Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 112*x^10 + 380*x^9 + 4719*x^8 - 13468*x^7 - 92386*x^6 + 222256*x^5 + 848264*x^4 - 1642580*x^3 - 3278996*x^2 + 3844396*x + 4689281)
gp: K = bnfinit(y^12 - 4*y^11 - 112*y^10 + 380*y^9 + 4719*y^8 - 13468*y^7 - 92386*y^6 + 222256*y^5 + 848264*y^4 - 1642580*y^3 - 3278996*y^2 + 3844396*y + 4689281, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 112*x^10 + 380*x^9 + 4719*x^8 - 13468*x^7 - 92386*x^6 + 222256*x^5 + 848264*x^4 - 1642580*x^3 - 3278996*x^2 + 3844396*x + 4689281);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 - 112*x^10 + 380*x^9 + 4719*x^8 - 13468*x^7 - 92386*x^6 + 222256*x^5 + 848264*x^4 - 1642580*x^3 - 3278996*x^2 + 3844396*x + 4689281)
\( x^{12} - 4 x^{11} - 112 x^{10} + 380 x^{9} + 4719 x^{8} - 13468 x^{7} - 92386 x^{6} + 222256 x^{5} + \cdots + 4689281 \)
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: |
\(25358702738605805230358528\)
\(\medspace = 2^{18}\cdot 13^{8}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(130.92\) | 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: | $2^{3/2}13^{2/3}17^{3/4}\approx 130.92138351184528$ | ||
| Ramified primes: |
\(2\), \(13\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1768=2^{3}\cdot 13\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1768}(1,·)$, $\chi_{1768}(1381,·)$, $\chi_{1768}(1665,·)$, $\chi_{1768}(1225,·)$, $\chi_{1768}(1509,·)$, $\chi_{1768}(1101,·)$, $\chi_{1768}(1517,·)$, $\chi_{1768}(1361,·)$, $\chi_{1768}(1257,·)$, $\chi_{1768}(1121,·)$, $\chi_{1768}(157,·)$, $\chi_{1768}(965,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{18090172}a^{10}+\frac{120199}{9045086}a^{9}-\frac{2073235}{9045086}a^{8}-\frac{2881033}{18090172}a^{7}+\frac{1081975}{9045086}a^{6}+\frac{2799737}{18090172}a^{5}+\frac{1050746}{4522543}a^{4}-\frac{285506}{4522543}a^{3}-\frac{7469271}{18090172}a^{2}-\frac{3086893}{18090172}a+\frac{6285}{341324}$, $\frac{1}{59\!\cdots\!12}a^{11}-\frac{13022460357}{59\!\cdots\!12}a^{10}-\frac{73\!\cdots\!81}{59\!\cdots\!12}a^{9}+\frac{14\!\cdots\!08}{14\!\cdots\!03}a^{8}-\frac{44\!\cdots\!27}{29\!\cdots\!06}a^{7}+\frac{59\!\cdots\!67}{59\!\cdots\!12}a^{6}+\frac{87\!\cdots\!13}{59\!\cdots\!12}a^{5}+\frac{78\!\cdots\!83}{59\!\cdots\!12}a^{4}+\frac{41\!\cdots\!63}{29\!\cdots\!06}a^{3}-\frac{26\!\cdots\!05}{59\!\cdots\!12}a^{2}+\frac{24\!\cdots\!97}{56\!\cdots\!02}a-\frac{851688712687466}{28\!\cdots\!51}$
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: |
$\frac{1485167878352}{14\!\cdots\!03}a^{11}-\frac{2855519834924}{14\!\cdots\!03}a^{10}-\frac{159474311445112}{14\!\cdots\!03}a^{9}+\frac{162597146685701}{14\!\cdots\!03}a^{8}+\frac{62\!\cdots\!64}{14\!\cdots\!03}a^{7}-\frac{16\!\cdots\!32}{14\!\cdots\!03}a^{6}-\frac{10\!\cdots\!20}{14\!\cdots\!03}a^{5}-\frac{29\!\cdots\!26}{14\!\cdots\!03}a^{4}+\frac{75\!\cdots\!36}{14\!\cdots\!03}a^{3}+\frac{50\!\cdots\!64}{14\!\cdots\!03}a^{2}-\frac{16\!\cdots\!80}{14\!\cdots\!03}a-\frac{36\!\cdots\!81}{28\!\cdots\!51}$, $\frac{1485167878352}{14\!\cdots\!03}a^{11}-\frac{2855519834924}{14\!\cdots\!03}a^{10}-\frac{159474311445112}{14\!\cdots\!03}a^{9}+\frac{162597146685701}{14\!\cdots\!03}a^{8}+\frac{62\!\cdots\!64}{14\!\cdots\!03}a^{7}-\frac{16\!\cdots\!32}{14\!\cdots\!03}a^{6}-\frac{10\!\cdots\!20}{14\!\cdots\!03}a^{5}-\frac{29\!\cdots\!26}{14\!\cdots\!03}a^{4}+\frac{75\!\cdots\!36}{14\!\cdots\!03}a^{3}+\frac{50\!\cdots\!64}{14\!\cdots\!03}a^{2}-\frac{16\!\cdots\!80}{14\!\cdots\!03}a-\frac{33\!\cdots\!30}{28\!\cdots\!51}$, $\frac{5761905866082}{14\!\cdots\!03}a^{11}+\frac{1980483889663}{14\!\cdots\!03}a^{10}-\frac{690941251381867}{14\!\cdots\!03}a^{9}-\frac{10\!\cdots\!23}{29\!\cdots\!06}a^{8}+\frac{29\!\cdots\!44}{14\!\cdots\!03}a^{7}+\frac{29\!\cdots\!49}{14\!\cdots\!03}a^{6}-\frac{55\!\cdots\!60}{14\!\cdots\!03}a^{5}-\frac{11\!\cdots\!27}{29\!\cdots\!06}a^{4}+\frac{43\!\cdots\!16}{14\!\cdots\!03}a^{3}+\frac{84\!\cdots\!47}{29\!\cdots\!06}a^{2}-\frac{10\!\cdots\!90}{14\!\cdots\!03}a-\frac{41\!\cdots\!11}{56\!\cdots\!02}$, $\frac{408397490886}{14\!\cdots\!03}a^{11}+\frac{221669773037}{28\!\cdots\!51}a^{10}-\frac{112592042794251}{14\!\cdots\!03}a^{9}-\frac{21\!\cdots\!73}{29\!\cdots\!06}a^{8}+\frac{69\!\cdots\!64}{14\!\cdots\!03}a^{7}+\frac{33\!\cdots\!55}{14\!\cdots\!03}a^{6}-\frac{16\!\cdots\!16}{14\!\cdots\!03}a^{5}-\frac{87\!\cdots\!41}{29\!\cdots\!06}a^{4}+\frac{14\!\cdots\!32}{14\!\cdots\!03}a^{3}+\frac{47\!\cdots\!07}{29\!\cdots\!06}a^{2}-\frac{40\!\cdots\!74}{14\!\cdots\!03}a-\frac{19\!\cdots\!33}{56\!\cdots\!02}$, $\frac{5356171394815}{29\!\cdots\!06}a^{11}-\frac{129376432929031}{59\!\cdots\!12}a^{10}-\frac{591314401130095}{59\!\cdots\!12}a^{9}+\frac{10\!\cdots\!95}{59\!\cdots\!12}a^{8}+\frac{10\!\cdots\!95}{29\!\cdots\!06}a^{7}-\frac{81\!\cdots\!62}{14\!\cdots\!03}a^{6}+\frac{31\!\cdots\!67}{59\!\cdots\!12}a^{5}+\frac{39\!\cdots\!75}{59\!\cdots\!12}a^{4}-\frac{46\!\cdots\!45}{59\!\cdots\!12}a^{3}-\frac{96\!\cdots\!23}{29\!\cdots\!06}a^{2}+\frac{14\!\cdots\!93}{59\!\cdots\!12}a+\frac{15\!\cdots\!21}{28\!\cdots\!51}$, $\frac{11296842908223}{29\!\cdots\!06}a^{11}-\frac{152220591608423}{59\!\cdots\!12}a^{10}-\frac{18\!\cdots\!91}{59\!\cdots\!12}a^{9}+\frac{12\!\cdots\!03}{59\!\cdots\!12}a^{8}+\frac{25\!\cdots\!51}{29\!\cdots\!06}a^{7}-\frac{84\!\cdots\!26}{14\!\cdots\!03}a^{6}-\frac{53\!\cdots\!93}{59\!\cdots\!12}a^{5}+\frac{37\!\cdots\!67}{59\!\cdots\!12}a^{4}+\frac{13\!\cdots\!43}{59\!\cdots\!12}a^{3}-\frac{76\!\cdots\!67}{29\!\cdots\!06}a^{2}+\frac{23\!\cdots\!77}{59\!\cdots\!12}a+\frac{67\!\cdots\!53}{28\!\cdots\!51}$, $\frac{260}{4522543}a^{11}+\frac{294}{4522543}a^{10}-\frac{32310}{4522543}a^{9}-\frac{41525}{4522543}a^{8}+\frac{1430160}{4522543}a^{7}+\frac{1877722}{4522543}a^{6}-\frac{27366480}{4522543}a^{5}-\frac{32882375}{4522543}a^{4}+\frac{220891560}{4522543}a^{3}+\frac{227006939}{4522543}a^{2}-\frac{565358020}{4522543}a-\frac{10924186}{85331}$, $\frac{398230543889921}{59\!\cdots\!12}a^{11}-\frac{16\!\cdots\!21}{59\!\cdots\!12}a^{10}-\frac{94\!\cdots\!65}{14\!\cdots\!03}a^{9}+\frac{11\!\cdots\!91}{59\!\cdots\!12}a^{8}+\frac{13\!\cdots\!73}{59\!\cdots\!12}a^{7}-\frac{25\!\cdots\!75}{59\!\cdots\!12}a^{6}-\frac{19\!\cdots\!75}{59\!\cdots\!12}a^{5}+\frac{42\!\cdots\!19}{14\!\cdots\!03}a^{4}+\frac{13\!\cdots\!29}{59\!\cdots\!12}a^{3}+\frac{91\!\cdots\!17}{29\!\cdots\!06}a^{2}-\frac{78\!\cdots\!70}{14\!\cdots\!03}a-\frac{42\!\cdots\!43}{11\!\cdots\!04}$, $\frac{228893676912526}{14\!\cdots\!03}a^{11}-\frac{16076923744775}{59\!\cdots\!12}a^{10}-\frac{27\!\cdots\!44}{14\!\cdots\!03}a^{9}-\frac{25\!\cdots\!17}{29\!\cdots\!06}a^{8}+\frac{45\!\cdots\!29}{59\!\cdots\!12}a^{7}+\frac{89\!\cdots\!77}{14\!\cdots\!03}a^{6}-\frac{84\!\cdots\!59}{59\!\cdots\!12}a^{5}-\frac{37\!\cdots\!59}{29\!\cdots\!06}a^{4}+\frac{61\!\cdots\!13}{56\!\cdots\!02}a^{3}+\frac{56\!\cdots\!65}{59\!\cdots\!12}a^{2}-\frac{16\!\cdots\!81}{59\!\cdots\!12}a-\frac{27\!\cdots\!41}{11\!\cdots\!04}$, $\frac{74769973724177}{59\!\cdots\!12}a^{11}+\frac{10\!\cdots\!45}{59\!\cdots\!12}a^{10}-\frac{14\!\cdots\!11}{59\!\cdots\!12}a^{9}-\frac{23\!\cdots\!76}{14\!\cdots\!03}a^{8}+\frac{40\!\cdots\!77}{29\!\cdots\!06}a^{7}+\frac{30\!\cdots\!63}{59\!\cdots\!12}a^{6}-\frac{18\!\cdots\!49}{59\!\cdots\!12}a^{5}-\frac{41\!\cdots\!89}{59\!\cdots\!12}a^{4}+\frac{40\!\cdots\!48}{14\!\cdots\!03}a^{3}+\frac{21\!\cdots\!41}{59\!\cdots\!12}a^{2}-\frac{10\!\cdots\!87}{14\!\cdots\!03}a-\frac{41\!\cdots\!25}{56\!\cdots\!02}$, $\frac{255705619664221}{59\!\cdots\!12}a^{11}-\frac{490948927883743}{59\!\cdots\!12}a^{10}-\frac{26\!\cdots\!89}{59\!\cdots\!12}a^{9}+\frac{60\!\cdots\!92}{14\!\cdots\!03}a^{8}+\frac{25\!\cdots\!05}{14\!\cdots\!03}a^{7}-\frac{72\!\cdots\!91}{59\!\cdots\!12}a^{6}-\frac{17\!\cdots\!21}{59\!\cdots\!12}a^{5}-\frac{11\!\cdots\!99}{59\!\cdots\!12}a^{4}+\frac{61\!\cdots\!45}{29\!\cdots\!06}a^{3}+\frac{14\!\cdots\!13}{59\!\cdots\!12}a^{2}-\frac{73\!\cdots\!17}{14\!\cdots\!03}a-\frac{20\!\cdots\!26}{28\!\cdots\!51}$
| 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: | \( 362884771.765 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 362884771.765 \cdot 2}{2\cdot\sqrt{25358702738605805230358528}}\cr\approx \mathstrut & 0.295165215244 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 112*x^10 + 380*x^9 + 4719*x^8 - 13468*x^7 - 92386*x^6 + 222256*x^5 + 848264*x^4 - 1642580*x^3 - 3278996*x^2 + 3844396*x + 4689281)
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 - 4*x^11 - 112*x^10 + 380*x^9 + 4719*x^8 - 13468*x^7 - 92386*x^6 + 222256*x^5 + 848264*x^4 - 1642580*x^3 - 3278996*x^2 + 3844396*x + 4689281, 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 - 4*x^11 - 112*x^10 + 380*x^9 + 4719*x^8 - 13468*x^7 - 92386*x^6 + 222256*x^5 + 848264*x^4 - 1642580*x^3 - 3278996*x^2 + 3844396*x + 4689281);
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 - 4*x^11 - 112*x^10 + 380*x^9 + 4719*x^8 - 13468*x^7 - 92386*x^6 + 222256*x^5 + 848264*x^4 - 1642580*x^3 - 3278996*x^2 + 3844396*x + 4689281);
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
| \(\Q(\sqrt{17}) \), 3.3.169.1, 4.4.314432.1, 6.6.140320193.1 |
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 | R | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.12.0.1}{12} }$ | R | R | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.1.0.1}{1} }^{12}$ | ${\href{/padicField/53.1.0.1}{1} }^{12}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.9.3 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ |
| 2.6.9.3 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
|
\(13\)
| 13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |