↑

Properties

Label 30.2.553...125.1
Degree $30$
Signature $[2, 14]$
Discriminant $5.540\times 10^{50}$
Root discriminant \(49.14\)
Ramified primes $5,751$
Class number $4$ (GRH)
Class group [4] (GRH)
Galois group $D_{30}$ (as 30T14)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 5*x^29 + 2*x^28 + 21*x^27 + 40*x^26 - 46*x^25 + 39*x^24 - 296*x^23 - 1612*x^22 - 2561*x^21 + 962*x^20 + 4669*x^19 + 19325*x^18 + 55780*x^17 + 98099*x^16 + 110152*x^15 + 94739*x^14 + 50734*x^13 + 40862*x^12 + 44242*x^11 + 25838*x^10 + 20656*x^9 + 3131*x^8 + 2527*x^7 + 6736*x^6 + 1305*x^5 + 3768*x^4 - 512*x^3 + 661*x^2 - 25*x - 1)
 
gp: K = bnfinit(y^30 - 5*y^29 + 2*y^28 + 21*y^27 + 40*y^26 - 46*y^25 + 39*y^24 - 296*y^23 - 1612*y^22 - 2561*y^21 + 962*y^20 + 4669*y^19 + 19325*y^18 + 55780*y^17 + 98099*y^16 + 110152*y^15 + 94739*y^14 + 50734*y^13 + 40862*y^12 + 44242*y^11 + 25838*y^10 + 20656*y^9 + 3131*y^8 + 2527*y^7 + 6736*y^6 + 1305*y^5 + 3768*y^4 - 512*y^3 + 661*y^2 - 25*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 5*x^29 + 2*x^28 + 21*x^27 + 40*x^26 - 46*x^25 + 39*x^24 - 296*x^23 - 1612*x^22 - 2561*x^21 + 962*x^20 + 4669*x^19 + 19325*x^18 + 55780*x^17 + 98099*x^16 + 110152*x^15 + 94739*x^14 + 50734*x^13 + 40862*x^12 + 44242*x^11 + 25838*x^10 + 20656*x^9 + 3131*x^8 + 2527*x^7 + 6736*x^6 + 1305*x^5 + 3768*x^4 - 512*x^3 + 661*x^2 - 25*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 5*x^29 + 2*x^28 + 21*x^27 + 40*x^26 - 46*x^25 + 39*x^24 - 296*x^23 - 1612*x^22 - 2561*x^21 + 962*x^20 + 4669*x^19 + 19325*x^18 + 55780*x^17 + 98099*x^16 + 110152*x^15 + 94739*x^14 + 50734*x^13 + 40862*x^12 + 44242*x^11 + 25838*x^10 + 20656*x^9 + 3131*x^8 + 2527*x^7 + 6736*x^6 + 1305*x^5 + 3768*x^4 - 512*x^3 + 661*x^2 - 25*x - 1)
 

\( x^{30} - 5 x^{29} + 2 x^{28} + 21 x^{27} + 40 x^{26} - 46 x^{25} + 39 x^{24} - 296 x^{23} - 1612 x^{22} + \cdots - 1 \) 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:  $[2, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(553999258054841955537532525076904905639678955078125\) \(\medspace = 5^{15}\cdot 751^{14}\) 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:  \(49.14\)
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:  $5^{1/2}751^{1/2}\approx 61.27805479941412$
Ramified primes:   \(5\), \(751\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{9}a^{15}-\frac{1}{9}a^{14}+\frac{1}{9}a^{13}-\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}-\frac{1}{9}a^{8}-\frac{4}{9}a^{7}+\frac{4}{9}a^{6}-\frac{4}{9}a^{5}+\frac{4}{9}a^{4}-\frac{4}{9}a^{3}+\frac{4}{9}a^{2}-\frac{4}{9}a+\frac{4}{9}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{8}-\frac{2}{9}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{9}-\frac{2}{9}a$, $\frac{1}{9}a^{18}+\frac{1}{9}a^{10}-\frac{2}{9}a^{2}$, $\frac{1}{9}a^{19}+\frac{1}{9}a^{11}-\frac{2}{9}a^{3}$, $\frac{1}{27}a^{20}+\frac{1}{27}a^{19}-\frac{1}{27}a^{18}+\frac{1}{27}a^{17}-\frac{1}{27}a^{15}-\frac{2}{27}a^{14}+\frac{2}{27}a^{13}-\frac{4}{27}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}-\frac{1}{9}a^{9}+\frac{1}{27}a^{8}+\frac{4}{27}a^{7}+\frac{8}{27}a^{6}+\frac{10}{27}a^{5}+\frac{1}{3}a^{4}-\frac{1}{27}a^{3}+\frac{4}{27}a^{2}-\frac{13}{27}a+\frac{5}{27}$, $\frac{1}{27}a^{21}+\frac{1}{27}a^{19}-\frac{1}{27}a^{18}-\frac{1}{27}a^{17}-\frac{1}{27}a^{16}-\frac{1}{27}a^{15}+\frac{4}{27}a^{14}+\frac{1}{9}a^{13}-\frac{2}{27}a^{12}+\frac{1}{9}a^{11}+\frac{4}{27}a^{9}+\frac{1}{9}a^{8}+\frac{4}{27}a^{7}+\frac{2}{27}a^{6}-\frac{10}{27}a^{5}-\frac{1}{27}a^{4}-\frac{1}{27}a^{3}+\frac{7}{27}a^{2}-\frac{1}{3}a-\frac{5}{27}$, $\frac{1}{27}a^{22}+\frac{1}{27}a^{19}+\frac{1}{27}a^{17}-\frac{1}{27}a^{16}-\frac{1}{27}a^{15}+\frac{2}{27}a^{14}-\frac{1}{27}a^{13}+\frac{4}{27}a^{12}+\frac{1}{9}a^{11}-\frac{2}{27}a^{10}+\frac{1}{9}a^{9}-\frac{5}{27}a^{7}-\frac{2}{9}a^{6}+\frac{4}{27}a^{5}+\frac{2}{27}a^{4}-\frac{10}{27}a^{3}-\frac{1}{27}a^{2}-\frac{1}{27}a+\frac{7}{27}$, $\frac{1}{27}a^{23}-\frac{1}{27}a^{19}-\frac{1}{27}a^{18}+\frac{1}{27}a^{17}-\frac{1}{27}a^{16}+\frac{4}{27}a^{14}-\frac{1}{27}a^{13}+\frac{1}{27}a^{12}+\frac{1}{27}a^{11}+\frac{1}{9}a^{9}-\frac{1}{9}a^{8}+\frac{2}{27}a^{7}+\frac{11}{27}a^{6}+\frac{4}{27}a^{5}+\frac{5}{27}a^{4}+\frac{1}{9}a^{3}-\frac{11}{27}a^{2}-\frac{1}{27}a+\frac{10}{27}$, $\frac{1}{5481}a^{24}+\frac{34}{5481}a^{23}-\frac{16}{5481}a^{22}+\frac{44}{5481}a^{21}+\frac{2}{5481}a^{20}-\frac{235}{5481}a^{19}+\frac{115}{5481}a^{18}+\frac{17}{1827}a^{17}-\frac{92}{5481}a^{16}+\frac{20}{609}a^{15}-\frac{260}{1827}a^{14}-\frac{128}{783}a^{13}+\frac{296}{1827}a^{12}-\frac{563}{5481}a^{11}-\frac{76}{5481}a^{10}-\frac{796}{5481}a^{9}+\frac{311}{5481}a^{8}-\frac{2263}{5481}a^{7}+\frac{19}{5481}a^{6}-\frac{20}{87}a^{5}-\frac{1298}{5481}a^{4}+\frac{215}{609}a^{3}+\frac{583}{1827}a^{2}-\frac{449}{5481}a-\frac{2140}{5481}$, $\frac{1}{16443}a^{25}-\frac{1}{16443}a^{24}-\frac{191}{16443}a^{23}+\frac{22}{1827}a^{22}+\frac{86}{16443}a^{21}+\frac{101}{16443}a^{20}+\frac{47}{1827}a^{19}+\frac{289}{16443}a^{18}-\frac{659}{16443}a^{17}-\frac{17}{5481}a^{16}-\frac{787}{16443}a^{15}+\frac{2}{2349}a^{14}-\frac{26}{189}a^{13}+\frac{634}{16443}a^{12}+\frac{1156}{16443}a^{11}+\frac{18}{203}a^{10}+\frac{1172}{16443}a^{9}-\frac{1780}{16443}a^{8}+\frac{1642}{5481}a^{7}-\frac{710}{2349}a^{6}-\frac{5918}{16443}a^{5}-\frac{1399}{5481}a^{4}-\frac{2437}{16443}a^{3}-\frac{2185}{16443}a^{2}-\frac{3883}{16443}a-\frac{175}{2349}$, $\frac{1}{49329}a^{26}-\frac{1}{16443}a^{24}-\frac{38}{7047}a^{23}-\frac{913}{49329}a^{22}+\frac{586}{49329}a^{21}+\frac{293}{49329}a^{20}+\frac{5}{1701}a^{19}+\frac{1877}{49329}a^{18}-\frac{1424}{49329}a^{17}+\frac{1871}{49329}a^{16}+\frac{1579}{49329}a^{15}-\frac{463}{49329}a^{14}+\frac{6247}{49329}a^{13}+\frac{7628}{49329}a^{12}+\frac{6436}{49329}a^{11}+\frac{2882}{49329}a^{10}-\frac{20}{49329}a^{9}+\frac{4679}{49329}a^{8}+\frac{14383}{49329}a^{7}+\frac{13409}{49329}a^{6}+\frac{2728}{7047}a^{5}+\frac{170}{49329}a^{4}+\frac{18835}{49329}a^{3}+\frac{24256}{49329}a^{2}-\frac{24197}{49329}a+\frac{74}{7047}$, $\frac{1}{49329}a^{27}+\frac{1}{49329}a^{24}+\frac{386}{49329}a^{23}+\frac{514}{49329}a^{22}-\frac{358}{49329}a^{21}-\frac{839}{49329}a^{20}+\frac{1814}{49329}a^{19}-\frac{2393}{49329}a^{18}-\frac{136}{7047}a^{17}-\frac{1490}{49329}a^{16}+\frac{1928}{49329}a^{15}-\frac{5168}{49329}a^{14}-\frac{5395}{49329}a^{13}-\frac{2201}{49329}a^{12}+\frac{5981}{49329}a^{11}+\frac{3931}{49329}a^{10}+\frac{5207}{49329}a^{9}+\frac{3490}{49329}a^{8}-\frac{10972}{49329}a^{7}-\frac{23570}{49329}a^{6}+\frac{3176}{7047}a^{5}-\frac{4394}{49329}a^{4}+\frac{7738}{49329}a^{3}-\frac{17099}{49329}a^{2}-\frac{22741}{49329}a+\frac{5927}{16443}$, $\frac{1}{1698525676071}a^{28}-\frac{2137156}{242646525153}a^{27}+\frac{267763}{58569850899}a^{26}-\frac{367088}{54791150841}a^{25}+\frac{1186888}{242646525153}a^{24}+\frac{11888307520}{1698525676071}a^{23}-\frac{3093067303}{1698525676071}a^{22}-\frac{4169648075}{242646525153}a^{21}-\frac{25503223408}{1698525676071}a^{20}-\frac{3114117326}{1698525676071}a^{19}+\frac{29428150796}{1698525676071}a^{18}+\frac{973512686}{58569850899}a^{17}+\frac{69276765407}{1698525676071}a^{16}+\frac{26119636432}{1698525676071}a^{15}-\frac{229960298653}{1698525676071}a^{14}+\frac{35976612691}{1698525676071}a^{13}+\frac{152462260997}{1698525676071}a^{12}+\frac{24182799313}{242646525153}a^{11}-\frac{125374199056}{1698525676071}a^{10}-\frac{259367916014}{1698525676071}a^{9}+\frac{101168133212}{1698525676071}a^{8}-\frac{322141988129}{1698525676071}a^{7}-\frac{204137227345}{1698525676071}a^{6}+\frac{1776112561}{242646525153}a^{5}-\frac{11353595342}{54791150841}a^{4}-\frac{602903428150}{1698525676071}a^{3}+\frac{33877139299}{188725075119}a^{2}+\frac{8980159702}{33304425021}a+\frac{118981838386}{1698525676071}$, $\frac{1}{46\!\cdots\!11}a^{29}+\frac{10\!\cdots\!04}{66\!\cdots\!73}a^{28}+\frac{19\!\cdots\!18}{46\!\cdots\!11}a^{27}+\frac{10\!\cdots\!96}{46\!\cdots\!11}a^{26}-\frac{25\!\cdots\!52}{46\!\cdots\!11}a^{25}+\frac{54\!\cdots\!48}{14\!\cdots\!81}a^{24}-\frac{35\!\cdots\!98}{46\!\cdots\!11}a^{23}+\frac{84\!\cdots\!32}{46\!\cdots\!11}a^{22}-\frac{38\!\cdots\!68}{46\!\cdots\!11}a^{21}-\frac{51\!\cdots\!35}{46\!\cdots\!11}a^{20}-\frac{11\!\cdots\!72}{46\!\cdots\!11}a^{19}-\frac{40\!\cdots\!61}{46\!\cdots\!11}a^{18}-\frac{85\!\cdots\!20}{94\!\cdots\!39}a^{17}+\frac{16\!\cdots\!94}{46\!\cdots\!11}a^{16}-\frac{23\!\cdots\!98}{66\!\cdots\!73}a^{15}+\frac{11\!\cdots\!29}{66\!\cdots\!73}a^{14}-\frac{25\!\cdots\!72}{46\!\cdots\!11}a^{13}+\frac{25\!\cdots\!35}{46\!\cdots\!11}a^{12}+\frac{54\!\cdots\!06}{46\!\cdots\!11}a^{11}+\frac{45\!\cdots\!81}{46\!\cdots\!11}a^{10}+\frac{59\!\cdots\!32}{46\!\cdots\!11}a^{9}-\frac{60\!\cdots\!83}{46\!\cdots\!11}a^{8}-\frac{18\!\cdots\!74}{94\!\cdots\!39}a^{7}+\frac{39\!\cdots\!56}{66\!\cdots\!73}a^{6}-\frac{76\!\cdots\!81}{46\!\cdots\!11}a^{5}-\frac{16\!\cdots\!82}{46\!\cdots\!11}a^{4}-\frac{18\!\cdots\!73}{15\!\cdots\!37}a^{3}-\frac{19\!\cdots\!44}{51\!\cdots\!79}a^{2}-\frac{23\!\cdots\!20}{66\!\cdots\!73}a-\frac{65\!\cdots\!84}{15\!\cdots\!37}$ 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

$C_{4}$, which has order $4$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (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:   $\frac{11\!\cdots\!81}{29\!\cdots\!31}a^{29}-\frac{81\!\cdots\!46}{41\!\cdots\!33}a^{28}+\frac{21\!\cdots\!84}{32\!\cdots\!59}a^{27}+\frac{24\!\cdots\!81}{29\!\cdots\!31}a^{26}+\frac{15\!\cdots\!70}{97\!\cdots\!77}a^{25}-\frac{20\!\cdots\!80}{12\!\cdots\!17}a^{24}+\frac{42\!\cdots\!17}{29\!\cdots\!31}a^{23}-\frac{34\!\cdots\!77}{29\!\cdots\!31}a^{22}-\frac{65\!\cdots\!63}{10\!\cdots\!39}a^{21}-\frac{30\!\cdots\!88}{29\!\cdots\!31}a^{20}+\frac{91\!\cdots\!70}{29\!\cdots\!31}a^{19}+\frac{54\!\cdots\!62}{29\!\cdots\!31}a^{18}+\frac{46\!\cdots\!56}{59\!\cdots\!19}a^{17}+\frac{22\!\cdots\!53}{10\!\cdots\!39}a^{16}+\frac{16\!\cdots\!02}{41\!\cdots\!33}a^{15}+\frac{19\!\cdots\!20}{41\!\cdots\!33}a^{14}+\frac{11\!\cdots\!98}{29\!\cdots\!31}a^{13}+\frac{66\!\cdots\!19}{29\!\cdots\!31}a^{12}+\frac{51\!\cdots\!19}{29\!\cdots\!31}a^{11}+\frac{54\!\cdots\!50}{29\!\cdots\!31}a^{10}+\frac{33\!\cdots\!52}{29\!\cdots\!31}a^{9}+\frac{26\!\cdots\!75}{29\!\cdots\!31}a^{8}+\frac{10\!\cdots\!35}{59\!\cdots\!19}a^{7}+\frac{46\!\cdots\!12}{41\!\cdots\!33}a^{6}+\frac{29\!\cdots\!44}{10\!\cdots\!53}a^{5}+\frac{20\!\cdots\!28}{29\!\cdots\!31}a^{4}+\frac{44\!\cdots\!63}{29\!\cdots\!31}a^{3}-\frac{10\!\cdots\!09}{97\!\cdots\!77}a^{2}+\frac{10\!\cdots\!86}{41\!\cdots\!33}a+\frac{19\!\cdots\!73}{29\!\cdots\!31}$, $\frac{73\!\cdots\!26}{15\!\cdots\!37}a^{29}-\frac{13\!\cdots\!69}{66\!\cdots\!73}a^{28}-\frac{50\!\cdots\!69}{46\!\cdots\!11}a^{27}+\frac{50\!\cdots\!41}{46\!\cdots\!11}a^{26}+\frac{13\!\cdots\!73}{46\!\cdots\!11}a^{25}-\frac{36\!\cdots\!64}{46\!\cdots\!11}a^{24}-\frac{27\!\cdots\!92}{46\!\cdots\!11}a^{23}-\frac{59\!\cdots\!27}{46\!\cdots\!11}a^{22}-\frac{40\!\cdots\!97}{46\!\cdots\!11}a^{21}-\frac{85\!\cdots\!97}{46\!\cdots\!11}a^{20}-\frac{18\!\cdots\!86}{46\!\cdots\!11}a^{19}+\frac{14\!\cdots\!21}{46\!\cdots\!11}a^{18}+\frac{63\!\cdots\!73}{55\!\cdots\!67}a^{17}+\frac{15\!\cdots\!76}{46\!\cdots\!11}a^{16}+\frac{44\!\cdots\!24}{66\!\cdots\!73}a^{15}+\frac{56\!\cdots\!72}{66\!\cdots\!73}a^{14}+\frac{34\!\cdots\!63}{46\!\cdots\!11}a^{13}+\frac{19\!\cdots\!81}{46\!\cdots\!11}a^{12}+\frac{10\!\cdots\!81}{46\!\cdots\!11}a^{11}+\frac{12\!\cdots\!51}{46\!\cdots\!11}a^{10}+\frac{12\!\cdots\!69}{46\!\cdots\!11}a^{9}+\frac{66\!\cdots\!92}{46\!\cdots\!11}a^{8}+\frac{26\!\cdots\!36}{94\!\cdots\!39}a^{7}+\frac{40\!\cdots\!13}{66\!\cdots\!73}a^{6}+\frac{14\!\cdots\!97}{46\!\cdots\!11}a^{5}+\frac{14\!\cdots\!65}{46\!\cdots\!11}a^{4}+\frac{32\!\cdots\!08}{14\!\cdots\!81}a^{3}+\frac{15\!\cdots\!75}{15\!\cdots\!37}a^{2}-\frac{81\!\cdots\!43}{73\!\cdots\!97}a+\frac{47\!\cdots\!33}{46\!\cdots\!11}$, $\frac{13\!\cdots\!32}{15\!\cdots\!37}a^{29}-\frac{31\!\cdots\!93}{66\!\cdots\!73}a^{28}+\frac{14\!\cdots\!46}{46\!\cdots\!11}a^{27}+\frac{87\!\cdots\!58}{46\!\cdots\!11}a^{26}+\frac{45\!\cdots\!28}{15\!\cdots\!59}a^{25}-\frac{26\!\cdots\!30}{46\!\cdots\!11}a^{24}+\frac{18\!\cdots\!78}{46\!\cdots\!11}a^{23}-\frac{12\!\cdots\!05}{46\!\cdots\!11}a^{22}-\frac{62\!\cdots\!95}{46\!\cdots\!11}a^{21}-\frac{79\!\cdots\!80}{46\!\cdots\!11}a^{20}+\frac{89\!\cdots\!32}{46\!\cdots\!11}a^{19}+\frac{20\!\cdots\!22}{46\!\cdots\!11}a^{18}+\frac{85\!\cdots\!39}{55\!\cdots\!67}a^{17}+\frac{19\!\cdots\!71}{46\!\cdots\!11}a^{16}+\frac{43\!\cdots\!35}{66\!\cdots\!73}a^{15}+\frac{37\!\cdots\!70}{66\!\cdots\!73}a^{14}+\frac{14\!\cdots\!33}{46\!\cdots\!11}a^{13}-\frac{16\!\cdots\!74}{46\!\cdots\!11}a^{12}+\frac{26\!\cdots\!44}{46\!\cdots\!11}a^{11}+\frac{98\!\cdots\!67}{46\!\cdots\!11}a^{10}+\frac{26\!\cdots\!09}{46\!\cdots\!11}a^{9}+\frac{20\!\cdots\!02}{46\!\cdots\!11}a^{8}-\frac{67\!\cdots\!87}{94\!\cdots\!39}a^{7}-\frac{97\!\cdots\!41}{66\!\cdots\!73}a^{6}+\frac{26\!\cdots\!87}{46\!\cdots\!11}a^{5}-\frac{30\!\cdots\!25}{46\!\cdots\!11}a^{4}+\frac{27\!\cdots\!67}{14\!\cdots\!81}a^{3}-\frac{29\!\cdots\!35}{15\!\cdots\!37}a^{2}+\frac{50\!\cdots\!89}{73\!\cdots\!97}a-\frac{15\!\cdots\!31}{46\!\cdots\!11}$, $\frac{48\!\cdots\!76}{15\!\cdots\!37}a^{29}-\frac{62\!\cdots\!91}{66\!\cdots\!73}a^{28}-\frac{11\!\cdots\!43}{46\!\cdots\!11}a^{27}+\frac{36\!\cdots\!20}{46\!\cdots\!11}a^{26}+\frac{40\!\cdots\!42}{15\!\cdots\!59}a^{25}+\frac{51\!\cdots\!07}{46\!\cdots\!11}a^{24}-\frac{65\!\cdots\!16}{46\!\cdots\!11}a^{23}-\frac{30\!\cdots\!53}{46\!\cdots\!11}a^{22}-\frac{32\!\cdots\!08}{46\!\cdots\!11}a^{21}-\frac{84\!\cdots\!77}{46\!\cdots\!11}a^{20}-\frac{62\!\cdots\!37}{46\!\cdots\!11}a^{19}+\frac{88\!\cdots\!75}{46\!\cdots\!11}a^{18}+\frac{49\!\cdots\!27}{55\!\cdots\!67}a^{17}+\frac{13\!\cdots\!17}{46\!\cdots\!11}a^{16}+\frac{43\!\cdots\!23}{66\!\cdots\!73}a^{15}+\frac{64\!\cdots\!97}{66\!\cdots\!73}a^{14}+\frac{48\!\cdots\!75}{46\!\cdots\!11}a^{13}+\frac{38\!\cdots\!16}{46\!\cdots\!11}a^{12}+\frac{24\!\cdots\!26}{46\!\cdots\!11}a^{11}+\frac{20\!\cdots\!64}{46\!\cdots\!11}a^{10}+\frac{17\!\cdots\!32}{46\!\cdots\!11}a^{9}+\frac{11\!\cdots\!56}{46\!\cdots\!11}a^{8}+\frac{16\!\cdots\!11}{94\!\cdots\!39}a^{7}+\frac{25\!\cdots\!97}{66\!\cdots\!73}a^{6}+\frac{18\!\cdots\!28}{46\!\cdots\!11}a^{5}+\frac{21\!\cdots\!81}{46\!\cdots\!11}a^{4}+\frac{32\!\cdots\!94}{14\!\cdots\!81}a^{3}+\frac{42\!\cdots\!05}{15\!\cdots\!37}a^{2}-\frac{28\!\cdots\!90}{24\!\cdots\!99}a+\frac{22\!\cdots\!06}{46\!\cdots\!11}$, $\frac{17\!\cdots\!13}{22\!\cdots\!37}a^{29}-\frac{12\!\cdots\!61}{94\!\cdots\!39}a^{28}+\frac{29\!\cdots\!68}{66\!\cdots\!73}a^{27}+\frac{26\!\cdots\!01}{22\!\cdots\!37}a^{26}-\frac{11\!\cdots\!19}{66\!\cdots\!73}a^{25}-\frac{30\!\cdots\!16}{66\!\cdots\!73}a^{24}+\frac{23\!\cdots\!93}{66\!\cdots\!73}a^{23}-\frac{30\!\cdots\!24}{66\!\cdots\!73}a^{22}+\frac{87\!\cdots\!87}{66\!\cdots\!73}a^{21}+\frac{89\!\cdots\!81}{66\!\cdots\!73}a^{20}+\frac{18\!\cdots\!81}{66\!\cdots\!73}a^{19}+\frac{73\!\cdots\!56}{66\!\cdots\!73}a^{18}-\frac{59\!\cdots\!22}{19\!\cdots\!23}a^{17}-\frac{96\!\cdots\!41}{66\!\cdots\!73}a^{16}-\frac{45\!\cdots\!93}{94\!\cdots\!39}a^{15}-\frac{90\!\cdots\!87}{94\!\cdots\!39}a^{14}-\frac{77\!\cdots\!14}{66\!\cdots\!73}a^{13}-\frac{71\!\cdots\!35}{66\!\cdots\!73}a^{12}-\frac{41\!\cdots\!17}{66\!\cdots\!73}a^{11}-\frac{23\!\cdots\!31}{66\!\cdots\!73}a^{10}-\frac{21\!\cdots\!06}{66\!\cdots\!73}a^{9}-\frac{10\!\cdots\!92}{66\!\cdots\!73}a^{8}-\frac{15\!\cdots\!51}{94\!\cdots\!39}a^{7}-\frac{91\!\cdots\!41}{94\!\cdots\!39}a^{6}-\frac{47\!\cdots\!54}{66\!\cdots\!73}a^{5}-\frac{40\!\cdots\!80}{66\!\cdots\!73}a^{4}+\frac{77\!\cdots\!50}{71\!\cdots\!61}a^{3}-\frac{67\!\cdots\!22}{24\!\cdots\!99}a^{2}+\frac{99\!\cdots\!58}{94\!\cdots\!39}a-\frac{81\!\cdots\!90}{73\!\cdots\!97}$, $\frac{23\!\cdots\!60}{51\!\cdots\!79}a^{29}+\frac{19\!\cdots\!86}{22\!\cdots\!91}a^{28}-\frac{88\!\cdots\!47}{15\!\cdots\!37}a^{27}+\frac{75\!\cdots\!46}{15\!\cdots\!37}a^{26}+\frac{37\!\cdots\!06}{15\!\cdots\!37}a^{25}+\frac{49\!\cdots\!61}{15\!\cdots\!37}a^{24}-\frac{97\!\cdots\!96}{15\!\cdots\!37}a^{23}+\frac{75\!\cdots\!76}{15\!\cdots\!37}a^{22}-\frac{61\!\cdots\!86}{15\!\cdots\!37}a^{21}-\frac{27\!\cdots\!20}{15\!\cdots\!37}a^{20}-\frac{32\!\cdots\!10}{15\!\cdots\!37}a^{19}+\frac{34\!\cdots\!01}{15\!\cdots\!37}a^{18}+\frac{14\!\cdots\!32}{26\!\cdots\!27}a^{17}+\frac{33\!\cdots\!30}{15\!\cdots\!37}a^{16}+\frac{12\!\cdots\!85}{22\!\cdots\!91}a^{15}+\frac{20\!\cdots\!32}{22\!\cdots\!91}a^{14}+\frac{13\!\cdots\!38}{15\!\cdots\!37}a^{13}+\frac{10\!\cdots\!71}{15\!\cdots\!37}a^{12}+\frac{41\!\cdots\!58}{15\!\cdots\!37}a^{11}+\frac{42\!\cdots\!46}{15\!\cdots\!37}a^{10}+\frac{53\!\cdots\!74}{15\!\cdots\!37}a^{9}+\frac{16\!\cdots\!78}{15\!\cdots\!37}a^{8}+\frac{54\!\cdots\!12}{31\!\cdots\!13}a^{7}+\frac{91\!\cdots\!06}{22\!\cdots\!91}a^{6}-\frac{23\!\cdots\!28}{15\!\cdots\!37}a^{5}+\frac{13\!\cdots\!42}{15\!\cdots\!37}a^{4}-\frac{78\!\cdots\!06}{49\!\cdots\!27}a^{3}+\frac{50\!\cdots\!76}{17\!\cdots\!93}a^{2}-\frac{83\!\cdots\!24}{73\!\cdots\!97}a-\frac{43\!\cdots\!63}{15\!\cdots\!37}$, $\frac{15\!\cdots\!57}{15\!\cdots\!37}a^{29}-\frac{42\!\cdots\!77}{27\!\cdots\!11}a^{28}+\frac{84\!\cdots\!15}{15\!\cdots\!37}a^{27}+\frac{29\!\cdots\!33}{51\!\cdots\!79}a^{26}-\frac{29\!\cdots\!71}{15\!\cdots\!37}a^{25}-\frac{78\!\cdots\!14}{15\!\cdots\!37}a^{24}+\frac{25\!\cdots\!64}{51\!\cdots\!79}a^{23}-\frac{33\!\cdots\!15}{51\!\cdots\!79}a^{22}+\frac{27\!\cdots\!77}{17\!\cdots\!93}a^{21}+\frac{79\!\cdots\!07}{51\!\cdots\!79}a^{20}+\frac{15\!\cdots\!71}{51\!\cdots\!79}a^{19}-\frac{14\!\cdots\!35}{51\!\cdots\!79}a^{18}-\frac{29\!\cdots\!05}{88\!\cdots\!09}a^{17}-\frac{83\!\cdots\!03}{51\!\cdots\!79}a^{16}-\frac{39\!\cdots\!89}{73\!\cdots\!97}a^{15}-\frac{84\!\cdots\!23}{81\!\cdots\!33}a^{14}-\frac{78\!\cdots\!77}{63\!\cdots\!59}a^{13}-\frac{57\!\cdots\!31}{51\!\cdots\!79}a^{12}-\frac{32\!\cdots\!15}{51\!\cdots\!79}a^{11}-\frac{23\!\cdots\!55}{51\!\cdots\!79}a^{10}-\frac{86\!\cdots\!70}{17\!\cdots\!93}a^{9}-\frac{16\!\cdots\!17}{51\!\cdots\!79}a^{8}-\frac{33\!\cdots\!49}{13\!\cdots\!37}a^{7}-\frac{42\!\cdots\!29}{73\!\cdots\!97}a^{6}-\frac{40\!\cdots\!38}{15\!\cdots\!37}a^{5}-\frac{12\!\cdots\!07}{17\!\cdots\!93}a^{4}-\frac{10\!\cdots\!72}{49\!\cdots\!27}a^{3}-\frac{21\!\cdots\!58}{51\!\cdots\!79}a^{2}+\frac{37\!\cdots\!12}{22\!\cdots\!91}a-\frac{10\!\cdots\!53}{15\!\cdots\!37}$, $\frac{18\!\cdots\!01}{73\!\cdots\!97}a^{29}-\frac{36\!\cdots\!40}{31\!\cdots\!13}a^{28}-\frac{35\!\cdots\!41}{22\!\cdots\!91}a^{27}+\frac{12\!\cdots\!48}{22\!\cdots\!91}a^{26}+\frac{28\!\cdots\!04}{22\!\cdots\!91}a^{25}-\frac{16\!\cdots\!39}{22\!\cdots\!91}a^{24}+\frac{58\!\cdots\!11}{22\!\cdots\!91}a^{23}-\frac{15\!\cdots\!87}{22\!\cdots\!91}a^{22}-\frac{98\!\cdots\!65}{22\!\cdots\!91}a^{21}-\frac{18\!\cdots\!32}{22\!\cdots\!91}a^{20}-\frac{34\!\cdots\!77}{22\!\cdots\!91}a^{19}+\frac{31\!\cdots\!02}{22\!\cdots\!91}a^{18}+\frac{10\!\cdots\!07}{18\!\cdots\!89}a^{17}+\frac{36\!\cdots\!85}{22\!\cdots\!91}a^{16}+\frac{98\!\cdots\!99}{31\!\cdots\!13}a^{15}+\frac{59\!\cdots\!91}{15\!\cdots\!71}a^{14}+\frac{75\!\cdots\!99}{22\!\cdots\!91}a^{13}+\frac{46\!\cdots\!70}{22\!\cdots\!91}a^{12}+\frac{30\!\cdots\!22}{22\!\cdots\!91}a^{11}+\frac{32\!\cdots\!92}{22\!\cdots\!91}a^{10}+\frac{23\!\cdots\!61}{22\!\cdots\!91}a^{9}+\frac{14\!\cdots\!98}{22\!\cdots\!91}a^{8}+\frac{12\!\cdots\!49}{45\!\cdots\!59}a^{7}+\frac{35\!\cdots\!14}{31\!\cdots\!13}a^{6}+\frac{34\!\cdots\!99}{22\!\cdots\!91}a^{5}+\frac{29\!\cdots\!61}{22\!\cdots\!91}a^{4}+\frac{57\!\cdots\!37}{71\!\cdots\!61}a^{3}+\frac{16\!\cdots\!11}{73\!\cdots\!97}a^{2}-\frac{12\!\cdots\!85}{10\!\cdots\!71}a-\frac{52\!\cdots\!68}{22\!\cdots\!91}$, $\frac{24\!\cdots\!81}{27\!\cdots\!83}a^{29}-\frac{30\!\cdots\!30}{66\!\cdots\!73}a^{28}+\frac{14\!\cdots\!80}{51\!\cdots\!79}a^{27}+\frac{86\!\cdots\!76}{46\!\cdots\!11}a^{26}+\frac{48\!\cdots\!68}{15\!\cdots\!37}a^{25}-\frac{77\!\cdots\!97}{15\!\cdots\!37}a^{24}+\frac{19\!\cdots\!04}{46\!\cdots\!11}a^{23}-\frac{12\!\cdots\!88}{46\!\cdots\!11}a^{22}-\frac{64\!\cdots\!50}{46\!\cdots\!11}a^{21}-\frac{89\!\cdots\!24}{46\!\cdots\!11}a^{20}+\frac{67\!\cdots\!39}{46\!\cdots\!11}a^{19}+\frac{18\!\cdots\!64}{46\!\cdots\!11}a^{18}+\frac{15\!\cdots\!62}{94\!\cdots\!39}a^{17}+\frac{21\!\cdots\!28}{46\!\cdots\!11}a^{16}+\frac{49\!\cdots\!46}{66\!\cdots\!73}a^{15}+\frac{50\!\cdots\!85}{66\!\cdots\!73}a^{14}+\frac{27\!\cdots\!84}{46\!\cdots\!11}a^{13}+\frac{11\!\cdots\!56}{46\!\cdots\!11}a^{12}+\frac{41\!\cdots\!80}{15\!\cdots\!59}a^{11}+\frac{15\!\cdots\!14}{46\!\cdots\!11}a^{10}+\frac{67\!\cdots\!91}{46\!\cdots\!11}a^{9}+\frac{58\!\cdots\!85}{46\!\cdots\!11}a^{8}-\frac{11\!\cdots\!44}{94\!\cdots\!39}a^{7}+\frac{12\!\cdots\!85}{66\!\cdots\!73}a^{6}+\frac{95\!\cdots\!10}{15\!\cdots\!37}a^{5}-\frac{26\!\cdots\!28}{46\!\cdots\!11}a^{4}+\frac{13\!\cdots\!06}{46\!\cdots\!11}a^{3}-\frac{17\!\cdots\!24}{15\!\cdots\!37}a^{2}+\frac{25\!\cdots\!20}{38\!\cdots\!69}a-\frac{12\!\cdots\!62}{46\!\cdots\!11}$, $\frac{50\!\cdots\!87}{66\!\cdots\!73}a^{29}-\frac{35\!\cdots\!76}{94\!\cdots\!39}a^{28}+\frac{98\!\cdots\!17}{73\!\cdots\!97}a^{27}+\frac{10\!\cdots\!55}{66\!\cdots\!73}a^{26}+\frac{23\!\cdots\!37}{73\!\cdots\!97}a^{25}-\frac{71\!\cdots\!02}{22\!\cdots\!91}a^{24}+\frac{18\!\cdots\!55}{66\!\cdots\!73}a^{23}-\frac{48\!\cdots\!79}{21\!\cdots\!83}a^{22}-\frac{82\!\cdots\!95}{66\!\cdots\!73}a^{21}-\frac{13\!\cdots\!89}{66\!\cdots\!73}a^{20}+\frac{36\!\cdots\!66}{66\!\cdots\!73}a^{19}+\frac{23\!\cdots\!92}{66\!\cdots\!73}a^{18}+\frac{14\!\cdots\!37}{94\!\cdots\!39}a^{17}+\frac{28\!\cdots\!84}{66\!\cdots\!73}a^{16}+\frac{74\!\cdots\!28}{94\!\cdots\!39}a^{15}+\frac{85\!\cdots\!94}{94\!\cdots\!39}a^{14}+\frac{31\!\cdots\!96}{38\!\cdots\!69}a^{13}+\frac{29\!\cdots\!33}{66\!\cdots\!73}a^{12}+\frac{22\!\cdots\!16}{66\!\cdots\!73}a^{11}+\frac{23\!\cdots\!19}{66\!\cdots\!73}a^{10}+\frac{15\!\cdots\!04}{66\!\cdots\!73}a^{9}+\frac{12\!\cdots\!27}{66\!\cdots\!73}a^{8}+\frac{26\!\cdots\!18}{94\!\cdots\!39}a^{7}+\frac{25\!\cdots\!53}{94\!\cdots\!39}a^{6}+\frac{11\!\cdots\!03}{22\!\cdots\!91}a^{5}+\frac{85\!\cdots\!33}{66\!\cdots\!73}a^{4}+\frac{22\!\cdots\!74}{66\!\cdots\!73}a^{3}-\frac{36\!\cdots\!20}{73\!\cdots\!97}a^{2}+\frac{59\!\cdots\!18}{94\!\cdots\!39}a-\frac{25\!\cdots\!93}{66\!\cdots\!73}$, $\frac{63\!\cdots\!67}{15\!\cdots\!37}a^{29}-\frac{13\!\cdots\!55}{66\!\cdots\!73}a^{28}+\frac{39\!\cdots\!50}{46\!\cdots\!11}a^{27}+\frac{39\!\cdots\!82}{46\!\cdots\!11}a^{26}+\frac{74\!\cdots\!88}{46\!\cdots\!11}a^{25}-\frac{89\!\cdots\!56}{46\!\cdots\!11}a^{24}+\frac{44\!\cdots\!46}{27\!\cdots\!83}a^{23}-\frac{33\!\cdots\!32}{27\!\cdots\!83}a^{22}-\frac{30\!\cdots\!69}{46\!\cdots\!11}a^{21}-\frac{28\!\cdots\!23}{27\!\cdots\!83}a^{20}+\frac{19\!\cdots\!10}{46\!\cdots\!11}a^{19}+\frac{88\!\cdots\!51}{46\!\cdots\!11}a^{18}+\frac{74\!\cdots\!74}{94\!\cdots\!39}a^{17}+\frac{10\!\cdots\!75}{46\!\cdots\!11}a^{16}+\frac{26\!\cdots\!13}{66\!\cdots\!73}a^{15}+\frac{28\!\cdots\!09}{66\!\cdots\!73}a^{14}+\frac{59\!\cdots\!95}{15\!\cdots\!59}a^{13}+\frac{89\!\cdots\!14}{46\!\cdots\!11}a^{12}+\frac{73\!\cdots\!44}{46\!\cdots\!11}a^{11}+\frac{81\!\cdots\!45}{46\!\cdots\!11}a^{10}+\frac{46\!\cdots\!49}{46\!\cdots\!11}a^{9}+\frac{36\!\cdots\!29}{46\!\cdots\!11}a^{8}+\frac{96\!\cdots\!49}{94\!\cdots\!39}a^{7}+\frac{64\!\cdots\!34}{66\!\cdots\!73}a^{6}+\frac{12\!\cdots\!37}{46\!\cdots\!11}a^{5}+\frac{21\!\cdots\!69}{46\!\cdots\!11}a^{4}+\frac{69\!\cdots\!56}{46\!\cdots\!11}a^{3}-\frac{12\!\cdots\!07}{49\!\cdots\!27}a^{2}+\frac{65\!\cdots\!41}{24\!\cdots\!99}a-\frac{76\!\cdots\!05}{46\!\cdots\!11}$, $\frac{12\!\cdots\!32}{46\!\cdots\!11}a^{29}-\frac{79\!\cdots\!87}{66\!\cdots\!73}a^{28}-\frac{15\!\cdots\!00}{51\!\cdots\!79}a^{27}+\frac{26\!\cdots\!90}{46\!\cdots\!11}a^{26}+\frac{19\!\cdots\!18}{15\!\cdots\!37}a^{25}-\frac{11\!\cdots\!47}{15\!\cdots\!37}a^{24}+\frac{20\!\cdots\!46}{46\!\cdots\!11}a^{23}-\frac{34\!\cdots\!55}{46\!\cdots\!11}a^{22}-\frac{20\!\cdots\!95}{46\!\cdots\!11}a^{21}-\frac{39\!\cdots\!45}{46\!\cdots\!11}a^{20}-\frac{94\!\cdots\!65}{46\!\cdots\!11}a^{19}+\frac{63\!\cdots\!05}{46\!\cdots\!11}a^{18}+\frac{53\!\cdots\!62}{94\!\cdots\!39}a^{17}+\frac{45\!\cdots\!73}{27\!\cdots\!83}a^{16}+\frac{20\!\cdots\!69}{66\!\cdots\!73}a^{15}+\frac{25\!\cdots\!28}{66\!\cdots\!73}a^{14}+\frac{16\!\cdots\!28}{46\!\cdots\!11}a^{13}+\frac{95\!\cdots\!81}{46\!\cdots\!11}a^{12}+\frac{59\!\cdots\!70}{46\!\cdots\!11}a^{11}+\frac{61\!\cdots\!58}{46\!\cdots\!11}a^{10}+\frac{28\!\cdots\!89}{27\!\cdots\!83}a^{9}+\frac{33\!\cdots\!05}{46\!\cdots\!11}a^{8}+\frac{17\!\cdots\!91}{94\!\cdots\!39}a^{7}+\frac{16\!\cdots\!40}{66\!\cdots\!73}a^{6}+\frac{24\!\cdots\!00}{15\!\cdots\!37}a^{5}+\frac{47\!\cdots\!49}{46\!\cdots\!11}a^{4}+\frac{52\!\cdots\!62}{46\!\cdots\!11}a^{3}+\frac{15\!\cdots\!86}{15\!\cdots\!37}a^{2}+\frac{23\!\cdots\!79}{66\!\cdots\!73}a-\frac{41\!\cdots\!12}{46\!\cdots\!11}$, $\frac{14\!\cdots\!88}{15\!\cdots\!37}a^{29}-\frac{98\!\cdots\!63}{22\!\cdots\!91}a^{28}+\frac{21\!\cdots\!85}{15\!\cdots\!37}a^{27}+\frac{32\!\cdots\!19}{15\!\cdots\!37}a^{26}+\frac{79\!\cdots\!79}{17\!\cdots\!93}a^{25}-\frac{45\!\cdots\!17}{15\!\cdots\!37}a^{24}+\frac{31\!\cdots\!97}{15\!\cdots\!37}a^{23}-\frac{42\!\cdots\!19}{15\!\cdots\!37}a^{22}-\frac{25\!\cdots\!66}{15\!\cdots\!37}a^{21}-\frac{47\!\cdots\!86}{15\!\cdots\!37}a^{20}-\frac{88\!\cdots\!99}{15\!\cdots\!37}a^{19}+\frac{76\!\cdots\!12}{15\!\cdots\!37}a^{18}+\frac{64\!\cdots\!48}{31\!\cdots\!13}a^{17}+\frac{93\!\cdots\!51}{15\!\cdots\!37}a^{16}+\frac{25\!\cdots\!65}{22\!\cdots\!91}a^{15}+\frac{31\!\cdots\!19}{22\!\cdots\!91}a^{14}+\frac{19\!\cdots\!76}{15\!\cdots\!37}a^{13}+\frac{11\!\cdots\!24}{15\!\cdots\!37}a^{12}+\frac{75\!\cdots\!03}{15\!\cdots\!37}a^{11}+\frac{76\!\cdots\!00}{15\!\cdots\!37}a^{10}+\frac{57\!\cdots\!83}{15\!\cdots\!37}a^{9}+\frac{42\!\cdots\!25}{15\!\cdots\!37}a^{8}+\frac{34\!\cdots\!94}{45\!\cdots\!59}a^{7}+\frac{14\!\cdots\!34}{12\!\cdots\!23}a^{6}+\frac{91\!\cdots\!17}{15\!\cdots\!37}a^{5}+\frac{18\!\cdots\!97}{51\!\cdots\!79}a^{4}+\frac{21\!\cdots\!72}{51\!\cdots\!79}a^{3}+\frac{71\!\cdots\!32}{15\!\cdots\!37}a^{2}+\frac{42\!\cdots\!61}{22\!\cdots\!91}a+\frac{18\!\cdots\!52}{15\!\cdots\!37}$, $\frac{99\!\cdots\!21}{46\!\cdots\!11}a^{29}-\frac{23\!\cdots\!42}{22\!\cdots\!91}a^{28}+\frac{21\!\cdots\!63}{46\!\cdots\!11}a^{27}+\frac{69\!\cdots\!70}{15\!\cdots\!37}a^{26}+\frac{39\!\cdots\!40}{46\!\cdots\!11}a^{25}-\frac{16\!\cdots\!25}{15\!\cdots\!59}a^{24}+\frac{13\!\cdots\!37}{15\!\cdots\!37}a^{23}-\frac{98\!\cdots\!37}{15\!\cdots\!37}a^{22}-\frac{17\!\cdots\!39}{51\!\cdots\!79}a^{21}-\frac{83\!\cdots\!47}{15\!\cdots\!37}a^{20}+\frac{35\!\cdots\!79}{15\!\cdots\!37}a^{19}+\frac{15\!\cdots\!28}{15\!\cdots\!37}a^{18}+\frac{13\!\cdots\!42}{31\!\cdots\!13}a^{17}+\frac{18\!\cdots\!15}{15\!\cdots\!37}a^{16}+\frac{45\!\cdots\!34}{22\!\cdots\!91}a^{15}+\frac{50\!\cdots\!18}{22\!\cdots\!91}a^{14}+\frac{30\!\cdots\!71}{15\!\cdots\!37}a^{13}+\frac{15\!\cdots\!76}{15\!\cdots\!37}a^{12}+\frac{43\!\cdots\!68}{51\!\cdots\!79}a^{11}+\frac{14\!\cdots\!01}{15\!\cdots\!37}a^{10}+\frac{38\!\cdots\!49}{73\!\cdots\!53}a^{9}+\frac{70\!\cdots\!30}{16\!\cdots\!09}a^{8}+\frac{24\!\cdots\!72}{45\!\cdots\!59}a^{7}+\frac{11\!\cdots\!35}{22\!\cdots\!91}a^{6}+\frac{66\!\cdots\!92}{46\!\cdots\!11}a^{5}+\frac{12\!\cdots\!13}{51\!\cdots\!79}a^{4}+\frac{36\!\cdots\!64}{46\!\cdots\!11}a^{3}-\frac{20\!\cdots\!79}{15\!\cdots\!37}a^{2}+\frac{30\!\cdots\!82}{21\!\cdots\!83}a-\frac{39\!\cdots\!52}{46\!\cdots\!11}$, $\frac{17\!\cdots\!89}{46\!\cdots\!11}a^{29}-\frac{12\!\cdots\!83}{66\!\cdots\!73}a^{28}+\frac{36\!\cdots\!65}{51\!\cdots\!79}a^{27}+\frac{35\!\cdots\!36}{46\!\cdots\!11}a^{26}+\frac{22\!\cdots\!98}{15\!\cdots\!37}a^{25}-\frac{87\!\cdots\!38}{51\!\cdots\!79}a^{24}+\frac{37\!\cdots\!00}{27\!\cdots\!83}a^{23}-\frac{29\!\cdots\!57}{27\!\cdots\!83}a^{22}-\frac{27\!\cdots\!76}{46\!\cdots\!11}a^{21}-\frac{25\!\cdots\!09}{27\!\cdots\!83}a^{20}+\frac{16\!\cdots\!44}{46\!\cdots\!11}a^{19}+\frac{80\!\cdots\!02}{46\!\cdots\!11}a^{18}+\frac{67\!\cdots\!34}{94\!\cdots\!39}a^{17}+\frac{95\!\cdots\!35}{46\!\cdots\!11}a^{16}+\frac{23\!\cdots\!63}{66\!\cdots\!73}a^{15}+\frac{26\!\cdots\!87}{66\!\cdots\!73}a^{14}+\frac{16\!\cdots\!76}{46\!\cdots\!11}a^{13}+\frac{29\!\cdots\!33}{15\!\cdots\!59}a^{12}+\frac{67\!\cdots\!20}{46\!\cdots\!11}a^{11}+\frac{74\!\cdots\!47}{46\!\cdots\!11}a^{10}+\frac{44\!\cdots\!58}{46\!\cdots\!11}a^{9}+\frac{34\!\cdots\!88}{46\!\cdots\!11}a^{8}+\frac{91\!\cdots\!04}{94\!\cdots\!39}a^{7}+\frac{60\!\cdots\!27}{66\!\cdots\!73}a^{6}+\frac{37\!\cdots\!63}{15\!\cdots\!37}a^{5}+\frac{23\!\cdots\!03}{46\!\cdots\!11}a^{4}+\frac{63\!\cdots\!32}{46\!\cdots\!11}a^{3}-\frac{38\!\cdots\!52}{16\!\cdots\!09}a^{2}+\frac{16\!\cdots\!09}{66\!\cdots\!73}a-\frac{71\!\cdots\!05}{46\!\cdots\!11}$ 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:  \( 26934148162576.97 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{14}\cdot 26934148162576.97 \cdot 4}{2\cdot\sqrt{553999258054841955537532525076904905639678955078125}}\cr\approx \mathstrut & 1.36822271826315 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^30 - 5*x^29 + 2*x^28 + 21*x^27 + 40*x^26 - 46*x^25 + 39*x^24 - 296*x^23 - 1612*x^22 - 2561*x^21 + 962*x^20 + 4669*x^19 + 19325*x^18 + 55780*x^17 + 98099*x^16 + 110152*x^15 + 94739*x^14 + 50734*x^13 + 40862*x^12 + 44242*x^11 + 25838*x^10 + 20656*x^9 + 3131*x^8 + 2527*x^7 + 6736*x^6 + 1305*x^5 + 3768*x^4 - 512*x^3 + 661*x^2 - 25*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 - 5*x^29 + 2*x^28 + 21*x^27 + 40*x^26 - 46*x^25 + 39*x^24 - 296*x^23 - 1612*x^22 - 2561*x^21 + 962*x^20 + 4669*x^19 + 19325*x^18 + 55780*x^17 + 98099*x^16 + 110152*x^15 + 94739*x^14 + 50734*x^13 + 40862*x^12 + 44242*x^11 + 25838*x^10 + 20656*x^9 + 3131*x^8 + 2527*x^7 + 6736*x^6 + 1305*x^5 + 3768*x^4 - 512*x^3 + 661*x^2 - 25*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 - 5*x^29 + 2*x^28 + 21*x^27 + 40*x^26 - 46*x^25 + 39*x^24 - 296*x^23 - 1612*x^22 - 2561*x^21 + 962*x^20 + 4669*x^19 + 19325*x^18 + 55780*x^17 + 98099*x^16 + 110152*x^15 + 94739*x^14 + 50734*x^13 + 40862*x^12 + 44242*x^11 + 25838*x^10 + 20656*x^9 + 3131*x^8 + 2527*x^7 + 6736*x^6 + 1305*x^5 + 3768*x^4 - 512*x^3 + 661*x^2 - 25*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 - 5*x^29 + 2*x^28 + 21*x^27 + 40*x^26 - 46*x^25 + 39*x^24 - 296*x^23 - 1612*x^22 - 2561*x^21 + 962*x^20 + 4669*x^19 + 19325*x^18 + 55780*x^17 + 98099*x^16 + 110152*x^15 + 94739*x^14 + 50734*x^13 + 40862*x^12 + 44242*x^11 + 25838*x^10 + 20656*x^9 + 3131*x^8 + 2527*x^7 + 6736*x^6 + 1305*x^5 + 3768*x^4 - 512*x^3 + 661*x^2 - 25*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

$D_{30}$ (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

\(\Q(\sqrt{5}) \), 3.1.751.1, 5.1.564001.1, 6.2.70500125.1, 10.2.994053525003125.2, 15.1.134734730815558692751.1

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: deg 30
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 $30$ ${\href{/padicField/3.2.0.1}{2} }^{15}$ R ${\href{/padicField/7.2.0.1}{2} }^{15}$ ${\href{/padicField/11.2.0.1}{2} }^{14}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ $30$ ${\href{/padicField/17.2.0.1}{2} }^{15}$ ${\href{/padicField/19.3.0.1}{3} }^{10}$ $30$ ${\href{/padicField/29.2.0.1}{2} }^{14}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{14}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ $30$ ${\href{/padicField/41.2.0.1}{2} }^{14}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }^{3}$ $30$ ${\href{/padicField/53.6.0.1}{6} }^{5}$ $15^{2}$

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
\(5\) Copy content Toggle raw display 5.10.5.1$x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(751\) Copy content Toggle raw display $\Q_{751}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{751}$$x$$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$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3755.2t1.a.a$1$ $ 5 \cdot 751 $ \(\Q(\sqrt{-3755}) \) $C_2$ (as 2T1) $1$ $-1$
1.751.2t1.a.a$1$ $ 751 $ \(\Q(\sqrt{-751}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.18775.6t3.a.a$2$ $ 5^{2} \cdot 751 $ 6.0.52945593875.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.751.3t2.a.a$2$ $ 751 $ 3.1.751.1 $S_3$ (as 3T2) $1$ $0$
* 2.751.5t2.a.b$2$ $ 751 $ 5.1.564001.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.751.5t2.a.a$2$ $ 751 $ 5.1.564001.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.18775.10t3.a.b$2$ $ 5^{2} \cdot 751 $ 10.0.746534197277346875.2 $D_{10}$ (as 10T3) $1$ $0$
* 2.18775.10t3.a.a$2$ $ 5^{2} \cdot 751 $ 10.0.746534197277346875.2 $D_{10}$ (as 10T3) $1$ $0$
* 2.751.15t2.a.a$2$ $ 751 $ 15.1.134734730815558692751.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.751.15t2.a.d$2$ $ 751 $ 15.1.134734730815558692751.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.751.15t2.a.c$2$ $ 751 $ 15.1.134734730815558692751.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.751.15t2.a.b$2$ $ 751 $ 15.1.134734730815558692751.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.18775.30t14.a.d$2$ $ 5^{2} \cdot 751 $ 30.2.553999258054841955537532525076904905639678955078125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.18775.30t14.a.c$2$ $ 5^{2} \cdot 751 $ 30.2.553999258054841955537532525076904905639678955078125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.18775.30t14.a.a$2$ $ 5^{2} \cdot 751 $ 30.2.553999258054841955537532525076904905639678955078125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.18775.30t14.a.b$2$ $ 5^{2} \cdot 751 $ 30.2.553999258054841955537532525076904905639678955078125.1 $D_{30}$ (as 30T14) $1$ $0$

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 *.