Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 9*x^29 + 29*x^28 - 62*x^27 + 140*x^26 - 289*x^25 + 749*x^24 - 2505*x^23 + 6556*x^22 - 13979*x^21 + 24020*x^20 - 33363*x^19 + 38542*x^18 - 41923*x^17 + 47672*x^16 - 60161*x^15 + 70650*x^14 - 64907*x^13 + 32243*x^12 + 11892*x^11 - 39347*x^10 + 39493*x^9 - 24068*x^8 + 9526*x^7 + 164*x^6 - 4686*x^5 + 4431*x^4 - 2269*x^3 + 729*x^2 - 213*x + 43)
gp: K = bnfinit(y^30 - 9*y^29 + 29*y^28 - 62*y^27 + 140*y^26 - 289*y^25 + 749*y^24 - 2505*y^23 + 6556*y^22 - 13979*y^21 + 24020*y^20 - 33363*y^19 + 38542*y^18 - 41923*y^17 + 47672*y^16 - 60161*y^15 + 70650*y^14 - 64907*y^13 + 32243*y^12 + 11892*y^11 - 39347*y^10 + 39493*y^9 - 24068*y^8 + 9526*y^7 + 164*y^6 - 4686*y^5 + 4431*y^4 - 2269*y^3 + 729*y^2 - 213*y + 43, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 9*x^29 + 29*x^28 - 62*x^27 + 140*x^26 - 289*x^25 + 749*x^24 - 2505*x^23 + 6556*x^22 - 13979*x^21 + 24020*x^20 - 33363*x^19 + 38542*x^18 - 41923*x^17 + 47672*x^16 - 60161*x^15 + 70650*x^14 - 64907*x^13 + 32243*x^12 + 11892*x^11 - 39347*x^10 + 39493*x^9 - 24068*x^8 + 9526*x^7 + 164*x^6 - 4686*x^5 + 4431*x^4 - 2269*x^3 + 729*x^2 - 213*x + 43);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 9*x^29 + 29*x^28 - 62*x^27 + 140*x^26 - 289*x^25 + 749*x^24 - 2505*x^23 + 6556*x^22 - 13979*x^21 + 24020*x^20 - 33363*x^19 + 38542*x^18 - 41923*x^17 + 47672*x^16 - 60161*x^15 + 70650*x^14 - 64907*x^13 + 32243*x^12 + 11892*x^11 - 39347*x^10 + 39493*x^9 - 24068*x^8 + 9526*x^7 + 164*x^6 - 4686*x^5 + 4431*x^4 - 2269*x^3 + 729*x^2 - 213*x + 43)
\( x^{30} - 9 x^{29} + 29 x^{28} - 62 x^{27} + 140 x^{26} - 289 x^{25} + 749 x^{24} - 2505 x^{23} + \cdots + 43 \)
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: |
\(812804153180660145912426894477473567856769041137\)
\(\medspace = 17^{15}\cdot 127^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(39.54\) | 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: | $17^{1/2}127^{1/2}\approx 46.46504062195577$ | ||
| Ramified primes: |
\(17\), \(127\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{17}a^{23}-\frac{4}{17}a^{22}+\frac{1}{17}a^{21}+\frac{6}{17}a^{20}-\frac{4}{17}a^{19}-\frac{8}{17}a^{18}-\frac{8}{17}a^{17}-\frac{7}{17}a^{16}-\frac{3}{17}a^{15}+\frac{8}{17}a^{14}+\frac{6}{17}a^{13}+\frac{1}{17}a^{12}-\frac{5}{17}a^{11}+\frac{5}{17}a^{10}+\frac{3}{17}a^{9}+\frac{8}{17}a^{8}-\frac{4}{17}a^{7}+\frac{1}{17}a^{6}+\frac{3}{17}a^{5}-\frac{4}{17}a^{4}-\frac{2}{17}a^{3}-\frac{2}{17}a^{2}-\frac{5}{17}a-\frac{4}{17}$, $\frac{1}{17}a^{24}+\frac{2}{17}a^{22}-\frac{7}{17}a^{21}+\frac{3}{17}a^{20}-\frac{7}{17}a^{19}-\frac{6}{17}a^{18}-\frac{5}{17}a^{17}+\frac{3}{17}a^{16}-\frac{4}{17}a^{15}+\frac{4}{17}a^{14}+\frac{8}{17}a^{13}-\frac{1}{17}a^{12}+\frac{2}{17}a^{11}+\frac{6}{17}a^{10}+\frac{3}{17}a^{9}-\frac{6}{17}a^{8}+\frac{2}{17}a^{7}+\frac{7}{17}a^{6}+\frac{8}{17}a^{5}-\frac{1}{17}a^{4}+\frac{7}{17}a^{3}+\frac{4}{17}a^{2}-\frac{7}{17}a+\frac{1}{17}$, $\frac{1}{17}a^{25}+\frac{1}{17}a^{22}+\frac{1}{17}a^{21}-\frac{2}{17}a^{20}+\frac{2}{17}a^{19}-\frac{6}{17}a^{18}+\frac{2}{17}a^{17}-\frac{7}{17}a^{16}-\frac{7}{17}a^{15}-\frac{8}{17}a^{14}+\frac{4}{17}a^{13}-\frac{1}{17}a^{11}-\frac{7}{17}a^{10}+\frac{5}{17}a^{9}+\frac{3}{17}a^{8}-\frac{2}{17}a^{7}+\frac{6}{17}a^{6}-\frac{7}{17}a^{5}-\frac{2}{17}a^{4}+\frac{8}{17}a^{3}-\frac{3}{17}a^{2}-\frac{6}{17}a+\frac{8}{17}$, $\frac{1}{17}a^{26}+\frac{5}{17}a^{22}-\frac{3}{17}a^{21}-\frac{4}{17}a^{20}-\frac{2}{17}a^{19}-\frac{7}{17}a^{18}+\frac{1}{17}a^{17}-\frac{5}{17}a^{15}-\frac{4}{17}a^{14}-\frac{6}{17}a^{13}-\frac{2}{17}a^{12}-\frac{2}{17}a^{11}+\frac{7}{17}a^{8}-\frac{7}{17}a^{7}-\frac{8}{17}a^{6}-\frac{5}{17}a^{5}-\frac{5}{17}a^{4}-\frac{1}{17}a^{3}-\frac{4}{17}a^{2}-\frac{4}{17}a+\frac{4}{17}$, $\frac{1}{731}a^{27}-\frac{16}{731}a^{26}-\frac{9}{731}a^{25}+\frac{16}{731}a^{24}+\frac{21}{731}a^{23}+\frac{182}{731}a^{22}+\frac{262}{731}a^{21}+\frac{105}{731}a^{20}+\frac{205}{731}a^{19}+\frac{232}{731}a^{18}+\frac{81}{731}a^{17}-\frac{74}{731}a^{16}-\frac{262}{731}a^{15}+\frac{50}{731}a^{14}-\frac{177}{731}a^{13}-\frac{208}{731}a^{12}-\frac{126}{731}a^{11}+\frac{256}{731}a^{10}-\frac{282}{731}a^{9}+\frac{260}{731}a^{8}+\frac{141}{731}a^{7}-\frac{143}{731}a^{6}+\frac{76}{731}a^{5}-\frac{4}{43}a^{4}-\frac{31}{731}a^{3}+\frac{15}{43}a^{2}+\frac{185}{731}a+\frac{6}{17}$, $\frac{1}{13211363}a^{28}-\frac{4584}{13211363}a^{27}+\frac{230029}{13211363}a^{26}-\frac{224053}{13211363}a^{25}+\frac{150533}{13211363}a^{24}+\frac{321612}{13211363}a^{23}-\frac{587863}{13211363}a^{22}+\frac{1078806}{13211363}a^{21}+\frac{5447513}{13211363}a^{20}-\frac{22515}{777139}a^{19}-\frac{3744787}{13211363}a^{18}+\frac{855246}{13211363}a^{17}-\frac{218220}{13211363}a^{16}+\frac{753493}{13211363}a^{15}+\frac{6070708}{13211363}a^{14}+\frac{3632482}{13211363}a^{13}+\frac{5541902}{13211363}a^{12}-\frac{5428309}{13211363}a^{11}-\frac{159675}{1201033}a^{10}-\frac{6234543}{13211363}a^{9}+\frac{4886985}{13211363}a^{8}+\frac{2383270}{13211363}a^{7}-\frac{2830001}{13211363}a^{6}-\frac{1389857}{13211363}a^{5}+\frac{1658084}{13211363}a^{4}+\frac{4053788}{13211363}a^{3}-\frac{152407}{307241}a^{2}+\frac{5228240}{13211363}a+\frac{92439}{307241}$, $\frac{1}{77\!\cdots\!19}a^{29}-\frac{43\!\cdots\!43}{25\!\cdots\!49}a^{28}-\frac{41\!\cdots\!89}{77\!\cdots\!19}a^{27}+\frac{79\!\cdots\!95}{18\!\cdots\!33}a^{26}-\frac{36\!\cdots\!73}{77\!\cdots\!19}a^{25}+\frac{18\!\cdots\!19}{77\!\cdots\!19}a^{24}-\frac{16\!\cdots\!37}{77\!\cdots\!19}a^{23}+\frac{27\!\cdots\!21}{70\!\cdots\!29}a^{22}-\frac{36\!\cdots\!23}{77\!\cdots\!19}a^{21}+\frac{29\!\cdots\!60}{77\!\cdots\!19}a^{20}+\frac{27\!\cdots\!97}{77\!\cdots\!19}a^{19}+\frac{36\!\cdots\!86}{77\!\cdots\!19}a^{18}+\frac{85\!\cdots\!87}{13\!\cdots\!41}a^{17}+\frac{25\!\cdots\!90}{77\!\cdots\!19}a^{16}+\frac{27\!\cdots\!89}{77\!\cdots\!19}a^{15}-\frac{33\!\cdots\!00}{77\!\cdots\!19}a^{14}-\frac{11\!\cdots\!01}{77\!\cdots\!19}a^{13}+\frac{60\!\cdots\!38}{77\!\cdots\!19}a^{12}+\frac{23\!\cdots\!96}{77\!\cdots\!19}a^{11}-\frac{16\!\cdots\!75}{77\!\cdots\!19}a^{10}+\frac{35\!\cdots\!19}{77\!\cdots\!19}a^{9}-\frac{17\!\cdots\!74}{70\!\cdots\!29}a^{8}-\frac{35\!\cdots\!52}{45\!\cdots\!07}a^{7}+\frac{31\!\cdots\!71}{77\!\cdots\!19}a^{6}+\frac{21\!\cdots\!06}{70\!\cdots\!29}a^{5}-\frac{19\!\cdots\!59}{77\!\cdots\!19}a^{4}-\frac{21\!\cdots\!83}{77\!\cdots\!19}a^{3}-\frac{22\!\cdots\!43}{77\!\cdots\!19}a^{2}+\frac{19\!\cdots\!03}{77\!\cdots\!19}a-\frac{47\!\cdots\!40}{18\!\cdots\!33}$
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: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{59\!\cdots\!48}{62\!\cdots\!21}a^{29}-\frac{30\!\cdots\!53}{36\!\cdots\!13}a^{28}+\frac{15\!\cdots\!01}{62\!\cdots\!21}a^{27}-\frac{31\!\cdots\!77}{62\!\cdots\!21}a^{26}+\frac{23\!\cdots\!00}{20\!\cdots\!91}a^{25}-\frac{14\!\cdots\!39}{62\!\cdots\!21}a^{24}+\frac{39\!\cdots\!09}{62\!\cdots\!21}a^{23}-\frac{79\!\cdots\!95}{36\!\cdots\!13}a^{22}+\frac{33\!\cdots\!77}{62\!\cdots\!21}a^{21}-\frac{70\!\cdots\!84}{62\!\cdots\!21}a^{20}+\frac{11\!\cdots\!96}{62\!\cdots\!21}a^{19}-\frac{15\!\cdots\!45}{62\!\cdots\!21}a^{18}+\frac{26\!\cdots\!62}{96\!\cdots\!29}a^{17}-\frac{18\!\cdots\!30}{62\!\cdots\!21}a^{16}+\frac{21\!\cdots\!25}{62\!\cdots\!21}a^{15}-\frac{27\!\cdots\!74}{62\!\cdots\!21}a^{14}+\frac{10\!\cdots\!91}{20\!\cdots\!91}a^{13}-\frac{26\!\cdots\!34}{62\!\cdots\!21}a^{12}+\frac{20\!\cdots\!47}{14\!\cdots\!47}a^{11}+\frac{10\!\cdots\!41}{62\!\cdots\!21}a^{10}-\frac{19\!\cdots\!60}{62\!\cdots\!21}a^{9}+\frac{16\!\cdots\!27}{62\!\cdots\!21}a^{8}-\frac{80\!\cdots\!97}{62\!\cdots\!21}a^{7}+\frac{25\!\cdots\!17}{62\!\cdots\!21}a^{6}+\frac{10\!\cdots\!95}{62\!\cdots\!21}a^{5}-\frac{23\!\cdots\!68}{62\!\cdots\!21}a^{4}+\frac{17\!\cdots\!48}{62\!\cdots\!21}a^{3}-\frac{40\!\cdots\!63}{36\!\cdots\!13}a^{2}+\frac{94\!\cdots\!59}{36\!\cdots\!13}a-\frac{14\!\cdots\!60}{14\!\cdots\!47}$, $\frac{37\!\cdots\!87}{62\!\cdots\!21}a^{29}-\frac{32\!\cdots\!39}{62\!\cdots\!21}a^{28}+\frac{97\!\cdots\!32}{62\!\cdots\!21}a^{27}-\frac{19\!\cdots\!44}{62\!\cdots\!21}a^{26}+\frac{45\!\cdots\!54}{62\!\cdots\!21}a^{25}-\frac{93\!\cdots\!72}{62\!\cdots\!21}a^{24}+\frac{25\!\cdots\!64}{62\!\cdots\!21}a^{23}-\frac{85\!\cdots\!00}{62\!\cdots\!21}a^{22}+\frac{21\!\cdots\!30}{62\!\cdots\!21}a^{21}-\frac{45\!\cdots\!72}{62\!\cdots\!21}a^{20}+\frac{75\!\cdots\!21}{62\!\cdots\!21}a^{19}-\frac{10\!\cdots\!78}{62\!\cdots\!21}a^{18}+\frac{44\!\cdots\!30}{24\!\cdots\!33}a^{17}-\frac{12\!\cdots\!06}{62\!\cdots\!21}a^{16}+\frac{14\!\cdots\!15}{62\!\cdots\!21}a^{15}-\frac{18\!\cdots\!16}{62\!\cdots\!21}a^{14}+\frac{20\!\cdots\!09}{62\!\cdots\!21}a^{13}-\frac{17\!\cdots\!00}{62\!\cdots\!21}a^{12}+\frac{66\!\cdots\!83}{62\!\cdots\!21}a^{11}+\frac{61\!\cdots\!64}{62\!\cdots\!21}a^{10}-\frac{11\!\cdots\!52}{56\!\cdots\!11}a^{9}+\frac{34\!\cdots\!07}{20\!\cdots\!91}a^{8}-\frac{56\!\cdots\!76}{62\!\cdots\!21}a^{7}+\frac{18\!\cdots\!47}{62\!\cdots\!21}a^{6}+\frac{62\!\cdots\!30}{62\!\cdots\!21}a^{5}-\frac{15\!\cdots\!42}{62\!\cdots\!21}a^{4}+\frac{19\!\cdots\!82}{10\!\cdots\!87}a^{3}-\frac{48\!\cdots\!05}{62\!\cdots\!21}a^{2}+\frac{12\!\cdots\!11}{62\!\cdots\!21}a-\frac{72\!\cdots\!04}{14\!\cdots\!47}$, $\frac{69\!\cdots\!60}{77\!\cdots\!19}a^{29}-\frac{58\!\cdots\!28}{77\!\cdots\!19}a^{28}+\frac{15\!\cdots\!30}{70\!\cdots\!29}a^{27}-\frac{33\!\cdots\!53}{77\!\cdots\!19}a^{26}+\frac{78\!\cdots\!23}{77\!\cdots\!19}a^{25}-\frac{15\!\cdots\!73}{77\!\cdots\!19}a^{24}+\frac{43\!\cdots\!21}{77\!\cdots\!19}a^{23}-\frac{15\!\cdots\!83}{77\!\cdots\!19}a^{22}+\frac{37\!\cdots\!98}{77\!\cdots\!19}a^{21}-\frac{76\!\cdots\!64}{77\!\cdots\!19}a^{20}+\frac{73\!\cdots\!75}{45\!\cdots\!07}a^{19}-\frac{16\!\cdots\!79}{77\!\cdots\!19}a^{18}+\frac{30\!\cdots\!96}{13\!\cdots\!41}a^{17}-\frac{19\!\cdots\!35}{77\!\cdots\!19}a^{16}+\frac{22\!\cdots\!88}{77\!\cdots\!19}a^{15}-\frac{29\!\cdots\!88}{77\!\cdots\!19}a^{14}+\frac{32\!\cdots\!56}{77\!\cdots\!19}a^{13}-\frac{27\!\cdots\!53}{77\!\cdots\!19}a^{12}+\frac{75\!\cdots\!72}{77\!\cdots\!19}a^{11}+\frac{40\!\cdots\!90}{25\!\cdots\!49}a^{10}-\frac{20\!\cdots\!13}{77\!\cdots\!19}a^{9}+\frac{16\!\cdots\!92}{77\!\cdots\!19}a^{8}-\frac{45\!\cdots\!66}{45\!\cdots\!07}a^{7}+\frac{21\!\cdots\!77}{77\!\cdots\!19}a^{6}+\frac{14\!\cdots\!08}{77\!\cdots\!19}a^{5}-\frac{25\!\cdots\!21}{77\!\cdots\!19}a^{4}+\frac{17\!\cdots\!06}{77\!\cdots\!19}a^{3}-\frac{11\!\cdots\!55}{14\!\cdots\!23}a^{2}+\frac{11\!\cdots\!32}{70\!\cdots\!29}a-\frac{10\!\cdots\!61}{18\!\cdots\!33}$, $\frac{24\!\cdots\!77}{77\!\cdots\!19}a^{29}-\frac{20\!\cdots\!53}{77\!\cdots\!19}a^{28}+\frac{32\!\cdots\!14}{45\!\cdots\!07}a^{27}-\frac{98\!\cdots\!37}{77\!\cdots\!19}a^{26}+\frac{23\!\cdots\!71}{77\!\cdots\!19}a^{25}-\frac{46\!\cdots\!19}{77\!\cdots\!19}a^{24}+\frac{13\!\cdots\!07}{77\!\cdots\!19}a^{23}-\frac{48\!\cdots\!40}{77\!\cdots\!19}a^{22}+\frac{11\!\cdots\!79}{77\!\cdots\!19}a^{21}-\frac{22\!\cdots\!06}{77\!\cdots\!19}a^{20}+\frac{34\!\cdots\!21}{77\!\cdots\!19}a^{19}-\frac{40\!\cdots\!69}{77\!\cdots\!19}a^{18}+\frac{65\!\cdots\!83}{13\!\cdots\!41}a^{17}-\frac{41\!\cdots\!74}{77\!\cdots\!19}a^{16}+\frac{11\!\cdots\!38}{18\!\cdots\!33}a^{15}-\frac{71\!\cdots\!80}{77\!\cdots\!19}a^{14}+\frac{75\!\cdots\!79}{77\!\cdots\!19}a^{13}-\frac{43\!\cdots\!72}{77\!\cdots\!19}a^{12}-\frac{18\!\cdots\!10}{77\!\cdots\!19}a^{11}+\frac{37\!\cdots\!63}{45\!\cdots\!07}a^{10}-\frac{51\!\cdots\!10}{70\!\cdots\!29}a^{9}+\frac{20\!\cdots\!25}{77\!\cdots\!19}a^{8}-\frac{64\!\cdots\!27}{77\!\cdots\!19}a^{7}-\frac{72\!\cdots\!27}{77\!\cdots\!19}a^{6}+\frac{72\!\cdots\!26}{77\!\cdots\!19}a^{5}-\frac{80\!\cdots\!56}{77\!\cdots\!19}a^{4}+\frac{19\!\cdots\!82}{70\!\cdots\!29}a^{3}+\frac{14\!\cdots\!79}{77\!\cdots\!19}a^{2}-\frac{46\!\cdots\!35}{77\!\cdots\!19}a+\frac{18\!\cdots\!92}{58\!\cdots\!43}$, $\frac{43\!\cdots\!97}{10\!\cdots\!03}a^{29}-\frac{33\!\cdots\!68}{96\!\cdots\!73}a^{28}+\frac{10\!\cdots\!31}{10\!\cdots\!03}a^{27}-\frac{22\!\cdots\!29}{10\!\cdots\!03}a^{26}+\frac{50\!\cdots\!30}{10\!\cdots\!03}a^{25}-\frac{93\!\cdots\!52}{96\!\cdots\!73}a^{24}+\frac{27\!\cdots\!61}{10\!\cdots\!03}a^{23}-\frac{96\!\cdots\!66}{10\!\cdots\!03}a^{22}+\frac{24\!\cdots\!16}{10\!\cdots\!03}a^{21}-\frac{49\!\cdots\!11}{10\!\cdots\!03}a^{20}+\frac{81\!\cdots\!06}{10\!\cdots\!03}a^{19}-\frac{10\!\cdots\!55}{10\!\cdots\!03}a^{18}+\frac{19\!\cdots\!45}{18\!\cdots\!17}a^{17}-\frac{12\!\cdots\!45}{10\!\cdots\!03}a^{16}+\frac{14\!\cdots\!17}{10\!\cdots\!03}a^{15}-\frac{17\!\cdots\!16}{96\!\cdots\!73}a^{14}+\frac{21\!\cdots\!33}{10\!\cdots\!03}a^{13}-\frac{17\!\cdots\!66}{10\!\cdots\!03}a^{12}+\frac{51\!\cdots\!73}{10\!\cdots\!03}a^{11}+\frac{84\!\cdots\!85}{10\!\cdots\!03}a^{10}-\frac{82\!\cdots\!40}{62\!\cdots\!59}a^{9}+\frac{10\!\cdots\!65}{10\!\cdots\!03}a^{8}-\frac{46\!\cdots\!68}{10\!\cdots\!03}a^{7}+\frac{54\!\cdots\!11}{62\!\cdots\!59}a^{6}+\frac{10\!\cdots\!36}{10\!\cdots\!03}a^{5}-\frac{15\!\cdots\!50}{10\!\cdots\!03}a^{4}+\frac{10\!\cdots\!77}{10\!\cdots\!03}a^{3}-\frac{19\!\cdots\!44}{56\!\cdots\!69}a^{2}+\frac{62\!\cdots\!72}{10\!\cdots\!03}a-\frac{63\!\cdots\!24}{24\!\cdots\!21}$, $\frac{55\!\cdots\!97}{70\!\cdots\!29}a^{29}-\frac{53\!\cdots\!04}{77\!\cdots\!19}a^{28}+\frac{16\!\cdots\!12}{77\!\cdots\!19}a^{27}-\frac{34\!\cdots\!31}{77\!\cdots\!19}a^{26}+\frac{77\!\cdots\!49}{77\!\cdots\!19}a^{25}-\frac{15\!\cdots\!07}{77\!\cdots\!19}a^{24}+\frac{41\!\cdots\!49}{77\!\cdots\!19}a^{23}-\frac{14\!\cdots\!86}{77\!\cdots\!19}a^{22}+\frac{36\!\cdots\!62}{77\!\cdots\!19}a^{21}-\frac{76\!\cdots\!86}{77\!\cdots\!19}a^{20}+\frac{12\!\cdots\!71}{77\!\cdots\!19}a^{19}-\frac{17\!\cdots\!33}{77\!\cdots\!19}a^{18}+\frac{32\!\cdots\!94}{13\!\cdots\!41}a^{17}-\frac{20\!\cdots\!38}{77\!\cdots\!19}a^{16}+\frac{24\!\cdots\!83}{77\!\cdots\!19}a^{15}-\frac{31\!\cdots\!03}{77\!\cdots\!19}a^{14}+\frac{36\!\cdots\!28}{77\!\cdots\!19}a^{13}-\frac{31\!\cdots\!96}{77\!\cdots\!19}a^{12}+\frac{12\!\cdots\!63}{77\!\cdots\!19}a^{11}+\frac{28\!\cdots\!25}{22\!\cdots\!59}a^{10}-\frac{19\!\cdots\!99}{77\!\cdots\!19}a^{9}+\frac{15\!\cdots\!01}{77\!\cdots\!19}a^{8}-\frac{69\!\cdots\!79}{77\!\cdots\!19}a^{7}+\frac{95\!\cdots\!84}{77\!\cdots\!19}a^{6}+\frac{25\!\cdots\!03}{77\!\cdots\!19}a^{5}-\frac{33\!\cdots\!36}{77\!\cdots\!19}a^{4}+\frac{21\!\cdots\!76}{77\!\cdots\!19}a^{3}-\frac{69\!\cdots\!75}{77\!\cdots\!19}a^{2}+\frac{11\!\cdots\!01}{77\!\cdots\!19}a-\frac{14\!\cdots\!97}{18\!\cdots\!33}$, $\frac{49\!\cdots\!92}{77\!\cdots\!19}a^{29}-\frac{41\!\cdots\!32}{77\!\cdots\!19}a^{28}+\frac{11\!\cdots\!80}{77\!\cdots\!19}a^{27}-\frac{23\!\cdots\!76}{77\!\cdots\!19}a^{26}+\frac{54\!\cdots\!62}{77\!\cdots\!19}a^{25}-\frac{10\!\cdots\!74}{77\!\cdots\!19}a^{24}+\frac{30\!\cdots\!37}{77\!\cdots\!19}a^{23}-\frac{10\!\cdots\!15}{77\!\cdots\!19}a^{22}+\frac{25\!\cdots\!01}{77\!\cdots\!19}a^{21}-\frac{52\!\cdots\!46}{77\!\cdots\!19}a^{20}+\frac{85\!\cdots\!91}{77\!\cdots\!19}a^{19}-\frac{11\!\cdots\!51}{77\!\cdots\!19}a^{18}+\frac{20\!\cdots\!67}{13\!\cdots\!41}a^{17}-\frac{13\!\cdots\!24}{77\!\cdots\!19}a^{16}+\frac{15\!\cdots\!96}{77\!\cdots\!19}a^{15}-\frac{20\!\cdots\!34}{77\!\cdots\!19}a^{14}+\frac{22\!\cdots\!60}{77\!\cdots\!19}a^{13}-\frac{18\!\cdots\!68}{77\!\cdots\!19}a^{12}+\frac{45\!\cdots\!35}{77\!\cdots\!19}a^{11}+\frac{86\!\cdots\!36}{77\!\cdots\!19}a^{10}-\frac{13\!\cdots\!16}{77\!\cdots\!19}a^{9}+\frac{10\!\cdots\!36}{77\!\cdots\!19}a^{8}-\frac{48\!\cdots\!59}{77\!\cdots\!19}a^{7}+\frac{14\!\cdots\!47}{77\!\cdots\!19}a^{6}+\frac{87\!\cdots\!92}{77\!\cdots\!19}a^{5}-\frac{15\!\cdots\!72}{77\!\cdots\!19}a^{4}+\frac{10\!\cdots\!79}{77\!\cdots\!19}a^{3}-\frac{38\!\cdots\!77}{77\!\cdots\!19}a^{2}+\frac{10\!\cdots\!93}{77\!\cdots\!19}a-\frac{93\!\cdots\!92}{18\!\cdots\!33}$, $\frac{11\!\cdots\!27}{77\!\cdots\!19}a^{29}-\frac{30\!\cdots\!39}{25\!\cdots\!49}a^{28}+\frac{24\!\cdots\!68}{70\!\cdots\!29}a^{27}-\frac{52\!\cdots\!44}{77\!\cdots\!19}a^{26}+\frac{12\!\cdots\!68}{77\!\cdots\!19}a^{25}-\frac{24\!\cdots\!75}{77\!\cdots\!19}a^{24}+\frac{68\!\cdots\!51}{77\!\cdots\!19}a^{23}-\frac{14\!\cdots\!65}{45\!\cdots\!07}a^{22}+\frac{58\!\cdots\!44}{77\!\cdots\!19}a^{21}-\frac{11\!\cdots\!53}{77\!\cdots\!19}a^{20}+\frac{19\!\cdots\!34}{77\!\cdots\!19}a^{19}-\frac{25\!\cdots\!82}{77\!\cdots\!19}a^{18}+\frac{46\!\cdots\!12}{13\!\cdots\!41}a^{17}-\frac{30\!\cdots\!12}{77\!\cdots\!19}a^{16}+\frac{35\!\cdots\!80}{77\!\cdots\!19}a^{15}-\frac{87\!\cdots\!31}{14\!\cdots\!23}a^{14}+\frac{51\!\cdots\!85}{77\!\cdots\!19}a^{13}-\frac{41\!\cdots\!37}{77\!\cdots\!19}a^{12}+\frac{11\!\cdots\!94}{77\!\cdots\!19}a^{11}+\frac{17\!\cdots\!25}{77\!\cdots\!19}a^{10}-\frac{28\!\cdots\!76}{77\!\cdots\!19}a^{9}+\frac{44\!\cdots\!97}{14\!\cdots\!23}a^{8}-\frac{13\!\cdots\!50}{77\!\cdots\!19}a^{7}+\frac{59\!\cdots\!19}{77\!\cdots\!19}a^{6}+\frac{23\!\cdots\!01}{18\!\cdots\!33}a^{5}-\frac{38\!\cdots\!14}{77\!\cdots\!19}a^{4}+\frac{29\!\cdots\!16}{77\!\cdots\!19}a^{3}-\frac{12\!\cdots\!47}{77\!\cdots\!19}a^{2}+\frac{24\!\cdots\!14}{70\!\cdots\!29}a-\frac{29\!\cdots\!61}{18\!\cdots\!33}$, $\frac{20\!\cdots\!04}{77\!\cdots\!19}a^{29}-\frac{18\!\cdots\!23}{77\!\cdots\!19}a^{28}+\frac{55\!\cdots\!29}{77\!\cdots\!19}a^{27}-\frac{11\!\cdots\!98}{77\!\cdots\!19}a^{26}+\frac{23\!\cdots\!45}{70\!\cdots\!29}a^{25}-\frac{53\!\cdots\!16}{77\!\cdots\!19}a^{24}+\frac{33\!\cdots\!02}{18\!\cdots\!33}a^{23}-\frac{48\!\cdots\!89}{77\!\cdots\!19}a^{22}+\frac{12\!\cdots\!21}{77\!\cdots\!19}a^{21}-\frac{26\!\cdots\!82}{77\!\cdots\!19}a^{20}+\frac{43\!\cdots\!72}{77\!\cdots\!19}a^{19}-\frac{34\!\cdots\!15}{45\!\cdots\!07}a^{18}+\frac{11\!\cdots\!50}{13\!\cdots\!41}a^{17}-\frac{73\!\cdots\!08}{77\!\cdots\!19}a^{16}+\frac{77\!\cdots\!23}{70\!\cdots\!29}a^{15}-\frac{10\!\cdots\!44}{77\!\cdots\!19}a^{14}+\frac{12\!\cdots\!89}{77\!\cdots\!19}a^{13}-\frac{11\!\cdots\!40}{77\!\cdots\!19}a^{12}+\frac{49\!\cdots\!63}{77\!\cdots\!19}a^{11}+\frac{24\!\cdots\!48}{77\!\cdots\!19}a^{10}-\frac{62\!\cdots\!30}{77\!\cdots\!19}a^{9}+\frac{55\!\cdots\!69}{77\!\cdots\!19}a^{8}-\frac{30\!\cdots\!19}{77\!\cdots\!19}a^{7}+\frac{20\!\cdots\!80}{14\!\cdots\!23}a^{6}+\frac{23\!\cdots\!60}{77\!\cdots\!19}a^{5}-\frac{17\!\cdots\!49}{18\!\cdots\!33}a^{4}+\frac{53\!\cdots\!14}{77\!\cdots\!19}a^{3}-\frac{11\!\cdots\!56}{45\!\cdots\!07}a^{2}+\frac{49\!\cdots\!80}{77\!\cdots\!19}a-\frac{45\!\cdots\!73}{18\!\cdots\!33}$, $\frac{81\!\cdots\!35}{77\!\cdots\!19}a^{29}-\frac{62\!\cdots\!24}{70\!\cdots\!29}a^{28}+\frac{20\!\cdots\!45}{77\!\cdots\!19}a^{27}-\frac{39\!\cdots\!92}{77\!\cdots\!19}a^{26}+\frac{89\!\cdots\!50}{77\!\cdots\!19}a^{25}-\frac{16\!\cdots\!97}{70\!\cdots\!29}a^{24}+\frac{49\!\cdots\!05}{77\!\cdots\!19}a^{23}-\frac{17\!\cdots\!64}{77\!\cdots\!19}a^{22}+\frac{25\!\cdots\!78}{45\!\cdots\!07}a^{21}-\frac{88\!\cdots\!92}{77\!\cdots\!19}a^{20}+\frac{83\!\cdots\!67}{45\!\cdots\!07}a^{19}-\frac{18\!\cdots\!60}{77\!\cdots\!19}a^{18}+\frac{32\!\cdots\!14}{13\!\cdots\!41}a^{17}-\frac{20\!\cdots\!03}{77\!\cdots\!19}a^{16}+\frac{23\!\cdots\!07}{77\!\cdots\!19}a^{15}-\frac{28\!\cdots\!18}{70\!\cdots\!29}a^{14}+\frac{35\!\cdots\!31}{77\!\cdots\!19}a^{13}-\frac{27\!\cdots\!38}{77\!\cdots\!19}a^{12}+\frac{33\!\cdots\!01}{77\!\cdots\!19}a^{11}+\frac{19\!\cdots\!51}{77\!\cdots\!19}a^{10}-\frac{24\!\cdots\!70}{77\!\cdots\!19}a^{9}+\frac{15\!\cdots\!95}{77\!\cdots\!19}a^{8}-\frac{51\!\cdots\!44}{77\!\cdots\!19}a^{7}+\frac{25\!\cdots\!99}{77\!\cdots\!19}a^{6}+\frac{22\!\cdots\!21}{77\!\cdots\!19}a^{5}-\frac{28\!\cdots\!49}{77\!\cdots\!19}a^{4}+\frac{35\!\cdots\!55}{18\!\cdots\!33}a^{3}-\frac{34\!\cdots\!00}{70\!\cdots\!29}a^{2}-\frac{39\!\cdots\!09}{77\!\cdots\!19}a-\frac{78\!\cdots\!35}{18\!\cdots\!33}$, $\frac{52\!\cdots\!44}{77\!\cdots\!19}a^{29}-\frac{43\!\cdots\!80}{77\!\cdots\!19}a^{28}+\frac{12\!\cdots\!62}{77\!\cdots\!19}a^{27}-\frac{24\!\cdots\!46}{77\!\cdots\!19}a^{26}+\frac{57\!\cdots\!36}{77\!\cdots\!19}a^{25}-\frac{11\!\cdots\!61}{77\!\cdots\!19}a^{24}+\frac{31\!\cdots\!37}{77\!\cdots\!19}a^{23}-\frac{65\!\cdots\!39}{45\!\cdots\!07}a^{22}+\frac{27\!\cdots\!77}{77\!\cdots\!19}a^{21}-\frac{56\!\cdots\!97}{77\!\cdots\!19}a^{20}+\frac{90\!\cdots\!76}{77\!\cdots\!19}a^{19}-\frac{11\!\cdots\!66}{77\!\cdots\!19}a^{18}+\frac{19\!\cdots\!48}{11\!\cdots\!31}a^{17}-\frac{13\!\cdots\!56}{77\!\cdots\!19}a^{16}+\frac{15\!\cdots\!54}{77\!\cdots\!19}a^{15}-\frac{21\!\cdots\!06}{77\!\cdots\!19}a^{14}+\frac{23\!\cdots\!51}{77\!\cdots\!19}a^{13}-\frac{18\!\cdots\!38}{77\!\cdots\!19}a^{12}+\frac{42\!\cdots\!54}{77\!\cdots\!19}a^{11}+\frac{99\!\cdots\!10}{77\!\cdots\!19}a^{10}-\frac{15\!\cdots\!17}{77\!\cdots\!19}a^{9}+\frac{11\!\cdots\!84}{77\!\cdots\!19}a^{8}-\frac{49\!\cdots\!10}{77\!\cdots\!19}a^{7}+\frac{11\!\cdots\!36}{77\!\cdots\!19}a^{6}+\frac{11\!\cdots\!68}{77\!\cdots\!19}a^{5}-\frac{17\!\cdots\!96}{77\!\cdots\!19}a^{4}+\frac{11\!\cdots\!65}{77\!\cdots\!19}a^{3}-\frac{40\!\cdots\!39}{77\!\cdots\!19}a^{2}+\frac{10\!\cdots\!75}{77\!\cdots\!19}a-\frac{65\!\cdots\!33}{18\!\cdots\!33}$, $\frac{20\!\cdots\!59}{77\!\cdots\!19}a^{29}-\frac{17\!\cdots\!44}{77\!\cdots\!19}a^{28}+\frac{52\!\cdots\!68}{77\!\cdots\!19}a^{27}-\frac{10\!\cdots\!60}{77\!\cdots\!19}a^{26}+\frac{24\!\cdots\!14}{77\!\cdots\!19}a^{25}-\frac{49\!\cdots\!46}{77\!\cdots\!19}a^{24}+\frac{13\!\cdots\!04}{77\!\cdots\!19}a^{23}-\frac{45\!\cdots\!29}{77\!\cdots\!19}a^{22}+\frac{11\!\cdots\!42}{77\!\cdots\!19}a^{21}-\frac{23\!\cdots\!87}{77\!\cdots\!19}a^{20}+\frac{39\!\cdots\!59}{77\!\cdots\!19}a^{19}-\frac{51\!\cdots\!97}{77\!\cdots\!19}a^{18}+\frac{18\!\cdots\!81}{24\!\cdots\!97}a^{17}-\frac{61\!\cdots\!59}{77\!\cdots\!19}a^{16}+\frac{71\!\cdots\!74}{77\!\cdots\!19}a^{15}-\frac{92\!\cdots\!82}{77\!\cdots\!19}a^{14}+\frac{10\!\cdots\!73}{77\!\cdots\!19}a^{13}-\frac{88\!\cdots\!46}{77\!\cdots\!19}a^{12}+\frac{28\!\cdots\!31}{77\!\cdots\!19}a^{11}+\frac{36\!\cdots\!16}{77\!\cdots\!19}a^{10}-\frac{58\!\cdots\!41}{70\!\cdots\!29}a^{9}+\frac{53\!\cdots\!52}{77\!\cdots\!19}a^{8}-\frac{26\!\cdots\!87}{77\!\cdots\!19}a^{7}+\frac{81\!\cdots\!41}{77\!\cdots\!19}a^{6}+\frac{38\!\cdots\!81}{77\!\cdots\!19}a^{5}-\frac{79\!\cdots\!15}{77\!\cdots\!19}a^{4}+\frac{51\!\cdots\!89}{70\!\cdots\!29}a^{3}-\frac{22\!\cdots\!92}{77\!\cdots\!19}a^{2}+\frac{53\!\cdots\!10}{77\!\cdots\!19}a-\frac{47\!\cdots\!13}{18\!\cdots\!33}$, $\frac{24\!\cdots\!77}{77\!\cdots\!19}a^{29}-\frac{20\!\cdots\!58}{77\!\cdots\!19}a^{28}+\frac{60\!\cdots\!66}{77\!\cdots\!19}a^{27}-\frac{12\!\cdots\!75}{77\!\cdots\!19}a^{26}+\frac{28\!\cdots\!09}{77\!\cdots\!19}a^{25}-\frac{56\!\cdots\!68}{77\!\cdots\!19}a^{24}+\frac{15\!\cdots\!83}{77\!\cdots\!19}a^{23}-\frac{53\!\cdots\!10}{77\!\cdots\!19}a^{22}+\frac{13\!\cdots\!09}{77\!\cdots\!19}a^{21}-\frac{27\!\cdots\!03}{77\!\cdots\!19}a^{20}+\frac{44\!\cdots\!33}{77\!\cdots\!19}a^{19}-\frac{58\!\cdots\!48}{77\!\cdots\!19}a^{18}+\frac{10\!\cdots\!61}{13\!\cdots\!41}a^{17}-\frac{70\!\cdots\!22}{77\!\cdots\!19}a^{16}+\frac{82\!\cdots\!14}{77\!\cdots\!19}a^{15}-\frac{14\!\cdots\!55}{10\!\cdots\!49}a^{14}+\frac{11\!\cdots\!10}{77\!\cdots\!19}a^{13}-\frac{98\!\cdots\!86}{77\!\cdots\!19}a^{12}+\frac{52\!\cdots\!47}{13\!\cdots\!93}a^{11}+\frac{41\!\cdots\!60}{77\!\cdots\!19}a^{10}-\frac{71\!\cdots\!44}{77\!\cdots\!19}a^{9}+\frac{34\!\cdots\!46}{45\!\cdots\!07}a^{8}-\frac{31\!\cdots\!36}{77\!\cdots\!19}a^{7}+\frac{99\!\cdots\!73}{77\!\cdots\!19}a^{6}+\frac{39\!\cdots\!92}{77\!\cdots\!19}a^{5}-\frac{87\!\cdots\!09}{77\!\cdots\!19}a^{4}+\frac{61\!\cdots\!00}{77\!\cdots\!19}a^{3}-\frac{27\!\cdots\!41}{77\!\cdots\!19}a^{2}+\frac{82\!\cdots\!78}{77\!\cdots\!19}a-\frac{43\!\cdots\!44}{16\!\cdots\!03}$, $\frac{11\!\cdots\!94}{77\!\cdots\!19}a^{29}-\frac{18\!\cdots\!43}{77\!\cdots\!19}a^{28}+\frac{10\!\cdots\!29}{77\!\cdots\!19}a^{27}-\frac{29\!\cdots\!46}{77\!\cdots\!19}a^{26}+\frac{62\!\cdots\!44}{77\!\cdots\!19}a^{25}-\frac{13\!\cdots\!12}{77\!\cdots\!19}a^{24}+\frac{29\!\cdots\!72}{77\!\cdots\!19}a^{23}-\frac{85\!\cdots\!88}{77\!\cdots\!19}a^{22}+\frac{26\!\cdots\!92}{77\!\cdots\!19}a^{21}-\frac{38\!\cdots\!18}{45\!\cdots\!07}a^{20}+\frac{13\!\cdots\!36}{77\!\cdots\!19}a^{19}-\frac{21\!\cdots\!51}{77\!\cdots\!19}a^{18}+\frac{42\!\cdots\!89}{11\!\cdots\!31}a^{17}-\frac{30\!\cdots\!23}{77\!\cdots\!19}a^{16}+\frac{32\!\cdots\!91}{77\!\cdots\!19}a^{15}-\frac{70\!\cdots\!25}{14\!\cdots\!23}a^{14}+\frac{47\!\cdots\!28}{77\!\cdots\!19}a^{13}-\frac{53\!\cdots\!43}{77\!\cdots\!19}a^{12}+\frac{42\!\cdots\!68}{77\!\cdots\!19}a^{11}-\frac{11\!\cdots\!46}{77\!\cdots\!19}a^{10}-\frac{13\!\cdots\!70}{45\!\cdots\!07}a^{9}+\frac{70\!\cdots\!40}{14\!\cdots\!23}a^{8}-\frac{92\!\cdots\!84}{25\!\cdots\!49}a^{7}+\frac{13\!\cdots\!49}{77\!\cdots\!19}a^{6}-\frac{22\!\cdots\!45}{77\!\cdots\!19}a^{5}-\frac{30\!\cdots\!16}{77\!\cdots\!19}a^{4}+\frac{26\!\cdots\!56}{45\!\cdots\!07}a^{3}-\frac{71\!\cdots\!79}{18\!\cdots\!33}a^{2}+\frac{10\!\cdots\!67}{77\!\cdots\!19}a-\frac{18\!\cdots\!41}{18\!\cdots\!33}$, $\frac{94\!\cdots\!70}{77\!\cdots\!19}a^{29}-\frac{78\!\cdots\!67}{77\!\cdots\!19}a^{28}+\frac{20\!\cdots\!30}{70\!\cdots\!29}a^{27}-\frac{44\!\cdots\!73}{77\!\cdots\!19}a^{26}+\frac{10\!\cdots\!27}{77\!\cdots\!19}a^{25}-\frac{20\!\cdots\!65}{77\!\cdots\!19}a^{24}+\frac{57\!\cdots\!16}{77\!\cdots\!19}a^{23}-\frac{19\!\cdots\!70}{77\!\cdots\!19}a^{22}+\frac{49\!\cdots\!80}{77\!\cdots\!19}a^{21}-\frac{10\!\cdots\!10}{77\!\cdots\!19}a^{20}+\frac{16\!\cdots\!35}{77\!\cdots\!19}a^{19}-\frac{21\!\cdots\!88}{77\!\cdots\!19}a^{18}+\frac{40\!\cdots\!29}{13\!\cdots\!41}a^{17}-\frac{25\!\cdots\!11}{77\!\cdots\!19}a^{16}+\frac{29\!\cdots\!45}{77\!\cdots\!19}a^{15}-\frac{39\!\cdots\!69}{77\!\cdots\!19}a^{14}+\frac{43\!\cdots\!58}{77\!\cdots\!19}a^{13}-\frac{35\!\cdots\!61}{77\!\cdots\!19}a^{12}+\frac{10\!\cdots\!26}{77\!\cdots\!19}a^{11}+\frac{14\!\cdots\!96}{77\!\cdots\!19}a^{10}-\frac{26\!\cdots\!75}{77\!\cdots\!19}a^{9}+\frac{21\!\cdots\!67}{77\!\cdots\!19}a^{8}-\frac{11\!\cdots\!78}{77\!\cdots\!19}a^{7}+\frac{35\!\cdots\!00}{77\!\cdots\!19}a^{6}+\frac{17\!\cdots\!93}{77\!\cdots\!19}a^{5}-\frac{11\!\cdots\!28}{25\!\cdots\!49}a^{4}+\frac{23\!\cdots\!17}{77\!\cdots\!19}a^{3}-\frac{85\!\cdots\!44}{77\!\cdots\!19}a^{2}+\frac{15\!\cdots\!40}{70\!\cdots\!29}a-\frac{15\!\cdots\!76}{18\!\cdots\!33}$
| 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: | \( 1232398776399.036 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{14}\cdot 1232398776399.036 \cdot 1}{2\cdot\sqrt{812804153180660145912426894477473567856769041137}}\cr\approx \mathstrut & 0.408607613133020 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 - 9*x^29 + 29*x^28 - 62*x^27 + 140*x^26 - 289*x^25 + 749*x^24 - 2505*x^23 + 6556*x^22 - 13979*x^21 + 24020*x^20 - 33363*x^19 + 38542*x^18 - 41923*x^17 + 47672*x^16 - 60161*x^15 + 70650*x^14 - 64907*x^13 + 32243*x^12 + 11892*x^11 - 39347*x^10 + 39493*x^9 - 24068*x^8 + 9526*x^7 + 164*x^6 - 4686*x^5 + 4431*x^4 - 2269*x^3 + 729*x^2 - 213*x + 43)
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 - 9*x^29 + 29*x^28 - 62*x^27 + 140*x^26 - 289*x^25 + 749*x^24 - 2505*x^23 + 6556*x^22 - 13979*x^21 + 24020*x^20 - 33363*x^19 + 38542*x^18 - 41923*x^17 + 47672*x^16 - 60161*x^15 + 70650*x^14 - 64907*x^13 + 32243*x^12 + 11892*x^11 - 39347*x^10 + 39493*x^9 - 24068*x^8 + 9526*x^7 + 164*x^6 - 4686*x^5 + 4431*x^4 - 2269*x^3 + 729*x^2 - 213*x + 43, 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 - 9*x^29 + 29*x^28 - 62*x^27 + 140*x^26 - 289*x^25 + 749*x^24 - 2505*x^23 + 6556*x^22 - 13979*x^21 + 24020*x^20 - 33363*x^19 + 38542*x^18 - 41923*x^17 + 47672*x^16 - 60161*x^15 + 70650*x^14 - 64907*x^13 + 32243*x^12 + 11892*x^11 - 39347*x^10 + 39493*x^9 - 24068*x^8 + 9526*x^7 + 164*x^6 - 4686*x^5 + 4431*x^4 - 2269*x^3 + 729*x^2 - 213*x + 43);
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 - 9*x^29 + 29*x^28 - 62*x^27 + 140*x^26 - 289*x^25 + 749*x^24 - 2505*x^23 + 6556*x^22 - 13979*x^21 + 24020*x^20 - 33363*x^19 + 38542*x^18 - 41923*x^17 + 47672*x^16 - 60161*x^15 + 70650*x^14 - 64907*x^13 + 32243*x^12 + 11892*x^11 - 39347*x^10 + 39493*x^9 - 24068*x^8 + 9526*x^7 + 164*x^6 - 4686*x^5 + 4431*x^4 - 2269*x^3 + 729*x^2 - 213*x + 43);
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
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
| \(\Q(\sqrt{17}) \), 3.1.2159.1, 5.1.4661281.1, 6.2.79241777.1, 10.2.369368189536337.2, 15.1.218659573334046061397519.1 |
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: | 30.0.6072125144349637560639895035214067242224098130847.1 |
| 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 | $15^{2}$ | $30$ | ${\href{/padicField/5.6.0.1}{6} }^{5}$ | ${\href{/padicField/7.10.0.1}{10} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{15}$ | ${\href{/padicField/13.5.0.1}{5} }^{6}$ | R | $15^{2}$ | $30$ | $30$ | ${\href{/padicField/31.2.0.1}{2} }^{15}$ | ${\href{/padicField/37.2.0.1}{2} }^{15}$ | ${\href{/padicField/41.2.0.1}{2} }^{15}$ | ${\href{/padicField/43.2.0.1}{2} }^{14}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{14}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{14}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(127\)
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.127.2t1.a.a | $1$ | $ 127 $ | \(\Q(\sqrt{-127}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.2159.2t1.a.a | $1$ | $ 17 \cdot 127 $ | \(\Q(\sqrt{-2159}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 2.2159.6t3.a.a | $2$ | $ 17 \cdot 127 $ | 6.0.591982687.1 | $D_{6}$ (as 6T3) | $1$ | $0$ |
| * | 2.2159.3t2.a.a | $2$ | $ 17 \cdot 127 $ | 3.1.2159.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.2159.5t2.a.b | $2$ | $ 17 \cdot 127 $ | 5.1.4661281.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.2159.5t2.a.a | $2$ | $ 17 \cdot 127 $ | 5.1.4661281.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.2159.10t3.b.b | $2$ | $ 17 \cdot 127 $ | 10.0.2759397651242047.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |
| * | 2.2159.10t3.b.a | $2$ | $ 17 \cdot 127 $ | 10.0.2759397651242047.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |
| * | 2.2159.15t2.a.d | $2$ | $ 17 \cdot 127 $ | 15.1.218659573334046061397519.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.2159.30t14.b.a | $2$ | $ 17 \cdot 127 $ | 30.2.812804153180660145912426894477473567856769041137.1 | $D_{30}$ (as 30T14) | $1$ | $0$ |
| * | 2.2159.15t2.a.b | $2$ | $ 17 \cdot 127 $ | 15.1.218659573334046061397519.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.2159.30t14.b.c | $2$ | $ 17 \cdot 127 $ | 30.2.812804153180660145912426894477473567856769041137.1 | $D_{30}$ (as 30T14) | $1$ | $0$ |
| * | 2.2159.15t2.a.a | $2$ | $ 17 \cdot 127 $ | 15.1.218659573334046061397519.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.2159.30t14.b.b | $2$ | $ 17 \cdot 127 $ | 30.2.812804153180660145912426894477473567856769041137.1 | $D_{30}$ (as 30T14) | $1$ | $0$ |
| * | 2.2159.15t2.a.c | $2$ | $ 17 \cdot 127 $ | 15.1.218659573334046061397519.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.2159.30t14.b.d | $2$ | $ 17 \cdot 127 $ | 30.2.812804153180660145912426894477473567856769041137.1 | $D_{30}$ (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 *.