Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 104*x^10 + 364*x^9 + 4239*x^8 - 12292*x^7 - 86658*x^6 + 191056*x^5 + 934304*x^4 - 1352932*x^3 - 4997300*x^2 + 3448884*x + 10118401)
gp: K = bnfinit(y^12 - 4*y^11 - 104*y^10 + 364*y^9 + 4239*y^8 - 12292*y^7 - 86658*y^6 + 191056*y^5 + 934304*y^4 - 1352932*y^3 - 4997300*y^2 + 3448884*y + 10118401, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 104*x^10 + 364*x^9 + 4239*x^8 - 12292*x^7 - 86658*x^6 + 191056*x^5 + 934304*x^4 - 1352932*x^3 - 4997300*x^2 + 3448884*x + 10118401);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 - 104*x^10 + 364*x^9 + 4239*x^8 - 12292*x^7 - 86658*x^6 + 191056*x^5 + 934304*x^4 - 1352932*x^3 - 4997300*x^2 + 3448884*x + 10118401)
\( x^{12} - 4 x^{11} - 104 x^{10} + 364 x^{9} + 4239 x^{8} - 12292 x^{7} - 86658 x^{6} + 191056 x^{5} + \cdots + 10118401 \)
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: |
\(179210946875957470035968\)
\(\medspace = 2^{18}\cdot 7^{8}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(86.65\) | 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: | $2^{3/2}7^{2/3}17^{3/4}\approx 86.6523565155333$ | ||
| Ramified primes: |
\(2\), \(7\), \(17\)
| 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: | \(952=2^{3}\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{952}(1,·)$, $\chi_{952}(837,·)$, $\chi_{952}(849,·)$, $\chi_{952}(137,·)$, $\chi_{952}(429,·)$, $\chi_{952}(557,·)$, $\chi_{952}(305,·)$, $\chi_{952}(149,·)$, $\chi_{952}(681,·)$, $\chi_{952}(169,·)$, $\chi_{952}(701,·)$, $\chi_{952}(421,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{12182524}a^{10}-\frac{14322}{3045631}a^{9}-\frac{578503}{3045631}a^{8}+\frac{1692969}{12182524}a^{7}-\frac{688085}{6091262}a^{6}-\frac{1181959}{12182524}a^{5}-\frac{274419}{6091262}a^{4}+\frac{2281489}{6091262}a^{3}-\frac{3402713}{12182524}a^{2}-\frac{2420597}{12182524}a-\frac{5256713}{12182524}$, $\frac{1}{63\!\cdots\!56}a^{11}+\frac{12131242177}{63\!\cdots\!56}a^{10}+\frac{18\!\cdots\!37}{63\!\cdots\!56}a^{9}-\frac{60\!\cdots\!57}{15\!\cdots\!39}a^{8}+\frac{33\!\cdots\!25}{15\!\cdots\!39}a^{7}+\frac{48\!\cdots\!39}{63\!\cdots\!56}a^{6}-\frac{18\!\cdots\!51}{63\!\cdots\!56}a^{5}-\frac{62\!\cdots\!89}{63\!\cdots\!56}a^{4}+\frac{76\!\cdots\!83}{15\!\cdots\!39}a^{3}-\frac{29\!\cdots\!35}{63\!\cdots\!56}a^{2}+\frac{83\!\cdots\!93}{31\!\cdots\!78}a-\frac{12\!\cdots\!27}{31\!\cdots\!78}$
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: |
$\frac{242139114900}{15\!\cdots\!39}a^{11}-\frac{2837094928350}{15\!\cdots\!39}a^{10}-\frac{15876195420584}{15\!\cdots\!39}a^{9}+\frac{276472880605002}{15\!\cdots\!39}a^{8}+\frac{200557758206448}{15\!\cdots\!39}a^{7}-\frac{99\!\cdots\!50}{15\!\cdots\!39}a^{6}+\frac{47\!\cdots\!52}{15\!\cdots\!39}a^{5}+\frac{16\!\cdots\!59}{15\!\cdots\!39}a^{4}-\frac{10\!\cdots\!84}{15\!\cdots\!39}a^{3}-\frac{11\!\cdots\!12}{15\!\cdots\!39}a^{2}+\frac{41\!\cdots\!00}{15\!\cdots\!39}a+\frac{23\!\cdots\!64}{15\!\cdots\!39}$, $\frac{854191568792}{15\!\cdots\!39}a^{11}-\frac{3592958085288}{15\!\cdots\!39}a^{10}-\frac{76166832114504}{15\!\cdots\!39}a^{9}+\frac{293205885995987}{15\!\cdots\!39}a^{8}+\frac{24\!\cdots\!56}{15\!\cdots\!39}a^{7}-\frac{84\!\cdots\!84}{15\!\cdots\!39}a^{6}-\frac{35\!\cdots\!44}{15\!\cdots\!39}a^{5}+\frac{10\!\cdots\!68}{15\!\cdots\!39}a^{4}+\frac{20\!\cdots\!52}{15\!\cdots\!39}a^{3}-\frac{52\!\cdots\!48}{15\!\cdots\!39}a^{2}-\frac{27\!\cdots\!20}{15\!\cdots\!39}a+\frac{90\!\cdots\!81}{15\!\cdots\!39}$, $\frac{1335844678514}{15\!\cdots\!39}a^{11}-\frac{6368105095400}{15\!\cdots\!39}a^{10}-\frac{124220973777775}{15\!\cdots\!39}a^{9}+\frac{485613912392468}{15\!\cdots\!39}a^{8}+\frac{44\!\cdots\!22}{15\!\cdots\!39}a^{7}-\frac{23\!\cdots\!93}{31\!\cdots\!78}a^{6}-\frac{75\!\cdots\!98}{15\!\cdots\!39}a^{5}+\frac{18\!\cdots\!83}{31\!\cdots\!78}a^{4}+\frac{60\!\cdots\!52}{15\!\cdots\!39}a^{3}+\frac{86\!\cdots\!72}{15\!\cdots\!39}a^{2}-\frac{16\!\cdots\!17}{15\!\cdots\!39}a-\frac{35\!\cdots\!79}{31\!\cdots\!78}$, $\frac{1336409605858}{15\!\cdots\!39}a^{11}-\frac{16639980703621}{15\!\cdots\!39}a^{10}-\frac{88592305417163}{15\!\cdots\!39}a^{9}+\frac{29\!\cdots\!23}{31\!\cdots\!78}a^{8}+\frac{15\!\cdots\!80}{15\!\cdots\!39}a^{7}-\frac{46\!\cdots\!43}{15\!\cdots\!39}a^{6}+\frac{24\!\cdots\!42}{15\!\cdots\!39}a^{5}+\frac{13\!\cdots\!77}{31\!\cdots\!78}a^{4}-\frac{26\!\cdots\!76}{15\!\cdots\!39}a^{3}-\frac{86\!\cdots\!31}{31\!\cdots\!78}a^{2}+\frac{15\!\cdots\!06}{15\!\cdots\!39}a+\frac{19\!\cdots\!31}{31\!\cdots\!78}$, $\frac{614112646626}{15\!\cdots\!39}a^{11}+\frac{6616969604695}{15\!\cdots\!39}a^{10}-\frac{79617554232429}{15\!\cdots\!39}a^{9}-\frac{11\!\cdots\!71}{31\!\cdots\!78}a^{8}+\frac{35\!\cdots\!80}{15\!\cdots\!39}a^{7}+\frac{19\!\cdots\!25}{15\!\cdots\!39}a^{6}-\frac{68\!\cdots\!78}{15\!\cdots\!39}a^{5}-\frac{59\!\cdots\!87}{31\!\cdots\!78}a^{4}+\frac{57\!\cdots\!96}{15\!\cdots\!39}a^{3}+\frac{42\!\cdots\!15}{31\!\cdots\!78}a^{2}-\frac{17\!\cdots\!46}{15\!\cdots\!39}a-\frac{11\!\cdots\!35}{31\!\cdots\!78}$, $\frac{15917259379171}{31\!\cdots\!78}a^{11}-\frac{235701302278325}{63\!\cdots\!56}a^{10}-\frac{25\!\cdots\!11}{63\!\cdots\!56}a^{9}+\frac{20\!\cdots\!93}{63\!\cdots\!56}a^{8}+\frac{36\!\cdots\!63}{31\!\cdots\!78}a^{7}-\frac{32\!\cdots\!95}{31\!\cdots\!78}a^{6}-\frac{83\!\cdots\!21}{63\!\cdots\!56}a^{5}+\frac{91\!\cdots\!31}{63\!\cdots\!56}a^{4}+\frac{32\!\cdots\!39}{63\!\cdots\!56}a^{3}-\frac{28\!\cdots\!17}{31\!\cdots\!78}a^{2}-\frac{14\!\cdots\!23}{63\!\cdots\!56}a+\frac{60\!\cdots\!91}{31\!\cdots\!78}$, $\frac{13539168438611}{31\!\cdots\!78}a^{11}-\frac{10299086213701}{63\!\cdots\!56}a^{10}-\frac{27\!\cdots\!55}{63\!\cdots\!56}a^{9}-\frac{245962426685535}{63\!\cdots\!56}a^{8}+\frac{51\!\cdots\!11}{31\!\cdots\!78}a^{7}+\frac{27\!\cdots\!41}{31\!\cdots\!78}a^{6}-\frac{17\!\cdots\!81}{63\!\cdots\!56}a^{5}-\frac{16\!\cdots\!97}{63\!\cdots\!56}a^{4}+\frac{14\!\cdots\!47}{63\!\cdots\!56}a^{3}+\frac{88\!\cdots\!27}{31\!\cdots\!78}a^{2}-\frac{38\!\cdots\!87}{63\!\cdots\!56}a-\frac{28\!\cdots\!97}{31\!\cdots\!78}$, $\frac{104419444984633}{63\!\cdots\!56}a^{11}+\frac{1805055779862}{15\!\cdots\!39}a^{10}-\frac{10\!\cdots\!19}{63\!\cdots\!56}a^{9}-\frac{20\!\cdots\!59}{31\!\cdots\!78}a^{8}+\frac{35\!\cdots\!77}{63\!\cdots\!56}a^{7}+\frac{24\!\cdots\!73}{63\!\cdots\!56}a^{6}-\frac{27\!\cdots\!35}{31\!\cdots\!78}a^{5}-\frac{48\!\cdots\!97}{63\!\cdots\!56}a^{4}+\frac{19\!\cdots\!21}{31\!\cdots\!78}a^{3}+\frac{98\!\cdots\!12}{15\!\cdots\!39}a^{2}-\frac{92\!\cdots\!43}{63\!\cdots\!56}a-\frac{10\!\cdots\!27}{63\!\cdots\!56}$, $\frac{3610717200177}{63\!\cdots\!56}a^{11}+\frac{4932951030457}{15\!\cdots\!39}a^{10}-\frac{102348520043890}{15\!\cdots\!39}a^{9}-\frac{18\!\cdots\!43}{63\!\cdots\!56}a^{8}+\frac{80\!\cdots\!27}{31\!\cdots\!78}a^{7}+\frac{64\!\cdots\!29}{63\!\cdots\!56}a^{6}-\frac{13\!\cdots\!27}{31\!\cdots\!78}a^{5}-\frac{48\!\cdots\!35}{31\!\cdots\!78}a^{4}+\frac{20\!\cdots\!11}{63\!\cdots\!56}a^{3}+\frac{64\!\cdots\!99}{63\!\cdots\!56}a^{2}-\frac{56\!\cdots\!09}{63\!\cdots\!56}a-\frac{37\!\cdots\!15}{15\!\cdots\!39}$, $\frac{40804442566857}{63\!\cdots\!56}a^{11}-\frac{222935744453477}{63\!\cdots\!56}a^{10}-\frac{990045609526974}{15\!\cdots\!39}a^{9}+\frac{20\!\cdots\!31}{63\!\cdots\!56}a^{8}+\frac{14\!\cdots\!29}{63\!\cdots\!56}a^{7}-\frac{72\!\cdots\!87}{63\!\cdots\!56}a^{6}-\frac{25\!\cdots\!47}{63\!\cdots\!56}a^{5}+\frac{29\!\cdots\!95}{15\!\cdots\!39}a^{4}+\frac{21\!\cdots\!77}{63\!\cdots\!56}a^{3}-\frac{44\!\cdots\!49}{31\!\cdots\!78}a^{2}-\frac{16\!\cdots\!50}{15\!\cdots\!39}a+\frac{24\!\cdots\!93}{63\!\cdots\!56}$, $\frac{132330051360617}{31\!\cdots\!78}a^{11}-\frac{17\!\cdots\!85}{63\!\cdots\!56}a^{10}-\frac{10\!\cdots\!41}{31\!\cdots\!78}a^{9}+\frac{38\!\cdots\!42}{15\!\cdots\!39}a^{8}+\frac{61\!\cdots\!33}{63\!\cdots\!56}a^{7}-\frac{24\!\cdots\!77}{31\!\cdots\!78}a^{6}-\frac{69\!\cdots\!93}{63\!\cdots\!56}a^{5}+\frac{33\!\cdots\!07}{31\!\cdots\!78}a^{4}+\frac{12\!\cdots\!79}{31\!\cdots\!78}a^{3}-\frac{39\!\cdots\!03}{63\!\cdots\!56}a^{2}+\frac{93\!\cdots\!91}{63\!\cdots\!56}a+\frac{79\!\cdots\!61}{63\!\cdots\!56}$
| 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: | \( 21207640.0343 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 21207640.0343 \cdot 2}{2\cdot\sqrt{179210946875957470035968}}\cr\approx \mathstrut & 0.205196535553 \end{aligned}\] (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 - 104*x^10 + 364*x^9 + 4239*x^8 - 12292*x^7 - 86658*x^6 + 191056*x^5 + 934304*x^4 - 1352932*x^3 - 4997300*x^2 + 3448884*x + 10118401)
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 - 104*x^10 + 364*x^9 + 4239*x^8 - 12292*x^7 - 86658*x^6 + 191056*x^5 + 934304*x^4 - 1352932*x^3 - 4997300*x^2 + 3448884*x + 10118401, 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 - 104*x^10 + 364*x^9 + 4239*x^8 - 12292*x^7 - 86658*x^6 + 191056*x^5 + 934304*x^4 - 1352932*x^3 - 4997300*x^2 + 3448884*x + 10118401);
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 - 104*x^10 + 364*x^9 + 4239*x^8 - 12292*x^7 - 86658*x^6 + 191056*x^5 + 934304*x^4 - 1352932*x^3 - 4997300*x^2 + 3448884*x + 10118401);
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
| \(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.4.314432.1, 6.6.11796113.1 |
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 | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.12.0.1}{12} }$ | R | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.1.0.1}{1} }^{12}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
|
\(7\)
| 7.12.8.1 | $x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |