Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 12*x^18 - 36*x^16 + 44*x^15 + 17*x^14 - 60*x^13 + x^12 + 28*x^11 + 86*x^10 - 122*x^9 - 84*x^8 + 232*x^7 - 72*x^6 - 128*x^5 + 100*x^4 + 12*x^3 - 28*x^2 + 4)
gp: K = bnfinit(y^20 - 6*y^19 + 12*y^18 - 36*y^16 + 44*y^15 + 17*y^14 - 60*y^13 + y^12 + 28*y^11 + 86*y^10 - 122*y^9 - 84*y^8 + 232*y^7 - 72*y^6 - 128*y^5 + 100*y^4 + 12*y^3 - 28*y^2 + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 6*x^19 + 12*x^18 - 36*x^16 + 44*x^15 + 17*x^14 - 60*x^13 + x^12 + 28*x^11 + 86*x^10 - 122*x^9 - 84*x^8 + 232*x^7 - 72*x^6 - 128*x^5 + 100*x^4 + 12*x^3 - 28*x^2 + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 6*x^19 + 12*x^18 - 36*x^16 + 44*x^15 + 17*x^14 - 60*x^13 + x^12 + 28*x^11 + 86*x^10 - 122*x^9 - 84*x^8 + 232*x^7 - 72*x^6 - 128*x^5 + 100*x^4 + 12*x^3 - 28*x^2 + 4)
\( x^{20} - 6 x^{19} + 12 x^{18} - 36 x^{16} + 44 x^{15} + 17 x^{14} - 60 x^{13} + x^{12} + 28 x^{11} + \cdots + 4 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-164475020247040000000000\)
\(\medspace = -\,2^{28}\cdot 5^{10}\cdot 89^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.48\) | 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: | $2^{115/48}5^{1/2}89^{2/3}\approx 234.5827309266611$ | ||
| Ramified primes: |
\(2\), \(5\), \(89\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{11}-\frac{1}{2}a^{10}+\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{12}-\frac{1}{2}a^{11}+\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2990188}a^{19}+\frac{226311}{2990188}a^{18}-\frac{78053}{1495094}a^{17}-\frac{185191}{1495094}a^{16}+\frac{98537}{1495094}a^{15}-\frac{223853}{1495094}a^{14}-\frac{958405}{2990188}a^{13}-\frac{1087301}{2990188}a^{12}-\frac{916691}{2990188}a^{11}+\frac{971703}{2990188}a^{10}+\frac{69756}{747547}a^{9}-\frac{102698}{747547}a^{8}+\frac{108527}{1495094}a^{7}-\frac{323302}{747547}a^{6}+\frac{80575}{1495094}a^{5}-\frac{169307}{1495094}a^{4}-\frac{17845}{747547}a^{3}+\frac{369579}{747547}a^{2}-\frac{275747}{747547}a-\frac{262692}{747547}$
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 $1$
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: | $10$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{2446049}{2990188}a^{19}-\frac{3109281}{747547}a^{18}+\frac{4384563}{747547}a^{17}+\frac{9280513}{1495094}a^{16}-\frac{38140965}{1495094}a^{15}+\frac{19780515}{1495094}a^{14}+\frac{88515607}{2990188}a^{13}-\frac{40843373}{1495094}a^{12}-\frac{72651683}{2990188}a^{11}+\frac{3921991}{747547}a^{10}+\frac{56552413}{747547}a^{9}-\frac{48873387}{1495094}a^{8}-\frac{82131617}{747547}a^{7}+\frac{158220259}{1495094}a^{6}+\frac{62425573}{1495094}a^{5}-\frac{65238119}{747547}a^{4}+\frac{7747942}{747547}a^{3}+\frac{19940040}{747547}a^{2}-\frac{2937007}{747547}a-\frac{3480058}{747547}$, $\frac{597091}{2990188}a^{19}-\frac{2819701}{2990188}a^{18}+\frac{2895331}{2990188}a^{17}+\frac{1757750}{747547}a^{16}-\frac{8353785}{1495094}a^{15}-\frac{703117}{1495094}a^{14}+\frac{28110901}{2990188}a^{13}-\frac{4468865}{2990188}a^{12}-\frac{16078689}{1495094}a^{11}-\frac{13488077}{2990188}a^{10}+\frac{63450977}{2990188}a^{9}+\frac{4219080}{747547}a^{8}-\frac{24229873}{747547}a^{7}+\frac{7907861}{1495094}a^{6}+\frac{18303651}{747547}a^{5}-\frac{12865879}{1495094}a^{4}-\frac{14806401}{1495094}a^{3}+\frac{2400665}{747547}a^{2}+\frac{2669867}{747547}a-\frac{717432}{747547}$, $\frac{424385}{1495094}a^{19}-\frac{1844825}{1495094}a^{18}+\frac{780259}{747547}a^{17}+\frac{2566404}{747547}a^{16}-\frac{11522075}{1495094}a^{15}+\frac{572445}{1495094}a^{14}+\frac{18082433}{1495094}a^{13}-\frac{9354041}{1495094}a^{12}-\frac{13926705}{1495094}a^{11}-\frac{2144519}{1495094}a^{10}+\frac{32804867}{1495094}a^{9}+\frac{2220497}{1495094}a^{8}-\frac{61821045}{1495094}a^{7}+\frac{37658243}{1495094}a^{6}+\frac{18467629}{747547}a^{5}-\frac{22550153}{747547}a^{4}-\frac{250883}{747547}a^{3}+\frac{7876066}{747547}a^{2}-\frac{776242}{747547}a-\frac{1720620}{747547}$, $\frac{655083}{2990188}a^{19}-\frac{818717}{747547}a^{18}+\frac{2516495}{1495094}a^{17}+\frac{936589}{1495094}a^{16}-\frac{6750255}{1495094}a^{15}+\frac{2373869}{747547}a^{14}+\frac{5981181}{2990188}a^{13}-\frac{698183}{1495094}a^{12}-\frac{8670535}{2990188}a^{11}-\frac{4827121}{747547}a^{10}+\frac{23007421}{1495094}a^{9}-\frac{3211857}{747547}a^{8}-\frac{6612723}{747547}a^{7}+\frac{6119521}{747547}a^{6}-\frac{9951509}{1495094}a^{5}+\frac{4321869}{747547}a^{4}+\frac{2426492}{747547}a^{3}-\frac{4621927}{747547}a^{2}+\frac{832626}{747547}a+\frac{1003511}{747547}$, $\frac{4593599}{2990188}a^{19}-\frac{6225422}{747547}a^{18}+\frac{19725991}{1495094}a^{17}+\frac{7273022}{747547}a^{16}-\frac{39368136}{747547}a^{15}+\frac{25964917}{747547}a^{14}+\frac{170738221}{2990188}a^{13}-\frac{97626261}{1495094}a^{12}-\frac{138584967}{2990188}a^{11}+\frac{36419249}{1495094}a^{10}+\frac{116172908}{747547}a^{9}-\frac{140671101}{1495094}a^{8}-\frac{325040461}{1495094}a^{7}+\frac{182137949}{747547}a^{6}+\frac{103940083}{1495094}a^{5}-\frac{144652002}{747547}a^{4}+\frac{24908728}{747547}a^{3}+\frac{43904419}{747547}a^{2}-\frac{9565525}{747547}a-\frac{7603373}{747547}$, $\frac{2069977}{2990188}a^{19}-\frac{2986618}{747547}a^{18}+\frac{22010615}{2990188}a^{17}+\frac{2486087}{1495094}a^{16}-\frac{17752552}{747547}a^{15}+\frac{17285733}{747547}a^{14}+\frac{50637367}{2990188}a^{13}-\frac{47702319}{1495094}a^{12}-\frac{17236871}{1495094}a^{11}+\frac{16359431}{1495094}a^{10}+\frac{206084545}{2990188}a^{9}-\frac{92578111}{1495094}a^{8}-\frac{56412438}{747547}a^{7}+\frac{185227809}{1495094}a^{6}-\frac{7003988}{747547}a^{5}-\frac{51755310}{747547}a^{4}+\frac{45201059}{1495094}a^{3}+\frac{7341575}{747547}a^{2}-\frac{2678610}{747547}a-\frac{2205378}{747547}$, $\frac{37085}{747547}a^{19}-\frac{614483}{2990188}a^{18}-\frac{621}{2990188}a^{17}+\frac{1866857}{1495094}a^{16}-\frac{2798099}{1495094}a^{15}-\frac{905687}{747547}a^{14}+\frac{3410425}{747547}a^{13}-\frac{1732261}{2990188}a^{12}-\frac{14356845}{2990188}a^{11}-\frac{3327255}{2990188}a^{10}+\frac{21916727}{2990188}a^{9}+\frac{6805909}{1495094}a^{8}-\frac{23450969}{1495094}a^{7}+\frac{1006652}{747547}a^{6}+\frac{22392927}{1495094}a^{5}-\frac{8634385}{1495094}a^{4}-\frac{11339951}{1495094}a^{3}+\frac{2737162}{747547}a^{2}+\frac{1461860}{747547}a-\frac{348811}{747547}$, $\frac{241860}{747547}a^{19}-\frac{1307974}{747547}a^{18}+\frac{2054263}{747547}a^{17}+\frac{1704225}{747547}a^{16}-\frac{17246835}{1495094}a^{15}+\frac{9737691}{1495094}a^{14}+\frac{11501212}{747547}a^{13}-\frac{11506764}{747547}a^{12}-\frac{11619017}{747547}a^{11}+\frac{8242096}{747547}a^{10}+\frac{55944455}{1495094}a^{9}-\frac{34982691}{1495094}a^{8}-\frac{84088081}{1495094}a^{7}+\frac{81228651}{1495094}a^{6}+\frac{27045206}{747547}a^{5}-\frac{42280634}{747547}a^{4}-\frac{3854117}{747547}a^{3}+\frac{21436899}{747547}a^{2}-\frac{2792995}{747547}a-\frac{4165454}{747547}$, $\frac{1236229}{2990188}a^{19}-\frac{3480821}{1495094}a^{18}+\frac{12192263}{2990188}a^{17}+\frac{2269293}{1495094}a^{16}-\frac{20913087}{1495094}a^{15}+\frac{9023587}{747547}a^{14}+\frac{35999127}{2990188}a^{13}-\frac{13824728}{747547}a^{12}-\frac{14134055}{1495094}a^{11}+\frac{5594176}{747547}a^{10}+\frac{124778823}{2990188}a^{9}-\frac{50278031}{1495094}a^{8}-\frac{75780295}{1495094}a^{7}+\frac{105299817}{1495094}a^{6}+\frac{4863565}{747547}a^{5}-\frac{34525416}{747547}a^{4}+\frac{17899605}{1495094}a^{3}+\frac{7472695}{747547}a^{2}-\frac{1268328}{747547}a-\frac{1090916}{747547}$, $\frac{2499739}{2990188}a^{19}-\frac{13176019}{2990188}a^{18}+\frac{19946717}{2990188}a^{17}+\frac{4001638}{747547}a^{16}-\frac{38827101}{1495094}a^{15}+\frac{11208126}{747547}a^{14}+\frac{83916073}{2990188}a^{13}-\frac{76204283}{2990188}a^{12}-\frac{19506229}{747547}a^{11}+\frac{8388981}{2990188}a^{10}+\frac{244855007}{2990188}a^{9}-\frac{54751659}{1495094}a^{8}-\frac{79672320}{747547}a^{7}+\frac{74955581}{747547}a^{6}+\frac{28231229}{747547}a^{5}-\frac{110213779}{1495094}a^{4}+\frac{6292181}{1495094}a^{3}+\frac{16011247}{747547}a^{2}-\frac{382461}{747547}a-\frac{2696742}{747547}$
| 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: | \( 3246.68202617 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 3246.68202617 \cdot 1}{2\cdot\sqrt{164475020247040000000000}}\cr\approx \mathstrut & 0.244364769656 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 12*x^18 - 36*x^16 + 44*x^15 + 17*x^14 - 60*x^13 + x^12 + 28*x^11 + 86*x^10 - 122*x^9 - 84*x^8 + 232*x^7 - 72*x^6 - 128*x^5 + 100*x^4 + 12*x^3 - 28*x^2 + 4)
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^20 - 6*x^19 + 12*x^18 - 36*x^16 + 44*x^15 + 17*x^14 - 60*x^13 + x^12 + 28*x^11 + 86*x^10 - 122*x^9 - 84*x^8 + 232*x^7 - 72*x^6 - 128*x^5 + 100*x^4 + 12*x^3 - 28*x^2 + 4, 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^20 - 6*x^19 + 12*x^18 - 36*x^16 + 44*x^15 + 17*x^14 - 60*x^13 + x^12 + 28*x^11 + 86*x^10 - 122*x^9 - 84*x^8 + 232*x^7 - 72*x^6 - 128*x^5 + 100*x^4 + 12*x^3 - 28*x^2 + 4);
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^20 - 6*x^19 + 12*x^18 - 36*x^16 + 44*x^15 + 17*x^14 - 60*x^13 + x^12 + 28*x^11 + 86*x^10 - 122*x^9 - 84*x^8 + 232*x^7 - 72*x^6 - 128*x^5 + 100*x^4 + 12*x^3 - 28*x^2 + 4);
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
$C_2^{10}.\PGOPlus(4,5)$ (as 20T1036):
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 non-solvable group of order 14745600 |
| The 396 conjugacy class representatives for $C_2^{10}.\PGOPlus(4,5)$ |
| Character table for $C_2^{10}.\PGOPlus(4,5)$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.2.25347200000.1 |
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 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| 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 | R | $20$ | R | $20$ | ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
| 2.12.16.5 | $x^{12} + 2 x^{11} + 4 x^{10} - 2 x^{9} + 8 x^{8} + 4 x^{7} + 4 x^{6} + 8 x^{5} + 4 x^{4} + 12 x^{3} + 12$ | $6$ | $2$ | $16$ | 12T50 | $[4/3, 4/3, 2, 2]_{3}^{2}$ | |
|
\(5\)
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(89\)
| 89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.3.0.1 | $x^{3} + 3 x + 86$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 89.3.0.1 | $x^{3} + 3 x + 86$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.6.4.1 | $x^{6} + 246 x^{5} + 20181 x^{4} + 553022 x^{3} + 82437 x^{2} + 1795920 x + 49014018$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |