Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 202*x^10 + 74*x^9 + 15753*x^8 + 1798*x^7 - 599316*x^6 - 252696*x^5 + 11554141*x^4 + 6283413*x^3 - 104562232*x^2 - 45628823*x + 334724041)
gp: K = bnfinit(y^12 - y^11 - 202*y^10 + 74*y^9 + 15753*y^8 + 1798*y^7 - 599316*y^6 - 252696*y^5 + 11554141*y^4 + 6283413*y^3 - 104562232*y^2 - 45628823*y + 334724041, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 202*x^10 + 74*x^9 + 15753*x^8 + 1798*x^7 - 599316*x^6 - 252696*x^5 + 11554141*x^4 + 6283413*x^3 - 104562232*x^2 - 45628823*x + 334724041);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 202*x^10 + 74*x^9 + 15753*x^8 + 1798*x^7 - 599316*x^6 - 252696*x^5 + 11554141*x^4 + 6283413*x^3 - 104562232*x^2 - 45628823*x + 334724041)
\( x^{12} - x^{11} - 202 x^{10} + 74 x^{9} + 15753 x^{8} + 1798 x^{7} - 599316 x^{6} - 252696 x^{5} + \cdots + 334724041 \)
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: |
\(198123237327133986328125\)
\(\medspace = 3^{6}\cdot 5^{9}\cdot 7^{8}\cdot 17^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(87.38\) | 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: | $3^{1/2}5^{3/4}7^{2/3}17^{1/2}\approx 87.37984794739731$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1785=3\cdot 5\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1785}(256,·)$, $\chi_{1785}(1,·)$, $\chi_{1785}(1733,·)$, $\chi_{1785}(1478,·)$, $\chi_{1785}(968,·)$, $\chi_{1785}(919,·)$, $\chi_{1785}(1682,·)$, $\chi_{1785}(1684,·)$, $\chi_{1785}(1429,·)$, $\chi_{1785}(662,·)$, $\chi_{1785}(407,·)$, $\chi_{1785}(1276,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{131}a^{9}-\frac{39}{131}a^{8}+\frac{15}{131}a^{7}-\frac{62}{131}a^{6}-\frac{39}{131}a^{5}+\frac{51}{131}a^{4}-\frac{18}{131}a^{3}+\frac{10}{131}a^{2}-\frac{43}{131}a+\frac{15}{131}$, $\frac{1}{5371}a^{10}+\frac{13}{5371}a^{9}+\frac{1655}{5371}a^{8}-\frac{2295}{5371}a^{7}-\frac{1953}{5371}a^{6}+\frac{2084}{5371}a^{5}-\frac{248}{5371}a^{4}-\frac{1188}{5371}a^{3}-\frac{2667}{5371}a^{2}+\frac{6}{5371}a+\frac{35}{131}$, $\frac{1}{11\!\cdots\!61}a^{11}-\frac{73\!\cdots\!88}{11\!\cdots\!61}a^{10}+\frac{36\!\cdots\!02}{11\!\cdots\!61}a^{9}+\frac{42\!\cdots\!54}{11\!\cdots\!61}a^{8}-\frac{74\!\cdots\!05}{11\!\cdots\!61}a^{7}-\frac{18\!\cdots\!50}{11\!\cdots\!61}a^{6}+\frac{21\!\cdots\!80}{11\!\cdots\!61}a^{5}+\frac{30\!\cdots\!34}{11\!\cdots\!61}a^{4}-\frac{76\!\cdots\!76}{11\!\cdots\!61}a^{3}-\frac{20\!\cdots\!41}{11\!\cdots\!61}a^{2}+\frac{40\!\cdots\!23}{11\!\cdots\!61}a+\frac{35\!\cdots\!86}{28\!\cdots\!21}$
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: |
$\frac{426854156849}{28\!\cdots\!11}a^{11}-\frac{2107821774491}{28\!\cdots\!11}a^{10}-\frac{77170730135987}{28\!\cdots\!11}a^{9}+\frac{340364211813836}{28\!\cdots\!11}a^{8}+\frac{52\!\cdots\!72}{28\!\cdots\!11}a^{7}-\frac{20\!\cdots\!22}{28\!\cdots\!11}a^{6}-\frac{16\!\cdots\!09}{28\!\cdots\!11}a^{5}+\frac{57\!\cdots\!47}{28\!\cdots\!11}a^{4}+\frac{23\!\cdots\!60}{28\!\cdots\!11}a^{3}-\frac{71\!\cdots\!19}{28\!\cdots\!11}a^{2}-\frac{11\!\cdots\!90}{28\!\cdots\!11}a+\frac{75\!\cdots\!16}{69\!\cdots\!71}$, $\frac{17\!\cdots\!55}{88\!\cdots\!31}a^{11}-\frac{10\!\cdots\!91}{88\!\cdots\!31}a^{10}-\frac{29\!\cdots\!25}{88\!\cdots\!31}a^{9}+\frac{16\!\cdots\!15}{88\!\cdots\!31}a^{8}+\frac{19\!\cdots\!30}{88\!\cdots\!31}a^{7}-\frac{93\!\cdots\!80}{88\!\cdots\!31}a^{6}-\frac{57\!\cdots\!93}{88\!\cdots\!31}a^{5}+\frac{24\!\cdots\!45}{88\!\cdots\!31}a^{4}+\frac{80\!\cdots\!20}{88\!\cdots\!31}a^{3}-\frac{29\!\cdots\!30}{88\!\cdots\!31}a^{2}-\frac{40\!\cdots\!00}{88\!\cdots\!31}a+\frac{30\!\cdots\!91}{21\!\cdots\!91}$, $\frac{79\!\cdots\!20}{88\!\cdots\!31}a^{11}-\frac{46\!\cdots\!44}{88\!\cdots\!31}a^{10}-\frac{14\!\cdots\!80}{88\!\cdots\!31}a^{9}+\frac{76\!\cdots\!15}{88\!\cdots\!31}a^{8}+\frac{99\!\cdots\!00}{88\!\cdots\!31}a^{7}-\frac{46\!\cdots\!70}{88\!\cdots\!31}a^{6}-\frac{31\!\cdots\!18}{88\!\cdots\!31}a^{5}+\frac{13\!\cdots\!15}{88\!\cdots\!31}a^{4}+\frac{46\!\cdots\!70}{88\!\cdots\!31}a^{3}-\frac{16\!\cdots\!85}{88\!\cdots\!31}a^{2}-\frac{24\!\cdots\!65}{88\!\cdots\!31}a+\frac{16\!\cdots\!06}{21\!\cdots\!91}$, $\frac{67\!\cdots\!79}{11\!\cdots\!61}a^{11}-\frac{23\!\cdots\!77}{11\!\cdots\!61}a^{10}-\frac{12\!\cdots\!57}{11\!\cdots\!61}a^{9}+\frac{36\!\cdots\!71}{11\!\cdots\!61}a^{8}+\frac{80\!\cdots\!72}{11\!\cdots\!61}a^{7}-\frac{21\!\cdots\!52}{11\!\cdots\!61}a^{6}-\frac{23\!\cdots\!01}{11\!\cdots\!61}a^{5}+\frac{60\!\cdots\!32}{11\!\cdots\!61}a^{4}+\frac{31\!\cdots\!90}{11\!\cdots\!61}a^{3}-\frac{79\!\cdots\!34}{11\!\cdots\!61}a^{2}-\frac{14\!\cdots\!75}{11\!\cdots\!61}a+\frac{86\!\cdots\!30}{28\!\cdots\!21}$, $\frac{40\!\cdots\!04}{11\!\cdots\!61}a^{11}-\frac{22\!\cdots\!62}{11\!\cdots\!61}a^{10}-\frac{70\!\cdots\!12}{11\!\cdots\!61}a^{9}+\frac{35\!\cdots\!01}{11\!\cdots\!61}a^{8}+\frac{46\!\cdots\!02}{11\!\cdots\!61}a^{7}-\frac{20\!\cdots\!02}{11\!\cdots\!61}a^{6}-\frac{14\!\cdots\!42}{11\!\cdots\!61}a^{5}+\frac{55\!\cdots\!92}{11\!\cdots\!61}a^{4}+\frac{19\!\cdots\!80}{11\!\cdots\!61}a^{3}-\frac{67\!\cdots\!99}{11\!\cdots\!61}a^{2}-\frac{99\!\cdots\!90}{11\!\cdots\!61}a+\frac{69\!\cdots\!16}{28\!\cdots\!21}$, $\frac{28\!\cdots\!78}{11\!\cdots\!61}a^{11}-\frac{75\!\cdots\!72}{11\!\cdots\!61}a^{10}-\frac{51\!\cdots\!80}{11\!\cdots\!61}a^{9}+\frac{10\!\cdots\!40}{11\!\cdots\!61}a^{8}+\frac{33\!\cdots\!30}{11\!\cdots\!61}a^{7}-\frac{62\!\cdots\!24}{11\!\cdots\!61}a^{6}-\frac{99\!\cdots\!00}{11\!\cdots\!61}a^{5}+\frac{17\!\cdots\!45}{11\!\cdots\!61}a^{4}+\frac{13\!\cdots\!40}{11\!\cdots\!61}a^{3}-\frac{24\!\cdots\!39}{11\!\cdots\!61}a^{2}-\frac{59\!\cdots\!85}{11\!\cdots\!61}a+\frac{28\!\cdots\!75}{28\!\cdots\!21}$, $\frac{28\!\cdots\!21}{11\!\cdots\!61}a^{11}-\frac{14\!\cdots\!79}{11\!\cdots\!61}a^{10}-\frac{51\!\cdots\!91}{11\!\cdots\!61}a^{9}+\frac{24\!\cdots\!23}{11\!\cdots\!61}a^{8}+\frac{35\!\cdots\!96}{11\!\cdots\!61}a^{7}-\frac{14\!\cdots\!08}{11\!\cdots\!61}a^{6}-\frac{11\!\cdots\!67}{11\!\cdots\!61}a^{5}+\frac{41\!\cdots\!16}{11\!\cdots\!61}a^{4}+\frac{15\!\cdots\!50}{11\!\cdots\!61}a^{3}-\frac{51\!\cdots\!34}{11\!\cdots\!61}a^{2}-\frac{78\!\cdots\!20}{11\!\cdots\!61}a+\frac{53\!\cdots\!46}{28\!\cdots\!21}$, $\frac{41\!\cdots\!21}{11\!\cdots\!61}a^{11}-\frac{28\!\cdots\!67}{11\!\cdots\!61}a^{10}-\frac{69\!\cdots\!43}{11\!\cdots\!61}a^{9}+\frac{44\!\cdots\!60}{11\!\cdots\!61}a^{8}+\frac{43\!\cdots\!45}{11\!\cdots\!61}a^{7}-\frac{25\!\cdots\!57}{11\!\cdots\!61}a^{6}-\frac{12\!\cdots\!20}{11\!\cdots\!61}a^{5}+\frac{69\!\cdots\!75}{11\!\cdots\!61}a^{4}+\frac{16\!\cdots\!94}{11\!\cdots\!61}a^{3}-\frac{84\!\cdots\!46}{11\!\cdots\!61}a^{2}-\frac{79\!\cdots\!13}{11\!\cdots\!61}a+\frac{91\!\cdots\!99}{28\!\cdots\!21}$, $\frac{14\!\cdots\!36}{11\!\cdots\!61}a^{11}-\frac{10\!\cdots\!63}{11\!\cdots\!61}a^{10}-\frac{23\!\cdots\!17}{11\!\cdots\!61}a^{9}+\frac{15\!\cdots\!82}{11\!\cdots\!61}a^{8}+\frac{14\!\cdots\!21}{11\!\cdots\!61}a^{7}-\frac{82\!\cdots\!26}{11\!\cdots\!61}a^{6}-\frac{39\!\cdots\!78}{11\!\cdots\!61}a^{5}+\frac{19\!\cdots\!22}{11\!\cdots\!61}a^{4}+\frac{50\!\cdots\!12}{11\!\cdots\!61}a^{3}-\frac{21\!\cdots\!39}{11\!\cdots\!61}a^{2}-\frac{24\!\cdots\!55}{11\!\cdots\!61}a+\frac{19\!\cdots\!18}{28\!\cdots\!21}$, $\frac{62\!\cdots\!24}{11\!\cdots\!61}a^{11}-\frac{43\!\cdots\!36}{11\!\cdots\!61}a^{10}-\frac{99\!\cdots\!82}{11\!\cdots\!61}a^{9}+\frac{65\!\cdots\!69}{11\!\cdots\!61}a^{8}+\frac{59\!\cdots\!38}{11\!\cdots\!61}a^{7}-\frac{36\!\cdots\!62}{11\!\cdots\!61}a^{6}-\frac{16\!\cdots\!20}{11\!\cdots\!61}a^{5}+\frac{91\!\cdots\!07}{11\!\cdots\!61}a^{4}+\frac{22\!\cdots\!47}{11\!\cdots\!61}a^{3}-\frac{10\!\cdots\!15}{11\!\cdots\!61}a^{2}-\frac{11\!\cdots\!63}{11\!\cdots\!61}a+\frac{10\!\cdots\!43}{28\!\cdots\!21}$, $\frac{48\!\cdots\!20}{11\!\cdots\!61}a^{11}-\frac{23\!\cdots\!26}{11\!\cdots\!61}a^{10}-\frac{71\!\cdots\!29}{11\!\cdots\!61}a^{9}+\frac{41\!\cdots\!01}{11\!\cdots\!61}a^{8}+\frac{58\!\cdots\!73}{11\!\cdots\!61}a^{7}-\frac{26\!\cdots\!99}{11\!\cdots\!61}a^{6}-\frac{30\!\cdots\!42}{11\!\cdots\!61}a^{5}+\frac{72\!\cdots\!41}{11\!\cdots\!61}a^{4}+\frac{73\!\cdots\!42}{11\!\cdots\!61}a^{3}-\frac{83\!\cdots\!81}{11\!\cdots\!61}a^{2}-\frac{56\!\cdots\!66}{11\!\cdots\!61}a+\frac{77\!\cdots\!76}{28\!\cdots\!21}$
| 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: | \( 18345440.8395 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 18345440.8395 \cdot 2}{2\cdot\sqrt{198123237327133986328125}}\cr\approx \mathstrut & 0.168818638097 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 202*x^10 + 74*x^9 + 15753*x^8 + 1798*x^7 - 599316*x^6 - 252696*x^5 + 11554141*x^4 + 6283413*x^3 - 104562232*x^2 - 45628823*x + 334724041)
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 - 202*x^10 + 74*x^9 + 15753*x^8 + 1798*x^7 - 599316*x^6 - 252696*x^5 + 11554141*x^4 + 6283413*x^3 - 104562232*x^2 - 45628823*x + 334724041, 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 - 202*x^10 + 74*x^9 + 15753*x^8 + 1798*x^7 - 599316*x^6 - 252696*x^5 + 11554141*x^4 + 6283413*x^3 - 104562232*x^2 - 45628823*x + 334724041);
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 - 202*x^10 + 74*x^9 + 15753*x^8 + 1798*x^7 - 599316*x^6 - 252696*x^5 + 11554141*x^4 + 6283413*x^3 - 104562232*x^2 - 45628823*x + 334724041);
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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.4.325125.1, 6.6.300125.1 |
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 | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.1.0.1}{1} }^{12}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.12.6.1 | $x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
|
\(5\)
| 5.12.9.2 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(7\)
| 7.12.8.1 | $x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
|
\(17\)
| 17.12.6.2 | $x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |