Properties

Label 30.30.695...813.1
Degree $30$
Signature $[30, 0]$
Discriminant $6.950\times 10^{52}$
Root discriminant \(57.73\)
Ramified primes $11,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{30}$ (as 30T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 - 49*x^28 + 44*x^27 + 1012*x^26 - 813*x^25 - 11582*x^24 + 8267*x^23 + 81301*x^22 - 51172*x^21 - 366235*x^20 + 201754*x^19 + 1077688*x^18 - 515602*x^17 - 2071990*x^16 + 853441*x^15 + 2566790*x^14 - 898459*x^13 - 1995511*x^12 + 578875*x^11 + 941075*x^10 - 215958*x^9 - 261009*x^8 + 45664*x^7 + 40851*x^6 - 5360*x^5 - 3328*x^4 + 330*x^3 + 117*x^2 - 9*x - 1)
 
gp: K = bnfinit(y^30 - y^29 - 49*y^28 + 44*y^27 + 1012*y^26 - 813*y^25 - 11582*y^24 + 8267*y^23 + 81301*y^22 - 51172*y^21 - 366235*y^20 + 201754*y^19 + 1077688*y^18 - 515602*y^17 - 2071990*y^16 + 853441*y^15 + 2566790*y^14 - 898459*y^13 - 1995511*y^12 + 578875*y^11 + 941075*y^10 - 215958*y^9 - 261009*y^8 + 45664*y^7 + 40851*y^6 - 5360*y^5 - 3328*y^4 + 330*y^3 + 117*y^2 - 9*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - x^29 - 49*x^28 + 44*x^27 + 1012*x^26 - 813*x^25 - 11582*x^24 + 8267*x^23 + 81301*x^22 - 51172*x^21 - 366235*x^20 + 201754*x^19 + 1077688*x^18 - 515602*x^17 - 2071990*x^16 + 853441*x^15 + 2566790*x^14 - 898459*x^13 - 1995511*x^12 + 578875*x^11 + 941075*x^10 - 215958*x^9 - 261009*x^8 + 45664*x^7 + 40851*x^6 - 5360*x^5 - 3328*x^4 + 330*x^3 + 117*x^2 - 9*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - x^29 - 49*x^28 + 44*x^27 + 1012*x^26 - 813*x^25 - 11582*x^24 + 8267*x^23 + 81301*x^22 - 51172*x^21 - 366235*x^20 + 201754*x^19 + 1077688*x^18 - 515602*x^17 - 2071990*x^16 + 853441*x^15 + 2566790*x^14 - 898459*x^13 - 1995511*x^12 + 578875*x^11 + 941075*x^10 - 215958*x^9 - 261009*x^8 + 45664*x^7 + 40851*x^6 - 5360*x^5 - 3328*x^4 + 330*x^3 + 117*x^2 - 9*x - 1)
 

\( x^{30} - x^{29} - 49 x^{28} + 44 x^{27} + 1012 x^{26} - 813 x^{25} - 11582 x^{24} + 8267 x^{23} + 81301 x^{22} - 51172 x^{21} - 366235 x^{20} + 201754 x^{19} + 1077688 x^{18} + \cdots - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[30, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(69503752297329754905479727341904896738456941915804813\) \(\medspace = 11^{24}\cdot 13^{25}\) 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:  \(57.73\)
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:  $11^{4/5}13^{5/6}\approx 57.72982952647523$
Ramified primes:   \(11\), \(13\) 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:  \(\Q(\sqrt{13}) \)
$\card{ \Gal(K/\Q) }$:  $30$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(143=11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{143}(64,·)$, $\chi_{143}(1,·)$, $\chi_{143}(3,·)$, $\chi_{143}(4,·)$, $\chi_{143}(133,·)$, $\chi_{143}(100,·)$, $\chi_{143}(113,·)$, $\chi_{143}(9,·)$, $\chi_{143}(75,·)$, $\chi_{143}(12,·)$, $\chi_{143}(14,·)$, $\chi_{143}(16,·)$, $\chi_{143}(81,·)$, $\chi_{143}(82,·)$, $\chi_{143}(23,·)$, $\chi_{143}(25,·)$, $\chi_{143}(27,·)$, $\chi_{143}(92,·)$, $\chi_{143}(69,·)$, $\chi_{143}(36,·)$, $\chi_{143}(38,·)$, $\chi_{143}(103,·)$, $\chi_{143}(42,·)$, $\chi_{143}(108,·)$, $\chi_{143}(48,·)$, $\chi_{143}(49,·)$, $\chi_{143}(114,·)$, $\chi_{143}(53,·)$, $\chi_{143}(56,·)$, $\chi_{143}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{131}a^{27}+\frac{43}{131}a^{26}+\frac{15}{131}a^{25}-\frac{58}{131}a^{24}+\frac{16}{131}a^{23}-\frac{37}{131}a^{22}+\frac{65}{131}a^{21}-\frac{44}{131}a^{20}-\frac{12}{131}a^{19}+\frac{29}{131}a^{18}+\frac{13}{131}a^{17}+\frac{31}{131}a^{16}-\frac{22}{131}a^{15}-\frac{11}{131}a^{14}+\frac{27}{131}a^{13}-\frac{42}{131}a^{12}-\frac{53}{131}a^{11}-\frac{55}{131}a^{10}+\frac{28}{131}a^{9}+\frac{59}{131}a^{8}+\frac{16}{131}a^{7}-\frac{62}{131}a^{6}+\frac{11}{131}a^{5}-\frac{19}{131}a^{4}-\frac{38}{131}a^{3}-\frac{26}{131}a^{2}+\frac{8}{131}a-\frac{14}{131}$, $\frac{1}{131}a^{28}-\frac{48}{131}a^{25}+\frac{21}{131}a^{24}+\frac{61}{131}a^{23}-\frac{47}{131}a^{22}+\frac{43}{131}a^{21}+\frac{46}{131}a^{20}+\frac{21}{131}a^{19}-\frac{55}{131}a^{18}-\frac{4}{131}a^{17}-\frac{45}{131}a^{16}+\frac{18}{131}a^{15}-\frac{24}{131}a^{14}-\frac{24}{131}a^{13}+\frac{50}{131}a^{12}-\frac{3}{131}a^{11}+\frac{35}{131}a^{10}+\frac{34}{131}a^{9}-\frac{32}{131}a^{8}+\frac{36}{131}a^{7}+\frac{57}{131}a^{6}+\frac{32}{131}a^{5}-\frac{7}{131}a^{4}+\frac{36}{131}a^{3}-\frac{53}{131}a^{2}+\frac{35}{131}a-\frac{53}{131}$, $\frac{1}{61\!\cdots\!81}a^{29}-\frac{18\!\cdots\!99}{61\!\cdots\!81}a^{28}-\frac{39\!\cdots\!62}{61\!\cdots\!81}a^{27}+\frac{24\!\cdots\!93}{61\!\cdots\!81}a^{26}+\frac{76\!\cdots\!27}{61\!\cdots\!81}a^{25}-\frac{58\!\cdots\!62}{61\!\cdots\!81}a^{24}+\frac{17\!\cdots\!55}{61\!\cdots\!81}a^{23}+\frac{23\!\cdots\!92}{61\!\cdots\!81}a^{22}-\frac{19\!\cdots\!34}{61\!\cdots\!81}a^{21}-\frac{89\!\cdots\!20}{61\!\cdots\!81}a^{20}-\frac{53\!\cdots\!58}{61\!\cdots\!81}a^{19}+\frac{13\!\cdots\!85}{61\!\cdots\!81}a^{18}-\frac{17\!\cdots\!27}{61\!\cdots\!81}a^{17}-\frac{13\!\cdots\!20}{61\!\cdots\!81}a^{16}-\frac{14\!\cdots\!82}{61\!\cdots\!81}a^{15}-\frac{30\!\cdots\!31}{61\!\cdots\!81}a^{14}+\frac{40\!\cdots\!30}{61\!\cdots\!81}a^{13}+\frac{13\!\cdots\!83}{61\!\cdots\!81}a^{12}+\frac{40\!\cdots\!41}{61\!\cdots\!81}a^{11}-\frac{24\!\cdots\!38}{61\!\cdots\!81}a^{10}+\frac{30\!\cdots\!79}{61\!\cdots\!81}a^{9}+\frac{14\!\cdots\!57}{61\!\cdots\!81}a^{8}-\frac{62\!\cdots\!20}{61\!\cdots\!81}a^{7}+\frac{15\!\cdots\!50}{61\!\cdots\!81}a^{6}+\frac{13\!\cdots\!17}{61\!\cdots\!81}a^{5}-\frac{10\!\cdots\!27}{61\!\cdots\!81}a^{4}-\frac{46\!\cdots\!94}{61\!\cdots\!81}a^{3}-\frac{41\!\cdots\!42}{61\!\cdots\!81}a^{2}-\frac{57\!\cdots\!01}{61\!\cdots\!81}a+\frac{15\!\cdots\!01}{61\!\cdots\!81}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

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:  $29$
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:   $\frac{43\!\cdots\!00}{61\!\cdots\!81}a^{29}-\frac{24\!\cdots\!96}{61\!\cdots\!81}a^{28}-\frac{21\!\cdots\!74}{61\!\cdots\!81}a^{27}+\frac{98\!\cdots\!17}{61\!\cdots\!81}a^{26}+\frac{44\!\cdots\!62}{61\!\cdots\!81}a^{25}-\frac{16\!\cdots\!47}{61\!\cdots\!81}a^{24}-\frac{51\!\cdots\!17}{61\!\cdots\!81}a^{23}+\frac{14\!\cdots\!64}{61\!\cdots\!81}a^{22}+\frac{36\!\cdots\!34}{61\!\cdots\!81}a^{21}-\frac{69\!\cdots\!63}{61\!\cdots\!81}a^{20}-\frac{16\!\cdots\!79}{61\!\cdots\!81}a^{19}+\frac{19\!\cdots\!21}{61\!\cdots\!81}a^{18}+\frac{49\!\cdots\!55}{61\!\cdots\!81}a^{17}-\frac{23\!\cdots\!01}{61\!\cdots\!81}a^{16}-\frac{97\!\cdots\!59}{61\!\cdots\!81}a^{15}-\frac{10\!\cdots\!89}{61\!\cdots\!81}a^{14}+\frac{12\!\cdots\!73}{61\!\cdots\!81}a^{13}+\frac{73\!\cdots\!07}{61\!\cdots\!81}a^{12}-\frac{98\!\cdots\!70}{61\!\cdots\!81}a^{11}-\frac{98\!\cdots\!51}{61\!\cdots\!81}a^{10}+\frac{47\!\cdots\!27}{61\!\cdots\!81}a^{9}+\frac{63\!\cdots\!04}{61\!\cdots\!81}a^{8}-\frac{13\!\cdots\!38}{61\!\cdots\!81}a^{7}-\frac{20\!\cdots\!46}{61\!\cdots\!81}a^{6}+\frac{20\!\cdots\!47}{61\!\cdots\!81}a^{5}+\frac{31\!\cdots\!57}{61\!\cdots\!81}a^{4}-\frac{15\!\cdots\!65}{61\!\cdots\!81}a^{3}-\frac{20\!\cdots\!24}{61\!\cdots\!81}a^{2}+\frac{42\!\cdots\!46}{61\!\cdots\!81}a+\frac{33\!\cdots\!96}{61\!\cdots\!81}$, $\frac{17\!\cdots\!74}{61\!\cdots\!81}a^{29}-\frac{15\!\cdots\!18}{61\!\cdots\!81}a^{28}-\frac{85\!\cdots\!29}{61\!\cdots\!81}a^{27}+\frac{67\!\cdots\!07}{61\!\cdots\!81}a^{26}+\frac{17\!\cdots\!36}{61\!\cdots\!81}a^{25}-\frac{93\!\cdots\!86}{47\!\cdots\!51}a^{24}-\frac{20\!\cdots\!94}{61\!\cdots\!81}a^{23}+\frac{12\!\cdots\!15}{61\!\cdots\!81}a^{22}+\frac{14\!\cdots\!47}{61\!\cdots\!81}a^{21}-\frac{73\!\cdots\!63}{61\!\cdots\!81}a^{20}-\frac{63\!\cdots\!11}{61\!\cdots\!81}a^{19}+\frac{27\!\cdots\!45}{61\!\cdots\!81}a^{18}+\frac{18\!\cdots\!11}{61\!\cdots\!81}a^{17}-\frac{67\!\cdots\!13}{61\!\cdots\!81}a^{16}-\frac{35\!\cdots\!87}{61\!\cdots\!81}a^{15}+\frac{10\!\cdots\!73}{61\!\cdots\!81}a^{14}+\frac{44\!\cdots\!06}{61\!\cdots\!81}a^{13}-\frac{10\!\cdots\!73}{61\!\cdots\!81}a^{12}-\frac{34\!\cdots\!25}{61\!\cdots\!81}a^{11}+\frac{56\!\cdots\!14}{61\!\cdots\!81}a^{10}+\frac{15\!\cdots\!59}{61\!\cdots\!81}a^{9}-\frac{15\!\cdots\!15}{61\!\cdots\!81}a^{8}-\frac{41\!\cdots\!01}{61\!\cdots\!81}a^{7}+\frac{16\!\cdots\!15}{61\!\cdots\!81}a^{6}+\frac{60\!\cdots\!46}{61\!\cdots\!81}a^{5}+\frac{73\!\cdots\!40}{61\!\cdots\!81}a^{4}-\frac{41\!\cdots\!03}{61\!\cdots\!81}a^{3}-\frac{21\!\cdots\!92}{61\!\cdots\!81}a^{2}+\frac{95\!\cdots\!35}{61\!\cdots\!81}a+\frac{61\!\cdots\!40}{61\!\cdots\!81}$, $\frac{43\!\cdots\!35}{61\!\cdots\!81}a^{29}-\frac{54\!\cdots\!16}{61\!\cdots\!81}a^{28}-\frac{21\!\cdots\!32}{61\!\cdots\!81}a^{27}+\frac{24\!\cdots\!98}{61\!\cdots\!81}a^{26}+\frac{43\!\cdots\!77}{61\!\cdots\!81}a^{25}-\frac{46\!\cdots\!26}{61\!\cdots\!81}a^{24}-\frac{49\!\cdots\!63}{61\!\cdots\!81}a^{23}+\frac{48\!\cdots\!60}{61\!\cdots\!81}a^{22}+\frac{34\!\cdots\!26}{61\!\cdots\!81}a^{21}-\frac{30\!\cdots\!59}{61\!\cdots\!81}a^{20}-\frac{15\!\cdots\!44}{61\!\cdots\!81}a^{19}+\frac{12\!\cdots\!13}{61\!\cdots\!81}a^{18}+\frac{43\!\cdots\!63}{61\!\cdots\!81}a^{17}-\frac{33\!\cdots\!41}{61\!\cdots\!81}a^{16}-\frac{81\!\cdots\!78}{61\!\cdots\!81}a^{15}+\frac{56\!\cdots\!55}{61\!\cdots\!81}a^{14}+\frac{96\!\cdots\!40}{61\!\cdots\!81}a^{13}-\frac{61\!\cdots\!95}{61\!\cdots\!81}a^{12}-\frac{70\!\cdots\!34}{61\!\cdots\!81}a^{11}+\frac{41\!\cdots\!90}{61\!\cdots\!81}a^{10}+\frac{29\!\cdots\!51}{61\!\cdots\!81}a^{9}-\frac{15\!\cdots\!55}{61\!\cdots\!81}a^{8}-\frac{70\!\cdots\!64}{61\!\cdots\!81}a^{7}+\frac{33\!\cdots\!25}{61\!\cdots\!81}a^{6}+\frac{84\!\cdots\!08}{61\!\cdots\!81}a^{5}-\frac{34\!\cdots\!82}{61\!\cdots\!81}a^{4}-\frac{38\!\cdots\!08}{61\!\cdots\!81}a^{3}+\frac{14\!\cdots\!11}{61\!\cdots\!81}a^{2}-\frac{35\!\cdots\!07}{61\!\cdots\!81}a-\frac{68\!\cdots\!39}{61\!\cdots\!81}$, $\frac{64\!\cdots\!39}{61\!\cdots\!81}a^{29}-\frac{33\!\cdots\!17}{61\!\cdots\!81}a^{28}-\frac{31\!\cdots\!47}{61\!\cdots\!81}a^{27}+\frac{13\!\cdots\!99}{61\!\cdots\!81}a^{26}+\frac{66\!\cdots\!24}{61\!\cdots\!81}a^{25}-\frac{21\!\cdots\!86}{61\!\cdots\!81}a^{24}-\frac{76\!\cdots\!01}{61\!\cdots\!81}a^{23}+\frac{18\!\cdots\!85}{61\!\cdots\!81}a^{22}+\frac{54\!\cdots\!36}{61\!\cdots\!81}a^{21}-\frac{84\!\cdots\!89}{61\!\cdots\!81}a^{20}-\frac{25\!\cdots\!12}{61\!\cdots\!81}a^{19}+\frac{19\!\cdots\!87}{61\!\cdots\!81}a^{18}+\frac{75\!\cdots\!18}{61\!\cdots\!81}a^{17}-\frac{11\!\cdots\!97}{61\!\cdots\!81}a^{16}-\frac{14\!\cdots\!79}{61\!\cdots\!81}a^{15}-\frac{60\!\cdots\!12}{61\!\cdots\!81}a^{14}+\frac{18\!\cdots\!06}{61\!\cdots\!81}a^{13}+\frac{16\!\cdots\!02}{61\!\cdots\!81}a^{12}-\frac{15\!\cdots\!91}{61\!\cdots\!81}a^{11}-\frac{18\!\cdots\!91}{61\!\cdots\!81}a^{10}+\frac{76\!\cdots\!58}{61\!\cdots\!81}a^{9}+\frac{11\!\cdots\!30}{61\!\cdots\!81}a^{8}-\frac{22\!\cdots\!12}{61\!\cdots\!81}a^{7}-\frac{36\!\cdots\!00}{61\!\cdots\!81}a^{6}+\frac{35\!\cdots\!13}{61\!\cdots\!81}a^{5}+\frac{56\!\cdots\!17}{61\!\cdots\!81}a^{4}-\frac{27\!\cdots\!40}{61\!\cdots\!81}a^{3}-\frac{37\!\cdots\!43}{61\!\cdots\!81}a^{2}+\frac{77\!\cdots\!57}{61\!\cdots\!81}a+\frac{62\!\cdots\!44}{61\!\cdots\!81}$, $\frac{13\!\cdots\!78}{61\!\cdots\!81}a^{29}-\frac{12\!\cdots\!84}{61\!\cdots\!81}a^{28}-\frac{64\!\cdots\!25}{61\!\cdots\!81}a^{27}+\frac{56\!\cdots\!78}{61\!\cdots\!81}a^{26}+\frac{13\!\cdots\!38}{61\!\cdots\!81}a^{25}-\frac{10\!\cdots\!19}{61\!\cdots\!81}a^{24}-\frac{15\!\cdots\!16}{61\!\cdots\!81}a^{23}+\frac{10\!\cdots\!93}{61\!\cdots\!81}a^{22}+\frac{10\!\cdots\!69}{61\!\cdots\!81}a^{21}-\frac{63\!\cdots\!15}{61\!\cdots\!81}a^{20}-\frac{47\!\cdots\!70}{61\!\cdots\!81}a^{19}+\frac{24\!\cdots\!90}{61\!\cdots\!81}a^{18}+\frac{13\!\cdots\!57}{61\!\cdots\!81}a^{17}-\frac{62\!\cdots\!15}{61\!\cdots\!81}a^{16}-\frac{26\!\cdots\!22}{61\!\cdots\!81}a^{15}+\frac{10\!\cdots\!52}{61\!\cdots\!81}a^{14}+\frac{31\!\cdots\!82}{61\!\cdots\!81}a^{13}-\frac{10\!\cdots\!51}{61\!\cdots\!81}a^{12}-\frac{24\!\cdots\!59}{61\!\cdots\!81}a^{11}+\frac{60\!\cdots\!06}{61\!\cdots\!81}a^{10}+\frac{10\!\cdots\!46}{61\!\cdots\!81}a^{9}-\frac{19\!\cdots\!68}{61\!\cdots\!81}a^{8}-\frac{27\!\cdots\!09}{61\!\cdots\!81}a^{7}+\frac{31\!\cdots\!71}{61\!\cdots\!81}a^{6}+\frac{37\!\cdots\!11}{61\!\cdots\!81}a^{5}-\frac{20\!\cdots\!77}{61\!\cdots\!81}a^{4}-\frac{23\!\cdots\!93}{61\!\cdots\!81}a^{3}-\frac{27\!\cdots\!38}{61\!\cdots\!81}a^{2}+\frac{45\!\cdots\!70}{61\!\cdots\!81}a+\frac{25\!\cdots\!13}{61\!\cdots\!81}$, $\frac{47\!\cdots\!28}{61\!\cdots\!81}a^{29}-\frac{48\!\cdots\!15}{61\!\cdots\!81}a^{28}-\frac{23\!\cdots\!39}{61\!\cdots\!81}a^{27}+\frac{21\!\cdots\!23}{61\!\cdots\!81}a^{26}+\frac{47\!\cdots\!88}{61\!\cdots\!81}a^{25}-\frac{39\!\cdots\!45}{61\!\cdots\!81}a^{24}-\frac{54\!\cdots\!88}{61\!\cdots\!81}a^{23}+\frac{40\!\cdots\!06}{61\!\cdots\!81}a^{22}+\frac{38\!\cdots\!89}{61\!\cdots\!81}a^{21}-\frac{24\!\cdots\!92}{61\!\cdots\!81}a^{20}-\frac{17\!\cdots\!35}{61\!\cdots\!81}a^{19}+\frac{97\!\cdots\!85}{61\!\cdots\!81}a^{18}+\frac{49\!\cdots\!78}{61\!\cdots\!81}a^{17}-\frac{24\!\cdots\!50}{61\!\cdots\!81}a^{16}-\frac{93\!\cdots\!91}{61\!\cdots\!81}a^{15}+\frac{40\!\cdots\!40}{61\!\cdots\!81}a^{14}+\frac{11\!\cdots\!30}{61\!\cdots\!81}a^{13}-\frac{40\!\cdots\!60}{61\!\cdots\!81}a^{12}-\frac{84\!\cdots\!27}{61\!\cdots\!81}a^{11}+\frac{24\!\cdots\!52}{61\!\cdots\!81}a^{10}+\frac{36\!\cdots\!47}{61\!\cdots\!81}a^{9}-\frac{82\!\cdots\!98}{61\!\cdots\!81}a^{8}-\frac{91\!\cdots\!75}{61\!\cdots\!81}a^{7}+\frac{14\!\cdots\!63}{61\!\cdots\!81}a^{6}+\frac{11\!\cdots\!00}{61\!\cdots\!81}a^{5}-\frac{10\!\cdots\!79}{61\!\cdots\!81}a^{4}-\frac{71\!\cdots\!86}{61\!\cdots\!81}a^{3}+\frac{18\!\cdots\!99}{61\!\cdots\!81}a^{2}+\frac{12\!\cdots\!74}{61\!\cdots\!81}a+\frac{71\!\cdots\!61}{61\!\cdots\!81}$, $\frac{16\!\cdots\!43}{61\!\cdots\!81}a^{29}-\frac{13\!\cdots\!85}{61\!\cdots\!81}a^{28}-\frac{80\!\cdots\!53}{61\!\cdots\!81}a^{27}+\frac{60\!\cdots\!27}{61\!\cdots\!81}a^{26}+\frac{16\!\cdots\!36}{61\!\cdots\!81}a^{25}-\frac{10\!\cdots\!94}{61\!\cdots\!81}a^{24}-\frac{19\!\cdots\!94}{61\!\cdots\!81}a^{23}+\frac{10\!\cdots\!30}{61\!\cdots\!81}a^{22}+\frac{13\!\cdots\!04}{61\!\cdots\!81}a^{21}-\frac{63\!\cdots\!08}{61\!\cdots\!81}a^{20}-\frac{60\!\cdots\!68}{61\!\cdots\!81}a^{19}+\frac{23\!\cdots\!31}{61\!\cdots\!81}a^{18}+\frac{17\!\cdots\!53}{61\!\cdots\!81}a^{17}-\frac{56\!\cdots\!06}{61\!\cdots\!81}a^{16}-\frac{34\!\cdots\!99}{61\!\cdots\!81}a^{15}+\frac{86\!\cdots\!46}{61\!\cdots\!81}a^{14}+\frac{42\!\cdots\!49}{61\!\cdots\!81}a^{13}-\frac{79\!\cdots\!59}{61\!\cdots\!81}a^{12}-\frac{25\!\cdots\!78}{47\!\cdots\!51}a^{11}+\frac{41\!\cdots\!38}{61\!\cdots\!81}a^{10}+\frac{15\!\cdots\!12}{61\!\cdots\!81}a^{9}-\frac{94\!\cdots\!75}{61\!\cdots\!81}a^{8}-\frac{41\!\cdots\!74}{61\!\cdots\!81}a^{7}+\frac{20\!\cdots\!91}{61\!\cdots\!81}a^{6}+\frac{59\!\cdots\!97}{61\!\cdots\!81}a^{5}+\frac{23\!\cdots\!15}{61\!\cdots\!81}a^{4}-\frac{42\!\cdots\!42}{61\!\cdots\!81}a^{3}-\frac{29\!\cdots\!55}{61\!\cdots\!81}a^{2}+\frac{10\!\cdots\!20}{61\!\cdots\!81}a+\frac{70\!\cdots\!95}{61\!\cdots\!81}$, $\frac{27\!\cdots\!25}{47\!\cdots\!51}a^{29}-\frac{19\!\cdots\!27}{61\!\cdots\!81}a^{28}-\frac{17\!\cdots\!22}{61\!\cdots\!81}a^{27}+\frac{79\!\cdots\!45}{61\!\cdots\!81}a^{26}+\frac{37\!\cdots\!95}{61\!\cdots\!81}a^{25}-\frac{12\!\cdots\!73}{61\!\cdots\!81}a^{24}-\frac{42\!\cdots\!09}{61\!\cdots\!81}a^{23}+\frac{11\!\cdots\!92}{61\!\cdots\!81}a^{22}+\frac{30\!\cdots\!87}{61\!\cdots\!81}a^{21}-\frac{53\!\cdots\!55}{61\!\cdots\!81}a^{20}-\frac{13\!\cdots\!85}{61\!\cdots\!81}a^{19}+\frac{14\!\cdots\!84}{61\!\cdots\!81}a^{18}+\frac{41\!\cdots\!59}{61\!\cdots\!81}a^{17}-\frac{14\!\cdots\!16}{61\!\cdots\!81}a^{16}-\frac{81\!\cdots\!20}{61\!\cdots\!81}a^{15}-\frac{17\!\cdots\!21}{61\!\cdots\!81}a^{14}+\frac{10\!\cdots\!20}{61\!\cdots\!81}a^{13}+\frac{71\!\cdots\!19}{61\!\cdots\!81}a^{12}-\frac{83\!\cdots\!72}{61\!\cdots\!81}a^{11}-\frac{89\!\cdots\!14}{61\!\cdots\!81}a^{10}+\frac{40\!\cdots\!91}{61\!\cdots\!81}a^{9}+\frac{55\!\cdots\!02}{61\!\cdots\!81}a^{8}-\frac{11\!\cdots\!77}{61\!\cdots\!81}a^{7}-\frac{17\!\cdots\!56}{61\!\cdots\!81}a^{6}+\frac{17\!\cdots\!42}{61\!\cdots\!81}a^{5}+\frac{26\!\cdots\!49}{61\!\cdots\!81}a^{4}-\frac{13\!\cdots\!96}{61\!\cdots\!81}a^{3}-\frac{17\!\cdots\!92}{61\!\cdots\!81}a^{2}+\frac{35\!\cdots\!18}{61\!\cdots\!81}a+\frac{28\!\cdots\!55}{61\!\cdots\!81}$, $\frac{16\!\cdots\!43}{61\!\cdots\!81}a^{29}-\frac{13\!\cdots\!85}{61\!\cdots\!81}a^{28}-\frac{80\!\cdots\!53}{61\!\cdots\!81}a^{27}+\frac{60\!\cdots\!27}{61\!\cdots\!81}a^{26}+\frac{16\!\cdots\!36}{61\!\cdots\!81}a^{25}-\frac{10\!\cdots\!94}{61\!\cdots\!81}a^{24}-\frac{19\!\cdots\!94}{61\!\cdots\!81}a^{23}+\frac{10\!\cdots\!30}{61\!\cdots\!81}a^{22}+\frac{13\!\cdots\!04}{61\!\cdots\!81}a^{21}-\frac{63\!\cdots\!08}{61\!\cdots\!81}a^{20}-\frac{60\!\cdots\!68}{61\!\cdots\!81}a^{19}+\frac{23\!\cdots\!31}{61\!\cdots\!81}a^{18}+\frac{17\!\cdots\!53}{61\!\cdots\!81}a^{17}-\frac{56\!\cdots\!06}{61\!\cdots\!81}a^{16}-\frac{34\!\cdots\!99}{61\!\cdots\!81}a^{15}+\frac{86\!\cdots\!46}{61\!\cdots\!81}a^{14}+\frac{42\!\cdots\!49}{61\!\cdots\!81}a^{13}-\frac{79\!\cdots\!59}{61\!\cdots\!81}a^{12}-\frac{25\!\cdots\!78}{47\!\cdots\!51}a^{11}+\frac{41\!\cdots\!38}{61\!\cdots\!81}a^{10}+\frac{15\!\cdots\!12}{61\!\cdots\!81}a^{9}-\frac{94\!\cdots\!75}{61\!\cdots\!81}a^{8}-\frac{41\!\cdots\!74}{61\!\cdots\!81}a^{7}+\frac{20\!\cdots\!91}{61\!\cdots\!81}a^{6}+\frac{59\!\cdots\!97}{61\!\cdots\!81}a^{5}+\frac{23\!\cdots\!15}{61\!\cdots\!81}a^{4}-\frac{42\!\cdots\!42}{61\!\cdots\!81}a^{3}-\frac{29\!\cdots\!55}{61\!\cdots\!81}a^{2}+\frac{10\!\cdots\!20}{61\!\cdots\!81}a+\frac{70\!\cdots\!14}{61\!\cdots\!81}$, $\frac{22\!\cdots\!43}{61\!\cdots\!81}a^{29}-\frac{20\!\cdots\!70}{61\!\cdots\!81}a^{28}-\frac{11\!\cdots\!03}{61\!\cdots\!81}a^{27}+\frac{89\!\cdots\!37}{61\!\cdots\!81}a^{26}+\frac{23\!\cdots\!71}{61\!\cdots\!81}a^{25}-\frac{16\!\cdots\!48}{61\!\cdots\!81}a^{24}-\frac{26\!\cdots\!24}{61\!\cdots\!81}a^{23}+\frac{16\!\cdots\!00}{61\!\cdots\!81}a^{22}+\frac{18\!\cdots\!89}{61\!\cdots\!81}a^{21}-\frac{97\!\cdots\!23}{61\!\cdots\!81}a^{20}-\frac{83\!\cdots\!23}{61\!\cdots\!81}a^{19}+\frac{37\!\cdots\!76}{61\!\cdots\!81}a^{18}+\frac{24\!\cdots\!83}{61\!\cdots\!81}a^{17}-\frac{90\!\cdots\!81}{61\!\cdots\!81}a^{16}-\frac{47\!\cdots\!29}{61\!\cdots\!81}a^{15}+\frac{14\!\cdots\!76}{61\!\cdots\!81}a^{14}+\frac{58\!\cdots\!09}{61\!\cdots\!81}a^{13}-\frac{13\!\cdots\!39}{61\!\cdots\!81}a^{12}-\frac{44\!\cdots\!18}{61\!\cdots\!81}a^{11}+\frac{75\!\cdots\!53}{61\!\cdots\!81}a^{10}+\frac{20\!\cdots\!62}{61\!\cdots\!81}a^{9}-\frac{20\!\cdots\!65}{61\!\cdots\!81}a^{8}-\frac{53\!\cdots\!99}{61\!\cdots\!81}a^{7}+\frac{21\!\cdots\!81}{61\!\cdots\!81}a^{6}+\frac{76\!\cdots\!22}{61\!\cdots\!81}a^{5}+\frac{93\!\cdots\!20}{61\!\cdots\!81}a^{4}-\frac{52\!\cdots\!97}{61\!\cdots\!81}a^{3}-\frac{26\!\cdots\!15}{61\!\cdots\!81}a^{2}+\frac{11\!\cdots\!05}{61\!\cdots\!81}a+\frac{77\!\cdots\!30}{61\!\cdots\!81}$, $\frac{42\!\cdots\!99}{61\!\cdots\!81}a^{29}-\frac{67\!\cdots\!42}{61\!\cdots\!81}a^{28}-\frac{20\!\cdots\!59}{61\!\cdots\!81}a^{27}+\frac{30\!\cdots\!64}{61\!\cdots\!81}a^{26}+\frac{41\!\cdots\!14}{61\!\cdots\!81}a^{25}-\frac{59\!\cdots\!11}{61\!\cdots\!81}a^{24}-\frac{46\!\cdots\!43}{61\!\cdots\!81}a^{23}+\frac{63\!\cdots\!53}{61\!\cdots\!81}a^{22}+\frac{31\!\cdots\!73}{61\!\cdots\!81}a^{21}-\frac{41\!\cdots\!79}{61\!\cdots\!81}a^{20}-\frac{13\!\cdots\!25}{61\!\cdots\!81}a^{19}+\frac{17\!\cdots\!10}{61\!\cdots\!81}a^{18}+\frac{38\!\cdots\!26}{61\!\cdots\!81}a^{17}-\frac{46\!\cdots\!14}{61\!\cdots\!81}a^{16}-\frac{69\!\cdots\!47}{61\!\cdots\!81}a^{15}+\frac{82\!\cdots\!60}{61\!\cdots\!81}a^{14}+\frac{77\!\cdots\!66}{61\!\cdots\!81}a^{13}-\frac{92\!\cdots\!12}{61\!\cdots\!81}a^{12}-\frac{50\!\cdots\!98}{61\!\cdots\!81}a^{11}+\frac{63\!\cdots\!79}{61\!\cdots\!81}a^{10}+\frac{17\!\cdots\!54}{61\!\cdots\!81}a^{9}-\frac{25\!\cdots\!99}{61\!\cdots\!81}a^{8}-\frac{24\!\cdots\!61}{61\!\cdots\!81}a^{7}+\frac{57\!\cdots\!81}{61\!\cdots\!81}a^{6}-\frac{56\!\cdots\!39}{61\!\cdots\!81}a^{5}-\frac{66\!\cdots\!88}{61\!\cdots\!81}a^{4}+\frac{45\!\cdots\!69}{61\!\cdots\!81}a^{3}+\frac{32\!\cdots\!40}{61\!\cdots\!81}a^{2}-\frac{27\!\cdots\!30}{61\!\cdots\!81}a-\frac{32\!\cdots\!16}{61\!\cdots\!81}$, $\frac{10\!\cdots\!75}{61\!\cdots\!81}a^{29}-\frac{86\!\cdots\!12}{61\!\cdots\!81}a^{28}-\frac{49\!\cdots\!72}{61\!\cdots\!81}a^{27}+\frac{37\!\cdots\!55}{61\!\cdots\!81}a^{26}+\frac{10\!\cdots\!30}{61\!\cdots\!81}a^{25}-\frac{67\!\cdots\!27}{61\!\cdots\!81}a^{24}-\frac{11\!\cdots\!39}{61\!\cdots\!81}a^{23}+\frac{66\!\cdots\!62}{61\!\cdots\!81}a^{22}+\frac{83\!\cdots\!72}{61\!\cdots\!81}a^{21}-\frac{39\!\cdots\!70}{61\!\cdots\!81}a^{20}-\frac{37\!\cdots\!40}{61\!\cdots\!81}a^{19}+\frac{14\!\cdots\!29}{61\!\cdots\!81}a^{18}+\frac{11\!\cdots\!89}{61\!\cdots\!81}a^{17}-\frac{35\!\cdots\!91}{61\!\cdots\!81}a^{16}-\frac{21\!\cdots\!50}{61\!\cdots\!81}a^{15}+\frac{52\!\cdots\!09}{61\!\cdots\!81}a^{14}+\frac{26\!\cdots\!80}{61\!\cdots\!81}a^{13}-\frac{48\!\cdots\!61}{61\!\cdots\!81}a^{12}-\frac{19\!\cdots\!72}{61\!\cdots\!81}a^{11}+\frac{24\!\cdots\!01}{61\!\cdots\!81}a^{10}+\frac{91\!\cdots\!41}{61\!\cdots\!81}a^{9}-\frac{56\!\cdots\!88}{61\!\cdots\!81}a^{8}-\frac{24\!\cdots\!02}{61\!\cdots\!81}a^{7}+\frac{15\!\cdots\!34}{61\!\cdots\!81}a^{6}+\frac{34\!\cdots\!67}{61\!\cdots\!81}a^{5}+\frac{12\!\cdots\!54}{61\!\cdots\!81}a^{4}-\frac{23\!\cdots\!51}{61\!\cdots\!81}a^{3}-\frac{14\!\cdots\!52}{61\!\cdots\!81}a^{2}+\frac{53\!\cdots\!03}{61\!\cdots\!81}a+\frac{35\!\cdots\!90}{61\!\cdots\!81}$, $\frac{10\!\cdots\!07}{61\!\cdots\!81}a^{29}-\frac{96\!\cdots\!82}{61\!\cdots\!81}a^{28}-\frac{51\!\cdots\!17}{61\!\cdots\!81}a^{27}+\frac{42\!\cdots\!02}{61\!\cdots\!81}a^{26}+\frac{10\!\cdots\!86}{61\!\cdots\!81}a^{25}-\frac{76\!\cdots\!82}{61\!\cdots\!81}a^{24}-\frac{12\!\cdots\!22}{61\!\cdots\!81}a^{23}+\frac{77\!\cdots\!88}{61\!\cdots\!81}a^{22}+\frac{85\!\cdots\!56}{61\!\cdots\!81}a^{21}-\frac{46\!\cdots\!45}{61\!\cdots\!81}a^{20}-\frac{38\!\cdots\!43}{61\!\cdots\!81}a^{19}+\frac{17\!\cdots\!94}{61\!\cdots\!81}a^{18}+\frac{11\!\cdots\!70}{61\!\cdots\!81}a^{17}-\frac{44\!\cdots\!67}{61\!\cdots\!81}a^{16}-\frac{21\!\cdots\!76}{61\!\cdots\!81}a^{15}+\frac{70\!\cdots\!56}{61\!\cdots\!81}a^{14}+\frac{26\!\cdots\!43}{61\!\cdots\!81}a^{13}-\frac{69\!\cdots\!08}{61\!\cdots\!81}a^{12}-\frac{20\!\cdots\!77}{61\!\cdots\!81}a^{11}+\frac{40\!\cdots\!02}{61\!\cdots\!81}a^{10}+\frac{92\!\cdots\!81}{61\!\cdots\!81}a^{9}-\frac{12\!\cdots\!43}{61\!\cdots\!81}a^{8}-\frac{24\!\cdots\!52}{61\!\cdots\!81}a^{7}+\frac{17\!\cdots\!06}{61\!\cdots\!81}a^{6}+\frac{33\!\cdots\!80}{61\!\cdots\!81}a^{5}-\frac{72\!\cdots\!41}{61\!\cdots\!81}a^{4}-\frac{22\!\cdots\!42}{61\!\cdots\!81}a^{3}-\frac{55\!\cdots\!72}{61\!\cdots\!81}a^{2}+\frac{47\!\cdots\!50}{61\!\cdots\!81}a+\frac{26\!\cdots\!12}{61\!\cdots\!81}$, $\frac{40\!\cdots\!76}{61\!\cdots\!81}a^{29}-\frac{36\!\cdots\!51}{61\!\cdots\!81}a^{28}-\frac{19\!\cdots\!67}{61\!\cdots\!81}a^{27}+\frac{15\!\cdots\!35}{61\!\cdots\!81}a^{26}+\frac{40\!\cdots\!95}{61\!\cdots\!81}a^{25}-\frac{28\!\cdots\!66}{61\!\cdots\!81}a^{24}-\frac{47\!\cdots\!60}{61\!\cdots\!81}a^{23}+\frac{28\!\cdots\!50}{61\!\cdots\!81}a^{22}+\frac{33\!\cdots\!03}{61\!\cdots\!81}a^{21}-\frac{17\!\cdots\!81}{61\!\cdots\!81}a^{20}-\frac{14\!\cdots\!46}{61\!\cdots\!81}a^{19}+\frac{65\!\cdots\!69}{61\!\cdots\!81}a^{18}+\frac{44\!\cdots\!05}{61\!\cdots\!81}a^{17}-\frac{16\!\cdots\!70}{61\!\cdots\!81}a^{16}-\frac{85\!\cdots\!40}{61\!\cdots\!81}a^{15}+\frac{25\!\cdots\!15}{61\!\cdots\!81}a^{14}+\frac{10\!\cdots\!89}{61\!\cdots\!81}a^{13}-\frac{24\!\cdots\!94}{61\!\cdots\!81}a^{12}-\frac{83\!\cdots\!60}{61\!\cdots\!81}a^{11}+\frac{14\!\cdots\!61}{61\!\cdots\!81}a^{10}+\frac{39\!\cdots\!79}{61\!\cdots\!81}a^{9}-\frac{41\!\cdots\!78}{61\!\cdots\!81}a^{8}-\frac{10\!\cdots\!06}{61\!\cdots\!81}a^{7}+\frac{48\!\cdots\!15}{61\!\cdots\!81}a^{6}+\frac{16\!\cdots\!19}{61\!\cdots\!81}a^{5}+\frac{15\!\cdots\!85}{61\!\cdots\!81}a^{4}-\frac{12\!\cdots\!27}{61\!\cdots\!81}a^{3}-\frac{62\!\cdots\!52}{61\!\cdots\!81}a^{2}+\frac{30\!\cdots\!74}{61\!\cdots\!81}a+\frac{20\!\cdots\!50}{61\!\cdots\!81}$, $\frac{90\!\cdots\!23}{61\!\cdots\!81}a^{29}-\frac{72\!\cdots\!10}{61\!\cdots\!81}a^{28}-\frac{44\!\cdots\!35}{61\!\cdots\!81}a^{27}+\frac{30\!\cdots\!70}{61\!\cdots\!81}a^{26}+\frac{92\!\cdots\!19}{61\!\cdots\!81}a^{25}-\frac{55\!\cdots\!66}{61\!\cdots\!81}a^{24}-\frac{10\!\cdots\!51}{61\!\cdots\!81}a^{23}+\frac{53\!\cdots\!48}{61\!\cdots\!81}a^{22}+\frac{74\!\cdots\!96}{61\!\cdots\!81}a^{21}-\frac{31\!\cdots\!09}{61\!\cdots\!81}a^{20}-\frac{25\!\cdots\!08}{47\!\cdots\!51}a^{19}+\frac{11\!\cdots\!93}{61\!\cdots\!81}a^{18}+\frac{10\!\cdots\!53}{61\!\cdots\!81}a^{17}-\frac{27\!\cdots\!97}{61\!\cdots\!81}a^{16}-\frac{19\!\cdots\!84}{61\!\cdots\!81}a^{15}+\frac{39\!\cdots\!17}{61\!\cdots\!81}a^{14}+\frac{24\!\cdots\!02}{61\!\cdots\!81}a^{13}-\frac{26\!\cdots\!41}{47\!\cdots\!51}a^{12}-\frac{18\!\cdots\!30}{61\!\cdots\!81}a^{11}+\frac{17\!\cdots\!44}{61\!\cdots\!81}a^{10}+\frac{87\!\cdots\!25}{61\!\cdots\!81}a^{9}-\frac{34\!\cdots\!47}{61\!\cdots\!81}a^{8}-\frac{23\!\cdots\!96}{61\!\cdots\!81}a^{7}-\frac{14\!\cdots\!81}{61\!\cdots\!81}a^{6}+\frac{34\!\cdots\!58}{61\!\cdots\!81}a^{5}+\frac{13\!\cdots\!56}{61\!\cdots\!81}a^{4}-\frac{23\!\cdots\!14}{61\!\cdots\!81}a^{3}-\frac{15\!\cdots\!35}{61\!\cdots\!81}a^{2}+\frac{55\!\cdots\!49}{61\!\cdots\!81}a+\frac{37\!\cdots\!55}{61\!\cdots\!81}$, $\frac{15\!\cdots\!62}{61\!\cdots\!81}a^{29}-\frac{18\!\cdots\!29}{61\!\cdots\!81}a^{28}-\frac{74\!\cdots\!16}{61\!\cdots\!81}a^{27}+\frac{85\!\cdots\!61}{61\!\cdots\!81}a^{26}+\frac{15\!\cdots\!63}{61\!\cdots\!81}a^{25}-\frac{16\!\cdots\!65}{61\!\cdots\!81}a^{24}-\frac{17\!\cdots\!14}{61\!\cdots\!81}a^{23}+\frac{16\!\cdots\!68}{61\!\cdots\!81}a^{22}+\frac{12\!\cdots\!28}{61\!\cdots\!81}a^{21}-\frac{10\!\cdots\!99}{61\!\cdots\!81}a^{20}-\frac{53\!\cdots\!75}{61\!\cdots\!81}a^{19}+\frac{43\!\cdots\!99}{61\!\cdots\!81}a^{18}+\frac{15\!\cdots\!86}{61\!\cdots\!81}a^{17}-\frac{86\!\cdots\!70}{47\!\cdots\!51}a^{16}-\frac{28\!\cdots\!96}{61\!\cdots\!81}a^{15}+\frac{19\!\cdots\!64}{61\!\cdots\!81}a^{14}+\frac{33\!\cdots\!38}{61\!\cdots\!81}a^{13}-\frac{20\!\cdots\!84}{61\!\cdots\!81}a^{12}-\frac{23\!\cdots\!46}{61\!\cdots\!81}a^{11}+\frac{13\!\cdots\!83}{61\!\cdots\!81}a^{10}+\frac{98\!\cdots\!16}{61\!\cdots\!81}a^{9}-\frac{52\!\cdots\!43}{61\!\cdots\!81}a^{8}-\frac{22\!\cdots\!76}{61\!\cdots\!81}a^{7}+\frac{10\!\cdots\!25}{61\!\cdots\!81}a^{6}+\frac{23\!\cdots\!23}{61\!\cdots\!81}a^{5}-\frac{11\!\cdots\!56}{61\!\cdots\!81}a^{4}-\frac{87\!\cdots\!92}{61\!\cdots\!81}a^{3}+\frac{49\!\cdots\!57}{61\!\cdots\!81}a^{2}-\frac{10\!\cdots\!87}{61\!\cdots\!81}a-\frac{33\!\cdots\!52}{61\!\cdots\!81}$, $\frac{16\!\cdots\!83}{61\!\cdots\!81}a^{29}-\frac{16\!\cdots\!63}{61\!\cdots\!81}a^{28}-\frac{80\!\cdots\!89}{61\!\cdots\!81}a^{27}+\frac{73\!\cdots\!62}{61\!\cdots\!81}a^{26}+\frac{16\!\cdots\!31}{61\!\cdots\!81}a^{25}-\frac{13\!\cdots\!76}{61\!\cdots\!81}a^{24}-\frac{18\!\cdots\!15}{61\!\cdots\!81}a^{23}+\frac{13\!\cdots\!75}{61\!\cdots\!81}a^{22}+\frac{13\!\cdots\!93}{61\!\cdots\!81}a^{21}-\frac{85\!\cdots\!81}{61\!\cdots\!81}a^{20}-\frac{59\!\cdots\!66}{61\!\cdots\!81}a^{19}+\frac{33\!\cdots\!29}{61\!\cdots\!81}a^{18}+\frac{17\!\cdots\!93}{61\!\cdots\!81}a^{17}-\frac{85\!\cdots\!61}{61\!\cdots\!81}a^{16}-\frac{33\!\cdots\!12}{61\!\cdots\!81}a^{15}+\frac{13\!\cdots\!72}{61\!\cdots\!81}a^{14}+\frac{40\!\cdots\!30}{61\!\cdots\!81}a^{13}-\frac{14\!\cdots\!29}{61\!\cdots\!81}a^{12}-\frac{30\!\cdots\!31}{61\!\cdots\!81}a^{11}+\frac{88\!\cdots\!54}{61\!\cdots\!81}a^{10}+\frac{13\!\cdots\!32}{61\!\cdots\!81}a^{9}-\frac{30\!\cdots\!04}{61\!\cdots\!81}a^{8}-\frac{34\!\cdots\!57}{61\!\cdots\!81}a^{7}+\frac{52\!\cdots\!49}{61\!\cdots\!81}a^{6}+\frac{47\!\cdots\!91}{61\!\cdots\!81}a^{5}-\frac{40\!\cdots\!44}{61\!\cdots\!81}a^{4}-\frac{30\!\cdots\!54}{61\!\cdots\!81}a^{3}+\frac{58\!\cdots\!42}{61\!\cdots\!81}a^{2}+\frac{56\!\cdots\!32}{61\!\cdots\!81}a+\frac{26\!\cdots\!16}{61\!\cdots\!81}$, $\frac{68\!\cdots\!86}{61\!\cdots\!81}a^{29}-\frac{76\!\cdots\!96}{61\!\cdots\!81}a^{28}-\frac{33\!\cdots\!63}{61\!\cdots\!81}a^{27}+\frac{33\!\cdots\!52}{61\!\cdots\!81}a^{26}+\frac{68\!\cdots\!20}{61\!\cdots\!81}a^{25}-\frac{63\!\cdots\!11}{61\!\cdots\!81}a^{24}-\frac{78\!\cdots\!82}{61\!\cdots\!81}a^{23}+\frac{65\!\cdots\!82}{61\!\cdots\!81}a^{22}+\frac{55\!\cdots\!18}{61\!\cdots\!81}a^{21}-\frac{41\!\cdots\!18}{61\!\cdots\!81}a^{20}-\frac{24\!\cdots\!84}{61\!\cdots\!81}a^{19}+\frac{16\!\cdots\!77}{61\!\cdots\!81}a^{18}+\frac{72\!\cdots\!80}{61\!\cdots\!81}a^{17}-\frac{43\!\cdots\!67}{61\!\cdots\!81}a^{16}-\frac{14\!\cdots\!10}{61\!\cdots\!81}a^{15}+\frac{74\!\cdots\!46}{61\!\cdots\!81}a^{14}+\frac{17\!\cdots\!57}{61\!\cdots\!81}a^{13}-\frac{80\!\cdots\!14}{61\!\cdots\!81}a^{12}-\frac{13\!\cdots\!76}{61\!\cdots\!81}a^{11}+\frac{52\!\cdots\!12}{61\!\cdots\!81}a^{10}+\frac{64\!\cdots\!48}{61\!\cdots\!81}a^{9}-\frac{19\!\cdots\!19}{61\!\cdots\!81}a^{8}-\frac{17\!\cdots\!40}{61\!\cdots\!81}a^{7}+\frac{38\!\cdots\!52}{61\!\cdots\!81}a^{6}+\frac{26\!\cdots\!35}{61\!\cdots\!81}a^{5}-\frac{32\!\cdots\!32}{61\!\cdots\!81}a^{4}-\frac{18\!\cdots\!54}{61\!\cdots\!81}a^{3}+\frac{52\!\cdots\!57}{61\!\cdots\!81}a^{2}+\frac{39\!\cdots\!78}{61\!\cdots\!81}a+\frac{19\!\cdots\!13}{61\!\cdots\!81}$, $\frac{36\!\cdots\!06}{61\!\cdots\!81}a^{29}-\frac{29\!\cdots\!26}{61\!\cdots\!81}a^{28}-\frac{18\!\cdots\!56}{61\!\cdots\!81}a^{27}+\frac{12\!\cdots\!02}{61\!\cdots\!81}a^{26}+\frac{37\!\cdots\!55}{61\!\cdots\!81}a^{25}-\frac{22\!\cdots\!63}{61\!\cdots\!81}a^{24}-\frac{42\!\cdots\!33}{61\!\cdots\!81}a^{23}+\frac{22\!\cdots\!67}{61\!\cdots\!81}a^{22}+\frac{30\!\cdots\!05}{61\!\cdots\!81}a^{21}-\frac{13\!\cdots\!84}{61\!\cdots\!81}a^{20}-\frac{13\!\cdots\!68}{61\!\cdots\!81}a^{19}+\frac{48\!\cdots\!87}{61\!\cdots\!81}a^{18}+\frac{40\!\cdots\!38}{61\!\cdots\!81}a^{17}-\frac{11\!\cdots\!15}{61\!\cdots\!81}a^{16}-\frac{77\!\cdots\!09}{61\!\cdots\!81}a^{15}+\frac{16\!\cdots\!73}{61\!\cdots\!81}a^{14}+\frac{96\!\cdots\!65}{61\!\cdots\!81}a^{13}-\frac{14\!\cdots\!09}{61\!\cdots\!81}a^{12}-\frac{75\!\cdots\!37}{61\!\cdots\!81}a^{11}+\frac{70\!\cdots\!40}{61\!\cdots\!81}a^{10}+\frac{35\!\cdots\!61}{61\!\cdots\!81}a^{9}-\frac{11\!\cdots\!44}{61\!\cdots\!81}a^{8}-\frac{95\!\cdots\!64}{61\!\cdots\!81}a^{7}-\frac{17\!\cdots\!94}{61\!\cdots\!81}a^{6}+\frac{14\!\cdots\!54}{61\!\cdots\!81}a^{5}+\frac{81\!\cdots\!43}{61\!\cdots\!81}a^{4}-\frac{10\!\cdots\!36}{61\!\cdots\!81}a^{3}-\frac{81\!\cdots\!80}{61\!\cdots\!81}a^{2}+\frac{24\!\cdots\!14}{61\!\cdots\!81}a+\frac{17\!\cdots\!40}{61\!\cdots\!81}$, $\frac{17\!\cdots\!78}{61\!\cdots\!81}a^{29}-\frac{15\!\cdots\!80}{61\!\cdots\!81}a^{28}-\frac{85\!\cdots\!99}{61\!\cdots\!81}a^{27}+\frac{65\!\cdots\!95}{61\!\cdots\!81}a^{26}+\frac{17\!\cdots\!00}{61\!\cdots\!81}a^{25}-\frac{11\!\cdots\!66}{61\!\cdots\!81}a^{24}-\frac{20\!\cdots\!33}{61\!\cdots\!81}a^{23}+\frac{11\!\cdots\!57}{61\!\cdots\!81}a^{22}+\frac{14\!\cdots\!03}{61\!\cdots\!81}a^{21}-\frac{70\!\cdots\!78}{61\!\cdots\!81}a^{20}-\frac{64\!\cdots\!49}{61\!\cdots\!81}a^{19}+\frac{26\!\cdots\!11}{61\!\cdots\!81}a^{18}+\frac{18\!\cdots\!12}{61\!\cdots\!81}a^{17}-\frac{49\!\cdots\!36}{47\!\cdots\!51}a^{16}-\frac{36\!\cdots\!81}{61\!\cdots\!81}a^{15}+\frac{99\!\cdots\!63}{61\!\cdots\!81}a^{14}+\frac{44\!\cdots\!55}{61\!\cdots\!81}a^{13}-\frac{93\!\cdots\!44}{61\!\cdots\!81}a^{12}-\frac{25\!\cdots\!59}{47\!\cdots\!51}a^{11}+\frac{50\!\cdots\!55}{61\!\cdots\!81}a^{10}+\frac{15\!\cdots\!73}{61\!\cdots\!81}a^{9}-\frac{13\!\cdots\!64}{61\!\cdots\!81}a^{8}-\frac{40\!\cdots\!47}{61\!\cdots\!81}a^{7}+\frac{11\!\cdots\!25}{61\!\cdots\!81}a^{6}+\frac{57\!\cdots\!58}{61\!\cdots\!81}a^{5}+\frac{10\!\cdots\!80}{61\!\cdots\!81}a^{4}-\frac{38\!\cdots\!58}{61\!\cdots\!81}a^{3}-\frac{20\!\cdots\!62}{61\!\cdots\!81}a^{2}+\frac{87\!\cdots\!16}{61\!\cdots\!81}a+\frac{58\!\cdots\!09}{61\!\cdots\!81}$, $\frac{33\!\cdots\!25}{61\!\cdots\!81}a^{29}-\frac{20\!\cdots\!22}{61\!\cdots\!81}a^{28}-\frac{14\!\cdots\!34}{61\!\cdots\!81}a^{27}+\frac{98\!\cdots\!18}{61\!\cdots\!81}a^{26}+\frac{26\!\cdots\!28}{61\!\cdots\!81}a^{25}-\frac{19\!\cdots\!27}{61\!\cdots\!81}a^{24}-\frac{25\!\cdots\!48}{61\!\cdots\!81}a^{23}+\frac{22\!\cdots\!42}{61\!\cdots\!81}a^{22}+\frac{13\!\cdots\!89}{61\!\cdots\!81}a^{21}-\frac{15\!\cdots\!75}{61\!\cdots\!81}a^{20}-\frac{38\!\cdots\!24}{61\!\cdots\!81}a^{19}+\frac{67\!\cdots\!26}{61\!\cdots\!81}a^{18}+\frac{34\!\cdots\!47}{61\!\cdots\!81}a^{17}-\frac{19\!\cdots\!46}{61\!\cdots\!81}a^{16}+\frac{12\!\cdots\!17}{61\!\cdots\!81}a^{15}+\frac{35\!\cdots\!36}{61\!\cdots\!81}a^{14}-\frac{44\!\cdots\!10}{61\!\cdots\!81}a^{13}-\frac{42\!\cdots\!93}{61\!\cdots\!81}a^{12}+\frac{64\!\cdots\!10}{61\!\cdots\!81}a^{11}+\frac{30\!\cdots\!50}{61\!\cdots\!81}a^{10}-\frac{46\!\cdots\!43}{61\!\cdots\!81}a^{9}-\frac{12\!\cdots\!99}{61\!\cdots\!81}a^{8}+\frac{16\!\cdots\!12}{61\!\cdots\!81}a^{7}+\frac{29\!\cdots\!36}{61\!\cdots\!81}a^{6}-\frac{29\!\cdots\!82}{61\!\cdots\!81}a^{5}-\frac{33\!\cdots\!82}{61\!\cdots\!81}a^{4}+\frac{27\!\cdots\!85}{61\!\cdots\!81}a^{3}+\frac{15\!\cdots\!79}{61\!\cdots\!81}a^{2}-\frac{12\!\cdots\!65}{61\!\cdots\!81}a-\frac{15\!\cdots\!54}{61\!\cdots\!81}$, $\frac{38\!\cdots\!71}{61\!\cdots\!81}a^{29}-\frac{63\!\cdots\!12}{61\!\cdots\!81}a^{28}-\frac{19\!\cdots\!70}{61\!\cdots\!81}a^{27}+\frac{12\!\cdots\!11}{61\!\cdots\!81}a^{26}+\frac{40\!\cdots\!70}{61\!\cdots\!81}a^{25}+\frac{11\!\cdots\!80}{61\!\cdots\!81}a^{24}-\frac{47\!\cdots\!30}{61\!\cdots\!81}a^{23}-\frac{51\!\cdots\!20}{61\!\cdots\!81}a^{22}+\frac{34\!\cdots\!89}{61\!\cdots\!81}a^{21}+\frac{60\!\cdots\!27}{61\!\cdots\!81}a^{20}-\frac{15\!\cdots\!17}{61\!\cdots\!81}a^{19}-\frac{37\!\cdots\!71}{61\!\cdots\!81}a^{18}+\frac{48\!\cdots\!07}{61\!\cdots\!81}a^{17}+\frac{13\!\cdots\!48}{61\!\cdots\!81}a^{16}-\frac{97\!\cdots\!82}{61\!\cdots\!81}a^{15}-\frac{30\!\cdots\!15}{61\!\cdots\!81}a^{14}+\frac{12\!\cdots\!34}{61\!\cdots\!81}a^{13}+\frac{42\!\cdots\!71}{61\!\cdots\!81}a^{12}-\frac{10\!\cdots\!76}{61\!\cdots\!81}a^{11}-\frac{34\!\cdots\!42}{61\!\cdots\!81}a^{10}+\frac{54\!\cdots\!56}{61\!\cdots\!81}a^{9}+\frac{16\!\cdots\!52}{61\!\cdots\!81}a^{8}-\frac{16\!\cdots\!04}{61\!\cdots\!81}a^{7}-\frac{45\!\cdots\!92}{61\!\cdots\!81}a^{6}+\frac{26\!\cdots\!17}{61\!\cdots\!81}a^{5}+\frac{61\!\cdots\!99}{61\!\cdots\!81}a^{4}-\frac{21\!\cdots\!95}{61\!\cdots\!81}a^{3}-\frac{36\!\cdots\!33}{61\!\cdots\!81}a^{2}+\frac{61\!\cdots\!11}{61\!\cdots\!81}a+\frac{50\!\cdots\!00}{61\!\cdots\!81}$, $\frac{10\!\cdots\!20}{61\!\cdots\!81}a^{29}-\frac{83\!\cdots\!74}{61\!\cdots\!81}a^{28}-\frac{53\!\cdots\!18}{61\!\cdots\!81}a^{27}+\frac{35\!\cdots\!06}{61\!\cdots\!81}a^{26}+\frac{11\!\cdots\!11}{61\!\cdots\!81}a^{25}-\frac{62\!\cdots\!07}{61\!\cdots\!81}a^{24}-\frac{12\!\cdots\!08}{61\!\cdots\!81}a^{23}+\frac{60\!\cdots\!31}{61\!\cdots\!81}a^{22}+\frac{90\!\cdots\!87}{61\!\cdots\!81}a^{21}-\frac{34\!\cdots\!21}{61\!\cdots\!81}a^{20}-\frac{40\!\cdots\!80}{61\!\cdots\!81}a^{19}+\frac{12\!\cdots\!76}{61\!\cdots\!81}a^{18}+\frac{12\!\cdots\!69}{61\!\cdots\!81}a^{17}-\frac{28\!\cdots\!85}{61\!\cdots\!81}a^{16}-\frac{23\!\cdots\!25}{61\!\cdots\!81}a^{15}+\frac{38\!\cdots\!40}{61\!\cdots\!81}a^{14}+\frac{29\!\cdots\!43}{61\!\cdots\!81}a^{13}-\frac{30\!\cdots\!48}{61\!\cdots\!81}a^{12}-\frac{22\!\cdots\!63}{61\!\cdots\!81}a^{11}+\frac{11\!\cdots\!77}{61\!\cdots\!81}a^{10}+\frac{10\!\cdots\!62}{61\!\cdots\!81}a^{9}+\frac{68\!\cdots\!32}{61\!\cdots\!81}a^{8}-\frac{29\!\cdots\!48}{61\!\cdots\!81}a^{7}-\frac{15\!\cdots\!22}{61\!\cdots\!81}a^{6}+\frac{43\!\cdots\!52}{61\!\cdots\!81}a^{5}+\frac{35\!\cdots\!92}{61\!\cdots\!81}a^{4}-\frac{31\!\cdots\!70}{61\!\cdots\!81}a^{3}-\frac{29\!\cdots\!90}{61\!\cdots\!81}a^{2}+\frac{79\!\cdots\!07}{61\!\cdots\!81}a+\frac{58\!\cdots\!43}{61\!\cdots\!81}$, $\frac{28\!\cdots\!42}{61\!\cdots\!81}a^{29}+\frac{52\!\cdots\!08}{61\!\cdots\!81}a^{28}-\frac{20\!\cdots\!65}{61\!\cdots\!81}a^{27}-\frac{25\!\cdots\!59}{61\!\cdots\!81}a^{26}+\frac{57\!\cdots\!15}{61\!\cdots\!81}a^{25}+\frac{53\!\cdots\!59}{61\!\cdots\!81}a^{24}-\frac{87\!\cdots\!30}{61\!\cdots\!81}a^{23}-\frac{60\!\cdots\!53}{61\!\cdots\!81}a^{22}+\frac{79\!\cdots\!48}{61\!\cdots\!81}a^{21}+\frac{42\!\cdots\!65}{61\!\cdots\!81}a^{20}-\frac{46\!\cdots\!76}{61\!\cdots\!81}a^{19}-\frac{19\!\cdots\!06}{61\!\cdots\!81}a^{18}+\frac{17\!\cdots\!42}{61\!\cdots\!81}a^{17}+\frac{55\!\cdots\!71}{61\!\cdots\!81}a^{16}-\frac{44\!\cdots\!57}{61\!\cdots\!81}a^{15}-\frac{10\!\cdots\!28}{61\!\cdots\!81}a^{14}+\frac{73\!\cdots\!92}{61\!\cdots\!81}a^{13}+\frac{12\!\cdots\!02}{61\!\cdots\!81}a^{12}-\frac{77\!\cdots\!09}{61\!\cdots\!81}a^{11}-\frac{97\!\cdots\!89}{61\!\cdots\!81}a^{10}+\frac{51\!\cdots\!31}{61\!\cdots\!81}a^{9}+\frac{43\!\cdots\!86}{61\!\cdots\!81}a^{8}-\frac{19\!\cdots\!82}{61\!\cdots\!81}a^{7}-\frac{11\!\cdots\!40}{61\!\cdots\!81}a^{6}+\frac{40\!\cdots\!15}{61\!\cdots\!81}a^{5}+\frac{14\!\cdots\!42}{61\!\cdots\!81}a^{4}-\frac{40\!\cdots\!55}{61\!\cdots\!81}a^{3}-\frac{86\!\cdots\!21}{61\!\cdots\!81}a^{2}+\frac{14\!\cdots\!55}{61\!\cdots\!81}a+\frac{10\!\cdots\!08}{61\!\cdots\!81}$, $\frac{20\!\cdots\!63}{61\!\cdots\!81}a^{29}-\frac{19\!\cdots\!12}{61\!\cdots\!81}a^{28}-\frac{97\!\cdots\!95}{61\!\cdots\!81}a^{27}+\frac{85\!\cdots\!48}{61\!\cdots\!81}a^{26}+\frac{20\!\cdots\!62}{61\!\cdots\!81}a^{25}-\frac{15\!\cdots\!72}{61\!\cdots\!81}a^{24}-\frac{23\!\cdots\!98}{61\!\cdots\!81}a^{23}+\frac{15\!\cdots\!00}{61\!\cdots\!81}a^{22}+\frac{16\!\cdots\!34}{61\!\cdots\!81}a^{21}-\frac{96\!\cdots\!29}{61\!\cdots\!81}a^{20}-\frac{72\!\cdots\!13}{61\!\cdots\!81}a^{19}+\frac{37\!\cdots\!64}{61\!\cdots\!81}a^{18}+\frac{21\!\cdots\!26}{61\!\cdots\!81}a^{17}-\frac{93\!\cdots\!67}{61\!\cdots\!81}a^{16}-\frac{40\!\cdots\!58}{61\!\cdots\!81}a^{15}+\frac{15\!\cdots\!36}{61\!\cdots\!81}a^{14}+\frac{48\!\cdots\!03}{61\!\cdots\!81}a^{13}-\frac{15\!\cdots\!28}{61\!\cdots\!81}a^{12}-\frac{36\!\cdots\!31}{61\!\cdots\!81}a^{11}+\frac{90\!\cdots\!06}{61\!\cdots\!81}a^{10}+\frac{16\!\cdots\!75}{61\!\cdots\!81}a^{9}-\frac{29\!\cdots\!39}{61\!\cdots\!81}a^{8}-\frac{42\!\cdots\!02}{61\!\cdots\!81}a^{7}+\frac{46\!\cdots\!32}{61\!\cdots\!81}a^{6}+\frac{57\!\cdots\!80}{61\!\cdots\!81}a^{5}-\frac{28\!\cdots\!99}{61\!\cdots\!81}a^{4}-\frac{36\!\cdots\!14}{61\!\cdots\!81}a^{3}-\frac{18\!\cdots\!74}{61\!\cdots\!81}a^{2}+\frac{72\!\cdots\!98}{61\!\cdots\!81}a+\frac{40\!\cdots\!34}{61\!\cdots\!81}$, $\frac{66\!\cdots\!36}{61\!\cdots\!81}a^{29}-\frac{53\!\cdots\!59}{61\!\cdots\!81}a^{28}-\frac{32\!\cdots\!23}{61\!\cdots\!81}a^{27}+\frac{23\!\cdots\!44}{61\!\cdots\!81}a^{26}+\frac{67\!\cdots\!98}{61\!\cdots\!81}a^{25}-\frac{41\!\cdots\!69}{61\!\cdots\!81}a^{24}-\frac{77\!\cdots\!50}{61\!\cdots\!81}a^{23}+\frac{40\!\cdots\!77}{61\!\cdots\!81}a^{22}+\frac{54\!\cdots\!38}{61\!\cdots\!81}a^{21}-\frac{23\!\cdots\!95}{61\!\cdots\!81}a^{20}-\frac{24\!\cdots\!74}{61\!\cdots\!81}a^{19}+\frac{86\!\cdots\!48}{61\!\cdots\!81}a^{18}+\frac{73\!\cdots\!67}{61\!\cdots\!81}a^{17}-\frac{20\!\cdots\!02}{61\!\cdots\!81}a^{16}-\frac{14\!\cdots\!61}{61\!\cdots\!81}a^{15}+\frac{29\!\cdots\!25}{61\!\cdots\!81}a^{14}+\frac{17\!\cdots\!34}{61\!\cdots\!81}a^{13}-\frac{25\!\cdots\!63}{61\!\cdots\!81}a^{12}-\frac{13\!\cdots\!36}{61\!\cdots\!81}a^{11}+\frac{11\!\cdots\!26}{61\!\cdots\!81}a^{10}+\frac{63\!\cdots\!47}{61\!\cdots\!81}a^{9}-\frac{15\!\cdots\!46}{61\!\cdots\!81}a^{8}-\frac{17\!\cdots\!28}{61\!\cdots\!81}a^{7}-\frac{48\!\cdots\!66}{61\!\cdots\!81}a^{6}+\frac{25\!\cdots\!72}{61\!\cdots\!81}a^{5}+\frac{16\!\cdots\!11}{61\!\cdots\!81}a^{4}-\frac{18\!\cdots\!32}{61\!\cdots\!81}a^{3}-\frac{15\!\cdots\!89}{61\!\cdots\!81}a^{2}+\frac{45\!\cdots\!08}{61\!\cdots\!81}a+\frac{25\!\cdots\!03}{47\!\cdots\!51}$, $\frac{19\!\cdots\!78}{61\!\cdots\!81}a^{29}-\frac{19\!\cdots\!69}{61\!\cdots\!81}a^{28}-\frac{96\!\cdots\!75}{61\!\cdots\!81}a^{27}+\frac{85\!\cdots\!88}{61\!\cdots\!81}a^{26}+\frac{19\!\cdots\!73}{61\!\cdots\!81}a^{25}-\frac{15\!\cdots\!73}{61\!\cdots\!81}a^{24}-\frac{22\!\cdots\!46}{61\!\cdots\!81}a^{23}+\frac{15\!\cdots\!63}{61\!\cdots\!81}a^{22}+\frac{15\!\cdots\!54}{61\!\cdots\!81}a^{21}-\frac{97\!\cdots\!30}{61\!\cdots\!81}a^{20}-\frac{70\!\cdots\!25}{61\!\cdots\!81}a^{19}+\frac{38\!\cdots\!35}{61\!\cdots\!81}a^{18}+\frac{20\!\cdots\!87}{61\!\cdots\!81}a^{17}-\frac{95\!\cdots\!90}{61\!\cdots\!81}a^{16}-\frac{39\!\cdots\!52}{61\!\cdots\!81}a^{15}+\frac{15\!\cdots\!82}{61\!\cdots\!81}a^{14}+\frac{47\!\cdots\!42}{61\!\cdots\!81}a^{13}-\frac{15\!\cdots\!31}{61\!\cdots\!81}a^{12}-\frac{35\!\cdots\!59}{61\!\cdots\!81}a^{11}+\frac{94\!\cdots\!21}{61\!\cdots\!81}a^{10}+\frac{15\!\cdots\!96}{61\!\cdots\!81}a^{9}-\frac{30\!\cdots\!58}{61\!\cdots\!81}a^{8}-\frac{40\!\cdots\!34}{61\!\cdots\!81}a^{7}+\frac{50\!\cdots\!61}{61\!\cdots\!81}a^{6}+\frac{53\!\cdots\!36}{61\!\cdots\!81}a^{5}-\frac{34\!\cdots\!72}{61\!\cdots\!81}a^{4}-\frac{33\!\cdots\!48}{61\!\cdots\!81}a^{3}+\frac{22\!\cdots\!02}{61\!\cdots\!81}a^{2}+\frac{63\!\cdots\!55}{61\!\cdots\!81}a+\frac{32\!\cdots\!29}{61\!\cdots\!81}$, $\frac{11\!\cdots\!28}{61\!\cdots\!81}a^{29}-\frac{11\!\cdots\!00}{61\!\cdots\!81}a^{28}-\frac{55\!\cdots\!89}{61\!\cdots\!81}a^{27}+\frac{50\!\cdots\!33}{61\!\cdots\!81}a^{26}+\frac{11\!\cdots\!23}{61\!\cdots\!81}a^{25}-\frac{93\!\cdots\!99}{61\!\cdots\!81}a^{24}-\frac{13\!\cdots\!18}{61\!\cdots\!81}a^{23}+\frac{95\!\cdots\!76}{61\!\cdots\!81}a^{22}+\frac{90\!\cdots\!74}{61\!\cdots\!81}a^{21}-\frac{58\!\cdots\!07}{61\!\cdots\!81}a^{20}-\frac{40\!\cdots\!90}{61\!\cdots\!81}a^{19}+\frac{23\!\cdots\!30}{61\!\cdots\!81}a^{18}+\frac{11\!\cdots\!08}{61\!\cdots\!81}a^{17}-\frac{58\!\cdots\!25}{61\!\cdots\!81}a^{16}-\frac{22\!\cdots\!21}{61\!\cdots\!81}a^{15}+\frac{94\!\cdots\!70}{61\!\cdots\!81}a^{14}+\frac{26\!\cdots\!90}{61\!\cdots\!81}a^{13}-\frac{96\!\cdots\!40}{61\!\cdots\!81}a^{12}-\frac{20\!\cdots\!27}{61\!\cdots\!81}a^{11}+\frac{58\!\cdots\!67}{61\!\cdots\!81}a^{10}+\frac{88\!\cdots\!97}{61\!\cdots\!81}a^{9}-\frac{19\!\cdots\!88}{61\!\cdots\!81}a^{8}-\frac{21\!\cdots\!00}{61\!\cdots\!81}a^{7}+\frac{33\!\cdots\!53}{61\!\cdots\!81}a^{6}+\frac{28\!\cdots\!25}{61\!\cdots\!81}a^{5}-\frac{25\!\cdots\!74}{61\!\cdots\!81}a^{4}-\frac{17\!\cdots\!41}{61\!\cdots\!81}a^{3}+\frac{44\!\cdots\!39}{61\!\cdots\!81}a^{2}+\frac{30\!\cdots\!59}{61\!\cdots\!81}a+\frac{14\!\cdots\!77}{61\!\cdots\!81}$, $\frac{47\!\cdots\!24}{61\!\cdots\!81}a^{29}-\frac{23\!\cdots\!82}{61\!\cdots\!81}a^{28}-\frac{23\!\cdots\!19}{61\!\cdots\!81}a^{27}+\frac{94\!\cdots\!38}{61\!\cdots\!81}a^{26}+\frac{48\!\cdots\!07}{61\!\cdots\!81}a^{25}-\frac{14\!\cdots\!62}{61\!\cdots\!81}a^{24}-\frac{56\!\cdots\!83}{61\!\cdots\!81}a^{23}+\frac{12\!\cdots\!28}{61\!\cdots\!81}a^{22}+\frac{40\!\cdots\!14}{61\!\cdots\!81}a^{21}-\frac{55\!\cdots\!20}{61\!\cdots\!81}a^{20}-\frac{18\!\cdots\!27}{61\!\cdots\!81}a^{19}+\frac{11\!\cdots\!83}{61\!\cdots\!81}a^{18}+\frac{55\!\cdots\!07}{61\!\cdots\!81}a^{17}-\frac{53\!\cdots\!04}{61\!\cdots\!81}a^{16}-\frac{10\!\cdots\!44}{61\!\cdots\!81}a^{15}-\frac{56\!\cdots\!75}{61\!\cdots\!81}a^{14}+\frac{14\!\cdots\!83}{61\!\cdots\!81}a^{13}+\frac{13\!\cdots\!11}{61\!\cdots\!81}a^{12}-\frac{11\!\cdots\!57}{61\!\cdots\!81}a^{11}-\frac{14\!\cdots\!24}{61\!\cdots\!81}a^{10}+\frac{56\!\cdots\!04}{61\!\cdots\!81}a^{9}+\frac{82\!\cdots\!04}{61\!\cdots\!81}a^{8}-\frac{16\!\cdots\!57}{61\!\cdots\!81}a^{7}-\frac{25\!\cdots\!17}{61\!\cdots\!81}a^{6}+\frac{25\!\cdots\!86}{61\!\cdots\!81}a^{5}+\frac{38\!\cdots\!93}{61\!\cdots\!81}a^{4}-\frac{19\!\cdots\!40}{61\!\cdots\!81}a^{3}-\frac{25\!\cdots\!88}{61\!\cdots\!81}a^{2}+\frac{52\!\cdots\!91}{61\!\cdots\!81}a+\frac{41\!\cdots\!66}{61\!\cdots\!81}$ 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:  \( 59204301251900104 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{30}\cdot(2\pi)^{0}\cdot 59204301251900104 \cdot 1}{2\cdot\sqrt{69503752297329754905479727341904896738456941915804813}}\cr\approx \mathstrut & 0.120564377552318 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 - 49*x^28 + 44*x^27 + 1012*x^26 - 813*x^25 - 11582*x^24 + 8267*x^23 + 81301*x^22 - 51172*x^21 - 366235*x^20 + 201754*x^19 + 1077688*x^18 - 515602*x^17 - 2071990*x^16 + 853441*x^15 + 2566790*x^14 - 898459*x^13 - 1995511*x^12 + 578875*x^11 + 941075*x^10 - 215958*x^9 - 261009*x^8 + 45664*x^7 + 40851*x^6 - 5360*x^5 - 3328*x^4 + 330*x^3 + 117*x^2 - 9*x - 1)
 
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^30 - x^29 - 49*x^28 + 44*x^27 + 1012*x^26 - 813*x^25 - 11582*x^24 + 8267*x^23 + 81301*x^22 - 51172*x^21 - 366235*x^20 + 201754*x^19 + 1077688*x^18 - 515602*x^17 - 2071990*x^16 + 853441*x^15 + 2566790*x^14 - 898459*x^13 - 1995511*x^12 + 578875*x^11 + 941075*x^10 - 215958*x^9 - 261009*x^8 + 45664*x^7 + 40851*x^6 - 5360*x^5 - 3328*x^4 + 330*x^3 + 117*x^2 - 9*x - 1, 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^30 - x^29 - 49*x^28 + 44*x^27 + 1012*x^26 - 813*x^25 - 11582*x^24 + 8267*x^23 + 81301*x^22 - 51172*x^21 - 366235*x^20 + 201754*x^19 + 1077688*x^18 - 515602*x^17 - 2071990*x^16 + 853441*x^15 + 2566790*x^14 - 898459*x^13 - 1995511*x^12 + 578875*x^11 + 941075*x^10 - 215958*x^9 - 261009*x^8 + 45664*x^7 + 40851*x^6 - 5360*x^5 - 3328*x^4 + 330*x^3 + 117*x^2 - 9*x - 1);
 
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^30 - x^29 - 49*x^28 + 44*x^27 + 1012*x^26 - 813*x^25 - 11582*x^24 + 8267*x^23 + 81301*x^22 - 51172*x^21 - 366235*x^20 + 201754*x^19 + 1077688*x^18 - 515602*x^17 - 2071990*x^16 + 853441*x^15 + 2566790*x^14 - 898459*x^13 - 1995511*x^12 + 578875*x^11 + 941075*x^10 - 215958*x^9 - 261009*x^8 + 45664*x^7 + 40851*x^6 - 5360*x^5 - 3328*x^4 + 330*x^3 + 117*x^2 - 9*x - 1);
 
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

$C_{30}$ (as 30T1):

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 cyclic group of order 30
The 30 conjugacy class representatives for $C_{30}$
Character table for $C_{30}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{13})^+\), 10.10.79589952003133.1, 15.15.432659002790862279847129.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $30$ $15^{2}$ ${\href{/padicField/5.10.0.1}{10} }^{3}$ $30$ R R $15^{2}$ $30$ ${\href{/padicField/23.3.0.1}{3} }^{10}$ $15^{2}$ ${\href{/padicField/31.10.0.1}{10} }^{3}$ $30$ $30$ ${\href{/padicField/43.3.0.1}{3} }^{10}$ ${\href{/padicField/47.10.0.1}{10} }^{3}$ ${\href{/padicField/53.5.0.1}{5} }^{6}$ $30$

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
\(11\) Copy content Toggle raw display Deg $30$$5$$6$$24$
\(13\) Copy content Toggle raw display Deg $30$$6$$5$$25$