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Properties

Label 30.0.885...327.1
Degree $30$
Signature $[0, 15]$
Discriminant $-8.851\times 10^{51}$
Root discriminant \(53.90\)
Ramified primes $3,7,53$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 4*x^29 - 32*x^28 + 92*x^27 + 588*x^26 - 752*x^25 - 6717*x^24 - 805*x^23 + 44049*x^22 + 54263*x^21 - 93757*x^20 - 350454*x^19 - 591742*x^18 + 307201*x^17 + 3600412*x^16 + 5669890*x^15 - 632841*x^14 - 17605194*x^13 - 34361042*x^12 - 23087625*x^11 + 45815379*x^10 + 119941341*x^9 + 85868062*x^8 - 30516460*x^7 - 62011703*x^6 + 46150369*x^5 + 155790202*x^4 + 147686157*x^3 + 67768188*x^2 + 10788948*x + 751689)
 
gp: K = bnfinit(y^30 - 4*y^29 - 32*y^28 + 92*y^27 + 588*y^26 - 752*y^25 - 6717*y^24 - 805*y^23 + 44049*y^22 + 54263*y^21 - 93757*y^20 - 350454*y^19 - 591742*y^18 + 307201*y^17 + 3600412*y^16 + 5669890*y^15 - 632841*y^14 - 17605194*y^13 - 34361042*y^12 - 23087625*y^11 + 45815379*y^10 + 119941341*y^9 + 85868062*y^8 - 30516460*y^7 - 62011703*y^6 + 46150369*y^5 + 155790202*y^4 + 147686157*y^3 + 67768188*y^2 + 10788948*y + 751689, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 4*x^29 - 32*x^28 + 92*x^27 + 588*x^26 - 752*x^25 - 6717*x^24 - 805*x^23 + 44049*x^22 + 54263*x^21 - 93757*x^20 - 350454*x^19 - 591742*x^18 + 307201*x^17 + 3600412*x^16 + 5669890*x^15 - 632841*x^14 - 17605194*x^13 - 34361042*x^12 - 23087625*x^11 + 45815379*x^10 + 119941341*x^9 + 85868062*x^8 - 30516460*x^7 - 62011703*x^6 + 46150369*x^5 + 155790202*x^4 + 147686157*x^3 + 67768188*x^2 + 10788948*x + 751689);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 4*x^29 - 32*x^28 + 92*x^27 + 588*x^26 - 752*x^25 - 6717*x^24 - 805*x^23 + 44049*x^22 + 54263*x^21 - 93757*x^20 - 350454*x^19 - 591742*x^18 + 307201*x^17 + 3600412*x^16 + 5669890*x^15 - 632841*x^14 - 17605194*x^13 - 34361042*x^12 - 23087625*x^11 + 45815379*x^10 + 119941341*x^9 + 85868062*x^8 - 30516460*x^7 - 62011703*x^6 + 46150369*x^5 + 155790202*x^4 + 147686157*x^3 + 67768188*x^2 + 10788948*x + 751689)
 

\( x^{30} - 4 x^{29} - 32 x^{28} + 92 x^{27} + 588 x^{26} - 752 x^{25} - 6717 x^{24} - 805 x^{23} + \cdots + 751689 \) 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:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-8851375005191383462430321349782942769050796640392327\) \(\medspace = -\,3^{14}\cdot 7^{25}\cdot 53^{14}\) 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:  \(53.90\)
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:  $3^{1/2}7^{5/6}53^{1/2}\approx 63.81854946060407$
Ramified primes:   \(3\), \(7\), \(53\) 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:  \(\Q(\sqrt{-7}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{18}+\frac{1}{3}a^{10}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{19}+\frac{1}{3}a^{11}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{20}+\frac{1}{3}a^{12}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{21}+\frac{1}{3}a^{13}+\frac{1}{3}a^{5}$, $\frac{1}{51}a^{22}+\frac{4}{51}a^{21}+\frac{2}{17}a^{20}+\frac{1}{17}a^{19}+\frac{1}{17}a^{18}-\frac{4}{51}a^{17}-\frac{2}{17}a^{16}-\frac{11}{51}a^{15}-\frac{4}{17}a^{14}+\frac{4}{51}a^{13}-\frac{5}{51}a^{12}-\frac{19}{51}a^{11}-\frac{4}{17}a^{10}+\frac{2}{17}a^{9}+\frac{2}{51}a^{8}-\frac{20}{51}a^{7}+\frac{1}{51}a^{6}+\frac{8}{17}a^{5}-\frac{5}{51}a^{4}+\frac{8}{17}a^{3}-\frac{22}{51}a^{2}-\frac{2}{17}a$, $\frac{1}{8109}a^{23}+\frac{23}{8109}a^{22}-\frac{139}{8109}a^{21}+\frac{47}{901}a^{20}+\frac{485}{8109}a^{19}-\frac{355}{8109}a^{18}-\frac{407}{2703}a^{17}-\frac{992}{8109}a^{16}-\frac{26}{53}a^{15}+\frac{3839}{8109}a^{14}-\frac{968}{2703}a^{13}-\frac{23}{153}a^{12}-\frac{1648}{8109}a^{11}+\frac{691}{2703}a^{10}-\frac{2893}{8109}a^{9}+\frac{103}{8109}a^{8}-\frac{2198}{8109}a^{7}-\frac{2456}{8109}a^{6}+\frac{700}{2703}a^{5}+\frac{283}{901}a^{4}-\frac{722}{8109}a^{3}+\frac{427}{901}a^{2}+\frac{64}{2703}a-\frac{21}{53}$, $\frac{1}{1694781}a^{24}+\frac{5}{1694781}a^{23}-\frac{10570}{1694781}a^{22}+\frac{92135}{564927}a^{21}+\frac{105761}{1694781}a^{20}-\frac{1633}{89199}a^{19}+\frac{3751}{51357}a^{18}-\frac{68690}{1694781}a^{17}-\frac{17909}{188309}a^{16}+\frac{177521}{1694781}a^{15}+\frac{7358}{51357}a^{14}+\frac{332642}{1694781}a^{13}-\frac{462112}{1694781}a^{12}+\frac{15821}{51357}a^{11}-\frac{136243}{1694781}a^{10}+\frac{632686}{1694781}a^{9}+\frac{657070}{1694781}a^{8}-\frac{613997}{1694781}a^{7}-\frac{114944}{564927}a^{6}-\frac{122521}{564927}a^{5}-\frac{374903}{1694781}a^{4}+\frac{21110}{188309}a^{3}+\frac{83801}{564927}a^{2}+\frac{61799}{188309}a-\frac{2378}{11077}$, $\frac{1}{1694781}a^{25}+\frac{64}{1694781}a^{23}+\frac{9485}{1694781}a^{22}+\frac{6070}{154071}a^{21}-\frac{5564}{1694781}a^{20}-\frac{200737}{1694781}a^{19}+\frac{47866}{1694781}a^{18}+\frac{160951}{1694781}a^{17}+\frac{13457}{1694781}a^{16}+\frac{453086}{1694781}a^{15}-\frac{71344}{1694781}a^{14}-\frac{313292}{1694781}a^{13}+\frac{30155}{89199}a^{12}+\frac{589559}{1694781}a^{11}+\frac{82667}{564927}a^{10}-\frac{577081}{1694781}a^{9}-\frac{541762}{1694781}a^{8}+\frac{198970}{1694781}a^{7}+\frac{18176}{51357}a^{6}+\frac{10898}{154071}a^{5}+\frac{401701}{1694781}a^{4}+\frac{520}{11077}a^{3}-\frac{137677}{564927}a^{2}+\frac{66280}{188309}a-\frac{1486}{11077}$, $\frac{1}{1694781}a^{26}-\frac{31}{1694781}a^{23}+\frac{46}{1694781}a^{22}+\frac{98567}{1694781}a^{21}+\frac{24550}{564927}a^{20}-\frac{1315}{33231}a^{19}+\frac{122528}{1694781}a^{18}+\frac{251980}{1694781}a^{17}-\frac{7339}{154071}a^{16}+\frac{54622}{188309}a^{15}+\frac{417680}{1694781}a^{14}-\frac{219053}{564927}a^{13}+\frac{533752}{1694781}a^{12}+\frac{565186}{1694781}a^{11}+\frac{181286}{564927}a^{10}+\frac{457850}{1694781}a^{9}+\frac{200216}{1694781}a^{8}+\frac{126469}{1694781}a^{7}-\frac{7577}{33231}a^{6}+\frac{327334}{1694781}a^{5}-\frac{644869}{1694781}a^{4}-\frac{50992}{1694781}a^{3}-\frac{80128}{188309}a^{2}-\frac{167806}{564927}a+\frac{1921}{11077}$, $\frac{1}{5084343}a^{27}-\frac{1}{5084343}a^{26}+\frac{1}{5084343}a^{24}-\frac{181}{5084343}a^{23}+\frac{16582}{1694781}a^{22}-\frac{94225}{5084343}a^{21}-\frac{787973}{5084343}a^{20}-\frac{108764}{5084343}a^{19}+\frac{17264}{1694781}a^{18}+\frac{172835}{5084343}a^{17}+\frac{14642}{299079}a^{16}+\frac{385018}{1694781}a^{15}-\frac{11366}{299079}a^{14}+\frac{7204}{299079}a^{13}-\frac{488974}{5084343}a^{12}-\frac{32741}{89199}a^{11}+\frac{26019}{188309}a^{10}-\frac{56647}{1694781}a^{9}-\frac{363161}{1694781}a^{8}-\frac{15118}{51357}a^{7}+\frac{34087}{99693}a^{6}-\frac{1914707}{5084343}a^{5}-\frac{969739}{5084343}a^{4}+\frac{530224}{1694781}a^{3}+\frac{402124}{1694781}a^{2}-\frac{8411}{188309}a-\frac{2777}{33231}$, $\frac{1}{2506581099}a^{28}-\frac{157}{2506581099}a^{27}+\frac{131}{835527033}a^{26}-\frac{707}{2506581099}a^{25}+\frac{401}{2506581099}a^{24}-\frac{1933}{43975107}a^{23}-\frac{7616461}{2506581099}a^{22}-\frac{247049072}{2506581099}a^{21}+\frac{250713010}{2506581099}a^{20}+\frac{31436755}{278509011}a^{19}+\frac{315730934}{2506581099}a^{18}+\frac{431920}{2506581099}a^{17}-\frac{651784}{92836337}a^{16}+\frac{1044972518}{2506581099}a^{15}-\frac{941542927}{2506581099}a^{14}+\frac{718809554}{2506581099}a^{13}-\frac{236056894}{835527033}a^{12}+\frac{2417695}{5254887}a^{11}+\frac{204527228}{835527033}a^{10}+\frac{18497759}{835527033}a^{9}-\frac{121706132}{835527033}a^{8}+\frac{10692005}{49148649}a^{7}+\frac{317576617}{2506581099}a^{6}+\frac{1143896495}{2506581099}a^{5}-\frac{241321297}{835527033}a^{4}-\frac{5258386}{278509011}a^{3}-\frac{32497403}{278509011}a^{2}-\frac{103801}{862257}a-\frac{25055}{321233}$, $\frac{1}{10\!\cdots\!67}a^{29}+\frac{28\!\cdots\!71}{33\!\cdots\!89}a^{28}-\frac{19\!\cdots\!23}{30\!\cdots\!99}a^{27}+\frac{14\!\cdots\!57}{37\!\cdots\!21}a^{26}+\frac{28\!\cdots\!52}{11\!\cdots\!63}a^{25}-\frac{86\!\cdots\!39}{33\!\cdots\!89}a^{24}+\frac{30\!\cdots\!04}{10\!\cdots\!67}a^{23}-\frac{11\!\cdots\!85}{33\!\cdots\!89}a^{22}+\frac{12\!\cdots\!99}{18\!\cdots\!39}a^{21}-\frac{12\!\cdots\!06}{10\!\cdots\!67}a^{20}+\frac{51\!\cdots\!03}{33\!\cdots\!89}a^{19}-\frac{21\!\cdots\!59}{37\!\cdots\!21}a^{18}-\frac{46\!\cdots\!54}{37\!\cdots\!21}a^{17}-\frac{12\!\cdots\!15}{11\!\cdots\!41}a^{16}+\frac{52\!\cdots\!10}{34\!\cdots\!23}a^{15}-\frac{35\!\cdots\!65}{11\!\cdots\!63}a^{14}+\frac{65\!\cdots\!63}{10\!\cdots\!67}a^{13}-\frac{15\!\cdots\!39}{10\!\cdots\!67}a^{12}-\frac{12\!\cdots\!49}{33\!\cdots\!89}a^{11}-\frac{63\!\cdots\!48}{33\!\cdots\!89}a^{10}-\frac{23\!\cdots\!60}{11\!\cdots\!63}a^{9}+\frac{10\!\cdots\!56}{65\!\cdots\!39}a^{8}-\frac{43\!\cdots\!04}{10\!\cdots\!67}a^{7}-\frac{28\!\cdots\!06}{11\!\cdots\!63}a^{6}+\frac{46\!\cdots\!40}{33\!\cdots\!89}a^{5}+\frac{42\!\cdots\!13}{52\!\cdots\!93}a^{4}+\frac{36\!\cdots\!84}{33\!\cdots\!89}a^{3}+\frac{70\!\cdots\!78}{19\!\cdots\!17}a^{2}-\frac{18\!\cdots\!26}{38\!\cdots\!67}a+\frac{67\!\cdots\!42}{22\!\cdots\!51}$ 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:  $3$

Class group and class number

$C_{3}$, which has order $3$ (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:   $\frac{44\!\cdots\!98}{33\!\cdots\!89}a^{29}-\frac{11\!\cdots\!25}{17\!\cdots\!31}a^{28}-\frac{14\!\cdots\!69}{37\!\cdots\!21}a^{27}+\frac{16\!\cdots\!11}{11\!\cdots\!63}a^{26}+\frac{13\!\cdots\!81}{20\!\cdots\!13}a^{25}-\frac{45\!\cdots\!91}{30\!\cdots\!99}a^{24}-\frac{26\!\cdots\!81}{33\!\cdots\!89}a^{23}+\frac{16\!\cdots\!61}{33\!\cdots\!89}a^{22}+\frac{18\!\cdots\!69}{33\!\cdots\!89}a^{21}+\frac{10\!\cdots\!41}{33\!\cdots\!89}a^{20}-\frac{56\!\cdots\!47}{37\!\cdots\!21}a^{19}-\frac{11\!\cdots\!68}{33\!\cdots\!89}a^{18}-\frac{15\!\cdots\!32}{30\!\cdots\!99}a^{17}+\frac{26\!\cdots\!25}{33\!\cdots\!89}a^{16}+\frac{14\!\cdots\!63}{33\!\cdots\!89}a^{15}+\frac{14\!\cdots\!30}{33\!\cdots\!11}a^{14}-\frac{14\!\cdots\!51}{33\!\cdots\!89}a^{13}-\frac{20\!\cdots\!19}{10\!\cdots\!33}a^{12}-\frac{10\!\cdots\!46}{33\!\cdots\!89}a^{11}-\frac{13\!\cdots\!40}{17\!\cdots\!31}a^{10}+\frac{74\!\cdots\!83}{11\!\cdots\!63}a^{9}+\frac{19\!\cdots\!49}{17\!\cdots\!47}a^{8}+\frac{97\!\cdots\!71}{33\!\cdots\!89}a^{7}-\frac{21\!\cdots\!72}{33\!\cdots\!89}a^{6}-\frac{30\!\cdots\!94}{11\!\cdots\!63}a^{5}+\frac{28\!\cdots\!23}{33\!\cdots\!89}a^{4}+\frac{15\!\cdots\!39}{11\!\cdots\!63}a^{3}+\frac{64\!\cdots\!51}{65\!\cdots\!39}a^{2}+\frac{22\!\cdots\!77}{12\!\cdots\!89}a-\frac{68\!\cdots\!20}{75\!\cdots\!17}$, $\frac{21\!\cdots\!18}{33\!\cdots\!89}a^{29}-\frac{83\!\cdots\!46}{33\!\cdots\!89}a^{28}-\frac{23\!\cdots\!36}{11\!\cdots\!63}a^{27}+\frac{63\!\cdots\!28}{11\!\cdots\!63}a^{26}+\frac{76\!\cdots\!05}{19\!\cdots\!59}a^{25}-\frac{14\!\cdots\!51}{33\!\cdots\!89}a^{24}-\frac{15\!\cdots\!68}{33\!\cdots\!89}a^{23}-\frac{40\!\cdots\!12}{33\!\cdots\!89}a^{22}+\frac{89\!\cdots\!33}{30\!\cdots\!99}a^{21}+\frac{13\!\cdots\!55}{33\!\cdots\!89}a^{20}-\frac{22\!\cdots\!18}{37\!\cdots\!21}a^{19}-\frac{84\!\cdots\!10}{33\!\cdots\!89}a^{18}-\frac{13\!\cdots\!91}{33\!\cdots\!89}a^{17}+\frac{75\!\cdots\!19}{33\!\cdots\!89}a^{16}+\frac{82\!\cdots\!55}{33\!\cdots\!89}a^{15}+\frac{14\!\cdots\!50}{37\!\cdots\!21}a^{14}-\frac{13\!\cdots\!10}{33\!\cdots\!89}a^{13}-\frac{14\!\cdots\!27}{11\!\cdots\!63}a^{12}-\frac{81\!\cdots\!34}{33\!\cdots\!89}a^{11}-\frac{48\!\cdots\!34}{33\!\cdots\!89}a^{10}+\frac{39\!\cdots\!59}{11\!\cdots\!63}a^{9}+\frac{17\!\cdots\!68}{19\!\cdots\!17}a^{8}+\frac{20\!\cdots\!48}{33\!\cdots\!89}a^{7}-\frac{47\!\cdots\!27}{11\!\cdots\!41}a^{6}-\frac{82\!\cdots\!04}{11\!\cdots\!63}a^{5}+\frac{12\!\cdots\!16}{33\!\cdots\!89}a^{4}+\frac{15\!\cdots\!85}{11\!\cdots\!63}a^{3}+\frac{22\!\cdots\!79}{21\!\cdots\!13}a^{2}+\frac{20\!\cdots\!72}{11\!\cdots\!99}a-\frac{37\!\cdots\!14}{75\!\cdots\!17}$, $\frac{80\!\cdots\!41}{10\!\cdots\!67}a^{29}-\frac{32\!\cdots\!59}{10\!\cdots\!67}a^{28}-\frac{28\!\cdots\!36}{11\!\cdots\!63}a^{27}+\frac{75\!\cdots\!71}{10\!\cdots\!67}a^{26}+\frac{46\!\cdots\!75}{10\!\cdots\!67}a^{25}-\frac{21\!\cdots\!20}{33\!\cdots\!89}a^{24}-\frac{16\!\cdots\!24}{30\!\cdots\!99}a^{23}-\frac{35\!\cdots\!30}{10\!\cdots\!67}a^{22}+\frac{35\!\cdots\!95}{10\!\cdots\!67}a^{21}+\frac{41\!\cdots\!93}{10\!\cdots\!67}a^{20}-\frac{77\!\cdots\!29}{10\!\cdots\!67}a^{19}-\frac{14\!\cdots\!71}{52\!\cdots\!93}a^{18}-\frac{51\!\cdots\!64}{11\!\cdots\!63}a^{17}+\frac{27\!\cdots\!04}{10\!\cdots\!67}a^{16}+\frac{32\!\cdots\!79}{11\!\cdots\!63}a^{15}+\frac{43\!\cdots\!87}{10\!\cdots\!67}a^{14}-\frac{77\!\cdots\!23}{10\!\cdots\!67}a^{13}-\frac{14\!\cdots\!31}{10\!\cdots\!67}a^{12}-\frac{99\!\cdots\!52}{37\!\cdots\!21}a^{11}-\frac{56\!\cdots\!34}{33\!\cdots\!89}a^{10}+\frac{67\!\cdots\!87}{17\!\cdots\!31}a^{9}+\frac{18\!\cdots\!70}{19\!\cdots\!17}a^{8}+\frac{63\!\cdots\!20}{10\!\cdots\!67}a^{7}-\frac{29\!\cdots\!94}{10\!\cdots\!67}a^{6}-\frac{50\!\cdots\!01}{10\!\cdots\!33}a^{5}+\frac{40\!\cdots\!05}{10\!\cdots\!67}a^{4}+\frac{41\!\cdots\!75}{33\!\cdots\!89}a^{3}+\frac{21\!\cdots\!93}{19\!\cdots\!17}a^{2}+\frac{10\!\cdots\!64}{22\!\cdots\!51}a+\frac{88\!\cdots\!96}{22\!\cdots\!51}$, $\frac{35\!\cdots\!49}{11\!\cdots\!63}a^{29}-\frac{78\!\cdots\!14}{58\!\cdots\!77}a^{28}-\frac{27\!\cdots\!40}{33\!\cdots\!89}a^{27}+\frac{41\!\cdots\!87}{17\!\cdots\!31}a^{26}+\frac{94\!\cdots\!50}{70\!\cdots\!57}a^{25}-\frac{27\!\cdots\!25}{33\!\cdots\!89}a^{24}-\frac{25\!\cdots\!04}{20\!\cdots\!13}a^{23}-\frac{21\!\cdots\!53}{11\!\cdots\!09}a^{22}+\frac{13\!\cdots\!48}{33\!\cdots\!89}a^{21}+\frac{26\!\cdots\!26}{11\!\cdots\!63}a^{20}+\frac{14\!\cdots\!24}{33\!\cdots\!89}a^{19}-\frac{32\!\cdots\!46}{37\!\cdots\!21}a^{18}-\frac{13\!\cdots\!01}{33\!\cdots\!89}a^{17}-\frac{83\!\cdots\!66}{33\!\cdots\!89}a^{16}+\frac{24\!\cdots\!31}{33\!\cdots\!89}a^{15}+\frac{93\!\cdots\!67}{33\!\cdots\!89}a^{14}+\frac{14\!\cdots\!76}{30\!\cdots\!99}a^{13}-\frac{60\!\cdots\!18}{30\!\cdots\!99}a^{12}-\frac{64\!\cdots\!37}{33\!\cdots\!89}a^{11}-\frac{34\!\cdots\!50}{11\!\cdots\!41}a^{10}-\frac{36\!\cdots\!67}{33\!\cdots\!89}a^{9}+\frac{29\!\cdots\!31}{65\!\cdots\!39}a^{8}+\frac{38\!\cdots\!02}{33\!\cdots\!89}a^{7}+\frac{29\!\cdots\!43}{30\!\cdots\!99}a^{6}-\frac{73\!\cdots\!38}{11\!\cdots\!63}a^{5}-\frac{31\!\cdots\!02}{30\!\cdots\!99}a^{4}+\frac{29\!\cdots\!78}{33\!\cdots\!89}a^{3}+\frac{43\!\cdots\!46}{21\!\cdots\!13}a^{2}+\frac{36\!\cdots\!97}{35\!\cdots\!97}a+\frac{94\!\cdots\!10}{75\!\cdots\!17}$, $\frac{46\!\cdots\!50}{10\!\cdots\!67}a^{29}-\frac{19\!\cdots\!36}{10\!\cdots\!67}a^{28}-\frac{49\!\cdots\!58}{34\!\cdots\!23}a^{27}+\frac{52\!\cdots\!29}{11\!\cdots\!63}a^{26}+\frac{26\!\cdots\!22}{10\!\cdots\!67}a^{25}-\frac{42\!\cdots\!28}{10\!\cdots\!67}a^{24}-\frac{30\!\cdots\!31}{10\!\cdots\!67}a^{23}+\frac{47\!\cdots\!81}{10\!\cdots\!67}a^{22}+\frac{68\!\cdots\!46}{33\!\cdots\!89}a^{21}+\frac{17\!\cdots\!99}{91\!\cdots\!97}a^{20}-\frac{63\!\cdots\!09}{12\!\cdots\!49}a^{19}-\frac{14\!\cdots\!00}{10\!\cdots\!67}a^{18}-\frac{23\!\cdots\!90}{10\!\cdots\!67}a^{17}+\frac{70\!\cdots\!02}{33\!\cdots\!11}a^{16}+\frac{16\!\cdots\!92}{10\!\cdots\!67}a^{15}+\frac{21\!\cdots\!10}{10\!\cdots\!67}a^{14}-\frac{88\!\cdots\!40}{10\!\cdots\!67}a^{13}-\frac{42\!\cdots\!56}{52\!\cdots\!93}a^{12}-\frac{46\!\cdots\!16}{33\!\cdots\!89}a^{11}-\frac{22\!\cdots\!75}{33\!\cdots\!89}a^{10}+\frac{77\!\cdots\!23}{33\!\cdots\!89}a^{9}+\frac{97\!\cdots\!58}{19\!\cdots\!17}a^{8}+\frac{26\!\cdots\!93}{10\!\cdots\!67}a^{7}-\frac{19\!\cdots\!74}{91\!\cdots\!97}a^{6}-\frac{24\!\cdots\!49}{10\!\cdots\!67}a^{5}+\frac{29\!\cdots\!48}{10\!\cdots\!67}a^{4}+\frac{24\!\cdots\!27}{37\!\cdots\!21}a^{3}+\frac{99\!\cdots\!77}{19\!\cdots\!17}a^{2}+\frac{62\!\cdots\!79}{38\!\cdots\!67}a+\frac{29\!\cdots\!83}{22\!\cdots\!51}$, $\frac{82\!\cdots\!68}{91\!\cdots\!97}a^{29}-\frac{37\!\cdots\!63}{10\!\cdots\!67}a^{28}-\frac{84\!\cdots\!79}{18\!\cdots\!39}a^{27}-\frac{40\!\cdots\!39}{37\!\cdots\!21}a^{26}+\frac{91\!\cdots\!45}{10\!\cdots\!67}a^{25}+\frac{99\!\cdots\!81}{10\!\cdots\!67}a^{24}-\frac{10\!\cdots\!45}{10\!\cdots\!67}a^{23}-\frac{18\!\cdots\!50}{91\!\cdots\!97}a^{22}+\frac{61\!\cdots\!20}{11\!\cdots\!63}a^{21}+\frac{66\!\cdots\!66}{34\!\cdots\!23}a^{20}-\frac{11\!\cdots\!79}{37\!\cdots\!21}a^{19}-\frac{75\!\cdots\!57}{10\!\cdots\!67}a^{18}-\frac{12\!\cdots\!26}{91\!\cdots\!97}a^{17}-\frac{24\!\cdots\!70}{33\!\cdots\!89}a^{16}+\frac{55\!\cdots\!63}{10\!\cdots\!67}a^{15}+\frac{15\!\cdots\!49}{10\!\cdots\!67}a^{14}+\frac{82\!\cdots\!03}{10\!\cdots\!67}a^{13}-\frac{31\!\cdots\!13}{10\!\cdots\!67}a^{12}-\frac{30\!\cdots\!69}{37\!\cdots\!21}a^{11}-\frac{28\!\cdots\!74}{33\!\cdots\!89}a^{10}+\frac{15\!\cdots\!03}{33\!\cdots\!89}a^{9}+\frac{57\!\cdots\!91}{19\!\cdots\!17}a^{8}+\frac{32\!\cdots\!09}{10\!\cdots\!67}a^{7}-\frac{46\!\cdots\!96}{10\!\cdots\!67}a^{6}-\frac{27\!\cdots\!65}{10\!\cdots\!67}a^{5}+\frac{24\!\cdots\!78}{10\!\cdots\!67}a^{4}+\frac{15\!\cdots\!17}{37\!\cdots\!21}a^{3}+\frac{73\!\cdots\!34}{19\!\cdots\!17}a^{2}+\frac{25\!\cdots\!43}{18\!\cdots\!63}a+\frac{19\!\cdots\!93}{22\!\cdots\!51}$, $\frac{98\!\cdots\!17}{33\!\cdots\!89}a^{29}-\frac{42\!\cdots\!22}{34\!\cdots\!23}a^{28}-\frac{30\!\cdots\!56}{33\!\cdots\!89}a^{27}+\frac{25\!\cdots\!47}{91\!\cdots\!97}a^{26}+\frac{16\!\cdots\!14}{10\!\cdots\!67}a^{25}-\frac{81\!\cdots\!80}{33\!\cdots\!89}a^{24}-\frac{19\!\cdots\!29}{10\!\cdots\!67}a^{23}+\frac{30\!\cdots\!33}{91\!\cdots\!97}a^{22}+\frac{14\!\cdots\!07}{11\!\cdots\!41}a^{21}+\frac{14\!\cdots\!39}{10\!\cdots\!67}a^{20}-\frac{28\!\cdots\!95}{10\!\cdots\!67}a^{19}-\frac{97\!\cdots\!61}{10\!\cdots\!67}a^{18}-\frac{28\!\cdots\!93}{17\!\cdots\!31}a^{17}+\frac{10\!\cdots\!88}{10\!\cdots\!67}a^{16}+\frac{10\!\cdots\!22}{10\!\cdots\!67}a^{15}+\frac{15\!\cdots\!85}{10\!\cdots\!67}a^{14}-\frac{32\!\cdots\!44}{10\!\cdots\!67}a^{13}-\frac{26\!\cdots\!37}{52\!\cdots\!93}a^{12}-\frac{28\!\cdots\!26}{30\!\cdots\!99}a^{11}-\frac{19\!\cdots\!50}{33\!\cdots\!89}a^{10}+\frac{45\!\cdots\!89}{33\!\cdots\!89}a^{9}+\frac{22\!\cdots\!24}{67\!\cdots\!73}a^{8}+\frac{72\!\cdots\!65}{33\!\cdots\!89}a^{7}-\frac{94\!\cdots\!23}{10\!\cdots\!67}a^{6}-\frac{54\!\cdots\!25}{33\!\cdots\!89}a^{5}+\frac{13\!\cdots\!82}{10\!\cdots\!67}a^{4}+\frac{14\!\cdots\!64}{33\!\cdots\!89}a^{3}+\frac{76\!\cdots\!65}{19\!\cdots\!17}a^{2}+\frac{59\!\cdots\!67}{35\!\cdots\!97}a+\frac{55\!\cdots\!93}{22\!\cdots\!51}$, $\frac{25\!\cdots\!47}{34\!\cdots\!23}a^{29}-\frac{24\!\cdots\!55}{10\!\cdots\!67}a^{28}-\frac{26\!\cdots\!96}{10\!\cdots\!67}a^{27}+\frac{18\!\cdots\!06}{33\!\cdots\!89}a^{26}+\frac{49\!\cdots\!78}{10\!\cdots\!67}a^{25}-\frac{37\!\cdots\!64}{10\!\cdots\!67}a^{24}-\frac{57\!\cdots\!43}{10\!\cdots\!67}a^{23}-\frac{27\!\cdots\!63}{10\!\cdots\!67}a^{22}+\frac{36\!\cdots\!99}{10\!\cdots\!67}a^{21}+\frac{18\!\cdots\!26}{33\!\cdots\!89}a^{20}-\frac{87\!\cdots\!77}{11\!\cdots\!63}a^{19}-\frac{31\!\cdots\!24}{10\!\cdots\!67}a^{18}-\frac{49\!\cdots\!94}{10\!\cdots\!67}a^{17}+\frac{46\!\cdots\!26}{33\!\cdots\!89}a^{16}+\frac{10\!\cdots\!77}{33\!\cdots\!89}a^{15}+\frac{53\!\cdots\!73}{10\!\cdots\!67}a^{14}-\frac{23\!\cdots\!61}{63\!\cdots\!13}a^{13}-\frac{13\!\cdots\!82}{91\!\cdots\!97}a^{12}-\frac{10\!\cdots\!97}{33\!\cdots\!89}a^{11}-\frac{70\!\cdots\!75}{33\!\cdots\!89}a^{10}+\frac{11\!\cdots\!91}{30\!\cdots\!99}a^{9}+\frac{21\!\cdots\!40}{19\!\cdots\!17}a^{8}+\frac{76\!\cdots\!23}{10\!\cdots\!67}a^{7}-\frac{36\!\cdots\!24}{91\!\cdots\!97}a^{6}-\frac{62\!\cdots\!39}{10\!\cdots\!67}a^{5}+\frac{45\!\cdots\!50}{10\!\cdots\!67}a^{4}+\frac{47\!\cdots\!63}{33\!\cdots\!89}a^{3}+\frac{24\!\cdots\!76}{19\!\cdots\!17}a^{2}+\frac{17\!\cdots\!70}{38\!\cdots\!67}a+\frac{91\!\cdots\!51}{22\!\cdots\!51}$, $\frac{42\!\cdots\!32}{10\!\cdots\!67}a^{29}-\frac{17\!\cdots\!33}{10\!\cdots\!67}a^{28}-\frac{13\!\cdots\!09}{10\!\cdots\!67}a^{27}+\frac{42\!\cdots\!10}{11\!\cdots\!63}a^{26}+\frac{26\!\cdots\!61}{10\!\cdots\!67}a^{25}-\frac{30\!\cdots\!54}{10\!\cdots\!67}a^{24}-\frac{31\!\cdots\!05}{10\!\cdots\!67}a^{23}-\frac{25\!\cdots\!85}{34\!\cdots\!23}a^{22}+\frac{22\!\cdots\!63}{10\!\cdots\!67}a^{21}+\frac{96\!\cdots\!22}{33\!\cdots\!89}a^{20}-\frac{23\!\cdots\!09}{37\!\cdots\!21}a^{19}-\frac{20\!\cdots\!56}{10\!\cdots\!67}a^{18}-\frac{17\!\cdots\!68}{10\!\cdots\!67}a^{17}+\frac{23\!\cdots\!92}{70\!\cdots\!57}a^{16}+\frac{54\!\cdots\!81}{33\!\cdots\!89}a^{15}+\frac{25\!\cdots\!76}{10\!\cdots\!67}a^{14}-\frac{77\!\cdots\!07}{37\!\cdots\!21}a^{13}-\frac{11\!\cdots\!34}{10\!\cdots\!67}a^{12}-\frac{44\!\cdots\!72}{37\!\cdots\!21}a^{11}+\frac{59\!\cdots\!05}{33\!\cdots\!89}a^{10}+\frac{10\!\cdots\!91}{33\!\cdots\!89}a^{9}+\frac{10\!\cdots\!80}{19\!\cdots\!17}a^{8}+\frac{68\!\cdots\!74}{10\!\cdots\!67}a^{7}-\frac{85\!\cdots\!08}{10\!\cdots\!67}a^{6}-\frac{23\!\cdots\!20}{10\!\cdots\!67}a^{5}+\frac{12\!\cdots\!53}{10\!\cdots\!67}a^{4}+\frac{23\!\cdots\!35}{33\!\cdots\!89}a^{3}-\frac{52\!\cdots\!10}{19\!\cdots\!17}a^{2}-\frac{42\!\cdots\!02}{12\!\cdots\!89}a-\frac{41\!\cdots\!54}{22\!\cdots\!51}$, $\frac{16\!\cdots\!54}{10\!\cdots\!67}a^{29}-\frac{90\!\cdots\!90}{10\!\cdots\!67}a^{28}-\frac{39\!\cdots\!21}{91\!\cdots\!97}a^{27}+\frac{20\!\cdots\!33}{91\!\cdots\!97}a^{26}+\frac{73\!\cdots\!16}{10\!\cdots\!67}a^{25}-\frac{24\!\cdots\!00}{10\!\cdots\!67}a^{24}-\frac{96\!\cdots\!38}{11\!\cdots\!63}a^{23}+\frac{12\!\cdots\!15}{10\!\cdots\!67}a^{22}+\frac{64\!\cdots\!07}{10\!\cdots\!67}a^{21}-\frac{70\!\cdots\!74}{91\!\cdots\!97}a^{20}-\frac{20\!\cdots\!93}{10\!\cdots\!67}a^{19}-\frac{30\!\cdots\!91}{10\!\cdots\!67}a^{18}-\frac{13\!\cdots\!20}{34\!\cdots\!23}a^{17}+\frac{13\!\cdots\!56}{10\!\cdots\!67}a^{16}+\frac{46\!\cdots\!86}{10\!\cdots\!67}a^{15}+\frac{23\!\cdots\!23}{11\!\cdots\!63}a^{14}-\frac{79\!\cdots\!49}{10\!\cdots\!67}a^{13}-\frac{20\!\cdots\!38}{10\!\cdots\!67}a^{12}-\frac{75\!\cdots\!11}{33\!\cdots\!89}a^{11}+\frac{33\!\cdots\!14}{33\!\cdots\!89}a^{10}+\frac{92\!\cdots\!33}{11\!\cdots\!63}a^{9}+\frac{14\!\cdots\!58}{17\!\cdots\!47}a^{8}-\frac{38\!\cdots\!52}{10\!\cdots\!67}a^{7}-\frac{70\!\cdots\!50}{10\!\cdots\!67}a^{6}+\frac{19\!\cdots\!84}{91\!\cdots\!97}a^{5}+\frac{10\!\cdots\!71}{10\!\cdots\!67}a^{4}+\frac{36\!\cdots\!22}{37\!\cdots\!21}a^{3}+\frac{37\!\cdots\!98}{19\!\cdots\!17}a^{2}-\frac{91\!\cdots\!76}{38\!\cdots\!67}a-\frac{10\!\cdots\!03}{22\!\cdots\!51}$, $\frac{21\!\cdots\!84}{10\!\cdots\!67}a^{29}-\frac{10\!\cdots\!39}{10\!\cdots\!67}a^{28}-\frac{53\!\cdots\!51}{91\!\cdots\!97}a^{27}+\frac{13\!\cdots\!88}{58\!\cdots\!77}a^{26}+\frac{10\!\cdots\!69}{10\!\cdots\!67}a^{25}-\frac{22\!\cdots\!05}{10\!\cdots\!67}a^{24}-\frac{11\!\cdots\!47}{10\!\cdots\!67}a^{23}+\frac{57\!\cdots\!28}{91\!\cdots\!97}a^{22}+\frac{26\!\cdots\!83}{33\!\cdots\!89}a^{21}+\frac{53\!\cdots\!19}{10\!\cdots\!67}a^{20}-\frac{67\!\cdots\!49}{37\!\cdots\!21}a^{19}-\frac{53\!\cdots\!15}{10\!\cdots\!67}a^{18}-\frac{99\!\cdots\!11}{10\!\cdots\!67}a^{17}+\frac{31\!\cdots\!43}{33\!\cdots\!89}a^{16}+\frac{63\!\cdots\!95}{10\!\cdots\!67}a^{15}+\frac{78\!\cdots\!21}{10\!\cdots\!67}a^{14}-\frac{27\!\cdots\!52}{10\!\cdots\!67}a^{13}-\frac{28\!\cdots\!89}{10\!\cdots\!67}a^{12}-\frac{55\!\cdots\!82}{10\!\cdots\!33}a^{11}-\frac{97\!\cdots\!04}{30\!\cdots\!99}a^{10}+\frac{28\!\cdots\!75}{33\!\cdots\!89}a^{9}+\frac{34\!\cdots\!75}{19\!\cdots\!17}a^{8}+\frac{12\!\cdots\!56}{10\!\cdots\!67}a^{7}-\frac{47\!\cdots\!10}{34\!\cdots\!23}a^{6}-\frac{65\!\cdots\!94}{10\!\cdots\!67}a^{5}+\frac{34\!\cdots\!14}{10\!\cdots\!67}a^{4}+\frac{25\!\cdots\!96}{11\!\cdots\!63}a^{3}+\frac{53\!\cdots\!01}{19\!\cdots\!17}a^{2}+\frac{48\!\cdots\!75}{38\!\cdots\!67}a+\frac{13\!\cdots\!63}{22\!\cdots\!51}$, $\frac{81\!\cdots\!50}{59\!\cdots\!51}a^{29}+\frac{57\!\cdots\!19}{34\!\cdots\!03}a^{28}-\frac{56\!\cdots\!89}{65\!\cdots\!39}a^{27}-\frac{20\!\cdots\!89}{59\!\cdots\!51}a^{26}+\frac{10\!\cdots\!47}{59\!\cdots\!51}a^{25}+\frac{32\!\cdots\!25}{21\!\cdots\!13}a^{24}-\frac{40\!\cdots\!95}{19\!\cdots\!17}a^{23}-\frac{18\!\cdots\!62}{59\!\cdots\!51}a^{22}+\frac{73\!\cdots\!88}{59\!\cdots\!51}a^{21}+\frac{17\!\cdots\!67}{59\!\cdots\!51}a^{20}-\frac{14\!\cdots\!75}{53\!\cdots\!41}a^{19}-\frac{36\!\cdots\!65}{31\!\cdots\!29}a^{18}-\frac{32\!\cdots\!49}{19\!\cdots\!17}a^{17}-\frac{53\!\cdots\!33}{59\!\cdots\!51}a^{16}+\frac{63\!\cdots\!97}{59\!\cdots\!49}a^{15}+\frac{13\!\cdots\!54}{59\!\cdots\!51}a^{14}+\frac{92\!\cdots\!18}{59\!\cdots\!51}a^{13}-\frac{31\!\cdots\!52}{59\!\cdots\!51}a^{12}-\frac{21\!\cdots\!92}{19\!\cdots\!17}a^{11}-\frac{18\!\cdots\!77}{21\!\cdots\!13}a^{10}+\frac{81\!\cdots\!58}{65\!\cdots\!39}a^{9}+\frac{87\!\cdots\!36}{19\!\cdots\!17}a^{8}+\frac{14\!\cdots\!36}{53\!\cdots\!41}a^{7}-\frac{94\!\cdots\!46}{34\!\cdots\!03}a^{6}-\frac{14\!\cdots\!49}{65\!\cdots\!39}a^{5}+\frac{13\!\cdots\!13}{59\!\cdots\!51}a^{4}+\frac{34\!\cdots\!64}{65\!\cdots\!39}a^{3}+\frac{79\!\cdots\!79}{19\!\cdots\!17}a^{2}+\frac{75\!\cdots\!28}{12\!\cdots\!89}a+\frac{19\!\cdots\!58}{78\!\cdots\!19}$, $\frac{29\!\cdots\!89}{33\!\cdots\!89}a^{29}-\frac{36\!\cdots\!52}{10\!\cdots\!67}a^{28}-\frac{13\!\cdots\!25}{48\!\cdots\!63}a^{27}+\frac{28\!\cdots\!37}{33\!\cdots\!89}a^{26}+\frac{51\!\cdots\!35}{10\!\cdots\!67}a^{25}-\frac{73\!\cdots\!66}{10\!\cdots\!67}a^{24}-\frac{19\!\cdots\!18}{33\!\cdots\!89}a^{23}-\frac{10\!\cdots\!56}{10\!\cdots\!67}a^{22}+\frac{35\!\cdots\!47}{91\!\cdots\!97}a^{21}+\frac{43\!\cdots\!76}{10\!\cdots\!67}a^{20}-\frac{29\!\cdots\!00}{33\!\cdots\!89}a^{19}-\frac{30\!\cdots\!80}{10\!\cdots\!67}a^{18}-\frac{49\!\cdots\!55}{10\!\cdots\!67}a^{17}+\frac{11\!\cdots\!97}{33\!\cdots\!89}a^{16}+\frac{31\!\cdots\!96}{10\!\cdots\!67}a^{15}+\frac{46\!\cdots\!70}{10\!\cdots\!67}a^{14}-\frac{10\!\cdots\!32}{10\!\cdots\!67}a^{13}-\frac{51\!\cdots\!97}{33\!\cdots\!89}a^{12}-\frac{10\!\cdots\!77}{37\!\cdots\!21}a^{11}-\frac{57\!\cdots\!27}{33\!\cdots\!89}a^{10}+\frac{14\!\cdots\!56}{33\!\cdots\!89}a^{9}+\frac{66\!\cdots\!13}{65\!\cdots\!39}a^{8}+\frac{21\!\cdots\!98}{33\!\cdots\!89}a^{7}-\frac{33\!\cdots\!28}{10\!\cdots\!67}a^{6}-\frac{51\!\cdots\!30}{10\!\cdots\!67}a^{5}+\frac{15\!\cdots\!04}{33\!\cdots\!89}a^{4}+\frac{44\!\cdots\!98}{33\!\cdots\!89}a^{3}+\frac{76\!\cdots\!71}{65\!\cdots\!39}a^{2}+\frac{60\!\cdots\!84}{12\!\cdots\!89}a+\frac{33\!\cdots\!19}{75\!\cdots\!17}$, $\frac{95\!\cdots\!71}{11\!\cdots\!63}a^{29}-\frac{34\!\cdots\!71}{91\!\cdots\!97}a^{28}-\frac{25\!\cdots\!78}{10\!\cdots\!67}a^{27}+\frac{88\!\cdots\!19}{10\!\cdots\!67}a^{26}+\frac{42\!\cdots\!29}{91\!\cdots\!97}a^{25}-\frac{82\!\cdots\!94}{10\!\cdots\!67}a^{24}-\frac{53\!\cdots\!38}{10\!\cdots\!67}a^{23}+\frac{14\!\cdots\!43}{10\!\cdots\!67}a^{22}+\frac{33\!\cdots\!19}{91\!\cdots\!97}a^{21}+\frac{10\!\cdots\!33}{33\!\cdots\!89}a^{20}-\frac{90\!\cdots\!16}{10\!\cdots\!67}a^{19}-\frac{26\!\cdots\!25}{10\!\cdots\!67}a^{18}-\frac{40\!\cdots\!65}{10\!\cdots\!67}a^{17}+\frac{40\!\cdots\!84}{10\!\cdots\!67}a^{16}+\frac{28\!\cdots\!61}{10\!\cdots\!67}a^{15}+\frac{11\!\cdots\!34}{30\!\cdots\!99}a^{14}-\frac{54\!\cdots\!61}{30\!\cdots\!99}a^{13}-\frac{14\!\cdots\!00}{10\!\cdots\!67}a^{12}-\frac{26\!\cdots\!07}{11\!\cdots\!63}a^{11}-\frac{12\!\cdots\!73}{11\!\cdots\!63}a^{10}+\frac{13\!\cdots\!92}{33\!\cdots\!89}a^{9}+\frac{16\!\cdots\!27}{19\!\cdots\!17}a^{8}+\frac{47\!\cdots\!99}{11\!\cdots\!63}a^{7}-\frac{36\!\cdots\!41}{10\!\cdots\!67}a^{6}-\frac{36\!\cdots\!66}{10\!\cdots\!67}a^{5}+\frac{50\!\cdots\!37}{10\!\cdots\!67}a^{4}+\frac{36\!\cdots\!08}{33\!\cdots\!89}a^{3}+\frac{32\!\cdots\!13}{37\!\cdots\!89}a^{2}+\frac{36\!\cdots\!64}{11\!\cdots\!99}a+\frac{63\!\cdots\!00}{20\!\cdots\!41}$ 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:  \( 44958053496730.96 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 44958053496730.96 \cdot 3}{2\cdot\sqrt{8851375005191383462430321349782942769050796640392327}}\cr\approx \mathstrut & 0.673118324687278 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^30 - 4*x^29 - 32*x^28 + 92*x^27 + 588*x^26 - 752*x^25 - 6717*x^24 - 805*x^23 + 44049*x^22 + 54263*x^21 - 93757*x^20 - 350454*x^19 - 591742*x^18 + 307201*x^17 + 3600412*x^16 + 5669890*x^15 - 632841*x^14 - 17605194*x^13 - 34361042*x^12 - 23087625*x^11 + 45815379*x^10 + 119941341*x^9 + 85868062*x^8 - 30516460*x^7 - 62011703*x^6 + 46150369*x^5 + 155790202*x^4 + 147686157*x^3 + 67768188*x^2 + 10788948*x + 751689)
 
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 - 4*x^29 - 32*x^28 + 92*x^27 + 588*x^26 - 752*x^25 - 6717*x^24 - 805*x^23 + 44049*x^22 + 54263*x^21 - 93757*x^20 - 350454*x^19 - 591742*x^18 + 307201*x^17 + 3600412*x^16 + 5669890*x^15 - 632841*x^14 - 17605194*x^13 - 34361042*x^12 - 23087625*x^11 + 45815379*x^10 + 119941341*x^9 + 85868062*x^8 - 30516460*x^7 - 62011703*x^6 + 46150369*x^5 + 155790202*x^4 + 147686157*x^3 + 67768188*x^2 + 10788948*x + 751689, 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 - 4*x^29 - 32*x^28 + 92*x^27 + 588*x^26 - 752*x^25 - 6717*x^24 - 805*x^23 + 44049*x^22 + 54263*x^21 - 93757*x^20 - 350454*x^19 - 591742*x^18 + 307201*x^17 + 3600412*x^16 + 5669890*x^15 - 632841*x^14 - 17605194*x^13 - 34361042*x^12 - 23087625*x^11 + 45815379*x^10 + 119941341*x^9 + 85868062*x^8 - 30516460*x^7 - 62011703*x^6 + 46150369*x^5 + 155790202*x^4 + 147686157*x^3 + 67768188*x^2 + 10788948*x + 751689);
 
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 - 4*x^29 - 32*x^28 + 92*x^27 + 588*x^26 - 752*x^25 - 6717*x^24 - 805*x^23 + 44049*x^22 + 54263*x^21 - 93757*x^20 - 350454*x^19 - 591742*x^18 + 307201*x^17 + 3600412*x^16 + 5669890*x^15 - 632841*x^14 - 17605194*x^13 - 34361042*x^12 - 23087625*x^11 + 45815379*x^10 + 119941341*x^9 + 85868062*x^8 - 30516460*x^7 - 62011703*x^6 + 46150369*x^5 + 155790202*x^4 + 147686157*x^3 + 67768188*x^2 + 10788948*x + 751689);
 
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

$D_{30}$ (as 30T14):

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 60
The 18 conjugacy class representatives for $D_{30}$
Character table for $D_{30}$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.1.7791.1, 5.1.25281.1, 6.0.424897767.1, 10.0.10741840447527.1, 15.1.725705259120295972581231.1

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]
 

Sibling fields

Degree 30 sibling: 30.2.1407368625825429970526421094615487900279076665822379993.1
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 $15^{2}$ R $30$ R ${\href{/padicField/11.2.0.1}{2} }^{14}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ $30$ ${\href{/padicField/17.2.0.1}{2} }^{15}$ ${\href{/padicField/19.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{14}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{15}$ ${\href{/padicField/37.5.0.1}{5} }^{6}$ $30$ ${\href{/padicField/43.5.0.1}{5} }^{6}$ ${\href{/padicField/47.2.0.1}{2} }^{15}$ R ${\href{/padicField/59.2.0.1}{2} }^{15}$

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
\(3\) Copy content Toggle raw display 3.2.0.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(7\) Copy content Toggle raw display Deg $30$$6$$5$$25$
\(53\) Copy content Toggle raw display $\Q_{53}$$x + 51$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 51$$1$$1$$0$Trivial$[\ ]$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
1.159.2t1.a.a$1$ $ 3 \cdot 53 $ \(\Q(\sqrt{-159}) \) $C_2$ (as 2T1) $1$ $-1$
1.1113.2t1.a.a$1$ $ 3 \cdot 7 \cdot 53 $ \(\Q(\sqrt{1113}) \) $C_2$ (as 2T1) $1$ $1$
* 2.7791.6t3.a.a$2$ $ 3 \cdot 7^{2} \cdot 53 $ 6.0.424897767.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.7791.3t2.a.a$2$ $ 3 \cdot 7^{2} \cdot 53 $ 3.1.7791.1 $S_3$ (as 3T2) $1$ $0$
* 2.159.5t2.a.b$2$ $ 3 \cdot 53 $ 5.1.25281.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.159.5t2.a.a$2$ $ 3 \cdot 53 $ 5.1.25281.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.7791.10t3.b.b$2$ $ 3 \cdot 7^{2} \cdot 53 $ 10.0.10741840447527.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.7791.10t3.b.a$2$ $ 3 \cdot 7^{2} \cdot 53 $ 10.0.10741840447527.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.7791.15t2.a.b$2$ $ 3 \cdot 7^{2} \cdot 53 $ 15.1.725705259120295972581231.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.7791.30t14.b.c$2$ $ 3 \cdot 7^{2} \cdot 53 $ 30.0.8851375005191383462430321349782942769050796640392327.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.7791.15t2.a.a$2$ $ 3 \cdot 7^{2} \cdot 53 $ 15.1.725705259120295972581231.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.7791.30t14.b.d$2$ $ 3 \cdot 7^{2} \cdot 53 $ 30.0.8851375005191383462430321349782942769050796640392327.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.7791.30t14.b.a$2$ $ 3 \cdot 7^{2} \cdot 53 $ 30.0.8851375005191383462430321349782942769050796640392327.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.7791.30t14.b.b$2$ $ 3 \cdot 7^{2} \cdot 53 $ 30.0.8851375005191383462430321349782942769050796640392327.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.7791.15t2.a.d$2$ $ 3 \cdot 7^{2} \cdot 53 $ 15.1.725705259120295972581231.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.7791.15t2.a.c$2$ $ 3 \cdot 7^{2} \cdot 53 $ 15.1.725705259120295972581231.1 $D_{15}$ (as 15T2) $1$ $0$

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 *.