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Properties

Label 30.2.605...125.1
Degree $30$
Signature $[2, 14]$
Discriminant $6.055\times 10^{43}$
Root discriminant \(28.80\)
Ramified primes $5,239$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 4*x^29 - 4*x^28 + 32*x^27 + 29*x^26 - 207*x^25 - 80*x^24 + 808*x^23 + 324*x^22 - 2807*x^21 - 784*x^20 + 6744*x^19 + 2549*x^18 - 11743*x^17 - 5231*x^16 + 15441*x^15 + 10445*x^14 - 10233*x^13 - 7776*x^12 + 6706*x^11 + 5598*x^10 - 2410*x^9 - 2278*x^8 + 1045*x^7 + 940*x^6 - 185*x^5 - 176*x^4 + 44*x^3 + 30*x^2 - 6*x - 1)
 
gp: K = bnfinit(y^30 - 4*y^29 - 4*y^28 + 32*y^27 + 29*y^26 - 207*y^25 - 80*y^24 + 808*y^23 + 324*y^22 - 2807*y^21 - 784*y^20 + 6744*y^19 + 2549*y^18 - 11743*y^17 - 5231*y^16 + 15441*y^15 + 10445*y^14 - 10233*y^13 - 7776*y^12 + 6706*y^11 + 5598*y^10 - 2410*y^9 - 2278*y^8 + 1045*y^7 + 940*y^6 - 185*y^5 - 176*y^4 + 44*y^3 + 30*y^2 - 6*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 4*x^29 - 4*x^28 + 32*x^27 + 29*x^26 - 207*x^25 - 80*x^24 + 808*x^23 + 324*x^22 - 2807*x^21 - 784*x^20 + 6744*x^19 + 2549*x^18 - 11743*x^17 - 5231*x^16 + 15441*x^15 + 10445*x^14 - 10233*x^13 - 7776*x^12 + 6706*x^11 + 5598*x^10 - 2410*x^9 - 2278*x^8 + 1045*x^7 + 940*x^6 - 185*x^5 - 176*x^4 + 44*x^3 + 30*x^2 - 6*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 4*x^29 - 4*x^28 + 32*x^27 + 29*x^26 - 207*x^25 - 80*x^24 + 808*x^23 + 324*x^22 - 2807*x^21 - 784*x^20 + 6744*x^19 + 2549*x^18 - 11743*x^17 - 5231*x^16 + 15441*x^15 + 10445*x^14 - 10233*x^13 - 7776*x^12 + 6706*x^11 + 5598*x^10 - 2410*x^9 - 2278*x^8 + 1045*x^7 + 940*x^6 - 185*x^5 - 176*x^4 + 44*x^3 + 30*x^2 - 6*x - 1)
 

\( x^{30} - 4 x^{29} - 4 x^{28} + 32 x^{27} + 29 x^{26} - 207 x^{25} - 80 x^{24} + 808 x^{23} + 324 x^{22} + \cdots - 1 \) 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:  $[2, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(60550910545856385116788597847358428955078125\) \(\medspace = 5^{15}\cdot 239^{14}\) 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:  \(28.80\)
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:  $5^{1/2}239^{1/2}\approx 34.56877203488721$
Ramified primes:   \(5\), \(239\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{7}a^{24}-\frac{2}{7}a^{23}+\frac{3}{7}a^{22}+\frac{3}{7}a^{21}+\frac{2}{7}a^{20}-\frac{3}{7}a^{19}-\frac{2}{7}a^{18}+\frac{1}{7}a^{17}-\frac{3}{7}a^{16}-\frac{3}{7}a^{15}-\frac{2}{7}a^{14}-\frac{1}{7}a^{13}-\frac{3}{7}a^{12}-\frac{1}{7}a^{11}+\frac{2}{7}a^{9}-\frac{3}{7}a^{8}+\frac{1}{7}a^{6}+\frac{3}{7}a^{5}-\frac{1}{7}a^{4}+\frac{1}{7}a^{3}+\frac{1}{7}a^{2}-\frac{2}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{25}-\frac{1}{7}a^{23}+\frac{2}{7}a^{22}+\frac{1}{7}a^{21}+\frac{1}{7}a^{20}-\frac{1}{7}a^{19}-\frac{3}{7}a^{18}-\frac{1}{7}a^{17}-\frac{2}{7}a^{16}-\frac{1}{7}a^{15}+\frac{2}{7}a^{14}+\frac{2}{7}a^{13}-\frac{2}{7}a^{11}+\frac{2}{7}a^{10}+\frac{1}{7}a^{9}+\frac{1}{7}a^{8}+\frac{1}{7}a^{7}-\frac{2}{7}a^{6}-\frac{2}{7}a^{5}-\frac{1}{7}a^{4}+\frac{3}{7}a^{3}-\frac{3}{7}a+\frac{2}{7}$, $\frac{1}{91}a^{26}+\frac{4}{91}a^{25}-\frac{5}{91}a^{24}+\frac{1}{7}a^{23}-\frac{17}{91}a^{22}+\frac{1}{13}a^{21}+\frac{37}{91}a^{20}+\frac{40}{91}a^{19}+\frac{16}{91}a^{18}-\frac{17}{91}a^{17}-\frac{3}{7}a^{16}+\frac{38}{91}a^{15}+\frac{32}{91}a^{14}-\frac{37}{91}a^{13}+\frac{24}{91}a^{12}+\frac{5}{91}a^{11}+\frac{30}{91}a^{10}+\frac{32}{91}a^{9}+\frac{24}{91}a^{8}-\frac{40}{91}a^{7}-\frac{1}{13}a^{6}-\frac{1}{13}a^{5}-\frac{25}{91}a^{4}+\frac{22}{91}a^{3}+\frac{2}{13}a^{2}+\frac{19}{91}a-\frac{10}{91}$, $\frac{1}{91}a^{27}+\frac{5}{91}a^{25}-\frac{6}{91}a^{24}-\frac{17}{91}a^{23}+\frac{10}{91}a^{22}+\frac{9}{91}a^{21}+\frac{22}{91}a^{20}+\frac{38}{91}a^{19}+\frac{10}{91}a^{18}-\frac{36}{91}a^{17}-\frac{2}{13}a^{16}-\frac{29}{91}a^{15}-\frac{5}{13}a^{14}-\frac{10}{91}a^{13}+\frac{2}{7}a^{12}-\frac{3}{91}a^{11}-\frac{36}{91}a^{10}+\frac{2}{7}a^{9}+\frac{1}{13}a^{8}-\frac{3}{91}a^{7}+\frac{3}{13}a^{6}+\frac{16}{91}a^{5}+\frac{44}{91}a^{4}-\frac{5}{13}a^{3}+\frac{15}{91}a^{2}+\frac{5}{91}a-\frac{38}{91}$, $\frac{1}{14676773839}a^{28}+\frac{3707215}{14676773839}a^{27}-\frac{6303146}{2096681977}a^{26}-\frac{981496800}{14676773839}a^{25}+\frac{555778896}{14676773839}a^{24}+\frac{6249378762}{14676773839}a^{23}+\frac{1781905890}{14676773839}a^{22}-\frac{3354810841}{14676773839}a^{21}-\frac{313809588}{772461781}a^{20}+\frac{5130237142}{14676773839}a^{19}+\frac{2122701111}{14676773839}a^{18}+\frac{5849101380}{14676773839}a^{17}-\frac{6851951881}{14676773839}a^{16}+\frac{5459439599}{14676773839}a^{15}+\frac{4499370672}{14676773839}a^{14}+\frac{377439652}{1128982603}a^{13}+\frac{4334058208}{14676773839}a^{12}-\frac{13751700}{84836843}a^{11}+\frac{50944304}{110351683}a^{10}+\frac{1029226629}{2096681977}a^{9}-\frac{579245126}{14676773839}a^{8}+\frac{2131576129}{14676773839}a^{7}+\frac{3122865965}{14676773839}a^{6}+\frac{5744248316}{14676773839}a^{5}+\frac{834951158}{14676773839}a^{4}-\frac{409863199}{1128982603}a^{3}+\frac{6701117463}{14676773839}a^{2}-\frac{2631267697}{14676773839}a-\frac{6162129166}{14676773839}$, $\frac{1}{53\!\cdots\!81}a^{29}-\frac{21\!\cdots\!39}{53\!\cdots\!81}a^{28}+\frac{16\!\cdots\!51}{33\!\cdots\!21}a^{27}-\frac{26\!\cdots\!08}{53\!\cdots\!81}a^{26}-\frac{77\!\cdots\!89}{53\!\cdots\!81}a^{25}-\frac{34\!\cdots\!85}{53\!\cdots\!81}a^{24}+\frac{12\!\cdots\!14}{53\!\cdots\!81}a^{23}-\frac{26\!\cdots\!16}{53\!\cdots\!81}a^{22}-\frac{27\!\cdots\!63}{76\!\cdots\!83}a^{21}-\frac{60\!\cdots\!97}{53\!\cdots\!81}a^{20}+\frac{60\!\cdots\!93}{40\!\cdots\!37}a^{19}+\frac{19\!\cdots\!86}{53\!\cdots\!81}a^{18}+\frac{65\!\cdots\!56}{53\!\cdots\!81}a^{17}+\frac{15\!\cdots\!33}{53\!\cdots\!81}a^{16}-\frac{18\!\cdots\!48}{53\!\cdots\!81}a^{15}-\frac{51\!\cdots\!76}{53\!\cdots\!81}a^{14}+\frac{99\!\cdots\!26}{53\!\cdots\!81}a^{13}+\frac{22\!\cdots\!30}{53\!\cdots\!81}a^{12}+\frac{85\!\cdots\!45}{40\!\cdots\!37}a^{11}+\frac{14\!\cdots\!29}{76\!\cdots\!83}a^{10}+\frac{11\!\cdots\!03}{53\!\cdots\!81}a^{9}+\frac{69\!\cdots\!69}{53\!\cdots\!81}a^{8}+\frac{15\!\cdots\!50}{53\!\cdots\!81}a^{7}-\frac{95\!\cdots\!46}{40\!\cdots\!57}a^{6}-\frac{22\!\cdots\!40}{53\!\cdots\!81}a^{5}+\frac{12\!\cdots\!00}{53\!\cdots\!81}a^{4}-\frac{13\!\cdots\!12}{53\!\cdots\!81}a^{3}-\frac{20\!\cdots\!47}{53\!\cdots\!81}a^{2}+\frac{25\!\cdots\!44}{53\!\cdots\!81}a-\frac{15\!\cdots\!30}{53\!\cdots\!81}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{54\!\cdots\!54}{53\!\cdots\!81}a^{29}-\frac{21\!\cdots\!60}{53\!\cdots\!81}a^{28}-\frac{10\!\cdots\!21}{23\!\cdots\!47}a^{27}+\frac{16\!\cdots\!04}{53\!\cdots\!81}a^{26}+\frac{17\!\cdots\!92}{53\!\cdots\!81}a^{25}-\frac{41\!\cdots\!83}{20\!\cdots\!41}a^{24}-\frac{56\!\cdots\!05}{53\!\cdots\!81}a^{23}+\frac{42\!\cdots\!55}{53\!\cdots\!81}a^{22}+\frac{23\!\cdots\!70}{53\!\cdots\!81}a^{21}-\frac{14\!\cdots\!43}{53\!\cdots\!81}a^{20}-\frac{61\!\cdots\!95}{53\!\cdots\!81}a^{19}+\frac{34\!\cdots\!09}{53\!\cdots\!81}a^{18}+\frac{53\!\cdots\!89}{15\!\cdots\!77}a^{17}-\frac{30\!\cdots\!53}{28\!\cdots\!99}a^{16}-\frac{37\!\cdots\!03}{53\!\cdots\!81}a^{15}+\frac{73\!\cdots\!24}{53\!\cdots\!81}a^{14}+\frac{68\!\cdots\!90}{53\!\cdots\!81}a^{13}-\frac{38\!\cdots\!13}{53\!\cdots\!81}a^{12}-\frac{70\!\cdots\!11}{76\!\cdots\!83}a^{11}+\frac{15\!\cdots\!21}{40\!\cdots\!37}a^{10}+\frac{32\!\cdots\!18}{53\!\cdots\!81}a^{9}-\frac{30\!\cdots\!41}{76\!\cdots\!83}a^{8}-\frac{13\!\cdots\!01}{58\!\cdots\!91}a^{7}+\frac{11\!\cdots\!53}{53\!\cdots\!81}a^{6}+\frac{46\!\cdots\!13}{53\!\cdots\!81}a^{5}+\frac{11\!\cdots\!44}{53\!\cdots\!81}a^{4}-\frac{55\!\cdots\!63}{40\!\cdots\!57}a^{3}-\frac{26\!\cdots\!32}{40\!\cdots\!37}a^{2}+\frac{69\!\cdots\!42}{53\!\cdots\!81}a+\frac{62\!\cdots\!10}{53\!\cdots\!81}$, $\frac{31\!\cdots\!99}{53\!\cdots\!81}a^{29}-\frac{59\!\cdots\!19}{53\!\cdots\!81}a^{28}-\frac{16\!\cdots\!06}{23\!\cdots\!47}a^{27}+\frac{68\!\cdots\!17}{53\!\cdots\!81}a^{26}+\frac{42\!\cdots\!19}{76\!\cdots\!83}a^{25}-\frac{15\!\cdots\!28}{20\!\cdots\!41}a^{24}-\frac{15\!\cdots\!74}{53\!\cdots\!81}a^{23}+\frac{17\!\cdots\!42}{53\!\cdots\!81}a^{22}+\frac{62\!\cdots\!69}{53\!\cdots\!81}a^{21}-\frac{55\!\cdots\!20}{53\!\cdots\!81}a^{20}-\frac{20\!\cdots\!16}{53\!\cdots\!81}a^{19}+\frac{12\!\cdots\!79}{53\!\cdots\!81}a^{18}+\frac{73\!\cdots\!81}{76\!\cdots\!83}a^{17}-\frac{58\!\cdots\!13}{28\!\cdots\!99}a^{16}-\frac{90\!\cdots\!52}{53\!\cdots\!81}a^{15}-\frac{12\!\cdots\!80}{53\!\cdots\!81}a^{14}+\frac{10\!\cdots\!62}{43\!\cdots\!71}a^{13}+\frac{56\!\cdots\!73}{53\!\cdots\!81}a^{12}-\frac{78\!\cdots\!08}{53\!\cdots\!81}a^{11}-\frac{60\!\cdots\!59}{76\!\cdots\!83}a^{10}+\frac{50\!\cdots\!13}{53\!\cdots\!81}a^{9}+\frac{35\!\cdots\!42}{53\!\cdots\!81}a^{8}-\frac{15\!\cdots\!60}{53\!\cdots\!81}a^{7}-\frac{12\!\cdots\!84}{53\!\cdots\!81}a^{6}+\frac{63\!\cdots\!12}{53\!\cdots\!81}a^{5}+\frac{56\!\cdots\!55}{53\!\cdots\!81}a^{4}-\frac{25\!\cdots\!66}{28\!\cdots\!99}a^{3}-\frac{11\!\cdots\!29}{76\!\cdots\!83}a^{2}+\frac{50\!\cdots\!78}{40\!\cdots\!37}a+\frac{70\!\cdots\!70}{40\!\cdots\!37}$, $\frac{35\!\cdots\!94}{76\!\cdots\!83}a^{29}-\frac{10\!\cdots\!43}{53\!\cdots\!81}a^{28}-\frac{22\!\cdots\!63}{33\!\cdots\!21}a^{27}+\frac{72\!\cdots\!13}{53\!\cdots\!81}a^{26}+\frac{44\!\cdots\!70}{76\!\cdots\!83}a^{25}-\frac{24\!\cdots\!90}{28\!\cdots\!63}a^{24}+\frac{70\!\cdots\!70}{53\!\cdots\!81}a^{23}+\frac{21\!\cdots\!26}{76\!\cdots\!83}a^{22}-\frac{18\!\cdots\!78}{58\!\cdots\!91}a^{21}-\frac{38\!\cdots\!74}{40\!\cdots\!37}a^{20}+\frac{15\!\cdots\!61}{53\!\cdots\!81}a^{19}+\frac{98\!\cdots\!16}{53\!\cdots\!81}a^{18}-\frac{20\!\cdots\!74}{53\!\cdots\!81}a^{17}-\frac{68\!\cdots\!80}{28\!\cdots\!99}a^{16}+\frac{43\!\cdots\!51}{53\!\cdots\!81}a^{15}+\frac{11\!\cdots\!41}{53\!\cdots\!81}a^{14}-\frac{13\!\cdots\!59}{53\!\cdots\!81}a^{13}+\frac{66\!\cdots\!02}{53\!\cdots\!81}a^{12}+\frac{15\!\cdots\!63}{53\!\cdots\!81}a^{11}-\frac{24\!\cdots\!58}{40\!\cdots\!37}a^{10}-\frac{13\!\cdots\!50}{76\!\cdots\!83}a^{9}+\frac{65\!\cdots\!37}{53\!\cdots\!81}a^{8}+\frac{11\!\cdots\!05}{76\!\cdots\!83}a^{7}-\frac{15\!\cdots\!83}{53\!\cdots\!81}a^{6}-\frac{25\!\cdots\!30}{53\!\cdots\!81}a^{5}+\frac{12\!\cdots\!94}{53\!\cdots\!81}a^{4}+\frac{63\!\cdots\!48}{28\!\cdots\!99}a^{3}-\frac{21\!\cdots\!87}{53\!\cdots\!81}a^{2}-\frac{55\!\cdots\!61}{53\!\cdots\!81}a+\frac{55\!\cdots\!09}{53\!\cdots\!81}$, $\frac{56\!\cdots\!23}{53\!\cdots\!81}a^{29}+\frac{28\!\cdots\!16}{53\!\cdots\!81}a^{28}-\frac{10\!\cdots\!60}{23\!\cdots\!47}a^{27}+\frac{80\!\cdots\!54}{53\!\cdots\!81}a^{26}+\frac{17\!\cdots\!55}{53\!\cdots\!81}a^{25}-\frac{14\!\cdots\!41}{20\!\cdots\!41}a^{24}-\frac{11\!\cdots\!46}{53\!\cdots\!81}a^{23}+\frac{50\!\cdots\!10}{53\!\cdots\!81}a^{22}+\frac{40\!\cdots\!77}{53\!\cdots\!81}a^{21}-\frac{15\!\cdots\!16}{53\!\cdots\!81}a^{20}-\frac{20\!\cdots\!21}{76\!\cdots\!83}a^{19}+\frac{41\!\cdots\!16}{40\!\cdots\!37}a^{18}+\frac{25\!\cdots\!03}{40\!\cdots\!37}a^{17}-\frac{31\!\cdots\!47}{28\!\cdots\!99}a^{16}-\frac{57\!\cdots\!85}{53\!\cdots\!81}a^{15}+\frac{37\!\cdots\!11}{53\!\cdots\!81}a^{14}+\frac{76\!\cdots\!91}{53\!\cdots\!81}a^{13}+\frac{19\!\cdots\!83}{53\!\cdots\!81}a^{12}-\frac{74\!\cdots\!83}{76\!\cdots\!83}a^{11}-\frac{15\!\cdots\!13}{58\!\cdots\!91}a^{10}+\frac{45\!\cdots\!11}{76\!\cdots\!83}a^{9}+\frac{12\!\cdots\!09}{53\!\cdots\!81}a^{8}-\frac{11\!\cdots\!15}{53\!\cdots\!81}a^{7}-\frac{50\!\cdots\!62}{53\!\cdots\!81}a^{6}+\frac{43\!\cdots\!50}{53\!\cdots\!81}a^{5}+\frac{22\!\cdots\!49}{53\!\cdots\!81}a^{4}-\frac{40\!\cdots\!75}{40\!\cdots\!57}a^{3}-\frac{55\!\cdots\!64}{76\!\cdots\!83}a^{2}+\frac{38\!\cdots\!21}{40\!\cdots\!37}a+\frac{72\!\cdots\!36}{53\!\cdots\!81}$, $\frac{21\!\cdots\!40}{53\!\cdots\!81}a^{29}-\frac{11\!\cdots\!93}{53\!\cdots\!81}a^{28}+\frac{27\!\cdots\!16}{23\!\cdots\!47}a^{27}+\frac{69\!\cdots\!83}{53\!\cdots\!81}a^{26}-\frac{43\!\cdots\!82}{53\!\cdots\!81}a^{25}-\frac{17\!\cdots\!78}{20\!\cdots\!41}a^{24}+\frac{51\!\cdots\!59}{53\!\cdots\!81}a^{23}+\frac{14\!\cdots\!95}{53\!\cdots\!81}a^{22}-\frac{18\!\cdots\!50}{53\!\cdots\!81}a^{21}-\frac{50\!\cdots\!69}{53\!\cdots\!81}a^{20}+\frac{10\!\cdots\!82}{76\!\cdots\!83}a^{19}+\frac{99\!\cdots\!03}{53\!\cdots\!81}a^{18}-\frac{15\!\cdots\!89}{53\!\cdots\!81}a^{17}-\frac{89\!\cdots\!33}{28\!\cdots\!99}a^{16}+\frac{26\!\cdots\!99}{53\!\cdots\!81}a^{15}+\frac{21\!\cdots\!43}{53\!\cdots\!81}a^{14}-\frac{30\!\cdots\!09}{53\!\cdots\!81}a^{13}-\frac{18\!\cdots\!00}{53\!\cdots\!81}a^{12}+\frac{29\!\cdots\!40}{53\!\cdots\!81}a^{11}+\frac{13\!\cdots\!39}{53\!\cdots\!81}a^{10}-\frac{14\!\cdots\!29}{40\!\cdots\!37}a^{9}-\frac{68\!\cdots\!36}{53\!\cdots\!81}a^{8}+\frac{94\!\cdots\!94}{53\!\cdots\!81}a^{7}+\frac{26\!\cdots\!57}{40\!\cdots\!37}a^{6}-\frac{35\!\cdots\!09}{53\!\cdots\!81}a^{5}-\frac{10\!\cdots\!38}{53\!\cdots\!81}a^{4}+\frac{50\!\cdots\!63}{28\!\cdots\!99}a^{3}+\frac{45\!\cdots\!07}{76\!\cdots\!83}a^{2}-\frac{93\!\cdots\!57}{53\!\cdots\!81}a-\frac{30\!\cdots\!80}{40\!\cdots\!37}$, $\frac{53\!\cdots\!56}{53\!\cdots\!81}a^{29}-\frac{35\!\cdots\!28}{53\!\cdots\!81}a^{28}+\frac{21\!\cdots\!79}{23\!\cdots\!47}a^{27}+\frac{16\!\cdots\!55}{53\!\cdots\!81}a^{26}-\frac{47\!\cdots\!07}{76\!\cdots\!83}a^{25}-\frac{38\!\cdots\!61}{20\!\cdots\!41}a^{24}+\frac{27\!\cdots\!61}{53\!\cdots\!81}a^{23}+\frac{32\!\cdots\!17}{76\!\cdots\!83}a^{22}-\frac{98\!\cdots\!64}{53\!\cdots\!81}a^{21}-\frac{69\!\cdots\!08}{53\!\cdots\!81}a^{20}+\frac{27\!\cdots\!06}{40\!\cdots\!37}a^{19}+\frac{36\!\cdots\!60}{53\!\cdots\!81}a^{18}-\frac{11\!\cdots\!89}{76\!\cdots\!83}a^{17}+\frac{39\!\cdots\!95}{28\!\cdots\!99}a^{16}+\frac{13\!\cdots\!01}{53\!\cdots\!81}a^{15}-\frac{31\!\cdots\!96}{53\!\cdots\!81}a^{14}-\frac{16\!\cdots\!80}{53\!\cdots\!81}a^{13}+\frac{52\!\cdots\!53}{53\!\cdots\!81}a^{12}+\frac{21\!\cdots\!13}{76\!\cdots\!83}a^{11}-\frac{47\!\cdots\!45}{53\!\cdots\!81}a^{10}-\frac{83\!\cdots\!73}{53\!\cdots\!81}a^{9}+\frac{41\!\cdots\!60}{53\!\cdots\!81}a^{8}+\frac{34\!\cdots\!35}{53\!\cdots\!81}a^{7}-\frac{16\!\cdots\!11}{53\!\cdots\!81}a^{6}-\frac{12\!\cdots\!39}{76\!\cdots\!83}a^{5}+\frac{76\!\cdots\!73}{53\!\cdots\!81}a^{4}+\frac{12\!\cdots\!01}{28\!\cdots\!99}a^{3}-\frac{21\!\cdots\!15}{53\!\cdots\!81}a^{2}+\frac{15\!\cdots\!30}{53\!\cdots\!81}a+\frac{43\!\cdots\!89}{53\!\cdots\!81}$, $\frac{44\!\cdots\!16}{53\!\cdots\!81}a^{29}-\frac{55\!\cdots\!27}{53\!\cdots\!81}a^{28}+\frac{68\!\cdots\!81}{33\!\cdots\!21}a^{27}+\frac{37\!\cdots\!35}{53\!\cdots\!81}a^{26}-\frac{97\!\cdots\!42}{53\!\cdots\!81}a^{25}-\frac{10\!\cdots\!43}{20\!\cdots\!41}a^{24}+\frac{67\!\cdots\!59}{53\!\cdots\!81}a^{23}+\frac{10\!\cdots\!69}{53\!\cdots\!81}a^{22}-\frac{26\!\cdots\!70}{53\!\cdots\!81}a^{21}-\frac{40\!\cdots\!74}{53\!\cdots\!81}a^{20}+\frac{93\!\cdots\!59}{53\!\cdots\!81}a^{19}+\frac{11\!\cdots\!42}{53\!\cdots\!81}a^{18}-\frac{22\!\cdots\!58}{53\!\cdots\!81}a^{17}-\frac{14\!\cdots\!78}{28\!\cdots\!99}a^{16}+\frac{35\!\cdots\!63}{53\!\cdots\!81}a^{15}+\frac{48\!\cdots\!90}{53\!\cdots\!81}a^{14}-\frac{31\!\cdots\!88}{40\!\cdots\!37}a^{13}-\frac{72\!\cdots\!51}{53\!\cdots\!81}a^{12}+\frac{92\!\cdots\!04}{40\!\cdots\!37}a^{11}+\frac{47\!\cdots\!57}{53\!\cdots\!81}a^{10}-\frac{70\!\cdots\!63}{53\!\cdots\!81}a^{9}-\frac{31\!\cdots\!09}{53\!\cdots\!81}a^{8}-\frac{27\!\cdots\!09}{53\!\cdots\!81}a^{7}+\frac{11\!\cdots\!92}{53\!\cdots\!81}a^{6}+\frac{47\!\cdots\!34}{53\!\cdots\!81}a^{5}-\frac{43\!\cdots\!69}{53\!\cdots\!81}a^{4}-\frac{50\!\cdots\!60}{28\!\cdots\!99}a^{3}+\frac{49\!\cdots\!32}{53\!\cdots\!81}a^{2}+\frac{62\!\cdots\!77}{76\!\cdots\!83}a-\frac{41\!\cdots\!65}{53\!\cdots\!81}$, $\frac{51\!\cdots\!40}{28\!\cdots\!63}a^{29}-\frac{13\!\cdots\!48}{20\!\cdots\!41}a^{28}-\frac{79\!\cdots\!76}{87\!\cdots\!67}a^{27}+\frac{15\!\cdots\!64}{28\!\cdots\!63}a^{26}+\frac{19\!\cdots\!88}{28\!\cdots\!63}a^{25}-\frac{70\!\cdots\!80}{20\!\cdots\!41}a^{24}-\frac{49\!\cdots\!98}{20\!\cdots\!41}a^{23}+\frac{27\!\cdots\!72}{20\!\cdots\!41}a^{22}+\frac{19\!\cdots\!92}{20\!\cdots\!41}a^{21}-\frac{94\!\cdots\!64}{20\!\cdots\!41}a^{20}-\frac{56\!\cdots\!09}{20\!\cdots\!41}a^{19}+\frac{22\!\cdots\!71}{20\!\cdots\!41}a^{18}+\frac{22\!\cdots\!77}{28\!\cdots\!63}a^{17}-\frac{37\!\cdots\!01}{20\!\cdots\!41}a^{16}-\frac{23\!\cdots\!81}{15\!\cdots\!57}a^{15}+\frac{46\!\cdots\!27}{20\!\cdots\!41}a^{14}+\frac{51\!\cdots\!53}{20\!\cdots\!41}a^{13}-\frac{21\!\cdots\!11}{20\!\cdots\!41}a^{12}-\frac{34\!\cdots\!85}{20\!\cdots\!41}a^{11}+\frac{13\!\cdots\!00}{20\!\cdots\!41}a^{10}+\frac{24\!\cdots\!66}{20\!\cdots\!41}a^{9}-\frac{13\!\cdots\!82}{20\!\cdots\!41}a^{8}-\frac{85\!\cdots\!59}{20\!\cdots\!41}a^{7}+\frac{11\!\cdots\!39}{20\!\cdots\!41}a^{6}+\frac{37\!\cdots\!45}{20\!\cdots\!41}a^{5}+\frac{65\!\cdots\!33}{28\!\cdots\!63}a^{4}-\frac{49\!\cdots\!83}{20\!\cdots\!41}a^{3}+\frac{91\!\cdots\!57}{20\!\cdots\!41}a^{2}+\frac{11\!\cdots\!81}{20\!\cdots\!41}a-\frac{20\!\cdots\!56}{20\!\cdots\!41}$, $\frac{62\!\cdots\!09}{53\!\cdots\!81}a^{29}-\frac{28\!\cdots\!54}{53\!\cdots\!81}a^{28}-\frac{29\!\cdots\!20}{23\!\cdots\!47}a^{27}+\frac{20\!\cdots\!83}{53\!\cdots\!81}a^{26}+\frac{43\!\cdots\!95}{40\!\cdots\!37}a^{25}-\frac{10\!\cdots\!45}{40\!\cdots\!37}a^{24}+\frac{30\!\cdots\!68}{53\!\cdots\!81}a^{23}+\frac{46\!\cdots\!18}{53\!\cdots\!81}a^{22}-\frac{12\!\cdots\!98}{76\!\cdots\!83}a^{21}-\frac{23\!\cdots\!99}{76\!\cdots\!83}a^{20}+\frac{71\!\cdots\!53}{76\!\cdots\!83}a^{19}+\frac{52\!\cdots\!28}{76\!\cdots\!83}a^{18}-\frac{61\!\cdots\!06}{53\!\cdots\!81}a^{17}-\frac{62\!\cdots\!66}{53\!\cdots\!81}a^{16}+\frac{76\!\cdots\!04}{76\!\cdots\!83}a^{15}+\frac{79\!\cdots\!86}{53\!\cdots\!81}a^{14}+\frac{15\!\cdots\!12}{53\!\cdots\!81}a^{13}-\frac{53\!\cdots\!48}{53\!\cdots\!81}a^{12}-\frac{11\!\cdots\!35}{53\!\cdots\!81}a^{11}+\frac{40\!\cdots\!45}{76\!\cdots\!83}a^{10}+\frac{10\!\cdots\!65}{53\!\cdots\!81}a^{9}-\frac{76\!\cdots\!32}{53\!\cdots\!81}a^{8}-\frac{29\!\cdots\!09}{40\!\cdots\!37}a^{7}+\frac{24\!\cdots\!03}{53\!\cdots\!81}a^{6}+\frac{18\!\cdots\!62}{53\!\cdots\!81}a^{5}+\frac{44\!\cdots\!68}{53\!\cdots\!81}a^{4}-\frac{22\!\cdots\!24}{58\!\cdots\!91}a^{3}-\frac{24\!\cdots\!30}{53\!\cdots\!81}a^{2}+\frac{24\!\cdots\!60}{76\!\cdots\!83}a-\frac{75\!\cdots\!47}{53\!\cdots\!81}$, $\frac{62\!\cdots\!94}{23\!\cdots\!47}a^{29}-\frac{29\!\cdots\!42}{23\!\cdots\!47}a^{28}-\frac{56\!\cdots\!76}{17\!\cdots\!19}a^{27}+\frac{16\!\cdots\!07}{17\!\cdots\!19}a^{26}+\frac{63\!\cdots\!12}{33\!\cdots\!21}a^{25}-\frac{13\!\cdots\!56}{23\!\cdots\!47}a^{24}+\frac{38\!\cdots\!91}{23\!\cdots\!47}a^{23}+\frac{51\!\cdots\!68}{23\!\cdots\!47}a^{22}-\frac{19\!\cdots\!00}{33\!\cdots\!21}a^{21}-\frac{25\!\cdots\!09}{33\!\cdots\!21}a^{20}+\frac{35\!\cdots\!10}{12\!\cdots\!13}a^{19}+\frac{60\!\cdots\!66}{33\!\cdots\!21}a^{18}-\frac{16\!\cdots\!29}{33\!\cdots\!21}a^{17}-\frac{75\!\cdots\!08}{23\!\cdots\!47}a^{16}+\frac{14\!\cdots\!33}{23\!\cdots\!47}a^{15}+\frac{10\!\cdots\!29}{23\!\cdots\!47}a^{14}+\frac{27\!\cdots\!55}{23\!\cdots\!47}a^{13}-\frac{84\!\cdots\!43}{23\!\cdots\!47}a^{12}-\frac{58\!\cdots\!22}{23\!\cdots\!47}a^{11}+\frac{52\!\cdots\!30}{23\!\cdots\!47}a^{10}+\frac{63\!\cdots\!92}{23\!\cdots\!47}a^{9}-\frac{22\!\cdots\!55}{23\!\cdots\!47}a^{8}-\frac{54\!\cdots\!30}{33\!\cdots\!21}a^{7}+\frac{77\!\cdots\!21}{23\!\cdots\!47}a^{6}+\frac{13\!\cdots\!99}{23\!\cdots\!47}a^{5}-\frac{17\!\cdots\!76}{23\!\cdots\!47}a^{4}-\frac{30\!\cdots\!69}{23\!\cdots\!47}a^{3}-\frac{60\!\cdots\!00}{23\!\cdots\!47}a^{2}+\frac{32\!\cdots\!86}{93\!\cdots\!01}a+\frac{21\!\cdots\!70}{23\!\cdots\!47}$, $\frac{32\!\cdots\!85}{53\!\cdots\!81}a^{29}-\frac{13\!\cdots\!51}{53\!\cdots\!81}a^{28}-\frac{47\!\cdots\!27}{23\!\cdots\!47}a^{27}+\frac{10\!\cdots\!57}{53\!\cdots\!81}a^{26}+\frac{77\!\cdots\!15}{53\!\cdots\!81}a^{25}-\frac{69\!\cdots\!97}{53\!\cdots\!81}a^{24}-\frac{21\!\cdots\!87}{76\!\cdots\!83}a^{23}+\frac{38\!\cdots\!97}{76\!\cdots\!83}a^{22}+\frac{63\!\cdots\!55}{53\!\cdots\!81}a^{21}-\frac{94\!\cdots\!98}{53\!\cdots\!81}a^{20}-\frac{58\!\cdots\!23}{28\!\cdots\!99}a^{19}+\frac{32\!\cdots\!64}{76\!\cdots\!83}a^{18}+\frac{49\!\cdots\!49}{53\!\cdots\!81}a^{17}-\frac{40\!\cdots\!82}{53\!\cdots\!81}a^{16}-\frac{16\!\cdots\!75}{76\!\cdots\!83}a^{15}+\frac{54\!\cdots\!59}{53\!\cdots\!81}a^{14}+\frac{27\!\cdots\!30}{53\!\cdots\!81}a^{13}-\frac{41\!\cdots\!47}{53\!\cdots\!81}a^{12}-\frac{23\!\cdots\!51}{53\!\cdots\!81}a^{11}+\frac{26\!\cdots\!53}{53\!\cdots\!81}a^{10}+\frac{17\!\cdots\!98}{53\!\cdots\!81}a^{9}-\frac{15\!\cdots\!70}{76\!\cdots\!83}a^{8}-\frac{76\!\cdots\!36}{53\!\cdots\!81}a^{7}+\frac{59\!\cdots\!28}{76\!\cdots\!83}a^{6}+\frac{29\!\cdots\!99}{53\!\cdots\!81}a^{5}-\frac{93\!\cdots\!95}{53\!\cdots\!81}a^{4}-\frac{87\!\cdots\!17}{76\!\cdots\!83}a^{3}+\frac{17\!\cdots\!05}{53\!\cdots\!81}a^{2}+\frac{46\!\cdots\!49}{28\!\cdots\!99}a-\frac{24\!\cdots\!85}{53\!\cdots\!81}$, $\frac{15\!\cdots\!75}{53\!\cdots\!81}a^{29}-\frac{56\!\cdots\!64}{53\!\cdots\!81}a^{28}-\frac{34\!\cdots\!22}{23\!\cdots\!47}a^{27}+\frac{44\!\cdots\!45}{53\!\cdots\!81}a^{26}+\frac{62\!\cdots\!00}{53\!\cdots\!81}a^{25}-\frac{28\!\cdots\!74}{53\!\cdots\!81}a^{24}-\frac{23\!\cdots\!10}{53\!\cdots\!81}a^{23}+\frac{16\!\cdots\!86}{79\!\cdots\!83}a^{22}+\frac{13\!\cdots\!02}{76\!\cdots\!83}a^{21}-\frac{36\!\cdots\!90}{53\!\cdots\!81}a^{20}-\frac{26\!\cdots\!19}{53\!\cdots\!81}a^{19}+\frac{83\!\cdots\!16}{53\!\cdots\!81}a^{18}+\frac{74\!\cdots\!94}{53\!\cdots\!81}a^{17}-\frac{12\!\cdots\!29}{53\!\cdots\!81}a^{16}-\frac{18\!\cdots\!34}{76\!\cdots\!83}a^{15}+\frac{14\!\cdots\!22}{53\!\cdots\!81}a^{14}+\frac{21\!\cdots\!06}{53\!\cdots\!81}a^{13}-\frac{18\!\cdots\!51}{53\!\cdots\!81}a^{12}-\frac{10\!\cdots\!93}{53\!\cdots\!81}a^{11}+\frac{22\!\cdots\!35}{53\!\cdots\!81}a^{10}+\frac{78\!\cdots\!14}{53\!\cdots\!81}a^{9}+\frac{95\!\cdots\!46}{30\!\cdots\!97}a^{8}-\frac{14\!\cdots\!67}{53\!\cdots\!81}a^{7}+\frac{15\!\cdots\!45}{58\!\cdots\!91}a^{6}+\frac{91\!\cdots\!04}{53\!\cdots\!81}a^{5}+\frac{35\!\cdots\!93}{53\!\cdots\!81}a^{4}+\frac{99\!\cdots\!46}{53\!\cdots\!81}a^{3}+\frac{68\!\cdots\!72}{53\!\cdots\!81}a^{2}+\frac{20\!\cdots\!13}{53\!\cdots\!81}a-\frac{43\!\cdots\!63}{53\!\cdots\!81}$, $\frac{80\!\cdots\!38}{53\!\cdots\!81}a^{29}-\frac{26\!\cdots\!17}{53\!\cdots\!81}a^{28}-\frac{25\!\cdots\!84}{23\!\cdots\!47}a^{27}+\frac{24\!\cdots\!85}{53\!\cdots\!81}a^{26}+\frac{42\!\cdots\!13}{53\!\cdots\!81}a^{25}-\frac{15\!\cdots\!05}{53\!\cdots\!81}a^{24}-\frac{18\!\cdots\!76}{53\!\cdots\!81}a^{23}+\frac{62\!\cdots\!32}{53\!\cdots\!81}a^{22}+\frac{10\!\cdots\!26}{76\!\cdots\!83}a^{21}-\frac{21\!\cdots\!13}{53\!\cdots\!81}a^{20}-\frac{22\!\cdots\!69}{53\!\cdots\!81}a^{19}+\frac{52\!\cdots\!52}{53\!\cdots\!81}a^{18}+\frac{83\!\cdots\!92}{76\!\cdots\!83}a^{17}-\frac{85\!\cdots\!52}{53\!\cdots\!81}a^{16}-\frac{10\!\cdots\!25}{53\!\cdots\!81}a^{15}+\frac{10\!\cdots\!29}{53\!\cdots\!81}a^{14}+\frac{17\!\cdots\!02}{53\!\cdots\!81}a^{13}-\frac{25\!\cdots\!81}{40\!\cdots\!37}a^{12}-\frac{12\!\cdots\!64}{53\!\cdots\!81}a^{11}+\frac{18\!\cdots\!35}{53\!\cdots\!81}a^{10}+\frac{84\!\cdots\!40}{53\!\cdots\!81}a^{9}+\frac{56\!\cdots\!70}{28\!\cdots\!99}a^{8}-\frac{31\!\cdots\!50}{53\!\cdots\!81}a^{7}-\frac{34\!\cdots\!60}{53\!\cdots\!81}a^{6}+\frac{12\!\cdots\!68}{53\!\cdots\!81}a^{5}+\frac{37\!\cdots\!78}{53\!\cdots\!81}a^{4}-\frac{17\!\cdots\!35}{53\!\cdots\!81}a^{3}-\frac{35\!\cdots\!72}{53\!\cdots\!81}a^{2}+\frac{44\!\cdots\!27}{53\!\cdots\!81}a+\frac{66\!\cdots\!52}{53\!\cdots\!81}$, $\frac{84\!\cdots\!06}{28\!\cdots\!99}a^{29}-\frac{95\!\cdots\!12}{76\!\cdots\!83}a^{28}-\frac{23\!\cdots\!85}{23\!\cdots\!47}a^{27}+\frac{74\!\cdots\!50}{76\!\cdots\!83}a^{26}+\frac{39\!\cdots\!55}{53\!\cdots\!81}a^{25}-\frac{33\!\cdots\!15}{53\!\cdots\!81}a^{24}-\frac{82\!\cdots\!17}{53\!\cdots\!81}a^{23}+\frac{12\!\cdots\!66}{53\!\cdots\!81}a^{22}+\frac{34\!\cdots\!26}{53\!\cdots\!81}a^{21}-\frac{23\!\cdots\!84}{28\!\cdots\!99}a^{20}-\frac{65\!\cdots\!45}{53\!\cdots\!81}a^{19}+\frac{10\!\cdots\!09}{53\!\cdots\!81}a^{18}+\frac{27\!\cdots\!01}{53\!\cdots\!81}a^{17}-\frac{26\!\cdots\!44}{76\!\cdots\!83}a^{16}-\frac{59\!\cdots\!71}{53\!\cdots\!81}a^{15}+\frac{25\!\cdots\!17}{53\!\cdots\!81}a^{14}+\frac{13\!\cdots\!36}{53\!\cdots\!81}a^{13}-\frac{17\!\cdots\!73}{53\!\cdots\!81}a^{12}-\frac{14\!\cdots\!92}{76\!\cdots\!83}a^{11}+\frac{89\!\cdots\!57}{40\!\cdots\!57}a^{10}+\frac{73\!\cdots\!39}{53\!\cdots\!81}a^{9}-\frac{53\!\cdots\!34}{58\!\cdots\!91}a^{8}-\frac{28\!\cdots\!03}{53\!\cdots\!81}a^{7}+\frac{16\!\cdots\!08}{40\!\cdots\!37}a^{6}+\frac{16\!\cdots\!70}{76\!\cdots\!83}a^{5}-\frac{51\!\cdots\!09}{53\!\cdots\!81}a^{4}-\frac{19\!\cdots\!48}{53\!\cdots\!81}a^{3}+\frac{12\!\cdots\!96}{53\!\cdots\!81}a^{2}+\frac{31\!\cdots\!29}{53\!\cdots\!81}a-\frac{13\!\cdots\!72}{53\!\cdots\!81}$, $\frac{12\!\cdots\!51}{33\!\cdots\!21}a^{29}-\frac{37\!\cdots\!44}{23\!\cdots\!47}a^{28}-\frac{26\!\cdots\!02}{23\!\cdots\!47}a^{27}+\frac{21\!\cdots\!99}{17\!\cdots\!19}a^{26}+\frac{18\!\cdots\!01}{23\!\cdots\!47}a^{25}-\frac{18\!\cdots\!02}{23\!\cdots\!47}a^{24}-\frac{27\!\cdots\!92}{23\!\cdots\!47}a^{23}+\frac{70\!\cdots\!08}{23\!\cdots\!47}a^{22}+\frac{16\!\cdots\!98}{33\!\cdots\!21}a^{21}-\frac{34\!\cdots\!92}{33\!\cdots\!21}a^{20}-\frac{89\!\cdots\!62}{23\!\cdots\!47}a^{19}+\frac{81\!\cdots\!36}{33\!\cdots\!21}a^{18}+\frac{74\!\cdots\!24}{23\!\cdots\!47}a^{17}-\frac{98\!\cdots\!74}{23\!\cdots\!47}a^{16}-\frac{17\!\cdots\!52}{23\!\cdots\!47}a^{15}+\frac{12\!\cdots\!95}{23\!\cdots\!47}a^{14}+\frac{50\!\cdots\!87}{23\!\cdots\!47}a^{13}-\frac{44\!\cdots\!13}{12\!\cdots\!13}a^{12}-\frac{28\!\cdots\!40}{23\!\cdots\!47}a^{11}+\frac{54\!\cdots\!13}{23\!\cdots\!47}a^{10}+\frac{18\!\cdots\!93}{23\!\cdots\!47}a^{9}-\frac{19\!\cdots\!61}{23\!\cdots\!47}a^{8}-\frac{37\!\cdots\!56}{17\!\cdots\!19}a^{7}+\frac{78\!\cdots\!74}{23\!\cdots\!47}a^{6}+\frac{17\!\cdots\!86}{23\!\cdots\!47}a^{5}-\frac{13\!\cdots\!36}{23\!\cdots\!47}a^{4}+\frac{28\!\cdots\!62}{23\!\cdots\!47}a^{3}-\frac{53\!\cdots\!01}{23\!\cdots\!47}a^{2}-\frac{20\!\cdots\!31}{23\!\cdots\!47}a-\frac{10\!\cdots\!72}{23\!\cdots\!47}$ 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:  \( 5581738694.683278 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{14}\cdot 5581738694.683278 \cdot 1}{2\cdot\sqrt{60550910545856385116788597847358428955078125}}\cr\approx \mathstrut & 0.214416061763025 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^30 - 4*x^29 - 4*x^28 + 32*x^27 + 29*x^26 - 207*x^25 - 80*x^24 + 808*x^23 + 324*x^22 - 2807*x^21 - 784*x^20 + 6744*x^19 + 2549*x^18 - 11743*x^17 - 5231*x^16 + 15441*x^15 + 10445*x^14 - 10233*x^13 - 7776*x^12 + 6706*x^11 + 5598*x^10 - 2410*x^9 - 2278*x^8 + 1045*x^7 + 940*x^6 - 185*x^5 - 176*x^4 + 44*x^3 + 30*x^2 - 6*x - 1)
 
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 - 4*x^29 - 4*x^28 + 32*x^27 + 29*x^26 - 207*x^25 - 80*x^24 + 808*x^23 + 324*x^22 - 2807*x^21 - 784*x^20 + 6744*x^19 + 2549*x^18 - 11743*x^17 - 5231*x^16 + 15441*x^15 + 10445*x^14 - 10233*x^13 - 7776*x^12 + 6706*x^11 + 5598*x^10 - 2410*x^9 - 2278*x^8 + 1045*x^7 + 940*x^6 - 185*x^5 - 176*x^4 + 44*x^3 + 30*x^2 - 6*x - 1, 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 - 4*x^29 - 4*x^28 + 32*x^27 + 29*x^26 - 207*x^25 - 80*x^24 + 808*x^23 + 324*x^22 - 2807*x^21 - 784*x^20 + 6744*x^19 + 2549*x^18 - 11743*x^17 - 5231*x^16 + 15441*x^15 + 10445*x^14 - 10233*x^13 - 7776*x^12 + 6706*x^11 + 5598*x^10 - 2410*x^9 - 2278*x^8 + 1045*x^7 + 940*x^6 - 185*x^5 - 176*x^4 + 44*x^3 + 30*x^2 - 6*x - 1);
 
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 - 4*x^29 - 4*x^28 + 32*x^27 + 29*x^26 - 207*x^25 - 80*x^24 + 808*x^23 + 324*x^22 - 2807*x^21 - 784*x^20 + 6744*x^19 + 2549*x^18 - 11743*x^17 - 5231*x^16 + 15441*x^15 + 10445*x^14 - 10233*x^13 - 7776*x^12 + 6706*x^11 + 5598*x^10 - 2410*x^9 - 2278*x^8 + 1045*x^7 + 940*x^6 - 185*x^5 - 176*x^4 + 44*x^3 + 30*x^2 - 6*x - 1);
 
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

$D_{30}$ (as 30T14):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 60
The 18 conjugacy class representatives for $D_{30}$
Character table for $D_{30}$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.239.1, 5.1.57121.1, 6.2.7140125.1, 10.2.10196277003125.1, 15.1.44543599279432079.1

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: deg 30
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $30$ ${\href{/padicField/3.10.0.1}{10} }^{3}$ R ${\href{/padicField/7.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{15}$ ${\href{/padicField/17.6.0.1}{6} }^{5}$ ${\href{/padicField/19.2.0.1}{2} }^{14}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{15}$ $15^{2}$ $15^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{15}$ ${\href{/padicField/41.2.0.1}{2} }^{14}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{15}$ ${\href{/padicField/47.2.0.1}{2} }^{15}$ ${\href{/padicField/53.2.0.1}{2} }^{15}$ ${\href{/padicField/59.2.0.1}{2} }^{14}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display Deg $30$$2$$15$$15$
\(239\) Copy content Toggle raw display $\Q_{239}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{239}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1195.2t1.a.a$1$ $ 5 \cdot 239 $ \(\Q(\sqrt{-1195}) \) $C_2$ (as 2T1) $1$ $-1$
1.239.2t1.a.a$1$ $ 239 $ \(\Q(\sqrt{-239}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.5975.6t3.a.a$2$ $ 5^{2} \cdot 239 $ 6.0.1706489875.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.239.3t2.a.a$2$ $ 239 $ 3.1.239.1 $S_3$ (as 3T2) $1$ $0$
* 2.239.5t2.a.b$2$ $ 239 $ 5.1.57121.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.239.5t2.a.a$2$ $ 239 $ 5.1.57121.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.5975.10t3.a.b$2$ $ 5^{2} \cdot 239 $ 10.0.2436910203746875.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.5975.10t3.a.a$2$ $ 5^{2} \cdot 239 $ 10.0.2436910203746875.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.239.15t2.a.c$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.5975.30t14.a.a$2$ $ 5^{2} \cdot 239 $ 30.2.60550910545856385116788597847358428955078125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.239.15t2.a.a$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.5975.30t14.a.c$2$ $ 5^{2} \cdot 239 $ 30.2.60550910545856385116788597847358428955078125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.239.15t2.a.d$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.5975.30t14.a.d$2$ $ 5^{2} \cdot 239 $ 30.2.60550910545856385116788597847358428955078125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.239.15t2.a.b$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.5975.30t14.a.b$2$ $ 5^{2} \cdot 239 $ 30.2.60550910545856385116788597847358428955078125.1 $D_{30}$ (as 30T14) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.