdots\!49}{11\!\cdots\!25}a^{27}+\frac{95\!\cdots\!87}{20\!\cdots\!25}a^{26}-\frac{28\!\cdots\!88}{33\!\cdots\!75}a^{25}+\frac{29\!\cdots\!37}{40\!\cdots\!25}a^{24}-\frac{14\!\cdots\!59}{10\!\cdots\!25}a^{23}+\frac{13\!\cdots\!74}{20\!\cdots\!25}a^{22}-\frac{43\!\cdots\!49}{33\!\cdots\!75}a^{21}+\frac{75\!\cdots\!62}{20\!\cdots\!25}a^{20}-\frac{66\!\cdots\!33}{10\!\cdots\!25}a^{19}+\frac{28\!\cdots\!12}{20\!\cdots\!25}a^{18}-\frac{70\!\cdots\!47}{33\!\cdots\!75}a^{17}+\frac{33\!\cdots\!27}{10\!\cdots\!25}a^{16}-\frac{42\!\cdots\!42}{10\!\cdots\!25}a^{15}+\frac{46\!\cdots\!26}{10\!\cdots\!25}a^{14}-\frac{15\!\cdots\!11}{26\!\cdots\!35}a^{13}+\frac{35\!\cdots\!62}{10\!\cdots\!25}a^{12}-\frac{71\!\cdots\!01}{10\!\cdots\!25}a^{11}+\frac{56\!\cdots\!51}{10\!\cdots\!25}a^{10}-\frac{39\!\cdots\!34}{33\!\cdots\!75}a^{9}+\frac{22\!\cdots\!92}{14\!\cdots\!75}a^{8}-\frac{26\!\cdots\!91}{10\!\cdots\!25}a^{7}+\frac{34\!\cdots\!96}{99\!\cdots\!25}a^{6}-\frac{33\!\cdots\!82}{47\!\cdots\!25}a^{5}+\frac{73\!\cdots\!29}{10\!\cdots\!25}a^{4}-\frac{55\!\cdots\!51}{10\!\cdots\!25}a^{3}+\frac{15\!\cdots\!58}{19\!\cdots\!25}a^{2}-\frac{43\!\cdots\!06}{63\!\cdots\!75}a+\frac{43\!\cdots\!42}{19\!\cdots\!81}$, $\frac{10\!\cdots\!19}{67\!\cdots\!75}a^{28}+\frac{67\!\cdots\!02}{20\!\cdots\!25}a^{27}+\frac{16\!\cdots\!63}{28\!\cdots\!75}a^{26}+\frac{12\!\cdots\!17}{20\!\cdots\!25}a^{25}+\frac{16\!\cdots\!91}{20\!\cdots\!25}a^{24}+\frac{10\!\cdots\!41}{20\!\cdots\!25}a^{23}+\frac{42\!\cdots\!19}{67\!\cdots\!75}a^{22}+\frac{68\!\cdots\!27}{20\!\cdots\!25}a^{21}+\frac{64\!\cdots\!13}{20\!\cdots\!25}a^{20}+\frac{31\!\cdots\!84}{20\!\cdots\!25}a^{19}+\frac{69\!\cdots\!64}{67\!\cdots\!75}a^{18}+\frac{10\!\cdots\!56}{20\!\cdots\!25}a^{17}+\frac{42\!\cdots\!91}{20\!\cdots\!25}a^{16}+\frac{22\!\cdots\!99}{20\!\cdots\!25}a^{15}+\frac{14\!\cdots\!81}{67\!\cdots\!75}a^{14}-\frac{28\!\cdots\!43}{40\!\cdots\!25}a^{13}-\frac{72\!\cdots\!49}{20\!\cdots\!25}a^{12}-\frac{26\!\cdots\!84}{20\!\cdots\!25}a^{11}-\frac{17\!\cdots\!86}{67\!\cdots\!75}a^{10}-\frac{20\!\cdots\!22}{40\!\cdots\!25}a^{9}-\frac{18\!\cdots\!58}{28\!\cdots\!75}a^{8}-\frac{16\!\cdots\!87}{20\!\cdots\!25}a^{7}-\frac{41\!\cdots\!21}{66\!\cdots\!75}a^{6}-\frac{29\!\cdots\!46}{41\!\cdots\!25}a^{5}-\frac{86\!\cdots\!26}{20\!\cdots\!25}a^{4}-\frac{27\!\cdots\!28}{67\!\cdots\!75}a^{3}+\frac{28\!\cdots\!86}{82\!\cdots\!25}a^{2}+\frac{19\!\cdots\!01}{38\!\cdots\!05}a+\frac{34\!\cdots\!79}{17\!\cdots\!29}$, $\frac{25\!\cdots\!51}{33\!\cdots\!75}a^{28}-\frac{90\!\cdots\!06}{33\!\cdots\!75}a^{27}+\frac{58\!\cdots\!19}{22\!\cdots\!25}a^{26}-\frac{34\!\cdots\!52}{10\!\cdots\!25}a^{25}+\frac{81\!\cdots\!56}{20\!\cdots\!25}a^{24}-\frac{25\!\cdots\!31}{37\!\cdots\!75}a^{23}+\frac{75\!\cdots\!78}{20\!\cdots\!25}a^{22}-\frac{63\!\cdots\!31}{10\!\cdots\!25}a^{21}+\frac{90\!\cdots\!72}{40\!\cdots\!25}a^{20}-\frac{11\!\cdots\!93}{33\!\cdots\!75}a^{19}+\frac{18\!\cdots\!88}{20\!\cdots\!25}a^{18}-\frac{12\!\cdots\!63}{10\!\cdots\!25}a^{17}+\frac{24\!\cdots\!36}{10\!\cdots\!25}a^{16}-\frac{10\!\cdots\!03}{37\!\cdots\!75}a^{15}+\frac{40\!\cdots\!38}{10\!\cdots\!25}a^{14}-\frac{86\!\cdots\!79}{20\!\cdots\!25}a^{13}+\frac{32\!\cdots\!76}{10\!\cdots\!25}a^{12}-\frac{17\!\cdots\!06}{33\!\cdots\!75}a^{11}-\frac{18\!\cdots\!67}{10\!\cdots\!25}a^{10}-\frac{97\!\cdots\!91}{10\!\cdots\!25}a^{9}+\frac{95\!\cdots\!81}{14\!\cdots\!75}a^{8}-\frac{28\!\cdots\!62}{11\!\cdots\!25}a^{7}+\frac{16\!\cdots\!53}{99\!\cdots\!25}a^{6}-\frac{91\!\cdots\!03}{14\!\cdots\!75}a^{5}+\frac{40\!\cdots\!02}{10\!\cdots\!25}a^{4}-\frac{73\!\cdots\!72}{11\!\cdots\!25}a^{3}+\frac{13\!\cdots\!14}{28\!\cdots\!75}a^{2}-\frac{13\!\cdots\!22}{27\!\cdots\!75}a+\frac{99\!\cdots\!56}{89\!\cdots\!45}$, $\frac{34\!\cdots\!69}{10\!\cdots\!25}a^{28}+\frac{24\!\cdots\!11}{10\!\cdots\!25}a^{27}+\frac{30\!\cdots\!34}{29\!\cdots\!75}a^{26}-\frac{21\!\cdots\!71}{10\!\cdots\!25}a^{25}+\frac{26\!\cdots\!78}{20\!\cdots\!25}a^{24}-\frac{19\!\cdots\!32}{33\!\cdots\!75}a^{23}+\frac{18\!\cdots\!73}{20\!\cdots\!25}a^{22}-\frac{24\!\cdots\!28}{10\!\cdots\!25}a^{21}+\frac{68\!\cdots\!21}{20\!\cdots\!25}a^{20}+\frac{35\!\cdots\!41}{33\!\cdots\!75}a^{19}+\frac{10\!\cdots\!04}{20\!\cdots\!25}a^{18}+\frac{12\!\cdots\!36}{10\!\cdots\!25}a^{17}-\frac{54\!\cdots\!57}{10\!\cdots\!25}a^{16}+\frac{12\!\cdots\!49}{33\!\cdots\!75}a^{15}-\frac{22\!\cdots\!86}{10\!\cdots\!25}a^{14}+\frac{30\!\cdots\!94}{20\!\cdots\!25}a^{13}+\frac{19\!\cdots\!48}{10\!\cdots\!25}a^{12}-\frac{74\!\cdots\!93}{33\!\cdots\!75}a^{11}-\frac{34\!\cdots\!26}{10\!\cdots\!25}a^{10}-\frac{46\!\cdots\!18}{10\!\cdots\!25}a^{9}-\frac{53\!\cdots\!11}{20\!\cdots\!25}a^{8}-\frac{68\!\cdots\!18}{33\!\cdots\!75}a^{7}-\frac{11\!\cdots\!06}{99\!\cdots\!25}a^{6}-\frac{84\!\cdots\!19}{14\!\cdots\!75}a^{5}-\frac{40\!\cdots\!18}{33\!\cdots\!75}a^{4}-\frac{50\!\cdots\!99}{10\!\cdots\!25}a^{3}+\frac{93\!\cdots\!87}{95\!\cdots\!25}a^{2}-\frac{15\!\cdots\!06}{71\!\cdots\!75}a+\frac{25\!\cdots\!58}{29\!\cdots\!15}$, $\frac{22\!\cdots\!67}{14\!\cdots\!75}a^{28}-\frac{56\!\cdots\!41}{20\!\cdots\!25}a^{27}+\frac{24\!\cdots\!67}{47\!\cdots\!25}a^{26}-\frac{20\!\cdots\!33}{14\!\cdots\!75}a^{25}+\frac{81\!\cdots\!99}{95\!\cdots\!25}a^{24}-\frac{35\!\cdots\!53}{14\!\cdots\!75}a^{23}+\frac{24\!\cdots\!67}{28\!\cdots\!75}a^{22}-\frac{32\!\cdots\!29}{14\!\cdots\!75}a^{21}+\frac{59\!\cdots\!71}{10\!\cdots\!25}a^{20}-\frac{18\!\cdots\!86}{14\!\cdots\!75}a^{19}+\frac{68\!\cdots\!78}{28\!\cdots\!75}a^{18}-\frac{66\!\cdots\!77}{14\!\cdots\!75}a^{17}+\frac{45\!\cdots\!79}{68\!\cdots\!75}a^{16}-\frac{15\!\cdots\!49}{14\!\cdots\!75}a^{15}+\frac{23\!\cdots\!71}{20\!\cdots\!25}a^{14}-\frac{17\!\cdots\!51}{11\!\cdots\!15}a^{13}+\frac{51\!\cdots\!13}{47\!\cdots\!25}a^{12}-\frac{18\!\cdots\!67}{14\!\cdots\!75}a^{11}+\frac{11\!\cdots\!92}{14\!\cdots\!75}a^{10}-\frac{82\!\cdots\!84}{14\!\cdots\!75}a^{9}+\frac{20\!\cdots\!56}{47\!\cdots\!25}a^{8}-\frac{26\!\cdots\!37}{14\!\cdots\!75}a^{7}+\frac{16\!\cdots\!27}{14\!\cdots\!75}a^{6}-\frac{12\!\cdots\!79}{14\!\cdots\!75}a^{5}+\frac{38\!\cdots\!73}{14\!\cdots\!75}a^{4}-\frac{47\!\cdots\!74}{47\!\cdots\!25}a^{3}+\frac{16\!\cdots\!62}{58\!\cdots\!75}a^{2}-\frac{31\!\cdots\!17}{19\!\cdots\!25}a+\frac{22\!\cdots\!96}{12\!\cdots\!35}$, $\frac{23\!\cdots\!97}{67\!\cdots\!75}a^{28}-\frac{35\!\cdots\!24}{20\!\cdots\!25}a^{27}+\frac{40\!\cdots\!13}{35\!\cdots\!75}a^{26}-\frac{33\!\cdots\!23}{20\!\cdots\!25}a^{25}+\frac{36\!\cdots\!42}{20\!\cdots\!25}a^{24}-\frac{63\!\cdots\!89}{20\!\cdots\!25}a^{23}+\frac{11\!\cdots\!14}{67\!\cdots\!75}a^{22}-\frac{56\!\cdots\!17}{20\!\cdots\!25}a^{21}+\frac{19\!\cdots\!01}{20\!\cdots\!25}a^{20}-\frac{29\!\cdots\!33}{20\!\cdots\!25}a^{19}+\frac{24\!\cdots\!43}{67\!\cdots\!75}a^{18}-\frac{96\!\cdots\!66}{20\!\cdots\!25}a^{17}+\frac{35\!\cdots\!73}{40\!\cdots\!25}a^{16}-\frac{19\!\cdots\!78}{20\!\cdots\!25}a^{15}+\frac{86\!\cdots\!24}{67\!\cdots\!75}a^{14}-\frac{28\!\cdots\!58}{20\!\cdots\!25}a^{13}+\frac{19\!\cdots\!82}{20\!\cdots\!25}a^{12}-\frac{41\!\cdots\!47}{20\!\cdots\!25}a^{11}-\frac{29\!\cdots\!17}{67\!\cdots\!75}a^{10}-\frac{34\!\cdots\!42}{80\!\cdots\!05}a^{9}+\frac{70\!\cdots\!12}{57\!\cdots\!75}a^{8}-\frac{19\!\cdots\!73}{20\!\cdots\!25}a^{7}+\frac{35\!\cdots\!33}{66\!\cdots\!75}a^{6}-\frac{11\!\cdots\!67}{57\!\cdots\!75}a^{5}+\frac{56\!\cdots\!78}{40\!\cdots\!25}a^{4}-\frac{41\!\cdots\!53}{20\!\cdots\!25}a^{3}+\frac{15\!\cdots\!88}{95\!\cdots\!25}a^{2}-\frac{23\!\cdots\!79}{12\!\cdots\!35}a+\frac{95\!\cdots\!89}{99\!\cdots\!05}$, $\frac{88\!\cdots\!76}{10\!\cdots\!25}a^{28}-\frac{43\!\cdots\!81}{10\!\cdots\!25}a^{27}+\frac{13\!\cdots\!93}{47\!\cdots\!25}a^{26}-\frac{42\!\cdots\!14}{10\!\cdots\!25}a^{25}+\frac{11\!\cdots\!58}{24\!\cdots\!25}a^{24}-\frac{80\!\cdots\!69}{10\!\cdots\!25}a^{23}+\frac{16\!\cdots\!23}{40\!\cdots\!25}a^{22}-\frac{73\!\cdots\!62}{10\!\cdots\!25}a^{21}+\frac{16\!\cdots\!26}{67\!\cdots\!75}a^{20}-\frac{39\!\cdots\!68}{10\!\cdots\!25}a^{19}+\frac{18\!\cdots\!58}{20\!\cdots\!25}a^{18}-\frac{12\!\cdots\!01}{10\!\cdots\!25}a^{17}+\frac{25\!\cdots\!53}{11\!\cdots\!25}a^{16}-\frac{26\!\cdots\!27}{10\!\cdots\!25}a^{15}+\frac{31\!\cdots\!06}{10\!\cdots\!25}a^{14}-\frac{12\!\cdots\!57}{40\!\cdots\!25}a^{13}+\frac{60\!\cdots\!24}{33\!\cdots\!75}a^{12}-\frac{40\!\cdots\!06}{10\!\cdots\!25}a^{11}-\frac{14\!\cdots\!04}{10\!\cdots\!25}a^{10}-\frac{10\!\cdots\!72}{10\!\cdots\!25}a^{9}+\frac{22\!\cdots\!16}{53\!\cdots\!25}a^{8}-\frac{24\!\cdots\!01}{10\!\cdots\!25}a^{7}+\frac{17\!\cdots\!36}{99\!\cdots\!25}a^{6}-\frac{75\!\cdots\!81}{14\!\cdots\!75}a^{5}+\frac{31\!\cdots\!19}{10\!\cdots\!25}a^{4}-\frac{17\!\cdots\!62}{33\!\cdots\!75}a^{3}+\frac{92\!\cdots\!59}{28\!\cdots\!75}a^{2}-\frac{64\!\cdots\!73}{19\!\cdots\!25}a+\frac{82\!\cdots\!06}{89\!\cdots\!45}$, $\frac{51\!\cdots\!53}{33\!\cdots\!75}a^{28}-\frac{85\!\cdots\!04}{10\!\cdots\!25}a^{27}+\frac{21\!\cdots\!22}{47\!\cdots\!25}a^{26}-\frac{74\!\cdots\!36}{10\!\cdots\!25}a^{25}+\frac{42\!\cdots\!98}{67\!\cdots\!75}a^{24}-\frac{11\!\cdots\!66}{10\!\cdots\!25}a^{23}+\frac{10\!\cdots\!46}{20\!\cdots\!25}a^{22}-\frac{86\!\cdots\!08}{10\!\cdots\!25}a^{21}+\frac{30\!\cdots\!42}{13\!\cdots\!75}a^{20}-\frac{30\!\cdots\!82}{10\!\cdots\!25}a^{19}+\frac{10\!\cdots\!16}{20\!\cdots\!25}a^{18}-\frac{27\!\cdots\!49}{10\!\cdots\!25}a^{17}-\frac{74\!\cdots\!49}{33\!\cdots\!75}a^{16}+\frac{20\!\cdots\!52}{10\!\cdots\!25}a^{15}-\frac{49\!\cdots\!71}{10\!\cdots\!25}a^{14}+\frac{19\!\cdots\!43}{20\!\cdots\!25}a^{13}-\frac{48\!\cdots\!14}{33\!\cdots\!75}a^{12}+\frac{19\!\cdots\!16}{10\!\cdots\!25}a^{11}-\frac{24\!\cdots\!61}{10\!\cdots\!25}a^{10}+\frac{21\!\cdots\!52}{10\!\cdots\!25}a^{9}-\frac{39\!\cdots\!69}{47\!\cdots\!25}a^{8}+\frac{45\!\cdots\!71}{10\!\cdots\!25}a^{7}-\frac{24\!\cdots\!31}{99\!\cdots\!25}a^{6}-\frac{13\!\cdots\!02}{20\!\cdots\!25}a^{5}+\frac{13\!\cdots\!99}{11\!\cdots\!25}a^{4}-\frac{34\!\cdots\!04}{10\!\cdots\!25}a^{3}+\frac{52\!\cdots\!46}{28\!\cdots\!75}a^{2}-\frac{24\!\cdots\!02}{19\!\cdots\!25}a-\frac{71\!\cdots\!41}{89\!\cdots\!45}$, $\frac{14\!\cdots\!49}{10\!\cdots\!25}a^{28}+\frac{20\!\cdots\!57}{33\!\cdots\!75}a^{27}+\frac{48\!\cdots\!28}{97\!\cdots\!25}a^{26}+\frac{15\!\cdots\!64}{10\!\cdots\!25}a^{25}+\frac{37\!\cdots\!23}{67\!\cdots\!75}a^{24}+\frac{20\!\cdots\!19}{10\!\cdots\!25}a^{23}+\frac{57\!\cdots\!62}{20\!\cdots\!25}a^{22}+\frac{17\!\cdots\!92}{10\!\cdots\!25}a^{21}+\frac{15\!\cdots\!74}{44\!\cdots\!25}a^{20}+\frac{10\!\cdots\!48}{10\!\cdots\!25}a^{19}-\frac{70\!\cdots\!79}{20\!\cdots\!25}a^{18}+\frac{37\!\cdots\!01}{10\!\cdots\!25}a^{17}-\frac{65\!\cdots\!84}{33\!\cdots\!75}a^{16}+\frac{90\!\cdots\!02}{10\!\cdots\!25}a^{15}-\frac{52\!\cdots\!41}{10\!\cdots\!25}a^{14}+\frac{23\!\cdots\!07}{20\!\cdots\!25}a^{13}-\frac{38\!\cdots\!09}{33\!\cdots\!75}a^{12}-\frac{86\!\cdots\!19}{10\!\cdots\!25}a^{11}-\frac{36\!\cdots\!66}{10\!\cdots\!25}a^{10}-\frac{48\!\cdots\!88}{10\!\cdots\!25}a^{9}-\frac{46\!\cdots\!94}{47\!\cdots\!25}a^{8}-\frac{78\!\cdots\!39}{10\!\cdots\!25}a^{7}-\frac{18\!\cdots\!11}{99\!\cdots\!25}a^{6}-\frac{16\!\cdots\!19}{14\!\cdots\!75}a^{5}-\frac{34\!\cdots\!09}{10\!\cdots\!25}a^{4}-\frac{67\!\cdots\!24}{10\!\cdots\!25}a^{3}-\frac{88\!\cdots\!86}{28\!\cdots\!75}a^{2}-\frac{39\!\cdots\!61}{27\!\cdots\!75}a-\frac{23\!\cdots\!89}{89\!\cdots\!45}$, $\frac{41\!\cdots\!63}{53\!\cdots\!25}a^{28}-\frac{14\!\cdots\!71}{14\!\cdots\!75}a^{27}+\frac{35\!\cdots\!33}{14\!\cdots\!75}a^{26}-\frac{26\!\cdots\!43}{47\!\cdots\!25}a^{25}+\frac{38\!\cdots\!68}{95\!\cdots\!25}a^{24}-\frac{45\!\cdots\!68}{47\!\cdots\!25}a^{23}+\frac{36\!\cdots\!51}{95\!\cdots\!25}a^{22}-\frac{13\!\cdots\!28}{15\!\cdots\!75}a^{21}+\frac{21\!\cdots\!27}{95\!\cdots\!25}a^{20}-\frac{20\!\cdots\!56}{47\!\cdots\!25}a^{19}+\frac{16\!\cdots\!27}{19\!\cdots\!25}a^{18}-\frac{64\!\cdots\!07}{47\!\cdots\!25}a^{17}+\frac{96\!\cdots\!09}{47\!\cdots\!25}a^{16}-\frac{12\!\cdots\!19}{47\!\cdots\!25}a^{15}+\frac{13\!\cdots\!92}{47\!\cdots\!25}a^{14}-\frac{37\!\cdots\!33}{10\!\cdots\!25}a^{13}+\frac{16\!\cdots\!77}{68\!\cdots\!75}a^{12}-\frac{20\!\cdots\!87}{47\!\cdots\!25}a^{11}+\frac{83\!\cdots\!42}{47\!\cdots\!25}a^{10}-\frac{29\!\cdots\!04}{47\!\cdots\!25}a^{9}+\frac{56\!\cdots\!73}{47\!\cdots\!25}a^{8}-\frac{74\!\cdots\!82}{47\!\cdots\!25}a^{7}+\frac{12\!\cdots\!47}{47\!\cdots\!25}a^{6}-\frac{73\!\cdots\!94}{22\!\cdots\!25}a^{5}+\frac{27\!\cdots\!13}{47\!\cdots\!25}a^{4}-\frac{44\!\cdots\!76}{14\!\cdots\!75}a^{3}+\frac{14\!\cdots\!89}{28\!\cdots\!75}a^{2}-\frac{69\!\cdots\!06}{19\!\cdots\!25}a+\frac{49\!\cdots\!48}{12\!\cdots\!35}$ 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:  \( 254102374099303.56 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 254102374099303.56 \cdot 1}{2\cdot\sqrt{142449251725173555024565249558533387292386826365409}}\cr\approx \mathstrut & 3.18197522288265 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^29 - x^28 + 33*x^27 - 64*x^26 + 535*x^25 - 1144*x^24 + 5145*x^23 - 10327*x^22 + 31220*x^21 - 55583*x^20 + 124455*x^19 - 188951*x^18 + 323227*x^17 - 411652*x^16 + 521736*x^15 - 591430*x^14 + 489137*x^13 - 738131*x^12 + 175881*x^11 - 1211717*x^10 + 917449*x^9 - 3034006*x^8 + 2818536*x^7 - 6798367*x^6 + 6818314*x^5 - 8408506*x^4 + 7552950*x^3 - 8000650*x^2 + 5448750*x - 1749375)
 
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^29 - x^28 + 33*x^27 - 64*x^26 + 535*x^25 - 1144*x^24 + 5145*x^23 - 10327*x^22 + 31220*x^21 - 55583*x^20 + 124455*x^19 - 188951*x^18 + 323227*x^17 - 411652*x^16 + 521736*x^15 - 591430*x^14 + 489137*x^13 - 738131*x^12 + 175881*x^11 - 1211717*x^10 + 917449*x^9 - 3034006*x^8 + 2818536*x^7 - 6798367*x^6 + 6818314*x^5 - 8408506*x^4 + 7552950*x^3 - 8000650*x^2 + 5448750*x - 1749375, 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^29 - x^28 + 33*x^27 - 64*x^26 + 535*x^25 - 1144*x^24 + 5145*x^23 - 10327*x^22 + 31220*x^21 - 55583*x^20 + 124455*x^19 - 188951*x^18 + 323227*x^17 - 411652*x^16 + 521736*x^15 - 591430*x^14 + 489137*x^13 - 738131*x^12 + 175881*x^11 - 1211717*x^10 + 917449*x^9 - 3034006*x^8 + 2818536*x^7 - 6798367*x^6 + 6818314*x^5 - 8408506*x^4 + 7552950*x^3 - 8000650*x^2 + 5448750*x - 1749375);
 
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^29 - x^28 + 33*x^27 - 64*x^26 + 535*x^25 - 1144*x^24 + 5145*x^23 - 10327*x^22 + 31220*x^21 - 55583*x^20 + 124455*x^19 - 188951*x^18 + 323227*x^17 - 411652*x^16 + 521736*x^15 - 591430*x^14 + 489137*x^13 - 738131*x^12 + 175881*x^11 - 1211717*x^10 + 917449*x^9 - 3034006*x^8 + 2818536*x^7 - 6798367*x^6 + 6818314*x^5 - 8408506*x^4 + 7552950*x^3 - 8000650*x^2 + 5448750*x - 1749375);
 
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

$D_{29}$ (as 29T2):

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 58
The 16 conjugacy class representatives for $D_{29}$
Character table for $D_{29}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $29$ ${\href{/padicField/3.2.0.1}{2} }^{14}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.2.0.1}{2} }^{14}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.2.0.1}{2} }^{14}{,}\,{\href{/padicField/7.1.0.1}{1} }$ $29$ $29$ $29$ ${\href{/padicField/19.2.0.1}{2} }^{14}{,}\,{\href{/padicField/19.1.0.1}{1} }$ $29$ $29$ ${\href{/padicField/31.2.0.1}{2} }^{14}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $29$ $29$ $29$ ${\href{/padicField/47.2.0.1}{2} }^{14}{,}\,{\href{/padicField/47.1.0.1}{1} }$ $29$ $29$

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
\(3823\) Copy content Toggle raw display $\Q_{3823}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3823.2t1.a.a$1$ $ 3823 $ \(\Q(\sqrt{-3823}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3823.29t2.a.d$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.h$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.e$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.k$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.g$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.n$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.a$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.i$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.b$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.l$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.m$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.c$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.j$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.f$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.