Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 54*x^28 + 1233*x^26 - 15732*x^24 + 124974*x^22 - 651888*x^20 + 2294766*x^18 - 5517450*x^16 + 9065763*x^14 - 10076778*x^12 + 7410609*x^10 - 3473442*x^8 + 979290*x^6 - 151875*x^4 + 10935*x^2 - 243)
gp: K = bnfinit(y^30 - 54*y^28 + 1233*y^26 - 15732*y^24 + 124974*y^22 - 651888*y^20 + 2294766*y^18 - 5517450*y^16 + 9065763*y^14 - 10076778*y^12 + 7410609*y^10 - 3473442*y^8 + 979290*y^6 - 151875*y^4 + 10935*y^2 - 243, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 54*x^28 + 1233*x^26 - 15732*x^24 + 124974*x^22 - 651888*x^20 + 2294766*x^18 - 5517450*x^16 + 9065763*x^14 - 10076778*x^12 + 7410609*x^10 - 3473442*x^8 + 979290*x^6 - 151875*x^4 + 10935*x^2 - 243);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 54*x^28 + 1233*x^26 - 15732*x^24 + 124974*x^22 - 651888*x^20 + 2294766*x^18 - 5517450*x^16 + 9065763*x^14 - 10076778*x^12 + 7410609*x^10 - 3473442*x^8 + 979290*x^6 - 151875*x^4 + 10935*x^2 - 243)
\( x^{30} - 54 x^{28} + 1233 x^{26} - 15732 x^{24} + 124974 x^{22} - 651888 x^{20} + 2294766 x^{18} + \cdots - 243 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[30, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(31245017777306374823059337284350587021473200026746355712\)
\(\medspace = 2^{30}\cdot 3^{45}\cdot 11^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(70.77\) | 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: | $2\cdot 3^{3/2}11^{4/5}\approx 70.76622450090987$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{3}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $30$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(396=2^{2}\cdot 3^{2}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{396}(383,·)$, $\chi_{396}(1,·)$, $\chi_{396}(323,·)$, $\chi_{396}(133,·)$, $\chi_{396}(71,·)$, $\chi_{396}(265,·)$, $\chi_{396}(203,·)$, $\chi_{396}(335,·)$, $\chi_{396}(251,·)$, $\chi_{396}(23,·)$, $\chi_{396}(25,·)$, $\chi_{396}(155,·)$, $\chi_{396}(361,·)$, $\chi_{396}(157,·)$, $\chi_{396}(287,·)$, $\chi_{396}(289,·)$, $\chi_{396}(229,·)$, $\chi_{396}(37,·)$, $\chi_{396}(97,·)$, $\chi_{396}(119,·)$, $\chi_{396}(169,·)$, $\chi_{396}(301,·)$, $\chi_{396}(47,·)$, $\chi_{396}(49,·)$, $\chi_{396}(179,·)$, $\chi_{396}(181,·)$, $\chi_{396}(311,·)$, $\chi_{396}(313,·)$, $\chi_{396}(59,·)$, $\chi_{396}(191,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{3}a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{3}a^{10}$, $\frac{1}{3}a^{11}$, $\frac{1}{9}a^{12}$, $\frac{1}{9}a^{13}$, $\frac{1}{9}a^{14}$, $\frac{1}{9}a^{15}$, $\frac{1}{9}a^{16}$, $\frac{1}{9}a^{17}$, $\frac{1}{27}a^{18}$, $\frac{1}{27}a^{19}$, $\frac{1}{27}a^{20}$, $\frac{1}{27}a^{21}$, $\frac{1}{27}a^{22}$, $\frac{1}{27}a^{23}$, $\frac{1}{81}a^{24}$, $\frac{1}{81}a^{25}$, $\frac{1}{16119}a^{26}-\frac{94}{16119}a^{24}-\frac{61}{5373}a^{22}-\frac{88}{5373}a^{20}-\frac{8}{597}a^{18}+\frac{94}{1791}a^{16}-\frac{25}{597}a^{14}+\frac{95}{1791}a^{12}-\frac{11}{199}a^{10}+\frac{40}{597}a^{8}-\frac{95}{597}a^{6}-\frac{51}{199}a^{4}-\frac{2}{199}a^{2}+\frac{98}{199}$, $\frac{1}{16119}a^{27}-\frac{94}{16119}a^{25}-\frac{61}{5373}a^{23}-\frac{88}{5373}a^{21}-\frac{8}{597}a^{19}+\frac{94}{1791}a^{17}-\frac{25}{597}a^{15}+\frac{95}{1791}a^{13}-\frac{11}{199}a^{11}+\frac{40}{597}a^{9}-\frac{95}{597}a^{7}-\frac{51}{199}a^{5}-\frac{2}{199}a^{3}+\frac{98}{199}a$, $\frac{1}{21\!\cdots\!71}a^{28}+\frac{154081532849}{73\!\cdots\!57}a^{26}-\frac{131010207677525}{21\!\cdots\!71}a^{24}-\frac{85349501463760}{73\!\cdots\!57}a^{22}-\frac{59640972579644}{73\!\cdots\!57}a^{20}+\frac{1786084798342}{73\!\cdots\!57}a^{18}-\frac{128066173696048}{24\!\cdots\!19}a^{16}+\frac{102265829605355}{24\!\cdots\!19}a^{14}-\frac{24624299898893}{24\!\cdots\!19}a^{12}+\frac{111557765609594}{811239940350573}a^{10}-\frac{64682639803519}{811239940350573}a^{8}-\frac{4174442717978}{811239940350573}a^{6}-\frac{107202787826711}{270413313450191}a^{4}+\frac{48845719161674}{270413313450191}a^{2}+\frac{23632150421837}{270413313450191}$, $\frac{1}{21\!\cdots\!71}a^{29}+\frac{154081532849}{73\!\cdots\!57}a^{27}-\frac{131010207677525}{21\!\cdots\!71}a^{25}-\frac{85349501463760}{73\!\cdots\!57}a^{23}-\frac{59640972579644}{73\!\cdots\!57}a^{21}+\frac{1786084798342}{73\!\cdots\!57}a^{19}-\frac{128066173696048}{24\!\cdots\!19}a^{17}+\frac{102265829605355}{24\!\cdots\!19}a^{15}-\frac{24624299898893}{24\!\cdots\!19}a^{13}+\frac{111557765609594}{811239940350573}a^{11}-\frac{64682639803519}{811239940350573}a^{9}-\frac{4174442717978}{811239940350573}a^{7}-\frac{107202787826711}{270413313450191}a^{5}+\frac{48845719161674}{270413313450191}a^{3}+\frac{23632150421837}{270413313450191}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $29$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{35294206375009}{24\!\cdots\!19}a^{28}-\frac{18\!\cdots\!97}{24\!\cdots\!19}a^{26}+\frac{38\!\cdots\!68}{21\!\cdots\!71}a^{24}-\frac{16\!\cdots\!42}{73\!\cdots\!57}a^{22}+\frac{47\!\cdots\!13}{270413313450191}a^{20}-\frac{66\!\cdots\!60}{73\!\cdots\!57}a^{18}+\frac{75\!\cdots\!80}{24\!\cdots\!19}a^{16}-\frac{19\!\cdots\!60}{270413313450191}a^{14}+\frac{27\!\cdots\!94}{24\!\cdots\!19}a^{12}-\frac{31\!\cdots\!87}{270413313450191}a^{10}+\frac{20\!\cdots\!92}{270413313450191}a^{8}-\frac{24\!\cdots\!95}{811239940350573}a^{6}+\frac{16\!\cdots\!09}{270413313450191}a^{4}-\frac{13\!\cdots\!14}{270413313450191}a^{2}+\frac{37\!\cdots\!70}{270413313450191}$, $\frac{684463118391679}{21\!\cdots\!71}a^{28}-\frac{453693710492772}{270413313450191}a^{26}+\frac{92\!\cdots\!66}{24\!\cdots\!19}a^{24}-\frac{35\!\cdots\!30}{73\!\cdots\!57}a^{22}+\frac{91\!\cdots\!48}{24\!\cdots\!19}a^{20}-\frac{15\!\cdots\!28}{811239940350573}a^{18}+\frac{17\!\cdots\!01}{270413313450191}a^{16}-\frac{41\!\cdots\!37}{270413313450191}a^{14}+\frac{57\!\cdots\!40}{24\!\cdots\!19}a^{12}-\frac{64\!\cdots\!82}{270413313450191}a^{10}+\frac{41\!\cdots\!04}{270413313450191}a^{8}-\frac{47\!\cdots\!81}{811239940350573}a^{6}+\frac{31\!\cdots\!06}{270413313450191}a^{4}-\frac{26\!\cdots\!19}{270413313450191}a^{2}+\frac{64\!\cdots\!65}{270413313450191}$, $\frac{734173934220223}{21\!\cdots\!71}a^{28}-\frac{43\!\cdots\!67}{24\!\cdots\!19}a^{26}+\frac{29\!\cdots\!18}{73\!\cdots\!57}a^{24}-\frac{37\!\cdots\!44}{73\!\cdots\!57}a^{22}+\frac{96\!\cdots\!95}{24\!\cdots\!19}a^{20}-\frac{14\!\cdots\!36}{73\!\cdots\!57}a^{18}+\frac{16\!\cdots\!71}{24\!\cdots\!19}a^{16}-\frac{12\!\cdots\!75}{811239940350573}a^{14}+\frac{64\!\cdots\!58}{270413313450191}a^{12}-\frac{19\!\cdots\!25}{811239940350573}a^{10}+\frac{41\!\cdots\!40}{270413313450191}a^{8}-\frac{46\!\cdots\!52}{811239940350573}a^{6}+\frac{30\!\cdots\!13}{270413313450191}a^{4}-\frac{25\!\cdots\!63}{270413313450191}a^{2}+\frac{59\!\cdots\!69}{270413313450191}$, $\frac{125394564325046}{24\!\cdots\!19}a^{28}-\frac{22\!\cdots\!62}{811239940350573}a^{26}+\frac{15\!\cdots\!44}{24\!\cdots\!19}a^{24}-\frac{19\!\cdots\!64}{24\!\cdots\!19}a^{22}+\frac{50\!\cdots\!10}{811239940350573}a^{20}-\frac{23\!\cdots\!16}{73\!\cdots\!57}a^{18}+\frac{88\!\cdots\!80}{811239940350573}a^{16}-\frac{68\!\cdots\!15}{270413313450191}a^{14}+\frac{31\!\cdots\!16}{811239940350573}a^{12}-\frac{10\!\cdots\!78}{270413313450191}a^{10}+\frac{70\!\cdots\!16}{270413313450191}a^{8}-\frac{26\!\cdots\!54}{270413313450191}a^{6}+\frac{53\!\cdots\!86}{270413313450191}a^{4}-\frac{45\!\cdots\!64}{270413313450191}a^{2}+\frac{11\!\cdots\!63}{270413313450191}$, $\frac{11\!\cdots\!27}{21\!\cdots\!71}a^{28}-\frac{795012623105333}{270413313450191}a^{26}+\frac{16\!\cdots\!25}{24\!\cdots\!19}a^{24}-\frac{61\!\cdots\!61}{73\!\cdots\!57}a^{22}+\frac{48\!\cdots\!70}{73\!\cdots\!57}a^{20}-\frac{82\!\cdots\!36}{24\!\cdots\!19}a^{18}+\frac{28\!\cdots\!60}{24\!\cdots\!19}a^{16}-\frac{65\!\cdots\!00}{24\!\cdots\!19}a^{14}+\frac{33\!\cdots\!73}{811239940350573}a^{12}-\frac{11\!\cdots\!64}{270413313450191}a^{10}+\frac{22\!\cdots\!09}{811239940350573}a^{8}-\frac{28\!\cdots\!57}{270413313450191}a^{6}+\frac{56\!\cdots\!54}{270413313450191}a^{4}-\frac{45\!\cdots\!07}{270413313450191}a^{2}+\frac{10\!\cdots\!63}{270413313450191}$, $\frac{227948583636529}{24\!\cdots\!19}a^{28}-\frac{40\!\cdots\!68}{811239940350573}a^{26}+\frac{92\!\cdots\!80}{811239940350573}a^{24}-\frac{35\!\cdots\!81}{24\!\cdots\!19}a^{22}+\frac{82\!\cdots\!08}{73\!\cdots\!57}a^{20}-\frac{14\!\cdots\!29}{24\!\cdots\!19}a^{18}+\frac{15\!\cdots\!64}{811239940350573}a^{16}-\frac{11\!\cdots\!60}{24\!\cdots\!19}a^{14}+\frac{57\!\cdots\!59}{811239940350573}a^{12}-\frac{19\!\cdots\!82}{270413313450191}a^{10}+\frac{38\!\cdots\!04}{811239940350573}a^{8}-\frac{48\!\cdots\!83}{270413313450191}a^{6}+\frac{96\!\cdots\!71}{270413313450191}a^{4}-\frac{81\!\cdots\!66}{270413313450191}a^{2}+\frac{18\!\cdots\!48}{270413313450191}$, $\frac{171352314677216}{73\!\cdots\!57}a^{28}-\frac{341318912612561}{270413313450191}a^{26}+\frac{69\!\cdots\!59}{24\!\cdots\!19}a^{24}-\frac{88\!\cdots\!77}{24\!\cdots\!19}a^{22}+\frac{20\!\cdots\!26}{73\!\cdots\!57}a^{20}-\frac{35\!\cdots\!52}{24\!\cdots\!19}a^{18}+\frac{12\!\cdots\!51}{24\!\cdots\!19}a^{16}-\frac{28\!\cdots\!67}{24\!\cdots\!19}a^{14}+\frac{44\!\cdots\!79}{24\!\cdots\!19}a^{12}-\frac{51\!\cdots\!82}{270413313450191}a^{10}+\frac{99\!\cdots\!97}{811239940350573}a^{8}-\frac{37\!\cdots\!90}{811239940350573}a^{6}+\frac{24\!\cdots\!48}{270413313450191}a^{4}-\frac{19\!\cdots\!88}{270413313450191}a^{2}+\frac{38\!\cdots\!98}{270413313450191}$, $\frac{221401650240770}{73\!\cdots\!57}a^{28}-\frac{39\!\cdots\!56}{24\!\cdots\!19}a^{26}+\frac{26\!\cdots\!75}{73\!\cdots\!57}a^{24}-\frac{12\!\cdots\!25}{270413313450191}a^{22}+\frac{26\!\cdots\!05}{73\!\cdots\!57}a^{20}-\frac{14\!\cdots\!16}{811239940350573}a^{18}+\frac{16\!\cdots\!39}{270413313450191}a^{16}-\frac{33\!\cdots\!60}{24\!\cdots\!19}a^{14}+\frac{57\!\cdots\!39}{270413313450191}a^{12}-\frac{17\!\cdots\!67}{811239940350573}a^{10}+\frac{10\!\cdots\!59}{811239940350573}a^{8}-\frac{41\!\cdots\!43}{811239940350573}a^{6}+\frac{27\!\cdots\!45}{270413313450191}a^{4}-\frac{24\!\cdots\!20}{270413313450191}a^{2}+\frac{70\!\cdots\!69}{270413313450191}$, $\frac{39\!\cdots\!98}{21\!\cdots\!71}a^{28}-\frac{23\!\cdots\!57}{24\!\cdots\!19}a^{26}+\frac{15\!\cdots\!70}{73\!\cdots\!57}a^{24}-\frac{20\!\cdots\!79}{73\!\cdots\!57}a^{22}+\frac{15\!\cdots\!83}{73\!\cdots\!57}a^{20}-\frac{88\!\cdots\!71}{811239940350573}a^{18}+\frac{91\!\cdots\!03}{24\!\cdots\!19}a^{16}-\frac{21\!\cdots\!20}{24\!\cdots\!19}a^{14}+\frac{36\!\cdots\!83}{270413313450191}a^{12}-\frac{11\!\cdots\!05}{811239940350573}a^{10}+\frac{71\!\cdots\!72}{811239940350573}a^{8}-\frac{27\!\cdots\!63}{811239940350573}a^{6}+\frac{18\!\cdots\!70}{270413313450191}a^{4}-\frac{15\!\cdots\!93}{270413313450191}a^{2}+\frac{36\!\cdots\!80}{270413313450191}$, $\frac{18\!\cdots\!16}{21\!\cdots\!71}a^{28}-\frac{34\!\cdots\!52}{73\!\cdots\!57}a^{26}+\frac{23\!\cdots\!84}{21\!\cdots\!71}a^{24}-\frac{97\!\cdots\!74}{73\!\cdots\!57}a^{22}+\frac{76\!\cdots\!20}{73\!\cdots\!57}a^{20}-\frac{39\!\cdots\!75}{73\!\cdots\!57}a^{18}+\frac{44\!\cdots\!03}{24\!\cdots\!19}a^{16}-\frac{10\!\cdots\!39}{24\!\cdots\!19}a^{14}+\frac{53\!\cdots\!79}{811239940350573}a^{12}-\frac{54\!\cdots\!92}{811239940350573}a^{10}+\frac{35\!\cdots\!29}{811239940350573}a^{8}-\frac{13\!\cdots\!84}{811239940350573}a^{6}+\frac{89\!\cdots\!12}{270413313450191}a^{4}-\frac{76\!\cdots\!24}{270413313450191}a^{2}+\frac{18\!\cdots\!37}{270413313450191}$, $\frac{23\!\cdots\!41}{21\!\cdots\!71}a^{28}-\frac{46\!\cdots\!61}{811239940350573}a^{26}+\frac{34\!\cdots\!41}{270413313450191}a^{24}-\frac{11\!\cdots\!53}{73\!\cdots\!57}a^{22}+\frac{93\!\cdots\!60}{73\!\cdots\!57}a^{20}-\frac{47\!\cdots\!24}{73\!\cdots\!57}a^{18}+\frac{54\!\cdots\!00}{24\!\cdots\!19}a^{16}-\frac{12\!\cdots\!35}{24\!\cdots\!19}a^{14}+\frac{65\!\cdots\!89}{811239940350573}a^{12}-\frac{22\!\cdots\!42}{270413313450191}a^{10}+\frac{43\!\cdots\!57}{811239940350573}a^{8}-\frac{55\!\cdots\!11}{270413313450191}a^{6}+\frac{10\!\cdots\!40}{270413313450191}a^{4}-\frac{91\!\cdots\!71}{270413313450191}a^{2}+\frac{21\!\cdots\!26}{270413313450191}$, $\frac{47\!\cdots\!15}{21\!\cdots\!71}a^{28}-\frac{25\!\cdots\!97}{21\!\cdots\!71}a^{26}+\frac{57\!\cdots\!20}{21\!\cdots\!71}a^{24}-\frac{24\!\cdots\!55}{73\!\cdots\!57}a^{22}+\frac{18\!\cdots\!79}{73\!\cdots\!57}a^{20}-\frac{96\!\cdots\!84}{73\!\cdots\!57}a^{18}+\frac{10\!\cdots\!97}{24\!\cdots\!19}a^{16}-\frac{25\!\cdots\!02}{24\!\cdots\!19}a^{14}+\frac{39\!\cdots\!47}{24\!\cdots\!19}a^{12}-\frac{13\!\cdots\!11}{811239940350573}a^{10}+\frac{86\!\cdots\!44}{811239940350573}a^{8}-\frac{10\!\cdots\!38}{270413313450191}a^{6}+\frac{21\!\cdots\!85}{270413313450191}a^{4}-\frac{18\!\cdots\!72}{270413313450191}a^{2}+\frac{44\!\cdots\!72}{270413313450191}$, $\frac{64218125753840}{811239940350573}a^{28}-\frac{10\!\cdots\!07}{24\!\cdots\!19}a^{26}+\frac{21\!\cdots\!92}{21\!\cdots\!71}a^{24}-\frac{88\!\cdots\!01}{73\!\cdots\!57}a^{22}+\frac{69\!\cdots\!57}{73\!\cdots\!57}a^{20}-\frac{35\!\cdots\!27}{73\!\cdots\!57}a^{18}+\frac{40\!\cdots\!12}{24\!\cdots\!19}a^{16}-\frac{93\!\cdots\!20}{24\!\cdots\!19}a^{14}+\frac{14\!\cdots\!83}{24\!\cdots\!19}a^{12}-\frac{16\!\cdots\!95}{270413313450191}a^{10}+\frac{31\!\cdots\!28}{811239940350573}a^{8}-\frac{12\!\cdots\!54}{811239940350573}a^{6}+\frac{80\!\cdots\!62}{270413313450191}a^{4}-\frac{67\!\cdots\!52}{270413313450191}a^{2}+\frac{15\!\cdots\!78}{270413313450191}$, $\frac{978268324136770}{73\!\cdots\!57}a^{28}-\frac{17\!\cdots\!09}{24\!\cdots\!19}a^{26}+\frac{35\!\cdots\!41}{21\!\cdots\!71}a^{24}-\frac{15\!\cdots\!77}{73\!\cdots\!57}a^{22}+\frac{11\!\cdots\!50}{73\!\cdots\!57}a^{20}-\frac{60\!\cdots\!56}{73\!\cdots\!57}a^{18}+\frac{22\!\cdots\!40}{811239940350573}a^{16}-\frac{15\!\cdots\!98}{24\!\cdots\!19}a^{14}+\frac{81\!\cdots\!25}{811239940350573}a^{12}-\frac{83\!\cdots\!24}{811239940350573}a^{10}+\frac{53\!\cdots\!17}{811239940350573}a^{8}-\frac{20\!\cdots\!99}{811239940350573}a^{6}+\frac{13\!\cdots\!82}{270413313450191}a^{4}-\frac{11\!\cdots\!03}{270413313450191}a^{2}+\frac{26\!\cdots\!76}{270413313450191}$, $\frac{404515994257477}{24\!\cdots\!19}a^{29}-\frac{19\!\cdots\!02}{21\!\cdots\!71}a^{27}+\frac{44\!\cdots\!17}{21\!\cdots\!71}a^{25}-\frac{18\!\cdots\!46}{73\!\cdots\!57}a^{23}+\frac{14\!\cdots\!20}{73\!\cdots\!57}a^{21}-\frac{74\!\cdots\!62}{73\!\cdots\!57}a^{19}+\frac{85\!\cdots\!58}{24\!\cdots\!19}a^{17}-\frac{65\!\cdots\!70}{811239940350573}a^{15}+\frac{10\!\cdots\!00}{811239940350573}a^{13}-\frac{10\!\cdots\!41}{811239940350573}a^{11}+\frac{67\!\cdots\!37}{811239940350573}a^{9}-\frac{26\!\cdots\!30}{811239940350573}a^{7}+\frac{17\!\cdots\!04}{270413313450191}a^{5}-\frac{15\!\cdots\!16}{270413313450191}a^{3}+\frac{40\!\cdots\!33}{270413313450191}a+1$, $\frac{28\!\cdots\!97}{21\!\cdots\!71}a^{29}-\frac{17\!\cdots\!98}{24\!\cdots\!19}a^{27}+\frac{34\!\cdots\!12}{21\!\cdots\!71}a^{25}-\frac{14\!\cdots\!08}{73\!\cdots\!57}a^{23}+\frac{42\!\cdots\!10}{270413313450191}a^{21}-\frac{58\!\cdots\!64}{73\!\cdots\!57}a^{19}+\frac{66\!\cdots\!00}{24\!\cdots\!19}a^{17}-\frac{51\!\cdots\!11}{811239940350573}a^{15}+\frac{79\!\cdots\!68}{811239940350573}a^{13}-\frac{81\!\cdots\!66}{811239940350573}a^{11}+\frac{17\!\cdots\!52}{270413313450191}a^{9}-\frac{19\!\cdots\!90}{811239940350573}a^{7}+\frac{13\!\cdots\!14}{270413313450191}a^{5}-\frac{11\!\cdots\!60}{270413313450191}a^{3}+\frac{27\!\cdots\!67}{270413313450191}a+1$, $\frac{775807959404896}{21\!\cdots\!71}a^{29}-\frac{41\!\cdots\!20}{21\!\cdots\!71}a^{27}+\frac{94\!\cdots\!05}{21\!\cdots\!71}a^{25}-\frac{39\!\cdots\!38}{73\!\cdots\!57}a^{23}+\frac{31\!\cdots\!50}{73\!\cdots\!57}a^{21}-\frac{16\!\cdots\!98}{73\!\cdots\!57}a^{19}+\frac{61\!\cdots\!86}{811239940350573}a^{17}-\frac{14\!\cdots\!59}{811239940350573}a^{15}+\frac{75\!\cdots\!44}{270413313450191}a^{13}-\frac{23\!\cdots\!75}{811239940350573}a^{11}+\frac{15\!\cdots\!81}{811239940350573}a^{9}-\frac{61\!\cdots\!40}{811239940350573}a^{7}+\frac{44\!\cdots\!90}{270413313450191}a^{5}-\frac{42\!\cdots\!56}{270413313450191}a^{3}+\frac{12\!\cdots\!66}{270413313450191}a+1$, $\frac{404515994257477}{24\!\cdots\!19}a^{29}-\frac{18\!\cdots\!37}{21\!\cdots\!71}a^{28}-\frac{19\!\cdots\!02}{21\!\cdots\!71}a^{27}+\frac{11\!\cdots\!53}{24\!\cdots\!19}a^{26}+\frac{44\!\cdots\!17}{21\!\cdots\!71}a^{25}-\frac{75\!\cdots\!50}{73\!\cdots\!57}a^{24}-\frac{18\!\cdots\!46}{73\!\cdots\!57}a^{23}+\frac{95\!\cdots\!36}{73\!\cdots\!57}a^{22}+\frac{14\!\cdots\!20}{73\!\cdots\!57}a^{21}-\frac{24\!\cdots\!25}{24\!\cdots\!19}a^{20}-\frac{74\!\cdots\!62}{73\!\cdots\!57}a^{19}+\frac{12\!\cdots\!84}{24\!\cdots\!19}a^{18}+\frac{85\!\cdots\!58}{24\!\cdots\!19}a^{17}-\frac{43\!\cdots\!11}{24\!\cdots\!19}a^{16}-\frac{65\!\cdots\!70}{811239940350573}a^{15}+\frac{33\!\cdots\!20}{811239940350573}a^{14}+\frac{10\!\cdots\!00}{811239940350573}a^{13}-\frac{51\!\cdots\!90}{811239940350573}a^{12}-\frac{10\!\cdots\!41}{811239940350573}a^{11}+\frac{52\!\cdots\!59}{811239940350573}a^{10}+\frac{67\!\cdots\!37}{811239940350573}a^{9}-\frac{11\!\cdots\!56}{270413313450191}a^{8}-\frac{26\!\cdots\!30}{811239940350573}a^{7}+\frac{12\!\cdots\!14}{811239940350573}a^{6}+\frac{17\!\cdots\!04}{270413313450191}a^{5}-\frac{83\!\cdots\!99}{270413313450191}a^{4}-\frac{15\!\cdots\!16}{270413313450191}a^{3}+\frac{70\!\cdots\!27}{270413313450191}a^{2}+\frac{40\!\cdots\!33}{270413313450191}a-\frac{17\!\cdots\!32}{270413313450191}$, $\frac{775807959404896}{21\!\cdots\!71}a^{29}-\frac{18\!\cdots\!37}{21\!\cdots\!71}a^{28}-\frac{41\!\cdots\!20}{21\!\cdots\!71}a^{27}+\frac{11\!\cdots\!53}{24\!\cdots\!19}a^{26}+\frac{94\!\cdots\!05}{21\!\cdots\!71}a^{25}-\frac{75\!\cdots\!50}{73\!\cdots\!57}a^{24}-\frac{39\!\cdots\!38}{73\!\cdots\!57}a^{23}+\frac{95\!\cdots\!36}{73\!\cdots\!57}a^{22}+\frac{31\!\cdots\!50}{73\!\cdots\!57}a^{21}-\frac{24\!\cdots\!25}{24\!\cdots\!19}a^{20}-\frac{16\!\cdots\!98}{73\!\cdots\!57}a^{19}+\frac{12\!\cdots\!84}{24\!\cdots\!19}a^{18}+\frac{61\!\cdots\!86}{811239940350573}a^{17}-\frac{43\!\cdots\!11}{24\!\cdots\!19}a^{16}-\frac{14\!\cdots\!59}{811239940350573}a^{15}+\frac{33\!\cdots\!20}{811239940350573}a^{14}+\frac{75\!\cdots\!44}{270413313450191}a^{13}-\frac{51\!\cdots\!90}{811239940350573}a^{12}-\frac{23\!\cdots\!75}{811239940350573}a^{11}+\frac{52\!\cdots\!59}{811239940350573}a^{10}+\frac{15\!\cdots\!81}{811239940350573}a^{9}-\frac{11\!\cdots\!56}{270413313450191}a^{8}-\frac{61\!\cdots\!40}{811239940350573}a^{7}+\frac{12\!\cdots\!14}{811239940350573}a^{6}+\frac{44\!\cdots\!90}{270413313450191}a^{5}-\frac{83\!\cdots\!99}{270413313450191}a^{4}-\frac{42\!\cdots\!56}{270413313450191}a^{3}+\frac{70\!\cdots\!27}{270413313450191}a^{2}+\frac{12\!\cdots\!66}{270413313450191}a-\frac{17\!\cdots\!32}{270413313450191}$, $\frac{28\!\cdots\!97}{21\!\cdots\!71}a^{29}-\frac{18\!\cdots\!37}{21\!\cdots\!71}a^{28}-\frac{17\!\cdots\!98}{24\!\cdots\!19}a^{27}+\frac{11\!\cdots\!53}{24\!\cdots\!19}a^{26}+\frac{34\!\cdots\!12}{21\!\cdots\!71}a^{25}-\frac{75\!\cdots\!50}{73\!\cdots\!57}a^{24}-\frac{14\!\cdots\!08}{73\!\cdots\!57}a^{23}+\frac{95\!\cdots\!36}{73\!\cdots\!57}a^{22}+\frac{42\!\cdots\!10}{270413313450191}a^{21}-\frac{24\!\cdots\!25}{24\!\cdots\!19}a^{20}-\frac{58\!\cdots\!64}{73\!\cdots\!57}a^{19}+\frac{12\!\cdots\!84}{24\!\cdots\!19}a^{18}+\frac{66\!\cdots\!00}{24\!\cdots\!19}a^{17}-\frac{43\!\cdots\!11}{24\!\cdots\!19}a^{16}-\frac{51\!\cdots\!11}{811239940350573}a^{15}+\frac{33\!\cdots\!20}{811239940350573}a^{14}+\frac{79\!\cdots\!68}{811239940350573}a^{13}-\frac{51\!\cdots\!90}{811239940350573}a^{12}-\frac{81\!\cdots\!66}{811239940350573}a^{11}+\frac{52\!\cdots\!59}{811239940350573}a^{10}+\frac{17\!\cdots\!52}{270413313450191}a^{9}-\frac{11\!\cdots\!56}{270413313450191}a^{8}-\frac{19\!\cdots\!90}{811239940350573}a^{7}+\frac{12\!\cdots\!14}{811239940350573}a^{6}+\frac{13\!\cdots\!14}{270413313450191}a^{5}-\frac{83\!\cdots\!99}{270413313450191}a^{4}-\frac{11\!\cdots\!60}{270413313450191}a^{3}+\frac{70\!\cdots\!27}{270413313450191}a^{2}+\frac{27\!\cdots\!67}{270413313450191}a-\frac{17\!\cdots\!32}{270413313450191}$, $\frac{404515994257477}{24\!\cdots\!19}a^{29}-\frac{17\!\cdots\!83}{21\!\cdots\!71}a^{28}-\frac{19\!\cdots\!02}{21\!\cdots\!71}a^{27}+\frac{10\!\cdots\!12}{24\!\cdots\!19}a^{26}+\frac{44\!\cdots\!17}{21\!\cdots\!71}a^{25}-\frac{21\!\cdots\!16}{21\!\cdots\!71}a^{24}-\frac{18\!\cdots\!46}{73\!\cdots\!57}a^{23}+\frac{88\!\cdots\!16}{73\!\cdots\!57}a^{22}+\frac{14\!\cdots\!20}{73\!\cdots\!57}a^{21}-\frac{76\!\cdots\!20}{811239940350573}a^{20}-\frac{74\!\cdots\!62}{73\!\cdots\!57}a^{19}+\frac{35\!\cdots\!48}{73\!\cdots\!57}a^{18}+\frac{85\!\cdots\!58}{24\!\cdots\!19}a^{17}-\frac{40\!\cdots\!60}{24\!\cdots\!19}a^{16}-\frac{65\!\cdots\!70}{811239940350573}a^{15}+\frac{30\!\cdots\!66}{811239940350573}a^{14}+\frac{10\!\cdots\!00}{811239940350573}a^{13}-\frac{15\!\cdots\!84}{270413313450191}a^{12}-\frac{10\!\cdots\!41}{811239940350573}a^{11}+\frac{48\!\cdots\!32}{811239940350573}a^{10}+\frac{67\!\cdots\!37}{811239940350573}a^{9}-\frac{10\!\cdots\!36}{270413313450191}a^{8}-\frac{26\!\cdots\!30}{811239940350573}a^{7}+\frac{11\!\cdots\!28}{811239940350573}a^{6}+\frac{17\!\cdots\!04}{270413313450191}a^{5}-\frac{77\!\cdots\!28}{270413313450191}a^{4}-\frac{15\!\cdots\!16}{270413313450191}a^{3}+\frac{65\!\cdots\!96}{270413313450191}a^{2}+\frac{40\!\cdots\!33}{270413313450191}a-\frac{15\!\cdots\!13}{270413313450191}$, $\frac{404515994257477}{24\!\cdots\!19}a^{29}+\frac{125394564325046}{24\!\cdots\!19}a^{28}-\frac{19\!\cdots\!02}{21\!\cdots\!71}a^{27}-\frac{22\!\cdots\!62}{811239940350573}a^{26}+\frac{44\!\cdots\!17}{21\!\cdots\!71}a^{25}+\frac{15\!\cdots\!44}{24\!\cdots\!19}a^{24}-\frac{18\!\cdots\!46}{73\!\cdots\!57}a^{23}-\frac{19\!\cdots\!64}{24\!\cdots\!19}a^{22}+\frac{14\!\cdots\!20}{73\!\cdots\!57}a^{21}+\frac{50\!\cdots\!10}{811239940350573}a^{20}-\frac{74\!\cdots\!62}{73\!\cdots\!57}a^{19}-\frac{23\!\cdots\!16}{73\!\cdots\!57}a^{18}+\frac{85\!\cdots\!58}{24\!\cdots\!19}a^{17}+\frac{88\!\cdots\!80}{811239940350573}a^{16}-\frac{65\!\cdots\!70}{811239940350573}a^{15}-\frac{68\!\cdots\!15}{270413313450191}a^{14}+\frac{10\!\cdots\!00}{811239940350573}a^{13}+\frac{31\!\cdots\!16}{811239940350573}a^{12}-\frac{10\!\cdots\!41}{811239940350573}a^{11}-\frac{10\!\cdots\!78}{270413313450191}a^{10}+\frac{67\!\cdots\!37}{811239940350573}a^{9}+\frac{70\!\cdots\!16}{270413313450191}a^{8}-\frac{26\!\cdots\!30}{811239940350573}a^{7}-\frac{26\!\cdots\!54}{270413313450191}a^{6}+\frac{17\!\cdots\!04}{270413313450191}a^{5}+\frac{53\!\cdots\!86}{270413313450191}a^{4}-\frac{15\!\cdots\!16}{270413313450191}a^{3}-\frac{45\!\cdots\!64}{270413313450191}a^{2}+\frac{40\!\cdots\!33}{270413313450191}a+\frac{11\!\cdots\!63}{270413313450191}$, $\frac{404515994257477}{24\!\cdots\!19}a^{29}-\frac{35294206375009}{24\!\cdots\!19}a^{28}-\frac{19\!\cdots\!02}{21\!\cdots\!71}a^{27}+\frac{18\!\cdots\!97}{24\!\cdots\!19}a^{26}+\frac{44\!\cdots\!17}{21\!\cdots\!71}a^{25}-\frac{38\!\cdots\!68}{21\!\cdots\!71}a^{24}-\frac{18\!\cdots\!46}{73\!\cdots\!57}a^{23}+\frac{16\!\cdots\!42}{73\!\cdots\!57}a^{22}+\frac{14\!\cdots\!20}{73\!\cdots\!57}a^{21}-\frac{47\!\cdots\!13}{270413313450191}a^{20}-\frac{74\!\cdots\!62}{73\!\cdots\!57}a^{19}+\frac{66\!\cdots\!60}{73\!\cdots\!57}a^{18}+\frac{85\!\cdots\!58}{24\!\cdots\!19}a^{17}-\frac{75\!\cdots\!80}{24\!\cdots\!19}a^{16}-\frac{65\!\cdots\!70}{811239940350573}a^{15}+\frac{19\!\cdots\!60}{270413313450191}a^{14}+\frac{10\!\cdots\!00}{811239940350573}a^{13}-\frac{27\!\cdots\!94}{24\!\cdots\!19}a^{12}-\frac{10\!\cdots\!41}{811239940350573}a^{11}+\frac{31\!\cdots\!87}{270413313450191}a^{10}+\frac{67\!\cdots\!37}{811239940350573}a^{9}-\frac{20\!\cdots\!92}{270413313450191}a^{8}-\frac{26\!\cdots\!30}{811239940350573}a^{7}+\frac{24\!\cdots\!95}{811239940350573}a^{6}+\frac{17\!\cdots\!04}{270413313450191}a^{5}-\frac{16\!\cdots\!09}{270413313450191}a^{4}-\frac{15\!\cdots\!16}{270413313450191}a^{3}+\frac{13\!\cdots\!14}{270413313450191}a^{2}+\frac{40\!\cdots\!33}{270413313450191}a-\frac{37\!\cdots\!70}{270413313450191}$, $\frac{28\!\cdots\!97}{21\!\cdots\!71}a^{29}+\frac{17\!\cdots\!83}{21\!\cdots\!71}a^{28}-\frac{17\!\cdots\!98}{24\!\cdots\!19}a^{27}-\frac{10\!\cdots\!12}{24\!\cdots\!19}a^{26}+\frac{34\!\cdots\!12}{21\!\cdots\!71}a^{25}+\frac{21\!\cdots\!16}{21\!\cdots\!71}a^{24}-\frac{14\!\cdots\!08}{73\!\cdots\!57}a^{23}-\frac{88\!\cdots\!16}{73\!\cdots\!57}a^{22}+\frac{42\!\cdots\!10}{270413313450191}a^{21}+\frac{76\!\cdots\!20}{811239940350573}a^{20}-\frac{58\!\cdots\!64}{73\!\cdots\!57}a^{19}-\frac{35\!\cdots\!48}{73\!\cdots\!57}a^{18}+\frac{66\!\cdots\!00}{24\!\cdots\!19}a^{17}+\frac{40\!\cdots\!60}{24\!\cdots\!19}a^{16}-\frac{51\!\cdots\!11}{811239940350573}a^{15}-\frac{30\!\cdots\!66}{811239940350573}a^{14}+\frac{79\!\cdots\!68}{811239940350573}a^{13}+\frac{15\!\cdots\!84}{270413313450191}a^{12}-\frac{81\!\cdots\!66}{811239940350573}a^{11}-\frac{48\!\cdots\!32}{811239940350573}a^{10}+\frac{17\!\cdots\!52}{270413313450191}a^{9}+\frac{10\!\cdots\!36}{270413313450191}a^{8}-\frac{19\!\cdots\!90}{811239940350573}a^{7}-\frac{11\!\cdots\!28}{811239940350573}a^{6}+\frac{13\!\cdots\!14}{270413313450191}a^{5}+\frac{77\!\cdots\!28}{270413313450191}a^{4}-\frac{11\!\cdots\!60}{270413313450191}a^{3}-\frac{65\!\cdots\!96}{270413313450191}a^{2}+\frac{27\!\cdots\!67}{270413313450191}a+\frac{15\!\cdots\!13}{270413313450191}$, $\frac{28\!\cdots\!97}{21\!\cdots\!71}a^{29}+\frac{125394564325046}{24\!\cdots\!19}a^{28}-\frac{17\!\cdots\!98}{24\!\cdots\!19}a^{27}-\frac{22\!\cdots\!62}{811239940350573}a^{26}+\frac{34\!\cdots\!12}{21\!\cdots\!71}a^{25}+\frac{15\!\cdots\!44}{24\!\cdots\!19}a^{24}-\frac{14\!\cdots\!08}{73\!\cdots\!57}a^{23}-\frac{19\!\cdots\!64}{24\!\cdots\!19}a^{22}+\frac{42\!\cdots\!10}{270413313450191}a^{21}+\frac{50\!\cdots\!10}{811239940350573}a^{20}-\frac{58\!\cdots\!64}{73\!\cdots\!57}a^{19}-\frac{23\!\cdots\!16}{73\!\cdots\!57}a^{18}+\frac{66\!\cdots\!00}{24\!\cdots\!19}a^{17}+\frac{88\!\cdots\!80}{811239940350573}a^{16}-\frac{51\!\cdots\!11}{811239940350573}a^{15}-\frac{68\!\cdots\!15}{270413313450191}a^{14}+\frac{79\!\cdots\!68}{811239940350573}a^{13}+\frac{31\!\cdots\!16}{811239940350573}a^{12}-\frac{81\!\cdots\!66}{811239940350573}a^{11}-\frac{10\!\cdots\!78}{270413313450191}a^{10}+\frac{17\!\cdots\!52}{270413313450191}a^{9}+\frac{70\!\cdots\!16}{270413313450191}a^{8}-\frac{19\!\cdots\!90}{811239940350573}a^{7}-\frac{26\!\cdots\!54}{270413313450191}a^{6}+\frac{13\!\cdots\!14}{270413313450191}a^{5}+\frac{53\!\cdots\!86}{270413313450191}a^{4}-\frac{11\!\cdots\!60}{270413313450191}a^{3}-\frac{45\!\cdots\!64}{270413313450191}a^{2}+\frac{27\!\cdots\!67}{270413313450191}a+\frac{11\!\cdots\!63}{270413313450191}$, $\frac{28\!\cdots\!97}{21\!\cdots\!71}a^{29}-\frac{684463118391679}{21\!\cdots\!71}a^{28}-\frac{17\!\cdots\!98}{24\!\cdots\!19}a^{27}+\frac{453693710492772}{270413313450191}a^{26}+\frac{34\!\cdots\!12}{21\!\cdots\!71}a^{25}-\frac{92\!\cdots\!66}{24\!\cdots\!19}a^{24}-\frac{14\!\cdots\!08}{73\!\cdots\!57}a^{23}+\frac{35\!\cdots\!30}{73\!\cdots\!57}a^{22}+\frac{42\!\cdots\!10}{270413313450191}a^{21}-\frac{91\!\cdots\!48}{24\!\cdots\!19}a^{20}-\frac{58\!\cdots\!64}{73\!\cdots\!57}a^{19}+\frac{15\!\cdots\!28}{811239940350573}a^{18}+\frac{66\!\cdots\!00}{24\!\cdots\!19}a^{17}-\frac{17\!\cdots\!01}{270413313450191}a^{16}-\frac{51\!\cdots\!11}{811239940350573}a^{15}+\frac{41\!\cdots\!37}{270413313450191}a^{14}+\frac{79\!\cdots\!68}{811239940350573}a^{13}-\frac{57\!\cdots\!40}{24\!\cdots\!19}a^{12}-\frac{81\!\cdots\!66}{811239940350573}a^{11}+\frac{64\!\cdots\!82}{270413313450191}a^{10}+\frac{17\!\cdots\!52}{270413313450191}a^{9}-\frac{41\!\cdots\!04}{270413313450191}a^{8}-\frac{19\!\cdots\!90}{811239940350573}a^{7}+\frac{47\!\cdots\!81}{811239940350573}a^{6}+\frac{13\!\cdots\!14}{270413313450191}a^{5}-\frac{31\!\cdots\!06}{270413313450191}a^{4}-\frac{11\!\cdots\!60}{270413313450191}a^{3}+\frac{26\!\cdots\!19}{270413313450191}a^{2}+\frac{27\!\cdots\!67}{270413313450191}a-\frac{64\!\cdots\!65}{270413313450191}$, $\frac{775807959404896}{21\!\cdots\!71}a^{29}-\frac{684463118391679}{21\!\cdots\!71}a^{28}-\frac{41\!\cdots\!20}{21\!\cdots\!71}a^{27}+\frac{453693710492772}{270413313450191}a^{26}+\frac{94\!\cdots\!05}{21\!\cdots\!71}a^{25}-\frac{92\!\cdots\!66}{24\!\cdots\!19}a^{24}-\frac{39\!\cdots\!38}{73\!\cdots\!57}a^{23}+\frac{35\!\cdots\!30}{73\!\cdots\!57}a^{22}+\frac{31\!\cdots\!50}{73\!\cdots\!57}a^{21}-\frac{91\!\cdots\!48}{24\!\cdots\!19}a^{20}-\frac{16\!\cdots\!98}{73\!\cdots\!57}a^{19}+\frac{15\!\cdots\!28}{811239940350573}a^{18}+\frac{61\!\cdots\!86}{811239940350573}a^{17}-\frac{17\!\cdots\!01}{270413313450191}a^{16}-\frac{14\!\cdots\!59}{811239940350573}a^{15}+\frac{41\!\cdots\!37}{270413313450191}a^{14}+\frac{75\!\cdots\!44}{270413313450191}a^{13}-\frac{57\!\cdots\!40}{24\!\cdots\!19}a^{12}-\frac{23\!\cdots\!75}{811239940350573}a^{11}+\frac{64\!\cdots\!82}{270413313450191}a^{10}+\frac{15\!\cdots\!81}{811239940350573}a^{9}-\frac{41\!\cdots\!04}{270413313450191}a^{8}-\frac{61\!\cdots\!40}{811239940350573}a^{7}+\frac{47\!\cdots\!81}{811239940350573}a^{6}+\frac{44\!\cdots\!90}{270413313450191}a^{5}-\frac{31\!\cdots\!06}{270413313450191}a^{4}-\frac{42\!\cdots\!56}{270413313450191}a^{3}+\frac{26\!\cdots\!19}{270413313450191}a^{2}+\frac{12\!\cdots\!66}{270413313450191}a-\frac{64\!\cdots\!65}{270413313450191}$, $\frac{775807959404896}{21\!\cdots\!71}a^{29}+\frac{35294206375009}{24\!\cdots\!19}a^{28}-\frac{41\!\cdots\!20}{21\!\cdots\!71}a^{27}-\frac{18\!\cdots\!97}{24\!\cdots\!19}a^{26}+\frac{94\!\cdots\!05}{21\!\cdots\!71}a^{25}+\frac{38\!\cdots\!68}{21\!\cdots\!71}a^{24}-\frac{39\!\cdots\!38}{73\!\cdots\!57}a^{23}-\frac{16\!\cdots\!42}{73\!\cdots\!57}a^{22}+\frac{31\!\cdots\!50}{73\!\cdots\!57}a^{21}+\frac{47\!\cdots\!13}{270413313450191}a^{20}-\frac{16\!\cdots\!98}{73\!\cdots\!57}a^{19}-\frac{66\!\cdots\!60}{73\!\cdots\!57}a^{18}+\frac{61\!\cdots\!86}{811239940350573}a^{17}+\frac{75\!\cdots\!80}{24\!\cdots\!19}a^{16}-\frac{14\!\cdots\!59}{811239940350573}a^{15}-\frac{19\!\cdots\!60}{270413313450191}a^{14}+\frac{75\!\cdots\!44}{270413313450191}a^{13}+\frac{27\!\cdots\!94}{24\!\cdots\!19}a^{12}-\frac{23\!\cdots\!75}{811239940350573}a^{11}-\frac{31\!\cdots\!87}{270413313450191}a^{10}+\frac{15\!\cdots\!81}{811239940350573}a^{9}+\frac{20\!\cdots\!92}{270413313450191}a^{8}-\frac{61\!\cdots\!40}{811239940350573}a^{7}-\frac{24\!\cdots\!95}{811239940350573}a^{6}+\frac{44\!\cdots\!90}{270413313450191}a^{5}+\frac{16\!\cdots\!09}{270413313450191}a^{4}-\frac{42\!\cdots\!56}{270413313450191}a^{3}-\frac{13\!\cdots\!14}{270413313450191}a^{2}+\frac{12\!\cdots\!66}{270413313450191}a+\frac{37\!\cdots\!70}{270413313450191}$, $\frac{775807959404896}{21\!\cdots\!71}a^{29}+\frac{17\!\cdots\!83}{21\!\cdots\!71}a^{28}-\frac{41\!\cdots\!20}{21\!\cdots\!71}a^{27}-\frac{10\!\cdots\!12}{24\!\cdots\!19}a^{26}+\frac{94\!\cdots\!05}{21\!\cdots\!71}a^{25}+\frac{21\!\cdots\!16}{21\!\cdots\!71}a^{24}-\frac{39\!\cdots\!38}{73\!\cdots\!57}a^{23}-\frac{88\!\cdots\!16}{73\!\cdots\!57}a^{22}+\frac{31\!\cdots\!50}{73\!\cdots\!57}a^{21}+\frac{76\!\cdots\!20}{811239940350573}a^{20}-\frac{16\!\cdots\!98}{73\!\cdots\!57}a^{19}-\frac{35\!\cdots\!48}{73\!\cdots\!57}a^{18}+\frac{61\!\cdots\!86}{811239940350573}a^{17}+\frac{40\!\cdots\!60}{24\!\cdots\!19}a^{16}-\frac{14\!\cdots\!59}{811239940350573}a^{15}-\frac{30\!\cdots\!66}{811239940350573}a^{14}+\frac{75\!\cdots\!44}{270413313450191}a^{13}+\frac{15\!\cdots\!84}{270413313450191}a^{12}-\frac{23\!\cdots\!75}{811239940350573}a^{11}-\frac{48\!\cdots\!32}{811239940350573}a^{10}+\frac{15\!\cdots\!81}{811239940350573}a^{9}+\frac{10\!\cdots\!36}{270413313450191}a^{8}-\frac{61\!\cdots\!40}{811239940350573}a^{7}-\frac{11\!\cdots\!28}{811239940350573}a^{6}+\frac{44\!\cdots\!90}{270413313450191}a^{5}+\frac{77\!\cdots\!28}{270413313450191}a^{4}-\frac{42\!\cdots\!56}{270413313450191}a^{3}-\frac{65\!\cdots\!96}{270413313450191}a^{2}+\frac{12\!\cdots\!66}{270413313450191}a+\frac{15\!\cdots\!13}{270413313450191}$
| 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: | \( 1288437153447097900 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{30}\cdot(2\pi)^{0}\cdot 1288437153447097900 \cdot 1}{2\cdot\sqrt{31245017777306374823059337284350587021473200026746355712}}\cr\approx \mathstrut & 0.123749292845030 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 - 54*x^28 + 1233*x^26 - 15732*x^24 + 124974*x^22 - 651888*x^20 + 2294766*x^18 - 5517450*x^16 + 9065763*x^14 - 10076778*x^12 + 7410609*x^10 - 3473442*x^8 + 979290*x^6 - 151875*x^4 + 10935*x^2 - 243)
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 - 54*x^28 + 1233*x^26 - 15732*x^24 + 124974*x^22 - 651888*x^20 + 2294766*x^18 - 5517450*x^16 + 9065763*x^14 - 10076778*x^12 + 7410609*x^10 - 3473442*x^8 + 979290*x^6 - 151875*x^4 + 10935*x^2 - 243, 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 - 54*x^28 + 1233*x^26 - 15732*x^24 + 124974*x^22 - 651888*x^20 + 2294766*x^18 - 5517450*x^16 + 9065763*x^14 - 10076778*x^12 + 7410609*x^10 - 3473442*x^8 + 979290*x^6 - 151875*x^4 + 10935*x^2 - 243);
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 - 54*x^28 + 1233*x^26 - 15732*x^24 + 124974*x^22 - 651888*x^20 + 2294766*x^18 - 5517450*x^16 + 9065763*x^14 - 10076778*x^12 + 7410609*x^10 - 3473442*x^8 + 979290*x^6 - 151875*x^4 + 10935*x^2 - 243);
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 cyclic group of order 30 |
| The 30 conjugacy class representatives for $C_{30}$ |
| Character table for $C_{30}$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{36})^+\), 10.10.53339349076992.1, 15.15.10943023107606534329121.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | $30$ | $30$ | R | $15^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{3}$ | ${\href{/padicField/19.10.0.1}{10} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{10}$ | $30$ | $30$ | ${\href{/padicField/37.5.0.1}{5} }^{6}$ | $30$ | ${\href{/padicField/43.6.0.1}{6} }^{5}$ | $15^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{3}$ | $15^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $30$ | $2$ | $15$ | $30$ | |||
|
\(3\)
| Deg $30$ | $6$ | $5$ | $45$ | |||
|
\(11\)
| 11.15.12.1 | $x^{15} + 10 x^{13} + 45 x^{12} + 40 x^{11} + 393 x^{10} + 890 x^{9} + 750 x^{8} - 3970 x^{7} + 9610 x^{6} + 13085 x^{5} + 151045 x^{4} + 26525 x^{3} + 116170 x^{2} - 53795 x + 67662$ | $5$ | $3$ | $12$ | $C_{15}$ | $[\ ]_{5}^{3}$ |
| 11.15.12.1 | $x^{15} + 10 x^{13} + 45 x^{12} + 40 x^{11} + 393 x^{10} + 890 x^{9} + 750 x^{8} - 3970 x^{7} + 9610 x^{6} + 13085 x^{5} + 151045 x^{4} + 26525 x^{3} + 116170 x^{2} - 53795 x + 67662$ | $5$ | $3$ | $12$ | $C_{15}$ | $[\ ]_{5}^{3}$ |