LMFDB - Number field 30.10.14414591053629993500878399466353926864896.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 15*x^28 + 104*x^26 - 482*x^24 + 1708*x^22 - 4753*x^20 + 10608*x^18 - 18925*x^16 + 26563*x^14 - 28460*x^12 + 22463*x^10 - 12716*x^8 + 5049*x^6 - 1331*x^4 + 198*x^2 - 11)

gp: K = bnfinit(y^30 - 15*y^28 + 104*y^26 - 482*y^24 + 1708*y^22 - 4753*y^20 + 10608*y^18 - 18925*y^16 + 26563*y^14 - 28460*y^12 + 22463*y^10 - 12716*y^8 + 5049*y^6 - 1331*y^4 + 198*y^2 - 11, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 15*x^28 + 104*x^26 - 482*x^24 + 1708*x^22 - 4753*x^20 + 10608*x^18 - 18925*x^16 + 26563*x^14 - 28460*x^12 + 22463*x^10 - 12716*x^8 + 5049*x^6 - 1331*x^4 + 198*x^2 - 11);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 15*x^28 + 104*x^26 - 482*x^24 + 1708*x^22 - 4753*x^20 + 10608*x^18 - 18925*x^16 + 26563*x^14 - 28460*x^12 + 22463*x^10 - 12716*x^8 + 5049*x^6 - 1331*x^4 + 198*x^2 - 11)

$ x^{30} - 15 x^{28} + 104 x^{26} - 482 x^{24} + 1708 x^{22} - 4753 x^{20} + 10608 x^{18} - 18925 x^{16} + \cdots - 11 $ x^30 - 15*x^28 + 104*x^26 - 482*x^24 + 1708*x^22 - 4753*x^20 + 10608*x^18 - 18925*x^16 + 26563*x^14 - 28460*x^12 + 22463*x^10 - 12716*x^8 + 5049*x^6 - 1331*x^4 + 198*x^2 - 11

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:	$[10, 10]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$14414591053629993500878399466353926864896$ 14414591053629993500878399466353926864896 $\medspace = 2^{40}\cdot 11^{27}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$21.81$	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:	$2^{4/3}11^{9/10}\approx 21.808547634345086$
Ramified primes:	$2$, $11$ 2, 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{11}) $
$\card{ \Aut(K/\Q) }$:	$10$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{58\!\cdots\!18}a^{28}-\frac{10\!\cdots\!49}{58\!\cdots\!18}a^{26}-\frac{68\!\cdots\!34}{29\!\cdots\!59}a^{24}+\frac{149570912943989}{53\!\cdots\!02}a^{22}-\frac{47\!\cdots\!23}{29\!\cdots\!59}a^{20}+\frac{80\!\cdots\!67}{58\!\cdots\!18}a^{18}+\frac{12\!\cdots\!41}{58\!\cdots\!18}a^{16}-\frac{16\!\cdots\!72}{29\!\cdots\!59}a^{14}+\frac{12\!\cdots\!85}{58\!\cdots\!18}a^{12}-\frac{1}{2}a^{11}-\frac{76\!\cdots\!53}{58\!\cdots\!18}a^{10}-\frac{1}{2}a^{9}-\frac{23\!\cdots\!51}{58\!\cdots\!18}a^{8}-\frac{1}{2}a^{7}-\frac{12\!\cdots\!08}{29\!\cdots\!59}a^{6}-\frac{1}{2}a^{5}-\frac{14\!\cdots\!21}{29\!\cdots\!59}a^{4}-\frac{55\!\cdots\!67}{58\!\cdots\!18}a^{2}-\frac{1}{2}a+\frac{20\!\cdots\!41}{58\!\cdots\!18}$, $\frac{1}{58\!\cdots\!18}a^{29}-\frac{10\!\cdots\!49}{58\!\cdots\!18}a^{27}-\frac{68\!\cdots\!34}{29\!\cdots\!59}a^{25}+\frac{149570912943989}{53\!\cdots\!02}a^{23}-\frac{47\!\cdots\!23}{29\!\cdots\!59}a^{21}+\frac{80\!\cdots\!67}{58\!\cdots\!18}a^{19}+\frac{12\!\cdots\!41}{58\!\cdots\!18}a^{17}-\frac{16\!\cdots\!72}{29\!\cdots\!59}a^{15}+\frac{12\!\cdots\!85}{58\!\cdots\!18}a^{13}-\frac{1}{2}a^{12}-\frac{76\!\cdots\!53}{58\!\cdots\!18}a^{11}-\frac{1}{2}a^{10}-\frac{23\!\cdots\!51}{58\!\cdots\!18}a^{9}-\frac{1}{2}a^{8}-\frac{12\!\cdots\!08}{29\!\cdots\!59}a^{7}-\frac{1}{2}a^{6}-\frac{14\!\cdots\!21}{29\!\cdots\!59}a^{5}-\frac{55\!\cdots\!67}{58\!\cdots\!18}a^{3}-\frac{1}{2}a^{2}+\frac{20\!\cdots\!41}{58\!\cdots\!18}a$ 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, 1/2*a^15 - 1/2*a^14 - 1/2*a^10 - 1/2*a^8 - 1/2*a^7 - 1/2*a^5 - 1/2*a^3 - 1/2*a^2 - 1/2, 1/2*a^16 - 1/2*a^14 - 1/2*a^11 - 1/2*a^10 - 1/2*a^9 - 1/2*a^7 - 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^2 - 1/2*a - 1/2, 1/2*a^17 - 1/2*a^14 - 1/2*a^12 - 1/2*a^11 - 1/2*a^6 - 1/2*a - 1/2, 1/2*a^18 - 1/2*a^14 - 1/2*a^13 - 1/2*a^12 - 1/2*a^10 - 1/2*a^8 - 1/2*a^5 - 1/2*a^3 - 1/2*a - 1/2, 1/2*a^19 - 1/2*a^13 - 1/2*a^11 - 1/2*a^10 - 1/2*a^9 - 1/2*a^8 - 1/2*a^7 - 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a - 1/2, 1/2*a^20 - 1/2*a^14 - 1/2*a^12 - 1/2*a^11 - 1/2*a^10 - 1/2*a^9 - 1/2*a^8 - 1/2*a^7 - 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^2 - 1/2*a, 1/2*a^21 - 1/2*a^14 - 1/2*a^13 - 1/2*a^12 - 1/2*a^11 - 1/2*a^9 - 1/2*a^6 - 1/2, 1/2*a^22 - 1/2*a^13 - 1/2*a^12 - 1/2*a^8 - 1/2*a^5 - 1/2*a^3 - 1/2*a^2 - 1/2*a - 1/2, 1/2*a^23 - 1/2*a^14 - 1/2*a^13 - 1/2*a^9 - 1/2*a^6 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2*a, 1/2*a^24 - 1/2*a^8 - 1/2*a^4 - 1/2, 1/2*a^25 - 1/2*a^9 - 1/2*a^5 - 1/2*a, 1/2*a^26 - 1/2*a^10 - 1/2*a^6 - 1/2*a^2, 1/2*a^27 - 1/2*a^11 - 1/2*a^7 - 1/2*a^3, 1/587409483542016518*a^28 - 103020836976652849/587409483542016518*a^26 - 6816620393891934/293704741771008259*a^24 + 149570912943989/5389077830660702*a^22 - 47659317464097223/293704741771008259*a^20 + 8081375902375867/587409483542016518*a^18 + 123471128679091941/587409483542016518*a^16 - 16960726100570672/293704741771008259*a^14 + 126667923291359885/587409483542016518*a^12 - 1/2*a^11 - 76716716275769853/587409483542016518*a^10 - 1/2*a^9 - 237814649392225551/587409483542016518*a^8 - 1/2*a^7 - 127991483737504008/293704741771008259*a^6 - 1/2*a^5 - 143232585408898121/293704741771008259*a^4 - 55036218279611367/587409483542016518*a^2 - 1/2*a + 208974501016961141/587409483542016518, 1/587409483542016518*a^29 - 103020836976652849/587409483542016518*a^27 - 6816620393891934/293704741771008259*a^25 + 149570912943989/5389077830660702*a^23 - 47659317464097223/293704741771008259*a^21 + 8081375902375867/587409483542016518*a^19 + 123471128679091941/587409483542016518*a^17 - 16960726100570672/293704741771008259*a^15 + 126667923291359885/587409483542016518*a^13 - 1/2*a^12 - 76716716275769853/587409483542016518*a^11 - 1/2*a^10 - 237814649392225551/587409483542016518*a^9 - 1/2*a^8 - 127991483737504008/293704741771008259*a^7 - 1/2*a^6 - 143232585408898121/293704741771008259*a^5 - 55036218279611367/587409483542016518*a^3 - 1/2*a^2 + 208974501016961141/587409483542016518*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:	$19$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -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:	$\frac{14\!\cdots\!73}{58\!\cdots\!18}a^{28}-\frac{10\!\cdots\!72}{29\!\cdots\!59}a^{26}+\frac{72\!\cdots\!02}{29\!\cdots\!59}a^{24}-\frac{59\!\cdots\!91}{53\!\cdots\!02}a^{22}+\frac{11\!\cdots\!04}{29\!\cdots\!59}a^{20}-\frac{60\!\cdots\!05}{58\!\cdots\!18}a^{18}+\frac{65\!\cdots\!63}{29\!\cdots\!59}a^{16}-\frac{22\!\cdots\!47}{58\!\cdots\!18}a^{14}+\frac{29\!\cdots\!75}{58\!\cdots\!18}a^{12}-\frac{29\!\cdots\!75}{58\!\cdots\!18}a^{10}+\frac{20\!\cdots\!39}{58\!\cdots\!18}a^{8}-\frac{47\!\cdots\!59}{29\!\cdots\!59}a^{6}+\frac{29\!\cdots\!75}{58\!\cdots\!18}a^{4}-\frac{51\!\cdots\!61}{58\!\cdots\!18}a^{2}+\frac{12\!\cdots\!74}{29\!\cdots\!59}$, $\frac{38\!\cdots\!69}{58\!\cdots\!18}a^{28}-\frac{55\!\cdots\!67}{58\!\cdots\!18}a^{26}+\frac{18\!\cdots\!70}{29\!\cdots\!59}a^{24}-\frac{14\!\cdots\!27}{53\!\cdots\!02}a^{22}+\frac{27\!\cdots\!72}{29\!\cdots\!59}a^{20}-\frac{14\!\cdots\!61}{58\!\cdots\!18}a^{18}+\frac{15\!\cdots\!65}{29\!\cdots\!59}a^{16}-\frac{51\!\cdots\!33}{58\!\cdots\!18}a^{14}+\frac{66\!\cdots\!29}{58\!\cdots\!18}a^{12}-\frac{31\!\cdots\!25}{29\!\cdots\!59}a^{10}+\frac{41\!\cdots\!65}{58\!\cdots\!18}a^{8}-\frac{18\!\cdots\!25}{58\!\cdots\!18}a^{6}+\frac{54\!\cdots\!45}{58\!\cdots\!18}a^{4}-\frac{43\!\cdots\!86}{29\!\cdots\!59}a^{2}+\frac{18\!\cdots\!40}{29\!\cdots\!59}$, $\frac{13\!\cdots\!11}{29\!\cdots\!59}a^{28}-\frac{19\!\cdots\!71}{29\!\cdots\!59}a^{26}+\frac{12\!\cdots\!10}{29\!\cdots\!59}a^{24}-\frac{10\!\cdots\!15}{53\!\cdots\!02}a^{22}+\frac{38\!\cdots\!77}{58\!\cdots\!18}a^{20}-\frac{10\!\cdots\!51}{58\!\cdots\!18}a^{18}+\frac{21\!\cdots\!11}{58\!\cdots\!18}a^{16}-\frac{36\!\cdots\!73}{58\!\cdots\!18}a^{14}+\frac{47\!\cdots\!13}{58\!\cdots\!18}a^{12}-\frac{45\!\cdots\!91}{58\!\cdots\!18}a^{10}+\frac{31\!\cdots\!99}{58\!\cdots\!18}a^{8}-\frac{72\!\cdots\!75}{29\!\cdots\!59}a^{6}+\frac{22\!\cdots\!29}{29\!\cdots\!59}a^{4}-\frac{79\!\cdots\!19}{58\!\cdots\!18}a^{2}+\frac{45\!\cdots\!07}{58\!\cdots\!18}$, $\frac{12\!\cdots\!23}{58\!\cdots\!18}a^{28}-\frac{88\!\cdots\!63}{29\!\cdots\!59}a^{26}+\frac{59\!\cdots\!81}{29\!\cdots\!59}a^{24}-\frac{24\!\cdots\!39}{26\!\cdots\!51}a^{22}+\frac{18\!\cdots\!77}{58\!\cdots\!18}a^{20}-\frac{24\!\cdots\!82}{29\!\cdots\!59}a^{18}+\frac{10\!\cdots\!39}{58\!\cdots\!18}a^{16}-\frac{90\!\cdots\!58}{29\!\cdots\!59}a^{14}+\frac{11\!\cdots\!80}{29\!\cdots\!59}a^{12}-\frac{11\!\cdots\!60}{29\!\cdots\!59}a^{10}+\frac{84\!\cdots\!44}{29\!\cdots\!59}a^{8}-\frac{41\!\cdots\!66}{29\!\cdots\!59}a^{6}+\frac{28\!\cdots\!21}{58\!\cdots\!18}a^{4}-\frac{27\!\cdots\!65}{29\!\cdots\!59}a^{2}+\frac{44\!\cdots\!49}{58\!\cdots\!18}$, $\frac{4587349496807}{11180659399711}a^{28}-\frac{129897861582483}{22361318799422}a^{26}+\frac{844170364116451}{22361318799422}a^{24}-\frac{37\!\cdots\!75}{22361318799422}a^{22}+\frac{12\!\cdots\!73}{22361318799422}a^{20}-\frac{32\!\cdots\!67}{22361318799422}a^{18}+\frac{68\!\cdots\!43}{22361318799422}a^{16}-\frac{11\!\cdots\!53}{22361318799422}a^{14}+\frac{14\!\cdots\!05}{22361318799422}a^{12}-\frac{66\!\cdots\!03}{11180659399711}a^{10}+\frac{43\!\cdots\!07}{11180659399711}a^{8}-\frac{37\!\cdots\!95}{22361318799422}a^{6}+\frac{11\!\cdots\!37}{22361318799422}a^{4}-\frac{914057714124808}{11180659399711}a^{2}+\frac{60286987246998}{11180659399711}$, $\frac{20\!\cdots\!97}{29\!\cdots\!59}a^{28}-\frac{28\!\cdots\!46}{29\!\cdots\!59}a^{26}+\frac{36\!\cdots\!21}{58\!\cdots\!18}a^{24}-\frac{14\!\cdots\!55}{53\!\cdots\!02}a^{22}+\frac{52\!\cdots\!73}{58\!\cdots\!18}a^{20}-\frac{13\!\cdots\!23}{58\!\cdots\!18}a^{18}+\frac{27\!\cdots\!79}{58\!\cdots\!18}a^{16}-\frac{45\!\cdots\!51}{58\!\cdots\!18}a^{14}+\frac{55\!\cdots\!59}{58\!\cdots\!18}a^{12}-\frac{49\!\cdots\!75}{58\!\cdots\!18}a^{10}+\frac{15\!\cdots\!82}{29\!\cdots\!59}a^{8}-\frac{62\!\cdots\!44}{29\!\cdots\!59}a^{6}+\frac{33\!\cdots\!63}{58\!\cdots\!18}a^{4}-\frac{46\!\cdots\!05}{58\!\cdots\!18}a^{2}+\frac{12\!\cdots\!36}{29\!\cdots\!59}$, $\frac{29\!\cdots\!71}{58\!\cdots\!18}a^{28}-\frac{40\!\cdots\!85}{58\!\cdots\!18}a^{26}+\frac{12\!\cdots\!00}{29\!\cdots\!59}a^{24}-\frac{49\!\cdots\!63}{26\!\cdots\!51}a^{22}+\frac{35\!\cdots\!93}{58\!\cdots\!18}a^{20}-\frac{45\!\cdots\!47}{29\!\cdots\!59}a^{18}+\frac{18\!\cdots\!63}{58\!\cdots\!18}a^{16}-\frac{14\!\cdots\!67}{29\!\cdots\!59}a^{14}+\frac{17\!\cdots\!58}{29\!\cdots\!59}a^{12}-\frac{29\!\cdots\!01}{58\!\cdots\!18}a^{10}+\frac{83\!\cdots\!54}{29\!\cdots\!59}a^{8}-\frac{57\!\cdots\!33}{58\!\cdots\!18}a^{6}+\frac{10\!\cdots\!83}{58\!\cdots\!18}a^{4}+\frac{42\!\cdots\!31}{58\!\cdots\!18}a^{2}-\frac{15\!\cdots\!57}{58\!\cdots\!18}$, $\frac{56\!\cdots\!13}{29\!\cdots\!59}a^{28}-\frac{83\!\cdots\!57}{29\!\cdots\!59}a^{26}+\frac{56\!\cdots\!33}{29\!\cdots\!59}a^{24}-\frac{47\!\cdots\!15}{53\!\cdots\!02}a^{22}+\frac{17\!\cdots\!49}{58\!\cdots\!18}a^{20}-\frac{49\!\cdots\!43}{58\!\cdots\!18}a^{18}+\frac{10\!\cdots\!57}{58\!\cdots\!18}a^{16}-\frac{18\!\cdots\!41}{58\!\cdots\!18}a^{14}+\frac{25\!\cdots\!53}{58\!\cdots\!18}a^{12}-\frac{25\!\cdots\!31}{58\!\cdots\!18}a^{10}+\frac{18\!\cdots\!47}{58\!\cdots\!18}a^{8}-\frac{48\!\cdots\!85}{29\!\cdots\!59}a^{6}+\frac{16\!\cdots\!63}{29\!\cdots\!59}a^{4}-\frac{67\!\cdots\!19}{58\!\cdots\!18}a^{2}+\frac{57\!\cdots\!39}{58\!\cdots\!18}$, $\frac{63\!\cdots\!91}{58\!\cdots\!18}a^{28}-\frac{89\!\cdots\!63}{58\!\cdots\!18}a^{26}+\frac{29\!\cdots\!20}{29\!\cdots\!59}a^{24}-\frac{23\!\cdots\!23}{53\!\cdots\!02}a^{22}+\frac{87\!\cdots\!69}{58\!\cdots\!18}a^{20}-\frac{23\!\cdots\!69}{58\!\cdots\!18}a^{18}+\frac{49\!\cdots\!27}{58\!\cdots\!18}a^{16}-\frac{81\!\cdots\!31}{58\!\cdots\!18}a^{14}+\frac{52\!\cdots\!76}{29\!\cdots\!59}a^{12}-\frac{49\!\cdots\!14}{29\!\cdots\!59}a^{10}+\frac{32\!\cdots\!68}{29\!\cdots\!59}a^{8}-\frac{30\!\cdots\!85}{58\!\cdots\!18}a^{6}+\frac{91\!\cdots\!01}{58\!\cdots\!18}a^{4}-\frac{79\!\cdots\!54}{29\!\cdots\!59}a^{2}+\frac{10\!\cdots\!13}{58\!\cdots\!18}$, $\frac{18\!\cdots\!93}{29\!\cdots\!59}a^{29}+\frac{31\!\cdots\!63}{58\!\cdots\!18}a^{28}-\frac{52\!\cdots\!41}{58\!\cdots\!18}a^{27}-\frac{46\!\cdots\!67}{58\!\cdots\!18}a^{26}+\frac{16\!\cdots\!38}{29\!\cdots\!59}a^{25}+\frac{15\!\cdots\!04}{29\!\cdots\!59}a^{24}-\frac{65\!\cdots\!28}{26\!\cdots\!51}a^{23}-\frac{63\!\cdots\!11}{26\!\cdots\!51}a^{22}+\frac{23\!\cdots\!51}{29\!\cdots\!59}a^{21}+\frac{48\!\cdots\!23}{58\!\cdots\!18}a^{20}-\frac{11\!\cdots\!17}{58\!\cdots\!18}a^{19}-\frac{12\!\cdots\!23}{58\!\cdots\!18}a^{18}+\frac{12\!\cdots\!41}{29\!\cdots\!59}a^{17}+\frac{14\!\cdots\!81}{29\!\cdots\!59}a^{16}-\frac{19\!\cdots\!09}{29\!\cdots\!59}a^{15}-\frac{23\!\cdots\!19}{29\!\cdots\!59}a^{14}+\frac{23\!\cdots\!21}{29\!\cdots\!59}a^{13}+\frac{63\!\cdots\!99}{58\!\cdots\!18}a^{12}-\frac{39\!\cdots\!15}{58\!\cdots\!18}a^{11}-\frac{31\!\cdots\!82}{29\!\cdots\!59}a^{10}+\frac{10\!\cdots\!03}{29\!\cdots\!59}a^{9}+\frac{22\!\cdots\!39}{29\!\cdots\!59}a^{8}-\frac{73\!\cdots\!07}{58\!\cdots\!18}a^{7}-\frac{21\!\cdots\!33}{58\!\cdots\!18}a^{6}+\frac{12\!\cdots\!03}{58\!\cdots\!18}a^{5}+\frac{71\!\cdots\!27}{58\!\cdots\!18}a^{4}+\frac{14\!\cdots\!37}{58\!\cdots\!18}a^{3}-\frac{69\!\cdots\!12}{29\!\cdots\!59}a^{2}-\frac{65\!\cdots\!97}{58\!\cdots\!18}a+\frac{54\!\cdots\!35}{29\!\cdots\!59}$, $\frac{25\!\cdots\!79}{58\!\cdots\!18}a^{29}-\frac{11\!\cdots\!77}{29\!\cdots\!59}a^{28}-\frac{18\!\cdots\!32}{29\!\cdots\!59}a^{27}+\frac{16\!\cdots\!63}{29\!\cdots\!59}a^{26}+\frac{11\!\cdots\!36}{29\!\cdots\!59}a^{25}-\frac{10\!\cdots\!68}{29\!\cdots\!59}a^{24}-\frac{46\!\cdots\!36}{26\!\cdots\!51}a^{23}+\frac{83\!\cdots\!79}{53\!\cdots\!02}a^{22}+\frac{16\!\cdots\!92}{29\!\cdots\!59}a^{21}-\frac{30\!\cdots\!27}{58\!\cdots\!18}a^{20}-\frac{43\!\cdots\!35}{29\!\cdots\!59}a^{19}+\frac{39\!\cdots\!18}{29\!\cdots\!59}a^{18}+\frac{18\!\cdots\!25}{58\!\cdots\!18}a^{17}-\frac{81\!\cdots\!49}{29\!\cdots\!59}a^{16}-\frac{14\!\cdots\!69}{29\!\cdots\!59}a^{15}+\frac{13\!\cdots\!87}{29\!\cdots\!59}a^{14}+\frac{18\!\cdots\!13}{29\!\cdots\!59}a^{13}-\frac{33\!\cdots\!13}{58\!\cdots\!18}a^{12}-\frac{16\!\cdots\!76}{29\!\cdots\!59}a^{11}+\frac{29\!\cdots\!23}{58\!\cdots\!18}a^{10}+\frac{97\!\cdots\!69}{29\!\cdots\!59}a^{9}-\frac{93\!\cdots\!28}{29\!\cdots\!59}a^{8}-\frac{76\!\cdots\!21}{58\!\cdots\!18}a^{7}+\frac{39\!\cdots\!86}{29\!\cdots\!59}a^{6}+\frac{17\!\cdots\!49}{58\!\cdots\!18}a^{5}-\frac{20\!\cdots\!37}{58\!\cdots\!18}a^{4}-\frac{11\!\cdots\!41}{58\!\cdots\!18}a^{3}+\frac{14\!\cdots\!43}{29\!\cdots\!59}a^{2}-\frac{31\!\cdots\!81}{58\!\cdots\!18}a-\frac{37\!\cdots\!15}{29\!\cdots\!59}$, $\frac{40\!\cdots\!55}{68\!\cdots\!13}a^{29}-\frac{13\!\cdots\!21}{58\!\cdots\!18}a^{28}-\frac{58\!\cdots\!48}{68\!\cdots\!13}a^{27}+\frac{18\!\cdots\!31}{58\!\cdots\!18}a^{26}+\frac{78\!\cdots\!89}{13\!\cdots\!26}a^{25}-\frac{62\!\cdots\!31}{29\!\cdots\!59}a^{24}-\frac{32\!\cdots\!53}{125327391410714}a^{23}+\frac{51\!\cdots\!13}{53\!\cdots\!02}a^{22}+\frac{12\!\cdots\!71}{13\!\cdots\!26}a^{21}-\frac{19\!\cdots\!77}{58\!\cdots\!18}a^{20}-\frac{32\!\cdots\!51}{13\!\cdots\!26}a^{19}+\frac{25\!\cdots\!99}{29\!\cdots\!59}a^{18}+\frac{34\!\cdots\!68}{68\!\cdots\!13}a^{17}-\frac{10\!\cdots\!51}{58\!\cdots\!18}a^{16}-\frac{58\!\cdots\!86}{68\!\cdots\!13}a^{15}+\frac{92\!\cdots\!75}{29\!\cdots\!59}a^{14}+\frac{77\!\cdots\!91}{68\!\cdots\!13}a^{13}-\frac{12\!\cdots\!66}{29\!\cdots\!59}a^{12}-\frac{76\!\cdots\!08}{68\!\cdots\!13}a^{11}+\frac{11\!\cdots\!41}{29\!\cdots\!59}a^{10}+\frac{53\!\cdots\!63}{68\!\cdots\!13}a^{9}-\frac{82\!\cdots\!12}{29\!\cdots\!59}a^{8}-\frac{52\!\cdots\!93}{13\!\cdots\!26}a^{7}+\frac{39\!\cdots\!57}{29\!\cdots\!59}a^{6}+\frac{17\!\cdots\!17}{13\!\cdots\!26}a^{5}-\frac{25\!\cdots\!59}{58\!\cdots\!18}a^{4}-\frac{32\!\cdots\!89}{13\!\cdots\!26}a^{3}+\frac{49\!\cdots\!87}{58\!\cdots\!18}a^{2}+\frac{11\!\cdots\!70}{68\!\cdots\!13}a-\frac{20\!\cdots\!80}{29\!\cdots\!59}$, $\frac{59\!\cdots\!43}{58\!\cdots\!18}a^{29}-\frac{40\!\cdots\!55}{68\!\cdots\!13}a^{28}-\frac{42\!\cdots\!72}{29\!\cdots\!59}a^{27}+\frac{58\!\cdots\!48}{68\!\cdots\!13}a^{26}+\frac{27\!\cdots\!40}{29\!\cdots\!59}a^{25}-\frac{78\!\cdots\!89}{13\!\cdots\!26}a^{24}-\frac{22\!\cdots\!67}{53\!\cdots\!02}a^{23}+\frac{32\!\cdots\!53}{125327391410714}a^{22}+\frac{42\!\cdots\!05}{29\!\cdots\!59}a^{21}-\frac{12\!\cdots\!71}{13\!\cdots\!26}a^{20}-\frac{11\!\cdots\!19}{29\!\cdots\!59}a^{19}+\frac{32\!\cdots\!51}{13\!\cdots\!26}a^{18}+\frac{47\!\cdots\!39}{58\!\cdots\!18}a^{17}-\frac{34\!\cdots\!68}{68\!\cdots\!13}a^{16}-\frac{40\!\cdots\!51}{29\!\cdots\!59}a^{15}+\frac{58\!\cdots\!86}{68\!\cdots\!13}a^{14}+\frac{52\!\cdots\!41}{29\!\cdots\!59}a^{13}-\frac{77\!\cdots\!91}{68\!\cdots\!13}a^{12}-\frac{50\!\cdots\!81}{29\!\cdots\!59}a^{11}+\frac{76\!\cdots\!08}{68\!\cdots\!13}a^{10}+\frac{67\!\cdots\!09}{58\!\cdots\!18}a^{9}-\frac{53\!\cdots\!63}{68\!\cdots\!13}a^{8}-\frac{31\!\cdots\!29}{58\!\cdots\!18}a^{7}+\frac{52\!\cdots\!93}{13\!\cdots\!26}a^{6}+\frac{48\!\cdots\!55}{29\!\cdots\!59}a^{5}-\frac{17\!\cdots\!17}{13\!\cdots\!26}a^{4}-\frac{16\!\cdots\!11}{58\!\cdots\!18}a^{3}+\frac{32\!\cdots\!89}{13\!\cdots\!26}a^{2}+\frac{83\!\cdots\!25}{58\!\cdots\!18}a-\frac{11\!\cdots\!70}{68\!\cdots\!13}$, $\frac{58\!\cdots\!51}{58\!\cdots\!18}a^{29}+\frac{29\!\cdots\!47}{58\!\cdots\!18}a^{28}-\frac{41\!\cdots\!02}{29\!\cdots\!59}a^{27}-\frac{41\!\cdots\!81}{58\!\cdots\!18}a^{26}+\frac{27\!\cdots\!12}{29\!\cdots\!59}a^{25}+\frac{13\!\cdots\!13}{29\!\cdots\!59}a^{24}-\frac{22\!\cdots\!67}{53\!\cdots\!02}a^{23}-\frac{53\!\cdots\!65}{26\!\cdots\!51}a^{22}+\frac{41\!\cdots\!96}{29\!\cdots\!59}a^{21}+\frac{19\!\cdots\!88}{29\!\cdots\!59}a^{20}-\frac{21\!\cdots\!75}{58\!\cdots\!18}a^{19}-\frac{10\!\cdots\!15}{58\!\cdots\!18}a^{18}+\frac{46\!\cdots\!35}{58\!\cdots\!18}a^{17}+\frac{10\!\cdots\!46}{29\!\cdots\!59}a^{16}-\frac{76\!\cdots\!23}{58\!\cdots\!18}a^{15}-\frac{17\!\cdots\!01}{29\!\cdots\!59}a^{14}+\frac{49\!\cdots\!85}{29\!\cdots\!59}a^{13}+\frac{44\!\cdots\!53}{58\!\cdots\!18}a^{12}-\frac{47\!\cdots\!25}{29\!\cdots\!59}a^{11}-\frac{20\!\cdots\!88}{29\!\cdots\!59}a^{10}+\frac{62\!\cdots\!09}{58\!\cdots\!18}a^{9}+\frac{12\!\cdots\!94}{29\!\cdots\!59}a^{8}-\frac{14\!\cdots\!82}{29\!\cdots\!59}a^{7}-\frac{56\!\cdots\!02}{29\!\cdots\!59}a^{6}+\frac{42\!\cdots\!78}{29\!\cdots\!59}a^{5}+\frac{32\!\cdots\!97}{58\!\cdots\!18}a^{4}-\frac{67\!\cdots\!82}{29\!\cdots\!59}a^{3}-\frac{47\!\cdots\!51}{58\!\cdots\!18}a^{2}+\frac{30\!\cdots\!31}{29\!\cdots\!59}a+\frac{11\!\cdots\!69}{29\!\cdots\!59}$, $\frac{32\!\cdots\!27}{58\!\cdots\!18}a^{29}-\frac{44\!\cdots\!02}{29\!\cdots\!59}a^{28}-\frac{47\!\cdots\!67}{58\!\cdots\!18}a^{27}+\frac{12\!\cdots\!41}{58\!\cdots\!18}a^{26}+\frac{31\!\cdots\!89}{58\!\cdots\!18}a^{25}-\frac{83\!\cdots\!27}{58\!\cdots\!18}a^{24}-\frac{12\!\cdots\!17}{53\!\cdots\!02}a^{23}+\frac{16\!\cdots\!78}{26\!\cdots\!51}a^{22}+\frac{47\!\cdots\!07}{58\!\cdots\!18}a^{21}-\frac{12\!\cdots\!33}{58\!\cdots\!18}a^{20}-\frac{12\!\cdots\!25}{58\!\cdots\!18}a^{19}+\frac{33\!\cdots\!23}{58\!\cdots\!18}a^{18}+\frac{26\!\cdots\!81}{58\!\cdots\!18}a^{17}-\frac{70\!\cdots\!71}{58\!\cdots\!18}a^{16}-\frac{44\!\cdots\!87}{58\!\cdots\!18}a^{15}+\frac{11\!\cdots\!29}{58\!\cdots\!18}a^{14}+\frac{57\!\cdots\!77}{58\!\cdots\!18}a^{13}-\frac{75\!\cdots\!80}{29\!\cdots\!59}a^{12}-\frac{27\!\cdots\!96}{29\!\cdots\!59}a^{11}+\frac{71\!\cdots\!98}{29\!\cdots\!59}a^{10}+\frac{36\!\cdots\!05}{58\!\cdots\!18}a^{9}-\frac{96\!\cdots\!41}{58\!\cdots\!18}a^{8}-\frac{16\!\cdots\!69}{58\!\cdots\!18}a^{7}+\frac{44\!\cdots\!59}{58\!\cdots\!18}a^{6}+\frac{46\!\cdots\!59}{58\!\cdots\!18}a^{5}-\frac{13\!\cdots\!95}{58\!\cdots\!18}a^{4}-\frac{67\!\cdots\!31}{58\!\cdots\!18}a^{3}+\frac{23\!\cdots\!17}{58\!\cdots\!18}a^{2}+\frac{80\!\cdots\!75}{58\!\cdots\!18}a-\frac{14\!\cdots\!45}{58\!\cdots\!18}$, $\frac{13\!\cdots\!45}{29\!\cdots\!59}a^{29}+\frac{36\!\cdots\!25}{58\!\cdots\!18}a^{28}-\frac{37\!\cdots\!59}{58\!\cdots\!18}a^{27}-\frac{25\!\cdots\!74}{29\!\cdots\!59}a^{26}+\frac{24\!\cdots\!61}{58\!\cdots\!18}a^{25}+\frac{16\!\cdots\!20}{29\!\cdots\!59}a^{24}-\frac{48\!\cdots\!39}{26\!\cdots\!51}a^{23}-\frac{13\!\cdots\!47}{53\!\cdots\!02}a^{22}+\frac{17\!\cdots\!20}{29\!\cdots\!59}a^{21}+\frac{49\!\cdots\!87}{58\!\cdots\!18}a^{20}-\frac{94\!\cdots\!99}{58\!\cdots\!18}a^{19}-\frac{13\!\cdots\!29}{58\!\cdots\!18}a^{18}+\frac{99\!\cdots\!69}{29\!\cdots\!59}a^{17}+\frac{13\!\cdots\!52}{29\!\cdots\!59}a^{16}-\frac{32\!\cdots\!77}{58\!\cdots\!18}a^{15}-\frac{45\!\cdots\!93}{58\!\cdots\!18}a^{14}+\frac{41\!\cdots\!11}{58\!\cdots\!18}a^{13}+\frac{28\!\cdots\!37}{29\!\cdots\!59}a^{12}-\frac{39\!\cdots\!63}{58\!\cdots\!18}a^{11}-\frac{26\!\cdots\!22}{29\!\cdots\!59}a^{10}+\frac{25\!\cdots\!99}{58\!\cdots\!18}a^{9}+\frac{17\!\cdots\!35}{29\!\cdots\!59}a^{8}-\frac{58\!\cdots\!64}{29\!\cdots\!59}a^{7}-\frac{78\!\cdots\!43}{29\!\cdots\!59}a^{6}+\frac{17\!\cdots\!89}{29\!\cdots\!59}a^{5}+\frac{46\!\cdots\!17}{58\!\cdots\!18}a^{4}-\frac{54\!\cdots\!33}{58\!\cdots\!18}a^{3}-\frac{78\!\cdots\!39}{58\!\cdots\!18}a^{2}+\frac{34\!\cdots\!99}{58\!\cdots\!18}a+\frac{26\!\cdots\!95}{29\!\cdots\!59}$, $\frac{29\!\cdots\!01}{58\!\cdots\!18}a^{29}-\frac{22\!\cdots\!59}{58\!\cdots\!18}a^{28}-\frac{19\!\cdots\!18}{29\!\cdots\!59}a^{27}+\frac{31\!\cdots\!85}{58\!\cdots\!18}a^{26}+\frac{25\!\cdots\!35}{58\!\cdots\!18}a^{25}-\frac{10\!\cdots\!36}{29\!\cdots\!59}a^{24}-\frac{49\!\cdots\!50}{26\!\cdots\!51}a^{23}+\frac{85\!\cdots\!41}{53\!\cdots\!02}a^{22}+\frac{17\!\cdots\!71}{29\!\cdots\!59}a^{21}-\frac{15\!\cdots\!16}{29\!\cdots\!59}a^{20}-\frac{91\!\cdots\!89}{58\!\cdots\!18}a^{19}+\frac{84\!\cdots\!93}{58\!\cdots\!18}a^{18}+\frac{93\!\cdots\!30}{29\!\cdots\!59}a^{17}-\frac{17\!\cdots\!93}{58\!\cdots\!18}a^{16}-\frac{14\!\cdots\!82}{29\!\cdots\!59}a^{15}+\frac{14\!\cdots\!78}{29\!\cdots\!59}a^{14}+\frac{17\!\cdots\!44}{29\!\cdots\!59}a^{13}-\frac{38\!\cdots\!11}{58\!\cdots\!18}a^{12}-\frac{15\!\cdots\!80}{29\!\cdots\!59}a^{11}+\frac{18\!\cdots\!04}{29\!\cdots\!59}a^{10}+\frac{88\!\cdots\!37}{29\!\cdots\!59}a^{9}-\frac{11\!\cdots\!46}{29\!\cdots\!59}a^{8}-\frac{32\!\cdots\!77}{29\!\cdots\!59}a^{7}+\frac{10\!\cdots\!41}{58\!\cdots\!18}a^{6}+\frac{71\!\cdots\!19}{29\!\cdots\!59}a^{5}-\frac{31\!\cdots\!87}{58\!\cdots\!18}a^{4}-\frac{82\!\cdots\!47}{58\!\cdots\!18}a^{3}+\frac{49\!\cdots\!35}{58\!\cdots\!18}a^{2}-\frac{14\!\cdots\!05}{58\!\cdots\!18}a-\frac{13\!\cdots\!85}{29\!\cdots\!59}$, $\frac{17\!\cdots\!66}{29\!\cdots\!59}a^{29}-\frac{37\!\cdots\!17}{58\!\cdots\!18}a^{28}-\frac{25\!\cdots\!75}{29\!\cdots\!59}a^{27}+\frac{52\!\cdots\!09}{58\!\cdots\!18}a^{26}+\frac{17\!\cdots\!86}{29\!\cdots\!59}a^{25}-\frac{17\!\cdots\!08}{29\!\cdots\!59}a^{24}-\frac{70\!\cdots\!06}{26\!\cdots\!51}a^{23}+\frac{69\!\cdots\!81}{26\!\cdots\!51}a^{22}+\frac{26\!\cdots\!12}{29\!\cdots\!59}a^{21}-\frac{51\!\cdots\!63}{58\!\cdots\!18}a^{20}-\frac{72\!\cdots\!20}{29\!\cdots\!59}a^{19}+\frac{68\!\cdots\!58}{29\!\cdots\!59}a^{18}+\frac{31\!\cdots\!13}{58\!\cdots\!18}a^{17}-\frac{29\!\cdots\!15}{58\!\cdots\!18}a^{16}-\frac{27\!\cdots\!44}{29\!\cdots\!59}a^{15}+\frac{48\!\cdots\!37}{58\!\cdots\!18}a^{14}+\frac{36\!\cdots\!83}{29\!\cdots\!59}a^{13}-\frac{62\!\cdots\!49}{58\!\cdots\!18}a^{12}-\frac{74\!\cdots\!65}{58\!\cdots\!18}a^{11}+\frac{59\!\cdots\!55}{58\!\cdots\!18}a^{10}+\frac{26\!\cdots\!43}{29\!\cdots\!59}a^{9}-\frac{20\!\cdots\!78}{29\!\cdots\!59}a^{8}-\frac{13\!\cdots\!48}{29\!\cdots\!59}a^{7}+\frac{93\!\cdots\!45}{29\!\cdots\!59}a^{6}+\frac{47\!\cdots\!90}{29\!\cdots\!59}a^{5}-\frac{58\!\cdots\!67}{58\!\cdots\!18}a^{4}-\frac{99\!\cdots\!10}{29\!\cdots\!59}a^{3}+\frac{10\!\cdots\!87}{58\!\cdots\!18}a^{2}+\frac{18\!\cdots\!27}{58\!\cdots\!18}a-\frac{40\!\cdots\!27}{29\!\cdots\!59}$, $\frac{30\!\cdots\!23}{29\!\cdots\!59}a^{29}+\frac{43\!\cdots\!29}{68\!\cdots\!13}a^{28}-\frac{43\!\cdots\!51}{29\!\cdots\!59}a^{27}-\frac{60\!\cdots\!48}{68\!\cdots\!13}a^{26}+\frac{56\!\cdots\!23}{58\!\cdots\!18}a^{25}+\frac{77\!\cdots\!43}{13\!\cdots\!26}a^{24}-\frac{23\!\cdots\!31}{53\!\cdots\!02}a^{23}-\frac{30\!\cdots\!43}{125327391410714}a^{22}+\frac{42\!\cdots\!39}{29\!\cdots\!59}a^{21}+\frac{11\!\cdots\!53}{13\!\cdots\!26}a^{20}-\frac{11\!\cdots\!71}{29\!\cdots\!59}a^{19}-\frac{29\!\cdots\!13}{13\!\cdots\!26}a^{18}+\frac{48\!\cdots\!23}{58\!\cdots\!18}a^{17}+\frac{30\!\cdots\!97}{68\!\cdots\!13}a^{16}-\frac{40\!\cdots\!77}{29\!\cdots\!59}a^{15}-\frac{49\!\cdots\!72}{68\!\cdots\!13}a^{14}+\frac{10\!\cdots\!49}{58\!\cdots\!18}a^{13}+\frac{60\!\cdots\!05}{68\!\cdots\!13}a^{12}-\frac{50\!\cdots\!93}{29\!\cdots\!59}a^{11}-\frac{54\!\cdots\!54}{68\!\cdots\!13}a^{10}+\frac{67\!\cdots\!61}{58\!\cdots\!18}a^{9}+\frac{33\!\cdots\!66}{68\!\cdots\!13}a^{8}-\frac{31\!\cdots\!39}{58\!\cdots\!18}a^{7}-\frac{27\!\cdots\!83}{13\!\cdots\!26}a^{6}+\frac{49\!\cdots\!61}{29\!\cdots\!59}a^{5}+\frac{71\!\cdots\!27}{13\!\cdots\!26}a^{4}-\frac{18\!\cdots\!37}{58\!\cdots\!18}a^{3}-\frac{86\!\cdots\!39}{13\!\cdots\!26}a^{2}+\frac{62\!\cdots\!47}{29\!\cdots\!59}a+\frac{86\!\cdots\!95}{68\!\cdots\!13}$ 146223791304284573/587409483542016518a^28 - 1070330004368874872/293704741771008259a^26 + 7207210773528053002/293704741771008259a^24 - 596299495718022691/5389077830660702a^22 + 112274439108826352104/293704741771008259a^20 - 606598029314588360305/587409483542016518a^18 + 654089388726515610863/293704741771008259a^16 - 2234628771442202111447/587409483542016518a^14 + 2956211514244813935475/587409483542016518a^12 - 2906051460649545611475/587409483542016518a^10 + 2014801722918999083739/587409483542016518a^8 - 475175058964547649359/293704741771008259a^6 + 295027873964211941675/587409483542016518a^4 - 51504621502542917861/587409483542016518a^2 + 1249724555812875774/293704741771008259, 387674352536527469/587409483542016518a^28 - 5547014701314195767/587409483542016518a^26 + 18229142013140527670/293704741771008259a^24 - 1480055119534042727/5389077830660702a^22 + 274305297311438620672/293704741771008259a^20 - 1454919156369674656761/587409483542016518a^18 + 1539610344233565110565/293704741771008259a^16 - 5139112971136947150633/587409483542016518a^14 + 6606766974222205553129/587409483542016518a^12 - 3125300132680050017025/293704741771008259a^10 + 4133015638838096759165/587409483542016518a^8 - 1859324857559585050525/587409483542016518a^6 + 549335641142108185445/587409483542016518a^4 - 43873092723383601886/293704741771008259a^2 + 1896424501663396740/293704741771008259, 134791516719007811/293704741771008259a^28 - 1930585445651314171/293704741771008259a^26 + 12718597576955129510/293704741771008259a^24 - 1035975434562947815/5389077830660702a^22 + 385264330143544264777/587409483542016518a^20 - 1025846572563784379951/587409483542016518a^18 + 2182682975573895442511/587409483542016518a^16 - 3668094078234307500073/587409483542016518a^14 + 4765658034399284378813/587409483542016518a^12 - 4582361017379729286491/587409483542016518a^10 + 3112579051720655808899/587409483542016518a^8 - 725354346238014793075/293704741771008259a^6 + 224525025773869853529/293704741771008259a^4 - 79890354161345784519/587409483542016518a^2 + 4526897620868447707/587409483542016518, 123021580046728623/587409483542016518a^28 - 889160370446894163/293704741771008259a^26 + 5920392542484019481/293704741771008259a^24 - 243381300463135039/2694538915330351a^22 + 182561500109477495977/587409483542016518a^20 - 245541553968797478782/293704741771008259a^18 + 1056884874983121506639/587409483542016518a^16 - 901346059278982692558/293704741771008259a^14 + 1194874951668378900380/293704741771008259a^12 - 1183826530824417909660/293704741771008259a^10 + 840966640410029401144/293704741771008259a^8 - 417063535068825410566/293704741771008259a^6 + 280141393361766873421/587409483542016518a^4 - 27971040813791272465/293704741771008259a^2 + 4487219615969003849/587409483542016518, 4587349496807/11180659399711a^28 - 129897861582483/22361318799422a^26 + 844170364116451/22361318799422a^24 - 3702605420690075/22361318799422a^22 + 12497165859465773/22361318799422a^20 - 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6210129055067596047/293704741771008259a + 8688361313091195/6830342831883913 (assuming GRH)	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$ 397294072.2356694 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{10}\cdot 397294072.2356694 \cdot 1}{2\cdot\sqrt{14414591053629993500878399466353926864896}}\cr\approx \mathstrut & 0.162472389195522 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^30 - 15*x^28 + 104*x^26 - 482*x^24 + 1708*x^22 - 4753*x^20 + 10608*x^18 - 18925*x^16 + 26563*x^14 - 28460*x^12 + 22463*x^10 - 12716*x^8 + 5049*x^6 - 1331*x^4 + 198*x^2 - 11)

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 - 15*x^28 + 104*x^26 - 482*x^24 + 1708*x^22 - 4753*x^20 + 10608*x^18 - 18925*x^16 + 26563*x^14 - 28460*x^12 + 22463*x^10 - 12716*x^8 + 5049*x^6 - 1331*x^4 + 198*x^2 - 11, 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 - 15*x^28 + 104*x^26 - 482*x^24 + 1708*x^22 - 4753*x^20 + 10608*x^18 - 18925*x^16 + 26563*x^14 - 28460*x^12 + 22463*x^10 - 12716*x^8 + 5049*x^6 - 1331*x^4 + 198*x^2 - 11);

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 - 15*x^28 + 104*x^26 - 482*x^24 + 1708*x^22 - 4753*x^20 + 10608*x^18 - 18925*x^16 + 26563*x^14 - 28460*x^12 + 22463*x^10 - 12716*x^8 + 5049*x^6 - 1331*x^4 + 198*x^2 - 11);

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

$S_3\times C_{10}$ (as 30T12):

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 30 conjugacy class representatives for $S_3\times C_{10}$

Character table for $S_3\times C_{10}$

Intermediate fields

$\Q(\sqrt{11}) $, 3.1.44.1, $\Q(\zeta_{11})^+$, 6.2.340736.1, $\Q(\zeta_{44})^+$, 15.5.35351257235385344.1

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:	30.0.1310417368511817590988945406032175169536.1
Minimal sibling:	30.0.1310417368511817590988945406032175169536.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	$30$	$15^{2}$	${\href{/padicField/7.10.0.1}{10} }^{2}{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}$	R	${\href{/padicField/13.10.0.1}{10} }^{3}$	${\href{/padicField/17.10.0.1}{10} }^{3}$	${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$	${\href{/padicField/23.6.0.1}{6} }^{5}$	${\href{/padicField/29.10.0.1}{10} }^{3}$	$30$	$15^{2}$	${\href{/padicField/41.10.0.1}{10} }^{3}$	${\href{/padicField/43.2.0.1}{2} }^{10}{,}\,{\href{/padicField/43.1.0.1}{1} }^{10}$	${\href{/padicField/47.10.0.1}{10} }^{3}$	${\href{/padicField/53.5.0.1}{5} }^{6}$	$30$

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
$2$ 2		Deg $30$	$6$	$5$	$40$
$11$ 11	11.10.9.1	$x^{10} + 110$	$10$	$1$	$9$	$C_{10}$	$[\ ]_{10}$
$11$ 11	11.20.18.1	$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$	$10$	$2$	$18$	20T3	$[\ ]_{10}^{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
	1.4.2t1.a.a	$1$	$ 2^{2}$	$\Q(\sqrt{-1}) $	$C_2$ (as 2T1)	$1$	$-1$
*	1.44.2t1.a.a	$1$	$ 2^{2} \cdot 11 $	$\Q(\sqrt{11}) $	$C_2$ (as 2T1)	$1$	$1$
	1.11.2t1.a.a	$1$	$ 11 $	$\Q(\sqrt{-11}) $	$C_2$ (as 2T1)	$1$	$-1$
*	1.44.10t1.b.a	$1$	$ 2^{2} \cdot 11 $	$\Q(\zeta_{44})^+$	$C_{10}$ (as 10T1)	$0$	$1$
	1.44.10t1.a.a	$1$	$ 2^{2} \cdot 11 $	10.0.219503494144.1	$C_{10}$ (as 10T1)	$0$	$-1$
	1.11.10t1.a.a	$1$	$ 11 $	$\Q(\zeta_{11})$	$C_{10}$ (as 10T1)	$0$	$-1$
*	1.11.5t1.a.a	$1$	$ 11 $	$\Q(\zeta_{11})^+$	$C_5$ (as 5T1)	$0$	$1$
	1.11.10t1.a.b	$1$	$ 11 $	$\Q(\zeta_{11})$	$C_{10}$ (as 10T1)	$0$	$-1$
*	1.44.10t1.b.b	$1$	$ 2^{2} \cdot 11 $	$\Q(\zeta_{44})^+$	$C_{10}$ (as 10T1)	$0$	$1$
	1.11.10t1.a.c	$1$	$ 11 $	$\Q(\zeta_{11})$	$C_{10}$ (as 10T1)	$0$	$-1$
	1.44.10t1.a.b	$1$	$ 2^{2} \cdot 11 $	10.0.219503494144.1	$C_{10}$ (as 10T1)	$0$	$-1$
	1.11.10t1.a.d	$1$	$ 11 $	$\Q(\zeta_{11})$	$C_{10}$ (as 10T1)	$0$	$-1$
*	1.11.5t1.a.b	$1$	$ 11 $	$\Q(\zeta_{11})^+$	$C_5$ (as 5T1)	$0$	$1$
*	1.11.5t1.a.c	$1$	$ 11 $	$\Q(\zeta_{11})^+$	$C_5$ (as 5T1)	$0$	$1$
*	1.44.10t1.b.c	$1$	$ 2^{2} \cdot 11 $	$\Q(\zeta_{44})^+$	$C_{10}$ (as 10T1)	$0$	$1$
*	1.11.5t1.a.d	$1$	$ 11 $	$\Q(\zeta_{11})^+$	$C_5$ (as 5T1)	$0$	$1$
	1.44.10t1.a.c	$1$	$ 2^{2} \cdot 11 $	10.0.219503494144.1	$C_{10}$ (as 10T1)	$0$	$-1$
	1.44.10t1.a.d	$1$	$ 2^{2} \cdot 11 $	10.0.219503494144.1	$C_{10}$ (as 10T1)	$0$	$-1$
*	1.44.10t1.b.d	$1$	$ 2^{2} \cdot 11 $	$\Q(\zeta_{44})^+$	$C_{10}$ (as 10T1)	$0$	$1$
*	2.176.6t3.b.a	$2$	$ 2^{4} \cdot 11 $	6.0.30976.1	$D_{6}$ (as 6T3)	$1$	$0$
*	2.44.3t2.b.a	$2$	$ 2^{2} \cdot 11 $	3.1.44.1	$S_3$ (as 3T2)	$1$	$0$
*	2.484.15t4.a.a	$2$	$ 2^{2} \cdot 11^{2}$	15.5.35351257235385344.1	$S_3 \times C_5$ (as 15T4)	$0$	$0$
*	2.484.15t4.a.b	$2$	$ 2^{2} \cdot 11^{2}$	15.5.35351257235385344.1	$S_3 \times C_5$ (as 15T4)	$0$	$0$
*	2.484.15t4.a.c	$2$	$ 2^{2} \cdot 11^{2}$	15.5.35351257235385344.1	$S_3 \times C_5$ (as 15T4)	$0$	$0$
*	2.1936.30t12.b.a	$2$	$ 2^{4} \cdot 11^{2}$	30.10.14414591053629993500878399466353926864896.1	$S_3\times C_{10}$ (as 30T12)	$0$	$0$
*	2.1936.30t12.b.b	$2$	$ 2^{4} \cdot 11^{2}$	30.10.14414591053629993500878399466353926864896.1	$S_3\times C_{10}$ (as 30T12)	$0$	$0$
*	2.1936.30t12.b.c	$2$	$ 2^{4} \cdot 11^{2}$	30.10.14414591053629993500878399466353926864896.1	$S_3\times C_{10}$ (as 30T12)	$0$	$0$
*	2.484.15t4.a.d	$2$	$ 2^{2} \cdot 11^{2}$	15.5.35351257235385344.1	$S_3 \times C_5$ (as 15T4)	$0$	$0$
*	2.1936.30t12.b.d	$2$	$ 2^{4} \cdot 11^{2}$	30.10.14414591053629993500878399466353926864896.1	$S_3\times C_{10}$ (as 30T12)	$0$	$0$

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 *.

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