Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 - 4*x^28 + 2*x^27 + 14*x^26 - 18*x^25 - 22*x^24 + 80*x^23 + 2*x^22 - 292*x^21 + 107*x^20 + 966*x^19 - 524*x^18 - 1689*x^17 + 1837*x^16 + 1003*x^15 - 6747*x^14 + 4493*x^13 + 13850*x^12 - 15662*x^11 - 11779*x^10 + 28372*x^9 - 4752*x^8 - 31522*x^7 + 29972*x^6 + 5164*x^5 - 25670*x^4 + 15150*x^3 + 3000*x^2 - 8125*x + 3125)
gp: K = bnfinit(y^30 - y^29 - 4*y^28 + 2*y^27 + 14*y^26 - 18*y^25 - 22*y^24 + 80*y^23 + 2*y^22 - 292*y^21 + 107*y^20 + 966*y^19 - 524*y^18 - 1689*y^17 + 1837*y^16 + 1003*y^15 - 6747*y^14 + 4493*y^13 + 13850*y^12 - 15662*y^11 - 11779*y^10 + 28372*y^9 - 4752*y^8 - 31522*y^7 + 29972*y^6 + 5164*y^5 - 25670*y^4 + 15150*y^3 + 3000*y^2 - 8125*y + 3125, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - x^29 - 4*x^28 + 2*x^27 + 14*x^26 - 18*x^25 - 22*x^24 + 80*x^23 + 2*x^22 - 292*x^21 + 107*x^20 + 966*x^19 - 524*x^18 - 1689*x^17 + 1837*x^16 + 1003*x^15 - 6747*x^14 + 4493*x^13 + 13850*x^12 - 15662*x^11 - 11779*x^10 + 28372*x^9 - 4752*x^8 - 31522*x^7 + 29972*x^6 + 5164*x^5 - 25670*x^4 + 15150*x^3 + 3000*x^2 - 8125*x + 3125);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - x^29 - 4*x^28 + 2*x^27 + 14*x^26 - 18*x^25 - 22*x^24 + 80*x^23 + 2*x^22 - 292*x^21 + 107*x^20 + 966*x^19 - 524*x^18 - 1689*x^17 + 1837*x^16 + 1003*x^15 - 6747*x^14 + 4493*x^13 + 13850*x^12 - 15662*x^11 - 11779*x^10 + 28372*x^9 - 4752*x^8 - 31522*x^7 + 29972*x^6 + 5164*x^5 - 25670*x^4 + 15150*x^3 + 3000*x^2 - 8125*x + 3125)
\( x^{30} - x^{29} - 4 x^{28} + 2 x^{27} + 14 x^{26} - 18 x^{25} - 22 x^{24} + 80 x^{23} + 2 x^{22} + \cdots + 3125 \)
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: |
\(-10745322081507339108891258824483901499337171\)
\(\medspace = -\,11^{27}\cdot 31^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.19\) | 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: | $11^{9/10}31^{2/3}\approx 85.40721213930478$ | ||
| Ramified primes: |
\(11\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $15$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{10}a^{26}-\frac{1}{10}a^{25}+\frac{1}{10}a^{24}+\frac{1}{5}a^{23}-\frac{1}{10}a^{22}+\frac{1}{5}a^{21}-\frac{1}{5}a^{20}-\frac{1}{2}a^{19}-\frac{3}{10}a^{18}+\frac{3}{10}a^{17}+\frac{1}{5}a^{16}+\frac{1}{10}a^{15}-\frac{2}{5}a^{14}+\frac{1}{10}a^{13}+\frac{1}{5}a^{12}-\frac{1}{5}a^{11}+\frac{3}{10}a^{10}+\frac{3}{10}a^{9}-\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{3}{10}a^{5}+\frac{3}{10}a^{4}-\frac{1}{5}a^{3}-\frac{3}{10}a^{2}-\frac{1}{10}a$, $\frac{1}{17702350}a^{27}+\frac{326932}{8851175}a^{26}+\frac{1385153}{8851175}a^{25}-\frac{1802454}{8851175}a^{24}-\frac{1355128}{8851175}a^{23}+\frac{1515571}{8851175}a^{22}-\frac{4103717}{17702350}a^{21}+\frac{112357}{708094}a^{20}-\frac{699223}{17702350}a^{19}+\frac{3769319}{8851175}a^{18}+\frac{2831026}{8851175}a^{17}+\frac{1521221}{17702350}a^{16}+\frac{1104908}{8851175}a^{15}-\frac{3731887}{8851175}a^{14}+\frac{4149977}{17702350}a^{13}+\frac{4383054}{8851175}a^{12}-\frac{3034576}{8851175}a^{11}+\frac{3336613}{17702350}a^{10}-\frac{1010771}{3540470}a^{9}+\frac{7875663}{17702350}a^{8}+\frac{636683}{8851175}a^{7}-\frac{113344}{8851175}a^{6}+\frac{4099914}{8851175}a^{5}-\frac{2495226}{8851175}a^{4}-\frac{3485804}{8851175}a^{3}+\frac{744719}{17702350}a^{2}+\frac{170669}{1770235}a-\frac{24324}{354047}$, $\frac{1}{24\!\cdots\!50}a^{28}+\frac{48\!\cdots\!59}{24\!\cdots\!50}a^{27}+\frac{22\!\cdots\!11}{24\!\cdots\!50}a^{26}+\frac{20\!\cdots\!06}{12\!\cdots\!25}a^{25}-\frac{20\!\cdots\!83}{12\!\cdots\!25}a^{24}-\frac{18\!\cdots\!14}{12\!\cdots\!25}a^{23}+\frac{17\!\cdots\!74}{12\!\cdots\!25}a^{22}-\frac{55\!\cdots\!84}{24\!\cdots\!25}a^{21}-\frac{86\!\cdots\!36}{11\!\cdots\!25}a^{20}+\frac{34\!\cdots\!64}{12\!\cdots\!25}a^{19}-\frac{36\!\cdots\!63}{24\!\cdots\!50}a^{18}-\frac{56\!\cdots\!07}{12\!\cdots\!25}a^{17}-\frac{55\!\cdots\!07}{12\!\cdots\!25}a^{16}+\frac{79\!\cdots\!71}{24\!\cdots\!50}a^{15}+\frac{11\!\cdots\!47}{24\!\cdots\!50}a^{14}+\frac{56\!\cdots\!24}{12\!\cdots\!25}a^{13}-\frac{93\!\cdots\!67}{24\!\cdots\!50}a^{12}+\frac{78\!\cdots\!73}{24\!\cdots\!50}a^{11}-\frac{26\!\cdots\!27}{24\!\cdots\!25}a^{10}+\frac{39\!\cdots\!94}{12\!\cdots\!25}a^{9}-\frac{35\!\cdots\!37}{12\!\cdots\!25}a^{8}-\frac{39\!\cdots\!93}{24\!\cdots\!50}a^{7}+\frac{10\!\cdots\!43}{24\!\cdots\!50}a^{6}-\frac{65\!\cdots\!67}{24\!\cdots\!50}a^{5}-\frac{36\!\cdots\!99}{12\!\cdots\!25}a^{4}-\frac{30\!\cdots\!41}{24\!\cdots\!50}a^{3}+\frac{19\!\cdots\!77}{24\!\cdots\!25}a^{2}-\frac{25\!\cdots\!73}{96\!\cdots\!90}a+\frac{37\!\cdots\!63}{19\!\cdots\!18}$, $\frac{1}{12\!\cdots\!50}a^{29}-\frac{1}{12\!\cdots\!50}a^{28}+\frac{24\!\cdots\!21}{12\!\cdots\!50}a^{27}+\frac{10\!\cdots\!76}{60\!\cdots\!25}a^{26}-\frac{19\!\cdots\!61}{12\!\cdots\!50}a^{25}+\frac{11\!\cdots\!16}{60\!\cdots\!25}a^{24}+\frac{76\!\cdots\!64}{60\!\cdots\!25}a^{23}+\frac{17\!\cdots\!88}{12\!\cdots\!25}a^{22}-\frac{96\!\cdots\!24}{60\!\cdots\!25}a^{21}-\frac{12\!\cdots\!21}{60\!\cdots\!25}a^{20}+\frac{21\!\cdots\!16}{60\!\cdots\!25}a^{19}-\frac{35\!\cdots\!59}{12\!\cdots\!50}a^{18}+\frac{20\!\cdots\!51}{12\!\cdots\!50}a^{17}+\frac{59\!\cdots\!61}{12\!\cdots\!50}a^{16}-\frac{36\!\cdots\!63}{12\!\cdots\!50}a^{15}+\frac{58\!\cdots\!53}{12\!\cdots\!50}a^{14}+\frac{25\!\cdots\!53}{12\!\cdots\!50}a^{13}+\frac{35\!\cdots\!93}{12\!\cdots\!50}a^{12}-\frac{15\!\cdots\!03}{48\!\cdots\!50}a^{11}-\frac{14\!\cdots\!81}{60\!\cdots\!25}a^{10}-\frac{23\!\cdots\!27}{60\!\cdots\!25}a^{9}-\frac{14\!\cdots\!89}{60\!\cdots\!25}a^{8}+\frac{13\!\cdots\!99}{60\!\cdots\!25}a^{7}-\frac{13\!\cdots\!36}{60\!\cdots\!25}a^{6}-\frac{10\!\cdots\!14}{60\!\cdots\!25}a^{5}-\frac{14\!\cdots\!61}{12\!\cdots\!50}a^{4}+\frac{12\!\cdots\!63}{12\!\cdots\!25}a^{3}+\frac{64\!\cdots\!69}{19\!\cdots\!18}a^{2}-\frac{19\!\cdots\!39}{96\!\cdots\!90}a+\frac{59\!\cdots\!97}{19\!\cdots\!18}$
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: | $14$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{9727516702630088716870914}{6030383388162466624111068125} a^{29} - \frac{2294911267042609066533144}{6030383388162466624111068125} a^{28} - \frac{39442099053684818060510601}{6030383388162466624111068125} a^{27} - \frac{108818645489696764161878}{55324618240022629579000625} a^{26} + \frac{121847934667376751086920911}{6030383388162466624111068125} a^{25} - \frac{80957803195482482152910022}{6030383388162466624111068125} a^{24} - \frac{256250868607174873205034543}{6030383388162466624111068125} a^{23} + \frac{113127532972992218921628141}{1206076677632493324822213625} a^{22} + \frac{419945592867374599325626653}{6030383388162466624111068125} a^{21} - \frac{2433186759714864829092541098}{6030383388162466624111068125} a^{20} - \frac{791567335824299188016029392}{6030383388162466624111068125} a^{19} + \frac{8440163662706098374989925414}{6030383388162466624111068125} a^{18} + \frac{1377580667579010033297816909}{6030383388162466624111068125} a^{17} - \frac{14105180527410398348374560651}{6030383388162466624111068125} a^{16} + \frac{6762372258392856416757180438}{6030383388162466624111068125} a^{15} + \frac{12594408990981787506055509057}{6030383388162466624111068125} a^{14} - \frac{54905549335169999056671421923}{6030383388162466624111068125} a^{13} + \frac{3680472257945871555141933762}{6030383388162466624111068125} a^{12} + \frac{26082899368030849007934349122}{1206076677632493324822213625} a^{11} - \frac{48743617508046519389628876468}{6030383388162466624111068125} a^{10} - \frac{133146732544870383695712037371}{6030383388162466624111068125} a^{9} + \frac{160791392265447600848846811153}{6030383388162466624111068125} a^{8} + \frac{55130958267088567882745538687}{6030383388162466624111068125} a^{7} - \frac{241468051137964941137737808748}{6030383388162466624111068125} a^{6} + \frac{113249275720342073531652311943}{6030383388162466624111068125} a^{5} + \frac{103291464352691597261385297186}{6030383388162466624111068125} a^{4} - \frac{5870674656092294547247421253}{241215335526498664964442725} a^{3} + \frac{324697725096020850405750948}{48243067105299732992888545} a^{2} + \frac{76808417406570367319664774}{9648613421059946598577709} a - \frac{56349166079551001431890390}{9648613421059946598577709} \)
(order $22$)
| 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: |
$\frac{50\!\cdots\!02}{60\!\cdots\!25}a^{29}-\frac{31\!\cdots\!02}{60\!\cdots\!25}a^{28}-\frac{19\!\cdots\!08}{60\!\cdots\!25}a^{27}-\frac{75\!\cdots\!46}{60\!\cdots\!25}a^{26}+\frac{54\!\cdots\!92}{55\!\cdots\!25}a^{25}-\frac{34\!\cdots\!29}{55\!\cdots\!25}a^{24}-\frac{13\!\cdots\!44}{60\!\cdots\!25}a^{23}+\frac{55\!\cdots\!12}{12\!\cdots\!25}a^{22}+\frac{23\!\cdots\!04}{60\!\cdots\!25}a^{21}-\frac{11\!\cdots\!84}{60\!\cdots\!25}a^{20}-\frac{52\!\cdots\!36}{60\!\cdots\!25}a^{19}+\frac{41\!\cdots\!57}{60\!\cdots\!25}a^{18}+\frac{10\!\cdots\!02}{60\!\cdots\!25}a^{17}-\frac{67\!\cdots\!28}{60\!\cdots\!25}a^{16}+\frac{32\!\cdots\!74}{60\!\cdots\!25}a^{15}+\frac{70\!\cdots\!56}{60\!\cdots\!25}a^{14}-\frac{26\!\cdots\!94}{60\!\cdots\!25}a^{13}-\frac{14\!\cdots\!14}{60\!\cdots\!25}a^{12}+\frac{24\!\cdots\!88}{24\!\cdots\!25}a^{11}-\frac{21\!\cdots\!24}{60\!\cdots\!25}a^{10}-\frac{65\!\cdots\!08}{60\!\cdots\!25}a^{9}+\frac{77\!\cdots\!44}{60\!\cdots\!25}a^{8}+\frac{41\!\cdots\!46}{60\!\cdots\!25}a^{7}-\frac{10\!\cdots\!44}{60\!\cdots\!25}a^{6}+\frac{45\!\cdots\!44}{60\!\cdots\!25}a^{5}+\frac{39\!\cdots\!28}{60\!\cdots\!25}a^{4}-\frac{10\!\cdots\!08}{12\!\cdots\!25}a^{3}+\frac{27\!\cdots\!88}{24\!\cdots\!25}a^{2}+\frac{25\!\cdots\!50}{96\!\cdots\!09}a-\frac{14\!\cdots\!32}{96\!\cdots\!09}$, $\frac{91\!\cdots\!53}{60\!\cdots\!25}a^{29}-\frac{32\!\cdots\!18}{60\!\cdots\!25}a^{28}-\frac{40\!\cdots\!72}{60\!\cdots\!25}a^{27}-\frac{51\!\cdots\!34}{60\!\cdots\!25}a^{26}+\frac{12\!\cdots\!12}{60\!\cdots\!25}a^{25}-\frac{86\!\cdots\!64}{60\!\cdots\!25}a^{24}-\frac{27\!\cdots\!96}{60\!\cdots\!25}a^{23}+\frac{11\!\cdots\!54}{12\!\cdots\!25}a^{22}+\frac{61\!\cdots\!68}{90\!\cdots\!75}a^{21}-\frac{25\!\cdots\!81}{60\!\cdots\!25}a^{20}-\frac{57\!\cdots\!24}{60\!\cdots\!25}a^{19}+\frac{88\!\cdots\!43}{60\!\cdots\!25}a^{18}+\frac{58\!\cdots\!63}{60\!\cdots\!25}a^{17}-\frac{16\!\cdots\!07}{60\!\cdots\!25}a^{16}+\frac{76\!\cdots\!21}{60\!\cdots\!25}a^{15}+\frac{15\!\cdots\!29}{60\!\cdots\!25}a^{14}-\frac{54\!\cdots\!61}{60\!\cdots\!25}a^{13}+\frac{63\!\cdots\!34}{60\!\cdots\!25}a^{12}+\frac{27\!\cdots\!26}{12\!\cdots\!25}a^{11}-\frac{63\!\cdots\!61}{60\!\cdots\!25}a^{10}-\frac{16\!\cdots\!82}{60\!\cdots\!25}a^{9}+\frac{26\!\cdots\!28}{90\!\cdots\!75}a^{8}+\frac{74\!\cdots\!14}{60\!\cdots\!25}a^{7}-\frac{27\!\cdots\!36}{60\!\cdots\!25}a^{6}+\frac{10\!\cdots\!96}{60\!\cdots\!25}a^{5}+\frac{14\!\cdots\!62}{60\!\cdots\!25}a^{4}-\frac{33\!\cdots\!94}{12\!\cdots\!25}a^{3}+\frac{11\!\cdots\!34}{24\!\cdots\!25}a^{2}+\frac{47\!\cdots\!21}{48\!\cdots\!45}a-\frac{65\!\cdots\!61}{96\!\cdots\!09}$, $\frac{70\!\cdots\!79}{12\!\cdots\!25}a^{29}-\frac{43\!\cdots\!36}{12\!\cdots\!25}a^{28}-\frac{25\!\cdots\!94}{12\!\cdots\!25}a^{27}-\frac{23\!\cdots\!79}{12\!\cdots\!25}a^{26}+\frac{85\!\cdots\!57}{12\!\cdots\!25}a^{25}-\frac{16\!\cdots\!98}{24\!\cdots\!25}a^{24}-\frac{14\!\cdots\!77}{12\!\cdots\!25}a^{23}+\frac{42\!\cdots\!79}{12\!\cdots\!25}a^{22}+\frac{15\!\cdots\!18}{12\!\cdots\!25}a^{21}-\frac{16\!\cdots\!82}{12\!\cdots\!25}a^{20}-\frac{73\!\cdots\!48}{12\!\cdots\!25}a^{19}+\frac{11\!\cdots\!07}{24\!\cdots\!25}a^{18}-\frac{66\!\cdots\!53}{12\!\cdots\!25}a^{17}-\frac{89\!\cdots\!98}{12\!\cdots\!25}a^{16}+\frac{63\!\cdots\!86}{12\!\cdots\!25}a^{15}+\frac{54\!\cdots\!18}{12\!\cdots\!25}a^{14}-\frac{38\!\cdots\!29}{12\!\cdots\!25}a^{13}+\frac{14\!\cdots\!21}{12\!\cdots\!25}a^{12}+\frac{81\!\cdots\!94}{12\!\cdots\!25}a^{11}-\frac{49\!\cdots\!28}{12\!\cdots\!25}a^{10}-\frac{66\!\cdots\!32}{12\!\cdots\!25}a^{9}+\frac{11\!\cdots\!11}{12\!\cdots\!25}a^{8}+\frac{67\!\cdots\!03}{12\!\cdots\!25}a^{7}-\frac{14\!\cdots\!94}{12\!\cdots\!25}a^{6}+\frac{93\!\cdots\!37}{12\!\cdots\!25}a^{5}+\frac{72\!\cdots\!47}{12\!\cdots\!25}a^{4}-\frac{54\!\cdots\!48}{12\!\cdots\!25}a^{3}+\frac{29\!\cdots\!42}{96\!\cdots\!09}a^{2}+\frac{92\!\cdots\!73}{48\!\cdots\!45}a+\frac{67\!\cdots\!18}{96\!\cdots\!09}$, $\frac{83\!\cdots\!32}{60\!\cdots\!25}a^{29}+\frac{43\!\cdots\!89}{90\!\cdots\!75}a^{28}-\frac{33\!\cdots\!48}{60\!\cdots\!25}a^{27}-\frac{28\!\cdots\!41}{60\!\cdots\!25}a^{26}+\frac{90\!\cdots\!38}{60\!\cdots\!25}a^{25}-\frac{11\!\cdots\!46}{60\!\cdots\!25}a^{24}-\frac{24\!\cdots\!64}{60\!\cdots\!25}a^{23}+\frac{71\!\cdots\!04}{12\!\cdots\!25}a^{22}+\frac{57\!\cdots\!39}{60\!\cdots\!25}a^{21}-\frac{17\!\cdots\!54}{60\!\cdots\!25}a^{20}-\frac{17\!\cdots\!41}{60\!\cdots\!25}a^{19}+\frac{64\!\cdots\!27}{60\!\cdots\!25}a^{18}+\frac{51\!\cdots\!02}{60\!\cdots\!25}a^{17}-\frac{99\!\cdots\!03}{60\!\cdots\!25}a^{16}-\frac{42\!\cdots\!46}{60\!\cdots\!25}a^{15}+\frac{11\!\cdots\!86}{60\!\cdots\!25}a^{14}-\frac{39\!\cdots\!44}{60\!\cdots\!25}a^{13}-\frac{21\!\cdots\!89}{60\!\cdots\!25}a^{12}+\frac{20\!\cdots\!67}{12\!\cdots\!25}a^{11}+\frac{18\!\cdots\!16}{60\!\cdots\!25}a^{10}-\frac{11\!\cdots\!18}{60\!\cdots\!25}a^{9}+\frac{63\!\cdots\!74}{60\!\cdots\!25}a^{8}+\frac{10\!\cdots\!76}{60\!\cdots\!25}a^{7}-\frac{14\!\cdots\!94}{60\!\cdots\!25}a^{6}-\frac{54\!\cdots\!36}{60\!\cdots\!25}a^{5}+\frac{10\!\cdots\!88}{60\!\cdots\!25}a^{4}-\frac{12\!\cdots\!67}{12\!\cdots\!25}a^{3}-\frac{37\!\cdots\!59}{24\!\cdots\!25}a^{2}+\frac{26\!\cdots\!83}{48\!\cdots\!45}a-\frac{16\!\cdots\!30}{96\!\cdots\!09}$, $\frac{10\!\cdots\!38}{60\!\cdots\!25}a^{29}+\frac{12\!\cdots\!82}{60\!\cdots\!25}a^{28}-\frac{46\!\cdots\!97}{60\!\cdots\!25}a^{27}-\frac{26\!\cdots\!79}{60\!\cdots\!25}a^{26}+\frac{13\!\cdots\!72}{60\!\cdots\!25}a^{25}-\frac{32\!\cdots\!29}{60\!\cdots\!25}a^{24}-\frac{66\!\cdots\!67}{12\!\cdots\!50}a^{23}+\frac{41\!\cdots\!41}{48\!\cdots\!50}a^{22}+\frac{70\!\cdots\!01}{60\!\cdots\!25}a^{21}-\frac{25\!\cdots\!06}{60\!\cdots\!25}a^{20}-\frac{18\!\cdots\!99}{60\!\cdots\!25}a^{19}+\frac{18\!\cdots\!21}{12\!\cdots\!50}a^{18}+\frac{49\!\cdots\!83}{60\!\cdots\!25}a^{17}-\frac{16\!\cdots\!12}{60\!\cdots\!25}a^{16}+\frac{91\!\cdots\!51}{60\!\cdots\!25}a^{15}+\frac{36\!\cdots\!83}{12\!\cdots\!50}a^{14}-\frac{10\!\cdots\!77}{12\!\cdots\!50}a^{13}-\frac{31\!\cdots\!37}{12\!\cdots\!50}a^{12}+\frac{60\!\cdots\!69}{24\!\cdots\!50}a^{11}-\frac{61\!\cdots\!56}{60\!\cdots\!25}a^{10}-\frac{18\!\cdots\!42}{60\!\cdots\!25}a^{9}+\frac{11\!\cdots\!06}{60\!\cdots\!25}a^{8}+\frac{25\!\cdots\!03}{12\!\cdots\!50}a^{7}-\frac{23\!\cdots\!01}{60\!\cdots\!25}a^{6}+\frac{36\!\cdots\!96}{60\!\cdots\!25}a^{5}+\frac{15\!\cdots\!22}{60\!\cdots\!25}a^{4}-\frac{52\!\cdots\!87}{24\!\cdots\!50}a^{3}+\frac{12\!\cdots\!81}{48\!\cdots\!50}a^{2}+\frac{40\!\cdots\!72}{48\!\cdots\!45}a-\frac{52\!\cdots\!27}{96\!\cdots\!09}$, $\frac{23\!\cdots\!08}{60\!\cdots\!25}a^{29}+\frac{15\!\cdots\!87}{60\!\cdots\!25}a^{28}-\frac{28\!\cdots\!52}{60\!\cdots\!25}a^{27}+\frac{63\!\cdots\!86}{60\!\cdots\!25}a^{26}+\frac{69\!\cdots\!77}{60\!\cdots\!25}a^{25}-\frac{14\!\cdots\!03}{12\!\cdots\!50}a^{24}-\frac{16\!\cdots\!61}{60\!\cdots\!25}a^{23}+\frac{24\!\cdots\!07}{48\!\cdots\!50}a^{22}+\frac{35\!\cdots\!16}{60\!\cdots\!25}a^{21}-\frac{30\!\cdots\!17}{12\!\cdots\!50}a^{20}-\frac{35\!\cdots\!43}{12\!\cdots\!50}a^{19}+\frac{90\!\cdots\!61}{12\!\cdots\!50}a^{18}+\frac{10\!\cdots\!78}{60\!\cdots\!25}a^{17}-\frac{11\!\cdots\!17}{60\!\cdots\!25}a^{16}+\frac{26\!\cdots\!91}{60\!\cdots\!25}a^{15}+\frac{16\!\cdots\!03}{12\!\cdots\!50}a^{14}-\frac{72\!\cdots\!07}{12\!\cdots\!50}a^{13}+\frac{23\!\cdots\!83}{12\!\cdots\!50}a^{12}+\frac{39\!\cdots\!89}{24\!\cdots\!50}a^{11}-\frac{82\!\cdots\!17}{12\!\cdots\!50}a^{10}-\frac{92\!\cdots\!47}{60\!\cdots\!25}a^{9}+\frac{12\!\cdots\!96}{60\!\cdots\!25}a^{8}-\frac{84\!\cdots\!26}{60\!\cdots\!25}a^{7}-\frac{31\!\cdots\!07}{12\!\cdots\!50}a^{6}+\frac{22\!\cdots\!47}{12\!\cdots\!50}a^{5}-\frac{95\!\cdots\!21}{12\!\cdots\!50}a^{4}-\frac{14\!\cdots\!86}{12\!\cdots\!25}a^{3}+\frac{25\!\cdots\!51}{19\!\cdots\!18}a^{2}-\frac{59\!\cdots\!27}{48\!\cdots\!45}a+\frac{69\!\cdots\!51}{19\!\cdots\!18}$, $\frac{14\!\cdots\!67}{60\!\cdots\!25}a^{29}+\frac{31\!\cdots\!73}{60\!\cdots\!25}a^{28}-\frac{86\!\cdots\!83}{60\!\cdots\!25}a^{27}-\frac{16\!\cdots\!26}{60\!\cdots\!25}a^{26}+\frac{48\!\cdots\!11}{12\!\cdots\!50}a^{25}-\frac{22\!\cdots\!21}{60\!\cdots\!25}a^{24}-\frac{15\!\cdots\!63}{12\!\cdots\!50}a^{23}+\frac{46\!\cdots\!77}{24\!\cdots\!50}a^{22}+\frac{23\!\cdots\!93}{12\!\cdots\!50}a^{21}-\frac{90\!\cdots\!43}{12\!\cdots\!50}a^{20}-\frac{70\!\cdots\!97}{12\!\cdots\!50}a^{19}+\frac{18\!\cdots\!52}{60\!\cdots\!25}a^{18}+\frac{10\!\cdots\!89}{12\!\cdots\!50}a^{17}-\frac{77\!\cdots\!71}{12\!\cdots\!50}a^{16}+\frac{62\!\cdots\!44}{60\!\cdots\!25}a^{15}+\frac{53\!\cdots\!56}{60\!\cdots\!25}a^{14}-\frac{20\!\cdots\!33}{12\!\cdots\!50}a^{13}-\frac{47\!\cdots\!99}{60\!\cdots\!25}a^{12}+\frac{13\!\cdots\!63}{24\!\cdots\!50}a^{11}-\frac{16\!\cdots\!33}{12\!\cdots\!50}a^{10}-\frac{92\!\cdots\!21}{12\!\cdots\!50}a^{9}+\frac{69\!\cdots\!03}{12\!\cdots\!50}a^{8}+\frac{41\!\cdots\!46}{60\!\cdots\!25}a^{7}-\frac{16\!\cdots\!33}{12\!\cdots\!50}a^{6}+\frac{29\!\cdots\!13}{12\!\cdots\!50}a^{5}+\frac{57\!\cdots\!68}{60\!\cdots\!25}a^{4}-\frac{18\!\cdots\!67}{24\!\cdots\!50}a^{3}-\frac{95\!\cdots\!77}{48\!\cdots\!50}a^{2}+\frac{53\!\cdots\!29}{19\!\cdots\!18}a-\frac{81\!\cdots\!11}{96\!\cdots\!09}$, $\frac{78\!\cdots\!61}{60\!\cdots\!25}a^{29}-\frac{49\!\cdots\!97}{12\!\cdots\!50}a^{28}-\frac{53\!\cdots\!13}{12\!\cdots\!50}a^{27}-\frac{87\!\cdots\!81}{12\!\cdots\!50}a^{26}+\frac{78\!\cdots\!79}{60\!\cdots\!25}a^{25}-\frac{19\!\cdots\!21}{12\!\cdots\!50}a^{24}-\frac{14\!\cdots\!42}{60\!\cdots\!25}a^{23}+\frac{10\!\cdots\!91}{12\!\cdots\!25}a^{22}+\frac{14\!\cdots\!47}{60\!\cdots\!25}a^{21}-\frac{18\!\cdots\!37}{60\!\cdots\!25}a^{20}-\frac{16\!\cdots\!71}{12\!\cdots\!50}a^{19}+\frac{59\!\cdots\!51}{60\!\cdots\!25}a^{18}-\frac{22\!\cdots\!53}{12\!\cdots\!50}a^{17}-\frac{15\!\cdots\!83}{12\!\cdots\!50}a^{16}+\frac{23\!\cdots\!89}{12\!\cdots\!50}a^{15}+\frac{34\!\cdots\!08}{60\!\cdots\!25}a^{14}-\frac{52\!\cdots\!17}{60\!\cdots\!25}a^{13}+\frac{15\!\cdots\!98}{60\!\cdots\!25}a^{12}+\frac{37\!\cdots\!53}{24\!\cdots\!25}a^{11}-\frac{14\!\cdots\!39}{12\!\cdots\!50}a^{10}-\frac{90\!\cdots\!63}{12\!\cdots\!50}a^{9}+\frac{19\!\cdots\!42}{60\!\cdots\!25}a^{8}-\frac{12\!\cdots\!19}{12\!\cdots\!50}a^{7}-\frac{48\!\cdots\!09}{12\!\cdots\!50}a^{6}+\frac{46\!\cdots\!09}{12\!\cdots\!50}a^{5}+\frac{12\!\cdots\!33}{12\!\cdots\!50}a^{4}-\frac{88\!\cdots\!23}{24\!\cdots\!50}a^{3}+\frac{77\!\cdots\!28}{48\!\cdots\!45}a^{2}+\frac{11\!\cdots\!51}{96\!\cdots\!90}a-\frac{20\!\cdots\!15}{19\!\cdots\!18}$, $\frac{12\!\cdots\!27}{60\!\cdots\!25}a^{29}-\frac{19\!\cdots\!39}{12\!\cdots\!50}a^{28}-\frac{49\!\cdots\!53}{60\!\cdots\!25}a^{27}-\frac{24\!\cdots\!51}{60\!\cdots\!25}a^{26}+\frac{14\!\cdots\!93}{60\!\cdots\!25}a^{25}-\frac{17\!\cdots\!37}{12\!\cdots\!50}a^{24}-\frac{65\!\cdots\!83}{12\!\cdots\!50}a^{23}+\frac{27\!\cdots\!83}{24\!\cdots\!50}a^{22}+\frac{12\!\cdots\!83}{12\!\cdots\!50}a^{21}-\frac{59\!\cdots\!13}{12\!\cdots\!50}a^{20}-\frac{29\!\cdots\!77}{12\!\cdots\!50}a^{19}+\frac{10\!\cdots\!72}{60\!\cdots\!25}a^{18}+\frac{65\!\cdots\!19}{12\!\cdots\!50}a^{17}-\frac{16\!\cdots\!33}{60\!\cdots\!25}a^{16}+\frac{13\!\cdots\!13}{12\!\cdots\!50}a^{15}+\frac{14\!\cdots\!96}{60\!\cdots\!25}a^{14}-\frac{13\!\cdots\!43}{12\!\cdots\!50}a^{13}-\frac{11\!\cdots\!33}{12\!\cdots\!50}a^{12}+\frac{65\!\cdots\!09}{24\!\cdots\!50}a^{11}-\frac{71\!\cdots\!23}{12\!\cdots\!50}a^{10}-\frac{31\!\cdots\!21}{12\!\cdots\!50}a^{9}+\frac{37\!\cdots\!03}{12\!\cdots\!50}a^{8}+\frac{76\!\cdots\!11}{60\!\cdots\!25}a^{7}-\frac{30\!\cdots\!84}{60\!\cdots\!25}a^{6}+\frac{11\!\cdots\!54}{60\!\cdots\!25}a^{5}+\frac{30\!\cdots\!61}{12\!\cdots\!50}a^{4}-\frac{36\!\cdots\!62}{12\!\cdots\!25}a^{3}+\frac{17\!\cdots\!22}{24\!\cdots\!25}a^{2}+\frac{50\!\cdots\!04}{48\!\cdots\!45}a-\frac{75\!\cdots\!92}{96\!\cdots\!09}$, $\frac{10\!\cdots\!43}{24\!\cdots\!50}a^{29}-\frac{13\!\cdots\!32}{12\!\cdots\!25}a^{28}-\frac{20\!\cdots\!18}{12\!\cdots\!25}a^{27}-\frac{16\!\cdots\!99}{48\!\cdots\!50}a^{26}+\frac{49\!\cdots\!23}{96\!\cdots\!90}a^{25}-\frac{16\!\cdots\!89}{36\!\cdots\!50}a^{24}-\frac{24\!\cdots\!33}{24\!\cdots\!50}a^{23}+\frac{32\!\cdots\!16}{12\!\cdots\!25}a^{22}+\frac{16\!\cdots\!13}{12\!\cdots\!25}a^{21}-\frac{26\!\cdots\!73}{24\!\cdots\!50}a^{20}-\frac{40\!\cdots\!37}{24\!\cdots\!50}a^{19}+\frac{43\!\cdots\!43}{12\!\cdots\!25}a^{18}+\frac{30\!\cdots\!37}{24\!\cdots\!50}a^{17}-\frac{13\!\cdots\!83}{24\!\cdots\!50}a^{16}+\frac{22\!\cdots\!26}{48\!\cdots\!45}a^{15}+\frac{47\!\cdots\!21}{12\!\cdots\!25}a^{14}-\frac{62\!\cdots\!79}{24\!\cdots\!50}a^{13}+\frac{75\!\cdots\!28}{12\!\cdots\!25}a^{12}+\frac{64\!\cdots\!96}{12\!\cdots\!25}a^{11}-\frac{75\!\cdots\!21}{24\!\cdots\!50}a^{10}-\frac{10\!\cdots\!77}{24\!\cdots\!25}a^{9}+\frac{39\!\cdots\!36}{48\!\cdots\!45}a^{8}-\frac{64\!\cdots\!79}{12\!\cdots\!25}a^{7}-\frac{11\!\cdots\!87}{12\!\cdots\!25}a^{6}+\frac{91\!\cdots\!39}{12\!\cdots\!25}a^{5}+\frac{22\!\cdots\!81}{24\!\cdots\!25}a^{4}-\frac{79\!\cdots\!68}{11\!\cdots\!25}a^{3}+\frac{13\!\cdots\!31}{48\!\cdots\!50}a^{2}-\frac{35\!\cdots\!32}{96\!\cdots\!09}a-\frac{53\!\cdots\!03}{19\!\cdots\!18}$, $\frac{33\!\cdots\!53}{12\!\cdots\!50}a^{29}-\frac{40\!\cdots\!99}{60\!\cdots\!25}a^{28}-\frac{66\!\cdots\!21}{60\!\cdots\!25}a^{27}-\frac{21\!\cdots\!82}{60\!\cdots\!25}a^{26}+\frac{20\!\cdots\!26}{60\!\cdots\!25}a^{25}-\frac{26\!\cdots\!09}{12\!\cdots\!50}a^{24}-\frac{43\!\cdots\!03}{60\!\cdots\!25}a^{23}+\frac{18\!\cdots\!23}{12\!\cdots\!25}a^{22}+\frac{71\!\cdots\!03}{60\!\cdots\!25}a^{21}-\frac{81\!\cdots\!41}{12\!\cdots\!50}a^{20}-\frac{28\!\cdots\!39}{12\!\cdots\!50}a^{19}+\frac{28\!\cdots\!83}{12\!\cdots\!50}a^{18}+\frac{54\!\cdots\!33}{12\!\cdots\!50}a^{17}-\frac{23\!\cdots\!56}{60\!\cdots\!25}a^{16}+\frac{20\!\cdots\!41}{12\!\cdots\!50}a^{15}+\frac{21\!\cdots\!97}{60\!\cdots\!25}a^{14}-\frac{17\!\cdots\!51}{12\!\cdots\!50}a^{13}+\frac{41\!\cdots\!72}{60\!\cdots\!25}a^{12}+\frac{43\!\cdots\!79}{12\!\cdots\!25}a^{11}-\frac{14\!\cdots\!11}{12\!\cdots\!50}a^{10}-\frac{23\!\cdots\!61}{60\!\cdots\!25}a^{9}+\frac{24\!\cdots\!73}{60\!\cdots\!25}a^{8}+\frac{23\!\cdots\!29}{12\!\cdots\!50}a^{7}-\frac{37\!\cdots\!38}{60\!\cdots\!25}a^{6}+\frac{30\!\cdots\!31}{12\!\cdots\!50}a^{5}+\frac{19\!\cdots\!01}{60\!\cdots\!25}a^{4}-\frac{90\!\cdots\!43}{24\!\cdots\!50}a^{3}+\frac{36\!\cdots\!87}{48\!\cdots\!50}a^{2}+\frac{13\!\cdots\!19}{96\!\cdots\!90}a-\frac{18\!\cdots\!91}{19\!\cdots\!18}$, $\frac{59\!\cdots\!81}{12\!\cdots\!50}a^{29}-\frac{24\!\cdots\!03}{60\!\cdots\!25}a^{28}-\frac{17\!\cdots\!49}{12\!\cdots\!50}a^{27}+\frac{29\!\cdots\!81}{60\!\cdots\!25}a^{26}+\frac{29\!\cdots\!42}{60\!\cdots\!25}a^{25}-\frac{46\!\cdots\!29}{60\!\cdots\!25}a^{24}-\frac{86\!\cdots\!57}{12\!\cdots\!50}a^{23}+\frac{38\!\cdots\!18}{12\!\cdots\!25}a^{22}-\frac{15\!\cdots\!44}{60\!\cdots\!25}a^{21}-\frac{65\!\cdots\!01}{60\!\cdots\!25}a^{20}+\frac{40\!\cdots\!67}{12\!\cdots\!50}a^{19}+\frac{43\!\cdots\!21}{12\!\cdots\!50}a^{18}-\frac{21\!\cdots\!69}{12\!\cdots\!50}a^{17}-\frac{29\!\cdots\!42}{60\!\cdots\!25}a^{16}+\frac{13\!\cdots\!41}{18\!\cdots\!50}a^{15}+\frac{72\!\cdots\!34}{60\!\cdots\!25}a^{14}-\frac{15\!\cdots\!41}{60\!\cdots\!25}a^{13}+\frac{18\!\cdots\!83}{12\!\cdots\!50}a^{12}+\frac{21\!\cdots\!17}{48\!\cdots\!50}a^{11}-\frac{29\!\cdots\!11}{60\!\cdots\!25}a^{10}-\frac{13\!\cdots\!62}{60\!\cdots\!25}a^{9}+\frac{11\!\cdots\!57}{12\!\cdots\!50}a^{8}-\frac{68\!\cdots\!56}{60\!\cdots\!25}a^{7}-\frac{11\!\cdots\!07}{12\!\cdots\!50}a^{6}+\frac{79\!\cdots\!07}{12\!\cdots\!50}a^{5}+\frac{38\!\cdots\!59}{12\!\cdots\!50}a^{4}-\frac{79\!\cdots\!17}{12\!\cdots\!25}a^{3}+\frac{44\!\cdots\!28}{24\!\cdots\!25}a^{2}+\frac{17\!\cdots\!99}{96\!\cdots\!09}a-\frac{20\!\cdots\!25}{19\!\cdots\!18}$, $\frac{96\!\cdots\!79}{24\!\cdots\!50}a^{29}-\frac{20\!\cdots\!22}{12\!\cdots\!25}a^{28}-\frac{37\!\cdots\!21}{24\!\cdots\!50}a^{27}-\frac{18\!\cdots\!87}{24\!\cdots\!50}a^{26}+\frac{62\!\cdots\!03}{12\!\cdots\!25}a^{25}-\frac{56\!\cdots\!97}{24\!\cdots\!50}a^{24}-\frac{21\!\cdots\!73}{24\!\cdots\!50}a^{23}+\frac{95\!\cdots\!77}{48\!\cdots\!50}a^{22}+\frac{40\!\cdots\!73}{24\!\cdots\!50}a^{21}-\frac{11\!\cdots\!49}{12\!\cdots\!25}a^{20}-\frac{10\!\cdots\!07}{24\!\cdots\!50}a^{19}+\frac{81\!\cdots\!49}{24\!\cdots\!50}a^{18}+\frac{24\!\cdots\!49}{24\!\cdots\!50}a^{17}-\frac{13\!\cdots\!41}{24\!\cdots\!50}a^{16}+\frac{33\!\cdots\!13}{24\!\cdots\!50}a^{15}+\frac{46\!\cdots\!06}{12\!\cdots\!25}a^{14}-\frac{22\!\cdots\!99}{12\!\cdots\!25}a^{13}+\frac{44\!\cdots\!17}{24\!\cdots\!50}a^{12}+\frac{25\!\cdots\!69}{48\!\cdots\!50}a^{11}-\frac{92\!\cdots\!83}{24\!\cdots\!50}a^{10}-\frac{69\!\cdots\!68}{12\!\cdots\!25}a^{9}+\frac{62\!\cdots\!63}{24\!\cdots\!50}a^{8}+\frac{35\!\cdots\!96}{12\!\cdots\!25}a^{7}-\frac{19\!\cdots\!23}{24\!\cdots\!50}a^{6}+\frac{23\!\cdots\!79}{12\!\cdots\!25}a^{5}+\frac{60\!\cdots\!08}{12\!\cdots\!25}a^{4}-\frac{58\!\cdots\!98}{24\!\cdots\!25}a^{3}-\frac{29\!\cdots\!11}{48\!\cdots\!50}a^{2}+\frac{14\!\cdots\!31}{48\!\cdots\!45}a-\frac{13\!\cdots\!76}{96\!\cdots\!09}$, $\frac{27\!\cdots\!34}{60\!\cdots\!25}a^{29}-\frac{92\!\cdots\!13}{12\!\cdots\!50}a^{28}-\frac{22\!\cdots\!77}{12\!\cdots\!50}a^{27}-\frac{37\!\cdots\!42}{60\!\cdots\!25}a^{26}+\frac{34\!\cdots\!81}{60\!\cdots\!25}a^{25}-\frac{39\!\cdots\!79}{12\!\cdots\!50}a^{24}-\frac{76\!\cdots\!68}{60\!\cdots\!25}a^{23}+\frac{30\!\cdots\!48}{12\!\cdots\!25}a^{22}+\frac{26\!\cdots\!11}{12\!\cdots\!50}a^{21}-\frac{13\!\cdots\!21}{12\!\cdots\!50}a^{20}-\frac{27\!\cdots\!67}{60\!\cdots\!25}a^{19}+\frac{47\!\cdots\!73}{12\!\cdots\!50}a^{18}+\frac{11\!\cdots\!73}{12\!\cdots\!50}a^{17}-\frac{41\!\cdots\!11}{60\!\cdots\!25}a^{16}+\frac{31\!\cdots\!21}{12\!\cdots\!50}a^{15}+\frac{11\!\cdots\!67}{18\!\cdots\!50}a^{14}-\frac{29\!\cdots\!81}{12\!\cdots\!50}a^{13}-\frac{19\!\cdots\!11}{12\!\cdots\!50}a^{12}+\frac{14\!\cdots\!63}{24\!\cdots\!50}a^{11}-\frac{23\!\cdots\!41}{12\!\cdots\!50}a^{10}-\frac{82\!\cdots\!57}{12\!\cdots\!50}a^{9}+\frac{42\!\cdots\!63}{60\!\cdots\!25}a^{8}+\frac{43\!\cdots\!49}{12\!\cdots\!50}a^{7}-\frac{65\!\cdots\!28}{60\!\cdots\!25}a^{6}+\frac{50\!\cdots\!61}{12\!\cdots\!50}a^{5}+\frac{70\!\cdots\!37}{12\!\cdots\!50}a^{4}-\frac{15\!\cdots\!03}{24\!\cdots\!50}a^{3}+\frac{29\!\cdots\!74}{24\!\cdots\!25}a^{2}+\frac{33\!\cdots\!11}{14\!\cdots\!70}a-\frac{31\!\cdots\!71}{19\!\cdots\!18}$
| 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: | \( 10842704727.78492 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 10842704727.78492 \cdot 1}{22\cdot\sqrt{10745322081507339108891258824483901499337171}}\cr\approx \mathstrut & 0.141189778474009 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 - 4*x^28 + 2*x^27 + 14*x^26 - 18*x^25 - 22*x^24 + 80*x^23 + 2*x^22 - 292*x^21 + 107*x^20 + 966*x^19 - 524*x^18 - 1689*x^17 + 1837*x^16 + 1003*x^15 - 6747*x^14 + 4493*x^13 + 13850*x^12 - 15662*x^11 - 11779*x^10 + 28372*x^9 - 4752*x^8 - 31522*x^7 + 29972*x^6 + 5164*x^5 - 25670*x^4 + 15150*x^3 + 3000*x^2 - 8125*x + 3125)
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 - 4*x^28 + 2*x^27 + 14*x^26 - 18*x^25 - 22*x^24 + 80*x^23 + 2*x^22 - 292*x^21 + 107*x^20 + 966*x^19 - 524*x^18 - 1689*x^17 + 1837*x^16 + 1003*x^15 - 6747*x^14 + 4493*x^13 + 13850*x^12 - 15662*x^11 - 11779*x^10 + 28372*x^9 - 4752*x^8 - 31522*x^7 + 29972*x^6 + 5164*x^5 - 25670*x^4 + 15150*x^3 + 3000*x^2 - 8125*x + 3125, 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 - 4*x^28 + 2*x^27 + 14*x^26 - 18*x^25 - 22*x^24 + 80*x^23 + 2*x^22 - 292*x^21 + 107*x^20 + 966*x^19 - 524*x^18 - 1689*x^17 + 1837*x^16 + 1003*x^15 - 6747*x^14 + 4493*x^13 + 13850*x^12 - 15662*x^11 - 11779*x^10 + 28372*x^9 - 4752*x^8 - 31522*x^7 + 29972*x^6 + 5164*x^5 - 25670*x^4 + 15150*x^3 + 3000*x^2 - 8125*x + 3125);
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 - 4*x^28 + 2*x^27 + 14*x^26 - 18*x^25 - 22*x^24 + 80*x^23 + 2*x^22 - 292*x^21 + 107*x^20 + 966*x^19 - 524*x^18 - 1689*x^17 + 1837*x^16 + 1003*x^15 - 6747*x^14 + 4493*x^13 + 13850*x^12 - 15662*x^11 - 11779*x^10 + 28372*x^9 - 4752*x^8 - 31522*x^7 + 29972*x^6 + 5164*x^5 - 25670*x^4 + 15150*x^3 + 3000*x^2 - 8125*x + 3125);
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
$S_3\times C_{15}$ (as 30T15):
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 90 |
| The 45 conjugacy class representatives for $S_3\times C_{15}$ |
| Character table for $S_3\times C_{15}$ |
Intermediate fields
| \(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), 6.0.1279091.2, \(\Q(\zeta_{11})\) |
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 45 sibling: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{3}$ | $15^{2}$ | $15{,}\,{\href{/padicField/5.5.0.1}{5} }^{3}$ | $30$ | R | $30$ | $30$ | $30$ | ${\href{/padicField/23.3.0.1}{3} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }^{3}$ | R | $15{,}\,{\href{/padicField/37.5.0.1}{5} }^{3}$ | $30$ | ${\href{/padicField/43.6.0.1}{6} }^{5}$ | $15^{2}$ | $15{,}\,{\href{/padicField/53.5.0.1}{5} }^{3}$ | $15{,}\,{\href{/padicField/59.5.0.1}{5} }^{3}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(11\)
| Deg $30$ | $10$ | $3$ | $27$ | |||
|
\(31\)
| 31.15.10.2 | $x^{15} + 13454 x^{9} + 45252529 x^{3} + 22445254384$ | $3$ | $5$ | $10$ | $C_{15}$ | $[\ ]_{3}^{5}$ |
| 31.15.0.1 | $x^{15} + 30 x^{6} + 29 x^{5} + 12 x^{4} + 13 x^{3} + 23 x^{2} + 25 x + 28$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |
Artin representations
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 *.