Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 210*x^10 + 70*x^9 + 16611*x^8 + 1360*x^7 - 624896*x^6 - 200970*x^5 + 11417901*x^4 + 4551265*x^3 - 90501020*x^2 - 23480651*x + 233001991)
gp: K = bnfinit(y^12 - y^11 - 210*y^10 + 70*y^9 + 16611*y^8 + 1360*y^7 - 624896*y^6 - 200970*y^5 + 11417901*y^4 + 4551265*y^3 - 90501020*y^2 - 23480651*y + 233001991, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 210*x^10 + 70*x^9 + 16611*x^8 + 1360*x^7 - 624896*x^6 - 200970*x^5 + 11417901*x^4 + 4551265*x^3 - 90501020*x^2 - 23480651*x + 233001991);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 210*x^10 + 70*x^9 + 16611*x^8 + 1360*x^7 - 624896*x^6 - 200970*x^5 + 11417901*x^4 + 4551265*x^3 - 90501020*x^2 - 23480651*x + 233001991)
\( x^{12} - x^{11} - 210 x^{10} + 70 x^{9} + 16611 x^{8} + 1360 x^{7} - 624896 x^{6} - 200970 x^{5} + \cdots + 233001991 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(28034829169596161173828125\)
\(\medspace = 3^{6}\cdot 5^{9}\cdot 13^{8}\cdot 17^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(132.02\) | 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: | $3^{1/2}5^{3/4}13^{2/3}17^{1/2}\approx 132.02053636333846$ | ||
| Ramified primes: |
\(3\), \(5\), \(13\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3315=3\cdot 5\cdot 13\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3315}(256,·)$, $\chi_{3315}(1,·)$, $\chi_{3315}(1478,·)$, $\chi_{3315}(1223,·)$, $\chi_{3315}(458,·)$, $\chi_{3315}(919,·)$, $\chi_{3315}(3212,·)$, $\chi_{3315}(3214,·)$, $\chi_{3315}(2447,·)$, $\chi_{3315}(664,·)$, $\chi_{3315}(2551,·)$, $\chi_{3315}(152,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{131}a^{9}-\frac{39}{131}a^{8}+\frac{9}{131}a^{7}-\frac{19}{131}a^{6}-\frac{63}{131}a^{5}+\frac{20}{131}a^{4}-\frac{20}{131}a^{3}-\frac{44}{131}a^{2}-\frac{51}{131}a+\frac{2}{131}$, $\frac{1}{131}a^{10}+\frac{60}{131}a^{8}-\frac{61}{131}a^{7}-\frac{18}{131}a^{6}+\frac{52}{131}a^{5}-\frac{26}{131}a^{4}-\frac{38}{131}a^{3}-\frac{64}{131}a^{2}-\frac{22}{131}a-\frac{53}{131}$, $\frac{1}{41\!\cdots\!59}a^{11}+\frac{95\!\cdots\!16}{41\!\cdots\!59}a^{10}-\frac{80\!\cdots\!98}{41\!\cdots\!59}a^{9}+\frac{12\!\cdots\!26}{41\!\cdots\!59}a^{8}+\frac{67\!\cdots\!24}{41\!\cdots\!59}a^{7}+\frac{17\!\cdots\!38}{41\!\cdots\!59}a^{6}-\frac{45\!\cdots\!35}{41\!\cdots\!59}a^{5}+\frac{18\!\cdots\!26}{41\!\cdots\!59}a^{4}-\frac{17\!\cdots\!07}{41\!\cdots\!59}a^{3}-\frac{97\!\cdots\!02}{41\!\cdots\!59}a^{2}+\frac{47\!\cdots\!89}{41\!\cdots\!59}a-\frac{16\!\cdots\!91}{41\!\cdots\!59}$
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
$C_{2}$, which has order $2$ (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: | $11$ | 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: |
$\frac{279906172817}{32\!\cdots\!29}a^{11}+\frac{2993842923683}{32\!\cdots\!29}a^{10}-\frac{59403210336275}{32\!\cdots\!29}a^{9}-\frac{521174098147090}{32\!\cdots\!29}a^{8}+\frac{40\!\cdots\!78}{32\!\cdots\!29}a^{7}+\frac{29\!\cdots\!52}{32\!\cdots\!29}a^{6}-\frac{11\!\cdots\!51}{32\!\cdots\!29}a^{5}-\frac{67\!\cdots\!25}{32\!\cdots\!29}a^{4}+\frac{11\!\cdots\!24}{32\!\cdots\!29}a^{3}+\frac{52\!\cdots\!09}{32\!\cdots\!29}a^{2}-\frac{30\!\cdots\!80}{32\!\cdots\!29}a-\frac{65\!\cdots\!79}{32\!\cdots\!29}$, $\frac{72\!\cdots\!05}{31\!\cdots\!89}a^{11}-\frac{65\!\cdots\!59}{31\!\cdots\!89}a^{10}-\frac{12\!\cdots\!75}{31\!\cdots\!89}a^{9}+\frac{10\!\cdots\!45}{31\!\cdots\!89}a^{8}+\frac{81\!\cdots\!20}{31\!\cdots\!89}a^{7}-\frac{55\!\cdots\!10}{31\!\cdots\!89}a^{6}-\frac{23\!\cdots\!93}{31\!\cdots\!89}a^{5}+\frac{12\!\cdots\!85}{31\!\cdots\!89}a^{4}+\frac{29\!\cdots\!60}{31\!\cdots\!89}a^{3}-\frac{11\!\cdots\!20}{31\!\cdots\!89}a^{2}-\frac{96\!\cdots\!00}{31\!\cdots\!89}a+\frac{28\!\cdots\!59}{31\!\cdots\!89}$, $\frac{69\!\cdots\!20}{31\!\cdots\!89}a^{11}+\frac{19\!\cdots\!20}{31\!\cdots\!89}a^{10}-\frac{12\!\cdots\!60}{31\!\cdots\!89}a^{9}-\frac{46\!\cdots\!55}{31\!\cdots\!89}a^{8}+\frac{71\!\cdots\!40}{31\!\cdots\!89}a^{7}+\frac{33\!\cdots\!10}{31\!\cdots\!89}a^{6}-\frac{16\!\cdots\!86}{31\!\cdots\!89}a^{5}-\frac{90\!\cdots\!55}{31\!\cdots\!89}a^{4}+\frac{14\!\cdots\!30}{31\!\cdots\!89}a^{3}+\frac{89\!\cdots\!65}{31\!\cdots\!89}a^{2}-\frac{61\!\cdots\!95}{31\!\cdots\!89}a-\frac{20\!\cdots\!65}{31\!\cdots\!89}$, $\frac{19\!\cdots\!09}{41\!\cdots\!59}a^{11}+\frac{14\!\cdots\!45}{41\!\cdots\!59}a^{10}-\frac{35\!\cdots\!39}{41\!\cdots\!59}a^{9}-\frac{56\!\cdots\!58}{41\!\cdots\!59}a^{8}+\frac{21\!\cdots\!49}{41\!\cdots\!59}a^{7}+\frac{43\!\cdots\!90}{41\!\cdots\!59}a^{6}-\frac{54\!\cdots\!46}{41\!\cdots\!59}a^{5}-\frac{11\!\cdots\!81}{41\!\cdots\!59}a^{4}+\frac{55\!\cdots\!90}{41\!\cdots\!59}a^{3}+\frac{79\!\cdots\!81}{41\!\cdots\!59}a^{2}-\frac{20\!\cdots\!12}{41\!\cdots\!59}a-\frac{29\!\cdots\!88}{41\!\cdots\!59}$, $\frac{99\!\cdots\!29}{41\!\cdots\!59}a^{11}-\frac{86\!\cdots\!58}{41\!\cdots\!59}a^{10}-\frac{18\!\cdots\!41}{41\!\cdots\!59}a^{9}+\frac{14\!\cdots\!14}{41\!\cdots\!59}a^{8}+\frac{13\!\cdots\!75}{41\!\cdots\!59}a^{7}-\frac{84\!\cdots\!49}{41\!\cdots\!59}a^{6}-\frac{43\!\cdots\!89}{41\!\cdots\!59}a^{5}+\frac{20\!\cdots\!03}{41\!\cdots\!59}a^{4}+\frac{61\!\cdots\!50}{41\!\cdots\!59}a^{3}-\frac{18\!\cdots\!11}{41\!\cdots\!59}a^{2}-\frac{22\!\cdots\!97}{41\!\cdots\!59}a+\frac{54\!\cdots\!85}{41\!\cdots\!59}$, $\frac{10\!\cdots\!01}{41\!\cdots\!59}a^{11}-\frac{27\!\cdots\!75}{41\!\cdots\!59}a^{10}-\frac{25\!\cdots\!45}{41\!\cdots\!59}a^{9}+\frac{37\!\cdots\!10}{41\!\cdots\!59}a^{8}-\frac{94\!\cdots\!56}{41\!\cdots\!59}a^{7}-\frac{16\!\cdots\!16}{41\!\cdots\!59}a^{6}+\frac{58\!\cdots\!03}{41\!\cdots\!59}a^{5}+\frac{26\!\cdots\!00}{41\!\cdots\!59}a^{4}-\frac{10\!\cdots\!68}{41\!\cdots\!59}a^{3}-\frac{52\!\cdots\!12}{41\!\cdots\!59}a^{2}+\frac{37\!\cdots\!89}{41\!\cdots\!59}a-\frac{32\!\cdots\!55}{41\!\cdots\!59}$, $\frac{15\!\cdots\!79}{41\!\cdots\!59}a^{11}-\frac{25\!\cdots\!27}{41\!\cdots\!59}a^{10}-\frac{80\!\cdots\!65}{41\!\cdots\!59}a^{9}+\frac{33\!\cdots\!55}{41\!\cdots\!59}a^{8}-\frac{87\!\cdots\!94}{41\!\cdots\!59}a^{7}-\frac{13\!\cdots\!70}{41\!\cdots\!59}a^{6}+\frac{69\!\cdots\!05}{41\!\cdots\!59}a^{5}+\frac{17\!\cdots\!00}{41\!\cdots\!59}a^{4}-\frac{14\!\cdots\!42}{41\!\cdots\!59}a^{3}+\frac{36\!\cdots\!30}{41\!\cdots\!59}a^{2}+\frac{93\!\cdots\!18}{41\!\cdots\!59}a-\frac{12\!\cdots\!50}{41\!\cdots\!59}$, $\frac{11\!\cdots\!07}{41\!\cdots\!59}a^{11}-\frac{54\!\cdots\!34}{41\!\cdots\!59}a^{10}-\frac{21\!\cdots\!69}{41\!\cdots\!59}a^{9}+\frac{90\!\cdots\!68}{41\!\cdots\!59}a^{8}+\frac{15\!\cdots\!39}{41\!\cdots\!59}a^{7}-\frac{56\!\cdots\!76}{41\!\cdots\!59}a^{6}-\frac{48\!\cdots\!01}{41\!\cdots\!59}a^{5}+\frac{16\!\cdots\!72}{41\!\cdots\!59}a^{4}+\frac{65\!\cdots\!71}{41\!\cdots\!59}a^{3}-\frac{20\!\cdots\!99}{41\!\cdots\!59}a^{2}-\frac{24\!\cdots\!77}{41\!\cdots\!59}a+\frac{68\!\cdots\!55}{41\!\cdots\!59}$, $\frac{16\!\cdots\!74}{41\!\cdots\!59}a^{11}-\frac{10\!\cdots\!23}{41\!\cdots\!59}a^{10}-\frac{28\!\cdots\!35}{41\!\cdots\!59}a^{9}+\frac{16\!\cdots\!71}{41\!\cdots\!59}a^{8}+\frac{18\!\cdots\!48}{41\!\cdots\!59}a^{7}-\frac{94\!\cdots\!67}{41\!\cdots\!59}a^{6}-\frac{50\!\cdots\!32}{41\!\cdots\!59}a^{5}+\frac{23\!\cdots\!02}{41\!\cdots\!59}a^{4}+\frac{58\!\cdots\!45}{41\!\cdots\!59}a^{3}-\frac{23\!\cdots\!26}{41\!\cdots\!59}a^{2}-\frac{20\!\cdots\!81}{41\!\cdots\!59}a+\frac{71\!\cdots\!15}{41\!\cdots\!59}$, $\frac{24\!\cdots\!07}{41\!\cdots\!59}a^{11}-\frac{18\!\cdots\!48}{41\!\cdots\!59}a^{10}-\frac{40\!\cdots\!54}{41\!\cdots\!59}a^{9}+\frac{27\!\cdots\!53}{41\!\cdots\!59}a^{8}+\frac{23\!\cdots\!81}{41\!\cdots\!59}a^{7}-\frac{14\!\cdots\!05}{41\!\cdots\!59}a^{6}-\frac{61\!\cdots\!65}{41\!\cdots\!59}a^{5}+\frac{34\!\cdots\!95}{41\!\cdots\!59}a^{4}+\frac{67\!\cdots\!97}{41\!\cdots\!59}a^{3}-\frac{31\!\cdots\!99}{41\!\cdots\!59}a^{2}-\frac{23\!\cdots\!87}{41\!\cdots\!59}a+\frac{91\!\cdots\!69}{41\!\cdots\!59}$, $\frac{16\!\cdots\!64}{41\!\cdots\!59}a^{11}-\frac{12\!\cdots\!11}{41\!\cdots\!59}a^{10}-\frac{25\!\cdots\!12}{41\!\cdots\!59}a^{9}+\frac{18\!\cdots\!16}{41\!\cdots\!59}a^{8}+\frac{14\!\cdots\!06}{41\!\cdots\!59}a^{7}-\frac{97\!\cdots\!55}{41\!\cdots\!59}a^{6}-\frac{37\!\cdots\!36}{41\!\cdots\!59}a^{5}+\frac{21\!\cdots\!01}{41\!\cdots\!59}a^{4}+\frac{40\!\cdots\!53}{41\!\cdots\!59}a^{3}-\frac{19\!\cdots\!34}{41\!\cdots\!59}a^{2}-\frac{14\!\cdots\!14}{41\!\cdots\!59}a+\frac{56\!\cdots\!42}{41\!\cdots\!59}$
| 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: | \( 427202136.726 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 427202136.726 \cdot 2}{2\cdot\sqrt{28034829169596161173828125}}\cr\approx \mathstrut & 0.330479410404 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 210*x^10 + 70*x^9 + 16611*x^8 + 1360*x^7 - 624896*x^6 - 200970*x^5 + 11417901*x^4 + 4551265*x^3 - 90501020*x^2 - 23480651*x + 233001991)
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^12 - x^11 - 210*x^10 + 70*x^9 + 16611*x^8 + 1360*x^7 - 624896*x^6 - 200970*x^5 + 11417901*x^4 + 4551265*x^3 - 90501020*x^2 - 23480651*x + 233001991, 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^12 - x^11 - 210*x^10 + 70*x^9 + 16611*x^8 + 1360*x^7 - 624896*x^6 - 200970*x^5 + 11417901*x^4 + 4551265*x^3 - 90501020*x^2 - 23480651*x + 233001991);
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^12 - x^11 - 210*x^10 + 70*x^9 + 16611*x^8 + 1360*x^7 - 624896*x^6 - 200970*x^5 + 11417901*x^4 + 4551265*x^3 - 90501020*x^2 - 23480651*x + 233001991);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A cyclic group of order 12 |
| The 12 conjugacy class representatives for $C_{12}$ |
| Character table for $C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 3.3.169.1, 4.4.325125.1, 6.6.3570125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.12.0.1}{12} }$ | R | R | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | R | R | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.12.6.1 | $x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
|
\(5\)
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(13\)
| 13.12.8.1 | $x^{12} + 9 x^{10} + 88 x^{9} + 33 x^{8} + 216 x^{7} - 1299 x^{6} - 78 x^{5} - 1797 x^{4} - 15494 x^{3} + 21687 x^{2} - 41586 x + 201846$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
|
\(17\)
| 17.12.6.2 | $x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |