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Properties

Label 30.0.266...291.1
Degree $30$
Signature $[0, 15]$
Discriminant $-2.667\times 10^{51}$
Root discriminant \(51.78\)
Ramified primes $11,19$
Class number $4$ (GRH)
Class group [2, 2] (GRH)
Galois group $C_3\times D_5$ (as 30T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 5*x^29 + 23*x^28 - 62*x^27 + 198*x^26 - 367*x^25 + 397*x^24 + 1188*x^23 - 7770*x^22 + 27186*x^21 - 70571*x^20 + 106780*x^19 + 113195*x^18 - 1691169*x^17 + 8127778*x^16 - 29421234*x^15 + 90493414*x^14 - 242325966*x^13 + 570478488*x^12 - 1182396448*x^11 + 2168615679*x^10 - 3480425904*x^9 + 4808261059*x^8 - 5571332106*x^7 + 5293750647*x^6 - 4028704993*x^5 + 2404982206*x^4 - 1088744723*x^3 + 357808383*x^2 - 76833878*x + 8961073)
 
gp: K = bnfinit(y^30 - 5*y^29 + 23*y^28 - 62*y^27 + 198*y^26 - 367*y^25 + 397*y^24 + 1188*y^23 - 7770*y^22 + 27186*y^21 - 70571*y^20 + 106780*y^19 + 113195*y^18 - 1691169*y^17 + 8127778*y^16 - 29421234*y^15 + 90493414*y^14 - 242325966*y^13 + 570478488*y^12 - 1182396448*y^11 + 2168615679*y^10 - 3480425904*y^9 + 4808261059*y^8 - 5571332106*y^7 + 5293750647*y^6 - 4028704993*y^5 + 2404982206*y^4 - 1088744723*y^3 + 357808383*y^2 - 76833878*y + 8961073, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 5*x^29 + 23*x^28 - 62*x^27 + 198*x^26 - 367*x^25 + 397*x^24 + 1188*x^23 - 7770*x^22 + 27186*x^21 - 70571*x^20 + 106780*x^19 + 113195*x^18 - 1691169*x^17 + 8127778*x^16 - 29421234*x^15 + 90493414*x^14 - 242325966*x^13 + 570478488*x^12 - 1182396448*x^11 + 2168615679*x^10 - 3480425904*x^9 + 4808261059*x^8 - 5571332106*x^7 + 5293750647*x^6 - 4028704993*x^5 + 2404982206*x^4 - 1088744723*x^3 + 357808383*x^2 - 76833878*x + 8961073);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 5*x^29 + 23*x^28 - 62*x^27 + 198*x^26 - 367*x^25 + 397*x^24 + 1188*x^23 - 7770*x^22 + 27186*x^21 - 70571*x^20 + 106780*x^19 + 113195*x^18 - 1691169*x^17 + 8127778*x^16 - 29421234*x^15 + 90493414*x^14 - 242325966*x^13 + 570478488*x^12 - 1182396448*x^11 + 2168615679*x^10 - 3480425904*x^9 + 4808261059*x^8 - 5571332106*x^7 + 5293750647*x^6 - 4028704993*x^5 + 2404982206*x^4 - 1088744723*x^3 + 357808383*x^2 - 76833878*x + 8961073)
 

\( x^{30} - 5 x^{29} + 23 x^{28} - 62 x^{27} + 198 x^{26} - 367 x^{25} + 397 x^{24} + 1188 x^{23} + \cdots + 8961073 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-2666804038012396184155132908419351509268871501674291\) \(\medspace = -\,11^{15}\cdot 19^{28}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(51.78\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $11^{1/2}19^{14/15}\approx 51.784490002099986$
Ramified primes:   \(11\), \(19\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-11}) \)
$\card{ \Gal(K/\Q) }$:  $30$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{22}a^{18}-\frac{1}{22}a^{17}+\frac{5}{11}a^{16}+\frac{9}{22}a^{15}+\frac{3}{11}a^{14}+\frac{3}{11}a^{13}-\frac{5}{22}a^{12}+\frac{1}{11}a^{11}+\frac{2}{11}a^{10}-\frac{1}{22}a^{9}-\frac{2}{11}a^{8}-\frac{9}{22}a^{7}+\frac{5}{11}a^{6}-\frac{1}{11}a^{5}-\frac{5}{22}a^{4}-\frac{3}{22}a^{3}+\frac{2}{11}a^{2}-\frac{1}{2}$, $\frac{1}{22}a^{19}+\frac{9}{22}a^{17}-\frac{3}{22}a^{16}-\frac{7}{22}a^{15}-\frac{5}{11}a^{14}+\frac{1}{22}a^{13}-\frac{3}{22}a^{12}+\frac{3}{11}a^{11}+\frac{3}{22}a^{10}-\frac{5}{22}a^{9}+\frac{9}{22}a^{8}+\frac{1}{22}a^{7}+\frac{4}{11}a^{6}-\frac{7}{22}a^{5}-\frac{4}{11}a^{4}+\frac{1}{22}a^{3}+\frac{2}{11}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{22}a^{20}+\frac{3}{11}a^{17}-\frac{9}{22}a^{16}-\frac{3}{22}a^{15}-\frac{9}{22}a^{14}+\frac{9}{22}a^{13}+\frac{7}{22}a^{12}+\frac{7}{22}a^{11}+\frac{3}{22}a^{10}-\frac{2}{11}a^{9}-\frac{7}{22}a^{8}+\frac{1}{22}a^{7}-\frac{9}{22}a^{6}+\frac{5}{11}a^{5}+\frac{1}{11}a^{4}+\frac{9}{22}a^{3}-\frac{3}{22}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{22}a^{21}-\frac{3}{22}a^{17}+\frac{3}{22}a^{16}+\frac{3}{22}a^{15}-\frac{5}{22}a^{14}-\frac{7}{22}a^{13}-\frac{7}{22}a^{12}-\frac{9}{22}a^{11}-\frac{3}{11}a^{10}-\frac{1}{22}a^{9}+\frac{3}{22}a^{8}+\frac{1}{22}a^{7}-\frac{3}{11}a^{6}-\frac{4}{11}a^{5}-\frac{5}{22}a^{4}-\frac{7}{22}a^{3}+\frac{9}{22}a^{2}-\frac{1}{2}a$, $\frac{1}{154}a^{22}-\frac{1}{77}a^{21}+\frac{1}{154}a^{20}+\frac{3}{154}a^{19}-\frac{27}{154}a^{17}+\frac{9}{154}a^{16}-\frac{15}{77}a^{15}-\frac{9}{77}a^{14}-\frac{73}{154}a^{13}+\frac{16}{77}a^{12}+\frac{65}{154}a^{11}-\frac{31}{154}a^{10}+\frac{7}{22}a^{9}-\frac{19}{154}a^{8}+\frac{5}{22}a^{7}-\frac{39}{154}a^{6}-\frac{2}{11}a^{5}+\frac{5}{77}a^{4}+\frac{5}{11}a^{3}-\frac{37}{77}a^{2}+\frac{1}{7}a+\frac{1}{14}$, $\frac{1}{154}a^{23}-\frac{3}{154}a^{21}-\frac{1}{77}a^{20}-\frac{1}{154}a^{19}+\frac{1}{154}a^{18}-\frac{12}{77}a^{17}+\frac{2}{7}a^{16}-\frac{32}{77}a^{15}+\frac{19}{77}a^{14}-\frac{8}{77}a^{13}-\frac{39}{154}a^{12}+\frac{32}{77}a^{11}+\frac{57}{154}a^{10}-\frac{20}{77}a^{9}+\frac{25}{154}a^{8}+\frac{73}{154}a^{7}+\frac{27}{154}a^{6}+\frac{31}{154}a^{5}-\frac{4}{77}a^{4}+\frac{3}{7}a^{3}-\frac{3}{22}a^{2}+\frac{5}{14}a+\frac{1}{7}$, $\frac{1}{308}a^{24}-\frac{1}{308}a^{22}+\frac{1}{308}a^{21}-\frac{3}{154}a^{20}-\frac{3}{308}a^{18}+\frac{151}{308}a^{17}-\frac{39}{308}a^{16}+\frac{26}{77}a^{15}+\frac{9}{154}a^{14}-\frac{6}{77}a^{13}+\frac{25}{77}a^{12}-\frac{79}{308}a^{11}+\frac{13}{77}a^{10}+\frac{1}{77}a^{9}+\frac{4}{11}a^{8}-\frac{9}{28}a^{7}-\frac{13}{154}a^{6}+\frac{125}{308}a^{5}-\frac{3}{77}a^{4}+\frac{13}{44}a^{3}-\frac{107}{308}a^{2}+\frac{13}{28}a-\frac{5}{28}$, $\frac{1}{31724}a^{25}+\frac{1}{1133}a^{24}-\frac{1}{2884}a^{23}-\frac{17}{31724}a^{22}-\frac{229}{15862}a^{21}+\frac{130}{7931}a^{20}-\frac{439}{31724}a^{19}-\frac{321}{31724}a^{18}-\frac{1413}{31724}a^{17}+\frac{157}{15862}a^{16}+\frac{1633}{7931}a^{15}-\frac{3730}{7931}a^{14}+\frac{7815}{15862}a^{13}+\frac{14169}{31724}a^{12}+\frac{1781}{15862}a^{11}-\frac{6885}{15862}a^{10}+\frac{1481}{15862}a^{9}-\frac{893}{4532}a^{8}-\frac{487}{1133}a^{7}-\frac{7213}{31724}a^{6}+\frac{325}{2266}a^{5}-\frac{1073}{31724}a^{4}-\frac{9377}{31724}a^{3}+\frac{12689}{31724}a^{2}+\frac{163}{412}a-\frac{705}{1442}$, $\frac{1}{348964}a^{26}-\frac{1}{87241}a^{25}+\frac{5}{87241}a^{24}+\frac{953}{348964}a^{23}-\frac{1047}{348964}a^{22}+\frac{241}{348964}a^{21}+\frac{1873}{348964}a^{20}+\frac{349}{31724}a^{19}-\frac{1931}{87241}a^{18}-\frac{5455}{348964}a^{17}+\frac{2973}{348964}a^{16}-\frac{2215}{7931}a^{15}+\frac{14551}{87241}a^{14}-\frac{41649}{348964}a^{13}-\frac{17650}{87241}a^{12}-\frac{5611}{49852}a^{11}+\frac{85017}{174482}a^{10}+\frac{121651}{348964}a^{9}+\frac{16111}{87241}a^{8}-\frac{6773}{24926}a^{7}+\frac{5750}{12463}a^{6}-\frac{4352}{87241}a^{5}-\frac{76805}{348964}a^{4}-\frac{235}{174482}a^{3}-\frac{4434}{12463}a^{2}-\frac{5889}{31724}a-\frac{1745}{31724}$, $\frac{1}{348964}a^{27}+\frac{1}{87241}a^{25}-\frac{25}{87241}a^{24}+\frac{499}{348964}a^{23}-\frac{137}{87241}a^{22}+\frac{1985}{174482}a^{21}-\frac{971}{49852}a^{20}+\frac{417}{174482}a^{19}-\frac{1747}{174482}a^{18}+\frac{9741}{24926}a^{17}-\frac{168277}{348964}a^{16}+\frac{59649}{174482}a^{15}-\frac{19571}{348964}a^{14}-\frac{25309}{87241}a^{13}+\frac{90735}{348964}a^{12}-\frac{21299}{49852}a^{11}-\frac{170327}{348964}a^{10}-\frac{31055}{87241}a^{9}-\frac{2528}{7931}a^{8}+\frac{172597}{348964}a^{7}-\frac{32269}{174482}a^{6}+\frac{3259}{174482}a^{5}+\frac{243}{174482}a^{4}+\frac{81307}{348964}a^{3}-\frac{26335}{174482}a^{2}+\frac{379}{15862}a+\frac{12281}{31724}$, $\frac{1}{766634475068}a^{28}-\frac{594263}{766634475068}a^{27}-\frac{117503}{383317237534}a^{26}+\frac{639241}{54759605362}a^{25}-\frac{300137163}{766634475068}a^{24}-\frac{778320623}{766634475068}a^{23}-\frac{77962936}{27379802681}a^{22}-\frac{8842599133}{766634475068}a^{21}-\frac{1944296693}{109519210724}a^{20}+\frac{20224901}{27379802681}a^{19}+\frac{2547196952}{191658618767}a^{18}-\frac{252355639149}{766634475068}a^{17}-\frac{44556317137}{109519210724}a^{16}+\frac{116472942301}{766634475068}a^{15}+\frac{365229356535}{766634475068}a^{14}-\frac{34701417409}{69694043188}a^{13}+\frac{1911008370}{27379802681}a^{12}-\frac{47283985773}{383317237534}a^{11}+\frac{381510091899}{766634475068}a^{10}+\frac{24613050991}{383317237534}a^{9}+\frac{307188795357}{766634475068}a^{8}-\frac{177381350007}{766634475068}a^{7}+\frac{183197880741}{383317237534}a^{6}-\frac{50593427}{4978145942}a^{5}+\frac{345457006239}{766634475068}a^{4}-\frac{310543049}{766634475068}a^{3}+\frac{76660817225}{191658618767}a^{2}-\frac{2310629469}{69694043188}a-\frac{19626810321}{69694043188}$, $\frac{1}{57\!\cdots\!76}a^{29}+\frac{11\!\cdots\!07}{28\!\cdots\!38}a^{28}-\frac{57\!\cdots\!09}{57\!\cdots\!76}a^{27}-\frac{49\!\cdots\!91}{52\!\cdots\!16}a^{26}-\frac{14\!\cdots\!20}{13\!\cdots\!79}a^{25}+\frac{12\!\cdots\!09}{14\!\cdots\!69}a^{24}-\frac{71\!\cdots\!23}{52\!\cdots\!16}a^{23}+\frac{73\!\cdots\!19}{52\!\cdots\!16}a^{22}-\frac{35\!\cdots\!19}{57\!\cdots\!76}a^{21}+\frac{84\!\cdots\!97}{41\!\cdots\!34}a^{20}-\frac{53\!\cdots\!67}{28\!\cdots\!38}a^{19}+\frac{67\!\cdots\!85}{20\!\cdots\!67}a^{18}-\frac{55\!\cdots\!21}{14\!\cdots\!69}a^{17}-\frac{54\!\cdots\!09}{57\!\cdots\!76}a^{16}-\frac{12\!\cdots\!17}{28\!\cdots\!38}a^{15}+\frac{12\!\cdots\!33}{29\!\cdots\!81}a^{14}+\frac{30\!\cdots\!43}{28\!\cdots\!38}a^{13}+\frac{22\!\cdots\!07}{57\!\cdots\!76}a^{12}+\frac{23\!\cdots\!21}{14\!\cdots\!69}a^{11}-\frac{83\!\cdots\!27}{57\!\cdots\!76}a^{10}+\frac{36\!\cdots\!25}{41\!\cdots\!34}a^{9}+\frac{44\!\cdots\!35}{57\!\cdots\!76}a^{8}-\frac{26\!\cdots\!95}{57\!\cdots\!76}a^{7}-\frac{10\!\cdots\!85}{57\!\cdots\!76}a^{6}+\frac{11\!\cdots\!07}{57\!\cdots\!76}a^{5}+\frac{28\!\cdots\!00}{14\!\cdots\!69}a^{4}+\frac{25\!\cdots\!85}{14\!\cdots\!69}a^{3}+\frac{69\!\cdots\!91}{28\!\cdots\!38}a^{2}-\frac{59\!\cdots\!57}{13\!\cdots\!79}a+\frac{60\!\cdots\!22}{13\!\cdots\!79}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $11$

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{58\!\cdots\!95}{10\!\cdots\!99}a^{29}-\frac{26\!\cdots\!20}{10\!\cdots\!99}a^{28}+\frac{11\!\cdots\!18}{10\!\cdots\!99}a^{27}-\frac{23\!\cdots\!67}{83\!\cdots\!19}a^{26}+\frac{86\!\cdots\!21}{91\!\cdots\!09}a^{25}-\frac{15\!\cdots\!64}{10\!\cdots\!99}a^{24}+\frac{98\!\cdots\!74}{91\!\cdots\!09}a^{23}+\frac{74\!\cdots\!60}{91\!\cdots\!09}a^{22}-\frac{41\!\cdots\!73}{10\!\cdots\!99}a^{21}+\frac{13\!\cdots\!27}{10\!\cdots\!99}a^{20}-\frac{32\!\cdots\!00}{10\!\cdots\!99}a^{19}+\frac{39\!\cdots\!61}{10\!\cdots\!99}a^{18}+\frac{10\!\cdots\!03}{10\!\cdots\!99}a^{17}-\frac{95\!\cdots\!86}{10\!\cdots\!99}a^{16}+\frac{41\!\cdots\!19}{10\!\cdots\!99}a^{15}-\frac{14\!\cdots\!12}{10\!\cdots\!99}a^{14}+\frac{43\!\cdots\!19}{10\!\cdots\!99}a^{13}-\frac{11\!\cdots\!93}{10\!\cdots\!99}a^{12}+\frac{25\!\cdots\!84}{10\!\cdots\!99}a^{11}-\frac{72\!\cdots\!46}{14\!\cdots\!57}a^{10}+\frac{12\!\cdots\!61}{14\!\cdots\!57}a^{9}-\frac{13\!\cdots\!69}{10\!\cdots\!99}a^{8}+\frac{17\!\cdots\!74}{10\!\cdots\!99}a^{7}-\frac{17\!\cdots\!35}{10\!\cdots\!99}a^{6}+\frac{14\!\cdots\!72}{10\!\cdots\!99}a^{5}-\frac{90\!\cdots\!14}{10\!\cdots\!99}a^{4}+\frac{40\!\cdots\!33}{10\!\cdots\!99}a^{3}-\frac{12\!\cdots\!52}{10\!\cdots\!99}a^{2}+\frac{22\!\cdots\!20}{91\!\cdots\!09}a-\frac{19\!\cdots\!14}{91\!\cdots\!09}$, $\frac{56\!\cdots\!31}{10\!\cdots\!99}a^{29}-\frac{31\!\cdots\!15}{10\!\cdots\!99}a^{28}+\frac{14\!\cdots\!73}{10\!\cdots\!99}a^{27}-\frac{37\!\cdots\!01}{91\!\cdots\!09}a^{26}+\frac{11\!\cdots\!37}{91\!\cdots\!09}a^{25}-\frac{25\!\cdots\!72}{10\!\cdots\!99}a^{24}+\frac{28\!\cdots\!73}{91\!\cdots\!09}a^{23}+\frac{55\!\cdots\!27}{91\!\cdots\!09}a^{22}-\frac{47\!\cdots\!10}{10\!\cdots\!99}a^{21}+\frac{17\!\cdots\!64}{10\!\cdots\!99}a^{20}-\frac{47\!\cdots\!99}{10\!\cdots\!99}a^{19}+\frac{78\!\cdots\!02}{10\!\cdots\!99}a^{18}+\frac{39\!\cdots\!80}{10\!\cdots\!99}a^{17}-\frac{10\!\cdots\!73}{10\!\cdots\!99}a^{16}+\frac{50\!\cdots\!08}{10\!\cdots\!99}a^{15}-\frac{18\!\cdots\!02}{10\!\cdots\!99}a^{14}+\frac{59\!\cdots\!94}{10\!\cdots\!99}a^{13}-\frac{16\!\cdots\!90}{10\!\cdots\!99}a^{12}+\frac{38\!\cdots\!76}{10\!\cdots\!99}a^{11}-\frac{11\!\cdots\!02}{14\!\cdots\!57}a^{10}+\frac{21\!\cdots\!19}{14\!\cdots\!57}a^{9}-\frac{24\!\cdots\!05}{10\!\cdots\!99}a^{8}+\frac{34\!\cdots\!10}{10\!\cdots\!99}a^{7}-\frac{40\!\cdots\!17}{10\!\cdots\!99}a^{6}+\frac{39\!\cdots\!47}{10\!\cdots\!99}a^{5}-\frac{29\!\cdots\!54}{10\!\cdots\!99}a^{4}+\frac{17\!\cdots\!66}{10\!\cdots\!99}a^{3}-\frac{68\!\cdots\!02}{10\!\cdots\!99}a^{2}+\frac{16\!\cdots\!88}{91\!\cdots\!09}a-\frac{18\!\cdots\!64}{91\!\cdots\!09}$, $\frac{17\!\cdots\!51}{57\!\cdots\!76}a^{29}-\frac{10\!\cdots\!23}{57\!\cdots\!76}a^{28}+\frac{62\!\cdots\!21}{57\!\cdots\!76}a^{27}+\frac{60\!\cdots\!35}{75\!\cdots\!88}a^{26}-\frac{26\!\cdots\!19}{52\!\cdots\!16}a^{25}+\frac{15\!\cdots\!48}{14\!\cdots\!69}a^{24}-\frac{83\!\cdots\!93}{37\!\cdots\!94}a^{23}+\frac{11\!\cdots\!53}{19\!\cdots\!08}a^{22}-\frac{41\!\cdots\!93}{82\!\cdots\!68}a^{21}-\frac{24\!\cdots\!45}{20\!\cdots\!67}a^{20}+\frac{51\!\cdots\!01}{57\!\cdots\!76}a^{19}-\frac{59\!\cdots\!13}{14\!\cdots\!69}a^{18}+\frac{14\!\cdots\!61}{11\!\cdots\!24}a^{17}-\frac{82\!\cdots\!51}{28\!\cdots\!38}a^{16}+\frac{17\!\cdots\!69}{57\!\cdots\!76}a^{15}+\frac{17\!\cdots\!77}{28\!\cdots\!38}a^{14}-\frac{65\!\cdots\!45}{11\!\cdots\!24}a^{13}+\frac{14\!\cdots\!09}{57\!\cdots\!76}a^{12}-\frac{11\!\cdots\!83}{14\!\cdots\!69}a^{11}+\frac{12\!\cdots\!91}{57\!\cdots\!76}a^{10}-\frac{14\!\cdots\!43}{28\!\cdots\!38}a^{9}+\frac{55\!\cdots\!87}{57\!\cdots\!76}a^{8}-\frac{45\!\cdots\!89}{28\!\cdots\!38}a^{7}+\frac{90\!\cdots\!01}{41\!\cdots\!34}a^{6}-\frac{69\!\cdots\!43}{28\!\cdots\!38}a^{5}+\frac{29\!\cdots\!39}{14\!\cdots\!69}a^{4}-\frac{17\!\cdots\!64}{14\!\cdots\!69}a^{3}+\frac{15\!\cdots\!67}{28\!\cdots\!38}a^{2}-\frac{72\!\cdots\!75}{52\!\cdots\!16}a+\frac{14\!\cdots\!95}{75\!\cdots\!88}$, $\frac{11\!\cdots\!07}{57\!\cdots\!76}a^{29}-\frac{49\!\cdots\!35}{57\!\cdots\!76}a^{28}+\frac{32\!\cdots\!69}{82\!\cdots\!68}a^{27}-\frac{12\!\cdots\!59}{13\!\cdots\!79}a^{26}+\frac{82\!\cdots\!87}{26\!\cdots\!58}a^{25}-\frac{39\!\cdots\!49}{82\!\cdots\!68}a^{24}+\frac{43\!\cdots\!81}{13\!\cdots\!79}a^{23}+\frac{72\!\cdots\!39}{26\!\cdots\!58}a^{22}-\frac{78\!\cdots\!87}{57\!\cdots\!76}a^{21}+\frac{35\!\cdots\!97}{82\!\cdots\!68}a^{20}-\frac{60\!\cdots\!61}{57\!\cdots\!76}a^{19}+\frac{35\!\cdots\!37}{28\!\cdots\!38}a^{18}+\frac{10\!\cdots\!07}{28\!\cdots\!38}a^{17}-\frac{45\!\cdots\!22}{14\!\cdots\!69}a^{16}+\frac{79\!\cdots\!87}{57\!\cdots\!76}a^{15}-\frac{13\!\cdots\!33}{28\!\cdots\!38}a^{14}+\frac{20\!\cdots\!50}{14\!\cdots\!69}a^{13}-\frac{52\!\cdots\!32}{14\!\cdots\!69}a^{12}+\frac{34\!\cdots\!85}{41\!\cdots\!34}a^{11}-\frac{13\!\cdots\!69}{82\!\cdots\!68}a^{10}+\frac{16\!\cdots\!47}{57\!\cdots\!76}a^{9}-\frac{12\!\cdots\!63}{28\!\cdots\!38}a^{8}+\frac{31\!\cdots\!43}{57\!\cdots\!76}a^{7}-\frac{32\!\cdots\!37}{57\!\cdots\!76}a^{6}+\frac{26\!\cdots\!51}{57\!\cdots\!76}a^{5}-\frac{41\!\cdots\!46}{14\!\cdots\!69}a^{4}+\frac{37\!\cdots\!09}{28\!\cdots\!38}a^{3}-\frac{11\!\cdots\!53}{28\!\cdots\!38}a^{2}+\frac{32\!\cdots\!69}{37\!\cdots\!94}a-\frac{50\!\cdots\!85}{52\!\cdots\!16}$, $\frac{43\!\cdots\!95}{57\!\cdots\!76}a^{29}-\frac{20\!\cdots\!69}{57\!\cdots\!76}a^{28}+\frac{46\!\cdots\!77}{28\!\cdots\!38}a^{27}-\frac{19\!\cdots\!69}{47\!\cdots\!56}a^{26}+\frac{34\!\cdots\!63}{26\!\cdots\!58}a^{25}-\frac{13\!\cdots\!15}{57\!\cdots\!76}a^{24}+\frac{51\!\cdots\!05}{26\!\cdots\!58}a^{23}+\frac{52\!\cdots\!31}{52\!\cdots\!16}a^{22}-\frac{16\!\cdots\!19}{28\!\cdots\!38}a^{21}+\frac{15\!\cdots\!85}{82\!\cdots\!68}a^{20}-\frac{66\!\cdots\!46}{14\!\cdots\!69}a^{19}+\frac{90\!\cdots\!15}{14\!\cdots\!69}a^{18}+\frac{65\!\cdots\!09}{57\!\cdots\!76}a^{17}-\frac{36\!\cdots\!19}{28\!\cdots\!38}a^{16}+\frac{47\!\cdots\!19}{82\!\cdots\!68}a^{15}-\frac{11\!\cdots\!15}{57\!\cdots\!76}a^{14}+\frac{35\!\cdots\!47}{57\!\cdots\!76}a^{13}-\frac{92\!\cdots\!17}{57\!\cdots\!76}a^{12}+\frac{10\!\cdots\!83}{28\!\cdots\!38}a^{11}-\frac{21\!\cdots\!27}{28\!\cdots\!38}a^{10}+\frac{38\!\cdots\!01}{28\!\cdots\!38}a^{9}-\frac{59\!\cdots\!61}{28\!\cdots\!38}a^{8}+\frac{39\!\cdots\!97}{14\!\cdots\!69}a^{7}-\frac{16\!\cdots\!97}{57\!\cdots\!76}a^{6}+\frac{14\!\cdots\!03}{57\!\cdots\!76}a^{5}-\frac{97\!\cdots\!89}{57\!\cdots\!76}a^{4}+\frac{48\!\cdots\!57}{57\!\cdots\!76}a^{3}-\frac{42\!\cdots\!98}{14\!\cdots\!69}a^{2}+\frac{34\!\cdots\!21}{52\!\cdots\!16}a-\frac{40\!\cdots\!29}{52\!\cdots\!16}$, $\frac{27\!\cdots\!19}{14\!\cdots\!69}a^{29}-\frac{91\!\cdots\!19}{82\!\cdots\!68}a^{28}+\frac{14\!\cdots\!27}{28\!\cdots\!38}a^{27}-\frac{74\!\cdots\!01}{52\!\cdots\!16}a^{26}+\frac{22\!\cdots\!57}{52\!\cdots\!16}a^{25}-\frac{25\!\cdots\!39}{28\!\cdots\!38}a^{24}+\frac{11\!\cdots\!48}{13\!\cdots\!79}a^{23}+\frac{12\!\cdots\!11}{52\!\cdots\!16}a^{22}-\frac{99\!\cdots\!05}{57\!\cdots\!76}a^{21}+\frac{49\!\cdots\!69}{82\!\cdots\!68}a^{20}-\frac{45\!\cdots\!71}{28\!\cdots\!38}a^{19}+\frac{73\!\cdots\!53}{28\!\cdots\!38}a^{18}+\frac{56\!\cdots\!17}{28\!\cdots\!38}a^{17}-\frac{10\!\cdots\!07}{28\!\cdots\!38}a^{16}+\frac{10\!\cdots\!63}{57\!\cdots\!76}a^{15}-\frac{18\!\cdots\!51}{28\!\cdots\!38}a^{14}+\frac{28\!\cdots\!48}{14\!\cdots\!69}a^{13}-\frac{77\!\cdots\!30}{14\!\cdots\!69}a^{12}+\frac{73\!\cdots\!91}{57\!\cdots\!76}a^{11}-\frac{76\!\cdots\!01}{28\!\cdots\!38}a^{10}+\frac{27\!\cdots\!01}{57\!\cdots\!76}a^{9}-\frac{45\!\cdots\!35}{59\!\cdots\!62}a^{8}+\frac{61\!\cdots\!11}{57\!\cdots\!76}a^{7}-\frac{69\!\cdots\!33}{57\!\cdots\!76}a^{6}+\frac{63\!\cdots\!25}{57\!\cdots\!76}a^{5}-\frac{44\!\cdots\!27}{57\!\cdots\!76}a^{4}+\frac{11\!\cdots\!45}{28\!\cdots\!38}a^{3}-\frac{44\!\cdots\!63}{28\!\cdots\!38}a^{2}+\frac{48\!\cdots\!44}{13\!\cdots\!79}a-\frac{62\!\cdots\!87}{13\!\cdots\!79}$, $\frac{12\!\cdots\!71}{57\!\cdots\!76}a^{29}-\frac{58\!\cdots\!29}{57\!\cdots\!76}a^{28}+\frac{26\!\cdots\!39}{57\!\cdots\!76}a^{27}-\frac{60\!\cdots\!85}{52\!\cdots\!16}a^{26}+\frac{19\!\cdots\!51}{52\!\cdots\!16}a^{25}-\frac{90\!\cdots\!21}{14\!\cdots\!69}a^{24}+\frac{13\!\cdots\!41}{26\!\cdots\!58}a^{23}+\frac{15\!\cdots\!51}{52\!\cdots\!16}a^{22}-\frac{92\!\cdots\!71}{57\!\cdots\!76}a^{21}+\frac{10\!\cdots\!27}{20\!\cdots\!67}a^{20}-\frac{74\!\cdots\!45}{57\!\cdots\!76}a^{19}+\frac{48\!\cdots\!37}{28\!\cdots\!38}a^{18}+\frac{20\!\cdots\!53}{57\!\cdots\!76}a^{17}-\frac{52\!\cdots\!25}{14\!\cdots\!69}a^{16}+\frac{93\!\cdots\!87}{57\!\cdots\!76}a^{15}-\frac{81\!\cdots\!80}{14\!\cdots\!69}a^{14}+\frac{98\!\cdots\!75}{57\!\cdots\!76}a^{13}-\frac{25\!\cdots\!55}{57\!\cdots\!76}a^{12}+\frac{14\!\cdots\!46}{14\!\cdots\!69}a^{11}-\frac{11\!\cdots\!93}{57\!\cdots\!76}a^{10}+\frac{52\!\cdots\!36}{14\!\cdots\!69}a^{9}-\frac{32\!\cdots\!47}{57\!\cdots\!76}a^{8}+\frac{10\!\cdots\!17}{14\!\cdots\!69}a^{7}-\frac{22\!\cdots\!33}{28\!\cdots\!38}a^{6}+\frac{18\!\cdots\!57}{28\!\cdots\!38}a^{5}-\frac{17\!\cdots\!11}{41\!\cdots\!34}a^{4}+\frac{56\!\cdots\!19}{28\!\cdots\!38}a^{3}-\frac{13\!\cdots\!45}{20\!\cdots\!67}a^{2}+\frac{70\!\cdots\!87}{52\!\cdots\!16}a-\frac{72\!\cdots\!61}{52\!\cdots\!16}$, $\frac{71\!\cdots\!51}{57\!\cdots\!76}a^{29}-\frac{24\!\cdots\!99}{57\!\cdots\!76}a^{28}+\frac{57\!\cdots\!93}{28\!\cdots\!38}a^{27}-\frac{99\!\cdots\!11}{26\!\cdots\!58}a^{26}+\frac{76\!\cdots\!27}{52\!\cdots\!16}a^{25}-\frac{19\!\cdots\!35}{14\!\cdots\!69}a^{24}-\frac{10\!\cdots\!35}{26\!\cdots\!58}a^{23}+\frac{24\!\cdots\!98}{13\!\cdots\!79}a^{22}-\frac{20\!\cdots\!21}{28\!\cdots\!38}a^{21}+\frac{41\!\cdots\!18}{20\!\cdots\!67}a^{20}-\frac{61\!\cdots\!53}{14\!\cdots\!69}a^{19}+\frac{65\!\cdots\!83}{28\!\cdots\!38}a^{18}+\frac{40\!\cdots\!76}{14\!\cdots\!69}a^{17}-\frac{25\!\cdots\!58}{14\!\cdots\!69}a^{16}+\frac{40\!\cdots\!45}{57\!\cdots\!76}a^{15}-\frac{12\!\cdots\!79}{57\!\cdots\!76}a^{14}+\frac{91\!\cdots\!32}{14\!\cdots\!69}a^{13}-\frac{22\!\cdots\!27}{14\!\cdots\!69}a^{12}+\frac{94\!\cdots\!27}{28\!\cdots\!38}a^{11}-\frac{17\!\cdots\!33}{28\!\cdots\!38}a^{10}+\frac{54\!\cdots\!51}{57\!\cdots\!76}a^{9}-\frac{70\!\cdots\!27}{57\!\cdots\!76}a^{8}+\frac{10\!\cdots\!57}{82\!\cdots\!68}a^{7}-\frac{10\!\cdots\!92}{14\!\cdots\!69}a^{6}+\frac{11\!\cdots\!45}{28\!\cdots\!38}a^{5}+\frac{27\!\cdots\!71}{57\!\cdots\!76}a^{4}-\frac{41\!\cdots\!13}{82\!\cdots\!68}a^{3}+\frac{40\!\cdots\!34}{14\!\cdots\!69}a^{2}-\frac{22\!\cdots\!17}{26\!\cdots\!58}a+\frac{73\!\cdots\!17}{52\!\cdots\!16}$, $\frac{16\!\cdots\!17}{57\!\cdots\!76}a^{29}-\frac{22\!\cdots\!26}{14\!\cdots\!69}a^{28}+\frac{10\!\cdots\!99}{14\!\cdots\!69}a^{27}-\frac{14\!\cdots\!63}{75\!\cdots\!88}a^{26}+\frac{15\!\cdots\!31}{26\!\cdots\!58}a^{25}-\frac{68\!\cdots\!39}{57\!\cdots\!76}a^{24}+\frac{57\!\cdots\!77}{48\!\cdots\!22}a^{23}+\frac{96\!\cdots\!25}{26\!\cdots\!58}a^{22}-\frac{51\!\cdots\!06}{20\!\cdots\!67}a^{21}+\frac{70\!\cdots\!53}{82\!\cdots\!68}a^{20}-\frac{63\!\cdots\!19}{28\!\cdots\!38}a^{19}+\frac{19\!\cdots\!43}{57\!\cdots\!76}a^{18}+\frac{28\!\cdots\!67}{82\!\cdots\!68}a^{17}-\frac{30\!\cdots\!95}{57\!\cdots\!76}a^{16}+\frac{36\!\cdots\!66}{14\!\cdots\!69}a^{15}-\frac{53\!\cdots\!65}{57\!\cdots\!76}a^{14}+\frac{58\!\cdots\!38}{20\!\cdots\!67}a^{13}-\frac{10\!\cdots\!35}{14\!\cdots\!69}a^{12}+\frac{51\!\cdots\!09}{28\!\cdots\!38}a^{11}-\frac{10\!\cdots\!37}{28\!\cdots\!38}a^{10}+\frac{96\!\cdots\!45}{14\!\cdots\!69}a^{9}-\frac{61\!\cdots\!95}{57\!\cdots\!76}a^{8}+\frac{82\!\cdots\!55}{57\!\cdots\!76}a^{7}-\frac{13\!\cdots\!55}{82\!\cdots\!68}a^{6}+\frac{41\!\cdots\!35}{28\!\cdots\!38}a^{5}-\frac{29\!\cdots\!07}{28\!\cdots\!38}a^{4}+\frac{76\!\cdots\!87}{14\!\cdots\!69}a^{3}-\frac{11\!\cdots\!83}{57\!\cdots\!76}a^{2}+\frac{24\!\cdots\!45}{52\!\cdots\!16}a-\frac{10\!\cdots\!88}{18\!\cdots\!97}$, $\frac{60\!\cdots\!17}{28\!\cdots\!38}a^{29}-\frac{47\!\cdots\!49}{82\!\cdots\!68}a^{28}+\frac{15\!\cdots\!97}{57\!\cdots\!76}a^{27}-\frac{16\!\cdots\!89}{52\!\cdots\!16}a^{26}+\frac{88\!\cdots\!21}{52\!\cdots\!16}a^{25}+\frac{24\!\cdots\!03}{57\!\cdots\!76}a^{24}-\frac{27\!\cdots\!65}{52\!\cdots\!16}a^{23}+\frac{47\!\cdots\!49}{13\!\cdots\!79}a^{22}-\frac{14\!\cdots\!78}{14\!\cdots\!69}a^{21}+\frac{46\!\cdots\!49}{20\!\cdots\!67}a^{20}-\frac{10\!\cdots\!91}{28\!\cdots\!38}a^{19}-\frac{32\!\cdots\!01}{57\!\cdots\!76}a^{18}+\frac{34\!\cdots\!97}{57\!\cdots\!76}a^{17}-\frac{80\!\cdots\!11}{28\!\cdots\!38}a^{16}+\frac{53\!\cdots\!81}{57\!\cdots\!76}a^{15}-\frac{15\!\cdots\!95}{57\!\cdots\!76}a^{14}+\frac{19\!\cdots\!37}{28\!\cdots\!38}a^{13}-\frac{80\!\cdots\!25}{57\!\cdots\!76}a^{12}+\frac{13\!\cdots\!43}{57\!\cdots\!76}a^{11}-\frac{15\!\cdots\!31}{57\!\cdots\!76}a^{10}+\frac{49\!\cdots\!87}{57\!\cdots\!76}a^{9}+\frac{36\!\cdots\!39}{59\!\cdots\!62}a^{8}-\frac{12\!\cdots\!77}{57\!\cdots\!76}a^{7}+\frac{24\!\cdots\!85}{57\!\cdots\!76}a^{6}-\frac{17\!\cdots\!69}{28\!\cdots\!38}a^{5}+\frac{35\!\cdots\!61}{57\!\cdots\!76}a^{4}-\frac{64\!\cdots\!30}{14\!\cdots\!69}a^{3}+\frac{13\!\cdots\!93}{57\!\cdots\!76}a^{2}-\frac{39\!\cdots\!05}{52\!\cdots\!16}a+\frac{16\!\cdots\!13}{13\!\cdots\!79}$, $\frac{42\!\cdots\!76}{14\!\cdots\!69}a^{29}-\frac{81\!\cdots\!45}{57\!\cdots\!76}a^{28}+\frac{36\!\cdots\!81}{57\!\cdots\!76}a^{27}-\frac{19\!\cdots\!77}{11\!\cdots\!89}a^{26}+\frac{68\!\cdots\!39}{13\!\cdots\!79}a^{25}-\frac{52\!\cdots\!45}{57\!\cdots\!76}a^{24}+\frac{40\!\cdots\!31}{52\!\cdots\!16}a^{23}+\frac{51\!\cdots\!44}{13\!\cdots\!79}a^{22}-\frac{12\!\cdots\!49}{57\!\cdots\!76}a^{21}+\frac{60\!\cdots\!39}{82\!\cdots\!68}a^{20}-\frac{26\!\cdots\!26}{14\!\cdots\!69}a^{19}+\frac{10\!\cdots\!05}{41\!\cdots\!34}a^{18}+\frac{26\!\cdots\!77}{57\!\cdots\!76}a^{17}-\frac{28\!\cdots\!67}{57\!\cdots\!76}a^{16}+\frac{13\!\cdots\!93}{57\!\cdots\!76}a^{15}-\frac{65\!\cdots\!57}{82\!\cdots\!68}a^{14}+\frac{13\!\cdots\!83}{57\!\cdots\!76}a^{13}-\frac{18\!\cdots\!95}{28\!\cdots\!38}a^{12}+\frac{20\!\cdots\!79}{14\!\cdots\!69}a^{11}-\frac{16\!\cdots\!59}{57\!\cdots\!76}a^{10}+\frac{10\!\cdots\!29}{20\!\cdots\!67}a^{9}-\frac{46\!\cdots\!89}{57\!\cdots\!76}a^{8}+\frac{61\!\cdots\!31}{57\!\cdots\!76}a^{7}-\frac{33\!\cdots\!77}{28\!\cdots\!38}a^{6}+\frac{28\!\cdots\!09}{28\!\cdots\!38}a^{5}-\frac{38\!\cdots\!51}{57\!\cdots\!76}a^{4}+\frac{19\!\cdots\!83}{57\!\cdots\!76}a^{3}-\frac{16\!\cdots\!58}{14\!\cdots\!69}a^{2}+\frac{13\!\cdots\!87}{52\!\cdots\!16}a-\frac{16\!\cdots\!45}{52\!\cdots\!16}$, $\frac{15\!\cdots\!81}{57\!\cdots\!76}a^{29}-\frac{69\!\cdots\!09}{57\!\cdots\!76}a^{28}+\frac{31\!\cdots\!11}{57\!\cdots\!76}a^{27}-\frac{36\!\cdots\!13}{26\!\cdots\!58}a^{26}+\frac{11\!\cdots\!45}{26\!\cdots\!58}a^{25}-\frac{42\!\cdots\!75}{57\!\cdots\!76}a^{24}+\frac{28\!\cdots\!81}{47\!\cdots\!27}a^{23}+\frac{94\!\cdots\!81}{26\!\cdots\!58}a^{22}-\frac{11\!\cdots\!49}{57\!\cdots\!76}a^{21}+\frac{51\!\cdots\!39}{82\!\cdots\!68}a^{20}-\frac{89\!\cdots\!41}{57\!\cdots\!76}a^{19}+\frac{11\!\cdots\!85}{59\!\cdots\!62}a^{18}+\frac{61\!\cdots\!17}{14\!\cdots\!69}a^{17}-\frac{12\!\cdots\!13}{28\!\cdots\!38}a^{16}+\frac{11\!\cdots\!87}{57\!\cdots\!76}a^{15}-\frac{14\!\cdots\!97}{20\!\cdots\!67}a^{14}+\frac{59\!\cdots\!73}{28\!\cdots\!38}a^{13}-\frac{77\!\cdots\!32}{14\!\cdots\!69}a^{12}+\frac{17\!\cdots\!41}{14\!\cdots\!69}a^{11}-\frac{14\!\cdots\!47}{57\!\cdots\!76}a^{10}+\frac{35\!\cdots\!59}{82\!\cdots\!68}a^{9}-\frac{19\!\cdots\!21}{28\!\cdots\!38}a^{8}+\frac{50\!\cdots\!43}{57\!\cdots\!76}a^{7}-\frac{53\!\cdots\!43}{57\!\cdots\!76}a^{6}+\frac{44\!\cdots\!91}{57\!\cdots\!76}a^{5}-\frac{71\!\cdots\!38}{14\!\cdots\!69}a^{4}+\frac{33\!\cdots\!46}{14\!\cdots\!69}a^{3}-\frac{11\!\cdots\!02}{14\!\cdots\!69}a^{2}+\frac{41\!\cdots\!65}{26\!\cdots\!58}a-\frac{79\!\cdots\!75}{52\!\cdots\!16}$, $\frac{50\!\cdots\!79}{26\!\cdots\!58}a^{29}-\frac{19\!\cdots\!33}{52\!\cdots\!16}a^{28}+\frac{27\!\cdots\!13}{13\!\cdots\!79}a^{27}-\frac{59\!\cdots\!39}{52\!\cdots\!16}a^{26}+\frac{18\!\cdots\!38}{13\!\cdots\!79}a^{25}+\frac{40\!\cdots\!27}{26\!\cdots\!58}a^{24}-\frac{18\!\cdots\!79}{52\!\cdots\!16}a^{23}+\frac{78\!\cdots\!99}{26\!\cdots\!58}a^{22}-\frac{31\!\cdots\!83}{47\!\cdots\!56}a^{21}+\frac{11\!\cdots\!59}{75\!\cdots\!88}a^{20}-\frac{94\!\cdots\!57}{47\!\cdots\!56}a^{19}-\frac{35\!\cdots\!41}{52\!\cdots\!16}a^{18}+\frac{25\!\cdots\!71}{52\!\cdots\!16}a^{17}-\frac{28\!\cdots\!71}{13\!\cdots\!79}a^{16}+\frac{35\!\cdots\!11}{52\!\cdots\!16}a^{15}-\frac{49\!\cdots\!07}{26\!\cdots\!58}a^{14}+\frac{59\!\cdots\!22}{13\!\cdots\!79}a^{13}-\frac{46\!\cdots\!89}{52\!\cdots\!16}a^{12}+\frac{71\!\cdots\!33}{52\!\cdots\!16}a^{11}-\frac{16\!\cdots\!18}{13\!\cdots\!79}a^{10}-\frac{15\!\cdots\!47}{52\!\cdots\!16}a^{9}+\frac{29\!\cdots\!09}{52\!\cdots\!16}a^{8}-\frac{11\!\cdots\!35}{75\!\cdots\!88}a^{7}+\frac{35\!\cdots\!43}{13\!\cdots\!79}a^{6}-\frac{17\!\cdots\!61}{52\!\cdots\!16}a^{5}+\frac{80\!\cdots\!99}{26\!\cdots\!58}a^{4}-\frac{14\!\cdots\!05}{75\!\cdots\!88}a^{3}+\frac{45\!\cdots\!63}{52\!\cdots\!16}a^{2}-\frac{10\!\cdots\!49}{43\!\cdots\!96}a+\frac{43\!\cdots\!99}{11\!\cdots\!89}$, $\frac{98\!\cdots\!23}{57\!\cdots\!76}a^{29}-\frac{13\!\cdots\!52}{14\!\cdots\!69}a^{28}+\frac{16\!\cdots\!75}{41\!\cdots\!34}a^{27}-\frac{27\!\cdots\!61}{23\!\cdots\!78}a^{26}+\frac{91\!\cdots\!17}{26\!\cdots\!58}a^{25}-\frac{56\!\cdots\!71}{82\!\cdots\!68}a^{24}+\frac{34\!\cdots\!79}{52\!\cdots\!16}a^{23}+\frac{11\!\cdots\!57}{52\!\cdots\!16}a^{22}-\frac{82\!\cdots\!09}{57\!\cdots\!76}a^{21}+\frac{10\!\cdots\!72}{20\!\cdots\!67}a^{20}-\frac{73\!\cdots\!63}{57\!\cdots\!76}a^{19}+\frac{11\!\cdots\!53}{57\!\cdots\!76}a^{18}+\frac{60\!\cdots\!99}{28\!\cdots\!38}a^{17}-\frac{88\!\cdots\!75}{28\!\cdots\!38}a^{16}+\frac{42\!\cdots\!09}{28\!\cdots\!38}a^{15}-\frac{30\!\cdots\!63}{57\!\cdots\!76}a^{14}+\frac{93\!\cdots\!99}{57\!\cdots\!76}a^{13}-\frac{12\!\cdots\!73}{28\!\cdots\!38}a^{12}+\frac{12\!\cdots\!33}{11\!\cdots\!24}a^{11}-\frac{43\!\cdots\!86}{20\!\cdots\!67}a^{10}+\frac{21\!\cdots\!77}{57\!\cdots\!76}a^{9}-\frac{34\!\cdots\!09}{57\!\cdots\!76}a^{8}+\frac{47\!\cdots\!83}{57\!\cdots\!76}a^{7}-\frac{52\!\cdots\!99}{57\!\cdots\!76}a^{6}+\frac{23\!\cdots\!01}{28\!\cdots\!38}a^{5}-\frac{33\!\cdots\!37}{57\!\cdots\!76}a^{4}+\frac{86\!\cdots\!09}{28\!\cdots\!38}a^{3}-\frac{63\!\cdots\!85}{57\!\cdots\!76}a^{2}+\frac{98\!\cdots\!01}{37\!\cdots\!94}a-\frac{17\!\cdots\!47}{52\!\cdots\!16}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 22790934759037.23 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 22790934759037.23 \cdot 4}{2\cdot\sqrt{2666804038012396184155132908419351509268871501674291}}\cr\approx \mathstrut & 0.828885623106817 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^30 - 5*x^29 + 23*x^28 - 62*x^27 + 198*x^26 - 367*x^25 + 397*x^24 + 1188*x^23 - 7770*x^22 + 27186*x^21 - 70571*x^20 + 106780*x^19 + 113195*x^18 - 1691169*x^17 + 8127778*x^16 - 29421234*x^15 + 90493414*x^14 - 242325966*x^13 + 570478488*x^12 - 1182396448*x^11 + 2168615679*x^10 - 3480425904*x^9 + 4808261059*x^8 - 5571332106*x^7 + 5293750647*x^6 - 4028704993*x^5 + 2404982206*x^4 - 1088744723*x^3 + 357808383*x^2 - 76833878*x + 8961073)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^30 - 5*x^29 + 23*x^28 - 62*x^27 + 198*x^26 - 367*x^25 + 397*x^24 + 1188*x^23 - 7770*x^22 + 27186*x^21 - 70571*x^20 + 106780*x^19 + 113195*x^18 - 1691169*x^17 + 8127778*x^16 - 29421234*x^15 + 90493414*x^14 - 242325966*x^13 + 570478488*x^12 - 1182396448*x^11 + 2168615679*x^10 - 3480425904*x^9 + 4808261059*x^8 - 5571332106*x^7 + 5293750647*x^6 - 4028704993*x^5 + 2404982206*x^4 - 1088744723*x^3 + 357808383*x^2 - 76833878*x + 8961073, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^30 - 5*x^29 + 23*x^28 - 62*x^27 + 198*x^26 - 367*x^25 + 397*x^24 + 1188*x^23 - 7770*x^22 + 27186*x^21 - 70571*x^20 + 106780*x^19 + 113195*x^18 - 1691169*x^17 + 8127778*x^16 - 29421234*x^15 + 90493414*x^14 - 242325966*x^13 + 570478488*x^12 - 1182396448*x^11 + 2168615679*x^10 - 3480425904*x^9 + 4808261059*x^8 - 5571332106*x^7 + 5293750647*x^6 - 4028704993*x^5 + 2404982206*x^4 - 1088744723*x^3 + 357808383*x^2 - 76833878*x + 8961073);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 5*x^29 + 23*x^28 - 62*x^27 + 198*x^26 - 367*x^25 + 397*x^24 + 1188*x^23 - 7770*x^22 + 27186*x^21 - 70571*x^20 + 106780*x^19 + 113195*x^18 - 1691169*x^17 + 8127778*x^16 - 29421234*x^15 + 90493414*x^14 - 242325966*x^13 + 570478488*x^12 - 1182396448*x^11 + 2168615679*x^10 - 3480425904*x^9 + 4808261059*x^8 - 5571332106*x^7 + 5293750647*x^6 - 4028704993*x^5 + 2404982206*x^4 - 1088744723*x^3 + 357808383*x^2 - 76833878*x + 8961073);
 
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

$C_3\times D_5$ (as 30T4):

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 30
The 12 conjugacy class representatives for $C_3\times D_5$
Character table for $C_3\times D_5$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.3.361.1, 5.1.15768841.1 x5, 6.0.173457251.1, 10.0.2735219811316091.1, 15.3.1415489083272211976282881.1 x5

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 15 sibling: 15.3.1415489083272211976282881.1
Minimal sibling: 15.3.1415489083272211976282881.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{5}$ $15^{2}$ $15^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{15}$ R ${\href{/padicField/13.6.0.1}{6} }^{5}$ ${\href{/padicField/17.6.0.1}{6} }^{5}$ R $15^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{5}$ ${\href{/padicField/31.5.0.1}{5} }^{6}$ ${\href{/padicField/37.5.0.1}{5} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{5}$ ${\href{/padicField/43.6.0.1}{6} }^{5}$ $15^{2}$ $15^{2}$ $15^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(11\) Copy content Toggle raw display 11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
\(19\) Copy content Toggle raw display Deg $30$$15$$2$$28$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.11.2t1.a.a$1$ $ 11 $ \(\Q(\sqrt{-11}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.19.3t1.a.b$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
* 1.19.3t1.a.a$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
* 1.209.6t1.b.b$1$ $ 11 \cdot 19 $ 6.0.173457251.1 $C_6$ (as 6T1) $0$ $-1$
* 1.209.6t1.b.a$1$ $ 11 \cdot 19 $ 6.0.173457251.1 $C_6$ (as 6T1) $0$ $-1$
*2 2.3971.5t2.a.b$2$ $ 11 \cdot 19^{2}$ 5.1.15768841.1 $D_{5}$ (as 5T2) $1$ $0$
*2 2.3971.5t2.a.a$2$ $ 11 \cdot 19^{2}$ 5.1.15768841.1 $D_{5}$ (as 5T2) $1$ $0$
*2 2.3971.15t3.b.c$2$ $ 11 \cdot 19^{2}$ 30.0.2666804038012396184155132908419351509268871501674291.1 $C_3\times D_5$ (as 30T4) $0$ $0$
*2 2.3971.15t3.b.d$2$ $ 11 \cdot 19^{2}$ 30.0.2666804038012396184155132908419351509268871501674291.1 $C_3\times D_5$ (as 30T4) $0$ $0$
*2 2.3971.15t3.b.b$2$ $ 11 \cdot 19^{2}$ 30.0.2666804038012396184155132908419351509268871501674291.1 $C_3\times D_5$ (as 30T4) $0$ $0$
*2 2.3971.15t3.b.a$2$ $ 11 \cdot 19^{2}$ 30.0.2666804038012396184155132908419351509268871501674291.1 $C_3\times D_5$ (as 30T4) $0$ $0$

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 *.