Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 194*x^10 + 194*x^9 + 12403*x^8 - 13703*x^7 - 305057*x^6 + 332682*x^5 + 2890746*x^4 - 2743521*x^3 - 8936524*x^2 + 4919524*x + 8816575)
gp: K = bnfinit(y^12 - y^11 - 194*y^10 + 194*y^9 + 12403*y^8 - 13703*y^7 - 305057*y^6 + 332682*y^5 + 2890746*y^4 - 2743521*y^3 - 8936524*y^2 + 4919524*y + 8816575, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 194*x^10 + 194*x^9 + 12403*x^8 - 13703*x^7 - 305057*x^6 + 332682*x^5 + 2890746*x^4 - 2743521*x^3 - 8936524*x^2 + 4919524*x + 8816575);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 194*x^10 + 194*x^9 + 12403*x^8 - 13703*x^7 - 305057*x^6 + 332682*x^5 + 2890746*x^4 - 2743521*x^3 - 8936524*x^2 + 4919524*x + 8816575)
\( x^{12} - x^{11} - 194 x^{10} + 194 x^{9} + 12403 x^{8} - 13703 x^{7} - 305057 x^{6} + 332682 x^{5} + \cdots + 8816575 \)
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: |
\(154933273198267591600075581\)
\(\medspace = 3^{6}\cdot 13^{11}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(152.23\) | 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: | $3^{1/2}13^{11/12}17^{3/4}\approx 152.2341399478133$ | ||
| Ramified primes: |
\(3\), \(13\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{221}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(663=3\cdot 13\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{663}(256,·)$, $\chi_{663}(1,·)$, $\chi_{663}(322,·)$, $\chi_{663}(548,·)$, $\chi_{663}(395,·)$, $\chi_{663}(98,·)$, $\chi_{663}(557,·)$, $\chi_{663}(562,·)$, $\chi_{663}(628,·)$, $\chi_{663}(344,·)$, $\chi_{663}(47,·)$, $\chi_{663}(220,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}-\frac{1}{5}a$, $\frac{1}{5}a^{6}-\frac{1}{5}a^{2}$, $\frac{1}{5}a^{7}-\frac{1}{5}a^{3}$, $\frac{1}{25}a^{8}-\frac{1}{25}a^{7}+\frac{1}{25}a^{6}-\frac{1}{25}a^{5}-\frac{11}{25}a^{4}+\frac{11}{25}a^{3}-\frac{11}{25}a^{2}+\frac{11}{25}a$, $\frac{1}{25}a^{9}-\frac{2}{25}a^{5}+\frac{1}{25}a$, $\frac{1}{483625}a^{10}+\frac{9519}{483625}a^{9}+\frac{8822}{483625}a^{8}+\frac{10528}{483625}a^{7}+\frac{29}{1325}a^{6}+\frac{74}{3869}a^{5}-\frac{120267}{483625}a^{4}+\frac{140242}{483625}a^{3}-\frac{140981}{483625}a^{2}+\frac{19376}{483625}a+\frac{84}{265}$, $\frac{1}{48\!\cdots\!25}a^{11}-\frac{15\!\cdots\!94}{96\!\cdots\!25}a^{10}+\frac{48\!\cdots\!06}{48\!\cdots\!25}a^{9}+\frac{59\!\cdots\!57}{96\!\cdots\!25}a^{8}-\frac{37\!\cdots\!97}{48\!\cdots\!25}a^{7}-\frac{18\!\cdots\!44}{38\!\cdots\!85}a^{6}-\frac{10\!\cdots\!32}{48\!\cdots\!25}a^{5}+\frac{70\!\cdots\!13}{96\!\cdots\!25}a^{4}-\frac{84\!\cdots\!79}{48\!\cdots\!25}a^{3}+\frac{31\!\cdots\!74}{96\!\cdots\!25}a^{2}+\frac{34\!\cdots\!26}{48\!\cdots\!25}a+\frac{12\!\cdots\!69}{26\!\cdots\!25}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
Class group and class number
$C_{2}\times C_{4}$, which has order $8$ (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: |
$\frac{35\!\cdots\!69}{48\!\cdots\!25}a^{11}+\frac{12\!\cdots\!73}{18\!\cdots\!25}a^{10}-\frac{60\!\cdots\!36}{48\!\cdots\!25}a^{9}-\frac{11\!\cdots\!97}{96\!\cdots\!25}a^{8}+\frac{28\!\cdots\!32}{48\!\cdots\!25}a^{7}+\frac{12\!\cdots\!87}{19\!\cdots\!25}a^{6}-\frac{29\!\cdots\!08}{48\!\cdots\!25}a^{5}-\frac{12\!\cdots\!48}{96\!\cdots\!25}a^{4}-\frac{40\!\cdots\!01}{48\!\cdots\!25}a^{3}+\frac{61\!\cdots\!16}{96\!\cdots\!25}a^{2}+\frac{55\!\cdots\!94}{48\!\cdots\!25}a+\frac{73\!\cdots\!11}{26\!\cdots\!25}$, $\frac{35\!\cdots\!69}{48\!\cdots\!25}a^{11}+\frac{12\!\cdots\!73}{18\!\cdots\!25}a^{10}-\frac{60\!\cdots\!36}{48\!\cdots\!25}a^{9}-\frac{11\!\cdots\!97}{96\!\cdots\!25}a^{8}+\frac{28\!\cdots\!32}{48\!\cdots\!25}a^{7}+\frac{12\!\cdots\!87}{19\!\cdots\!25}a^{6}-\frac{29\!\cdots\!08}{48\!\cdots\!25}a^{5}-\frac{12\!\cdots\!48}{96\!\cdots\!25}a^{4}-\frac{40\!\cdots\!01}{48\!\cdots\!25}a^{3}+\frac{61\!\cdots\!16}{96\!\cdots\!25}a^{2}+\frac{55\!\cdots\!94}{48\!\cdots\!25}a+\frac{33\!\cdots\!36}{26\!\cdots\!25}$, $\frac{7151357270743}{27\!\cdots\!75}a^{11}+\frac{1304380161959}{55\!\cdots\!75}a^{10}-\frac{13\!\cdots\!72}{27\!\cdots\!75}a^{9}-\frac{251169884701827}{55\!\cdots\!75}a^{8}+\frac{89\!\cdots\!44}{27\!\cdots\!75}a^{7}+\frac{27\!\cdots\!37}{11\!\cdots\!75}a^{6}-\frac{21\!\cdots\!51}{27\!\cdots\!75}a^{5}-\frac{33\!\cdots\!08}{55\!\cdots\!75}a^{4}+\frac{19\!\cdots\!38}{27\!\cdots\!75}a^{3}+\frac{34\!\cdots\!01}{55\!\cdots\!75}a^{2}-\frac{36\!\cdots\!27}{27\!\cdots\!75}a-\frac{42\!\cdots\!13}{15\!\cdots\!75}$, $\frac{95\!\cdots\!78}{96\!\cdots\!25}a^{11}+\frac{12\!\cdots\!23}{96\!\cdots\!25}a^{10}-\frac{36\!\cdots\!62}{19\!\cdots\!25}a^{9}-\frac{22\!\cdots\!69}{96\!\cdots\!25}a^{8}+\frac{11\!\cdots\!58}{96\!\cdots\!25}a^{7}+\frac{24\!\cdots\!56}{19\!\cdots\!25}a^{6}-\frac{27\!\cdots\!86}{96\!\cdots\!25}a^{5}-\frac{27\!\cdots\!66}{96\!\cdots\!25}a^{4}+\frac{23\!\cdots\!74}{96\!\cdots\!25}a^{3}+\frac{22\!\cdots\!87}{96\!\cdots\!25}a^{2}-\frac{42\!\cdots\!19}{96\!\cdots\!25}a-\frac{23\!\cdots\!21}{52\!\cdots\!45}$, $\frac{41\!\cdots\!14}{96\!\cdots\!25}a^{11}-\frac{47\!\cdots\!79}{96\!\cdots\!25}a^{10}-\frac{68\!\cdots\!97}{96\!\cdots\!25}a^{9}+\frac{86\!\cdots\!87}{96\!\cdots\!25}a^{8}+\frac{54\!\cdots\!69}{19\!\cdots\!25}a^{7}-\frac{98\!\cdots\!28}{19\!\cdots\!25}a^{6}+\frac{18\!\cdots\!77}{96\!\cdots\!25}a^{5}+\frac{88\!\cdots\!43}{96\!\cdots\!25}a^{4}-\frac{13\!\cdots\!89}{96\!\cdots\!25}a^{3}-\frac{35\!\cdots\!01}{96\!\cdots\!25}a^{2}+\frac{73\!\cdots\!23}{19\!\cdots\!25}a+\frac{52\!\cdots\!49}{10\!\cdots\!49}$, $\frac{30\!\cdots\!09}{48\!\cdots\!25}a^{11}-\frac{14\!\cdots\!96}{96\!\cdots\!25}a^{10}-\frac{58\!\cdots\!21}{48\!\cdots\!25}a^{9}+\frac{27\!\cdots\!58}{96\!\cdots\!25}a^{8}+\frac{36\!\cdots\!77}{48\!\cdots\!25}a^{7}-\frac{36\!\cdots\!36}{19\!\cdots\!25}a^{6}-\frac{16\!\cdots\!21}{90\!\cdots\!25}a^{5}+\frac{43\!\cdots\!22}{96\!\cdots\!25}a^{4}+\frac{67\!\cdots\!64}{48\!\cdots\!25}a^{3}-\frac{34\!\cdots\!79}{96\!\cdots\!25}a^{2}-\frac{11\!\cdots\!41}{48\!\cdots\!25}a+\frac{16\!\cdots\!96}{26\!\cdots\!25}$, $\frac{20\!\cdots\!97}{48\!\cdots\!25}a^{11}+\frac{42\!\cdots\!01}{96\!\cdots\!25}a^{10}-\frac{39\!\cdots\!88}{48\!\cdots\!25}a^{9}-\frac{85\!\cdots\!98}{96\!\cdots\!25}a^{8}+\frac{24\!\cdots\!76}{48\!\cdots\!25}a^{7}+\frac{10\!\cdots\!04}{19\!\cdots\!25}a^{6}-\frac{59\!\cdots\!79}{48\!\cdots\!25}a^{5}-\frac{13\!\cdots\!67}{96\!\cdots\!25}a^{4}+\frac{55\!\cdots\!27}{48\!\cdots\!25}a^{3}+\frac{13\!\cdots\!34}{96\!\cdots\!25}a^{2}-\frac{14\!\cdots\!08}{48\!\cdots\!25}a-\frac{10\!\cdots\!52}{26\!\cdots\!25}$, $\frac{18\!\cdots\!98}{96\!\cdots\!25}a^{11}+\frac{19\!\cdots\!28}{76\!\cdots\!77}a^{10}-\frac{36\!\cdots\!97}{96\!\cdots\!25}a^{9}-\frac{93\!\cdots\!64}{19\!\cdots\!25}a^{8}+\frac{22\!\cdots\!94}{96\!\cdots\!25}a^{7}+\frac{53\!\cdots\!26}{19\!\cdots\!25}a^{6}-\frac{54\!\cdots\!66}{96\!\cdots\!25}a^{5}-\frac{12\!\cdots\!51}{19\!\cdots\!25}a^{4}+\frac{47\!\cdots\!58}{96\!\cdots\!25}a^{3}+\frac{11\!\cdots\!64}{19\!\cdots\!25}a^{2}-\frac{10\!\cdots\!62}{96\!\cdots\!25}a-\frac{70\!\cdots\!08}{52\!\cdots\!45}$, $\frac{34\!\cdots\!14}{48\!\cdots\!25}a^{11}-\frac{21\!\cdots\!84}{96\!\cdots\!25}a^{10}-\frac{66\!\cdots\!76}{48\!\cdots\!25}a^{9}+\frac{41\!\cdots\!37}{96\!\cdots\!25}a^{8}+\frac{40\!\cdots\!72}{48\!\cdots\!25}a^{7}-\frac{53\!\cdots\!84}{19\!\cdots\!25}a^{6}-\frac{88\!\cdots\!48}{48\!\cdots\!25}a^{5}+\frac{61\!\cdots\!28}{96\!\cdots\!25}a^{4}+\frac{58\!\cdots\!64}{48\!\cdots\!25}a^{3}-\frac{47\!\cdots\!66}{96\!\cdots\!25}a^{2}+\frac{89\!\cdots\!49}{48\!\cdots\!25}a+\frac{15\!\cdots\!06}{26\!\cdots\!25}$, $\frac{32\!\cdots\!69}{19\!\cdots\!25}a^{11}+\frac{14\!\cdots\!88}{96\!\cdots\!25}a^{10}-\frac{31\!\cdots\!53}{96\!\cdots\!25}a^{9}-\frac{49\!\cdots\!34}{96\!\cdots\!25}a^{8}+\frac{19\!\cdots\!44}{96\!\cdots\!25}a^{7}-\frac{45\!\cdots\!93}{19\!\cdots\!25}a^{6}-\frac{95\!\cdots\!21}{19\!\cdots\!25}a^{5}+\frac{51\!\cdots\!99}{96\!\cdots\!25}a^{4}+\frac{41\!\cdots\!96}{96\!\cdots\!25}a^{3}+\frac{68\!\cdots\!42}{96\!\cdots\!25}a^{2}-\frac{95\!\cdots\!82}{96\!\cdots\!25}a-\frac{19\!\cdots\!28}{52\!\cdots\!45}$, $\frac{65\!\cdots\!44}{48\!\cdots\!25}a^{11}-\frac{57\!\cdots\!41}{96\!\cdots\!25}a^{10}-\frac{11\!\cdots\!11}{48\!\cdots\!25}a^{9}+\frac{10\!\cdots\!18}{96\!\cdots\!25}a^{8}+\frac{58\!\cdots\!57}{48\!\cdots\!25}a^{7}-\frac{11\!\cdots\!01}{19\!\cdots\!25}a^{6}-\frac{75\!\cdots\!83}{48\!\cdots\!25}a^{5}+\frac{98\!\cdots\!12}{96\!\cdots\!25}a^{4}-\frac{23\!\cdots\!26}{48\!\cdots\!25}a^{3}-\frac{35\!\cdots\!34}{96\!\cdots\!25}a^{2}+\frac{55\!\cdots\!44}{48\!\cdots\!25}a+\frac{92\!\cdots\!11}{26\!\cdots\!25}$
| 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: | \( 1968028336.0826359 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 1968028336.0826359 \cdot 8}{2\cdot\sqrt{154933273198267591600075581}}\cr\approx \mathstrut & 2.59047221485936 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 194*x^10 + 194*x^9 + 12403*x^8 - 13703*x^7 - 305057*x^6 + 332682*x^5 + 2890746*x^4 - 2743521*x^3 - 8936524*x^2 + 4919524*x + 8816575)
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 - 194*x^10 + 194*x^9 + 12403*x^8 - 13703*x^7 - 305057*x^6 + 332682*x^5 + 2890746*x^4 - 2743521*x^3 - 8936524*x^2 + 4919524*x + 8816575, 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 - 194*x^10 + 194*x^9 + 12403*x^8 - 13703*x^7 - 305057*x^6 + 332682*x^5 + 2890746*x^4 - 2743521*x^3 - 8936524*x^2 + 4919524*x + 8816575);
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 - 194*x^10 + 194*x^9 + 12403*x^8 - 13703*x^7 - 305057*x^6 + 332682*x^5 + 2890746*x^4 - 2743521*x^3 - 8936524*x^2 + 4919524*x + 8816575);
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
| \(\Q(\sqrt{221}) \), 3.3.169.1, 4.4.97144749.1, 6.6.1824162509.1 |
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 | ${\href{/padicField/5.1.0.1}{1} }^{12}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | R | R | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.1.0.1}{1} }^{12}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 | $x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
|
\(13\)
| 13.12.11.4 | $x^{12} + 13$ | $12$ | $1$ | $11$ | $C_{12}$ | $[\ ]_{12}$ |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |