Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 207*x^10 - 4*x^9 + 16434*x^8 + 36*x^7 - 637037*x^6 + 16272*x^5 + 12549855*x^4 - 755976*x^3 - 115834236*x^2 + 10394682*x + 366872221)
gp: K = bnfinit(y^12 - 207*y^10 - 4*y^9 + 16434*y^8 + 36*y^7 - 637037*y^6 + 16272*y^5 + 12549855*y^4 - 755976*y^3 - 115834236*y^2 + 10394682*y + 366872221, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 207*x^10 - 4*x^9 + 16434*x^8 + 36*x^7 - 637037*x^6 + 16272*x^5 + 12549855*x^4 - 755976*x^3 - 115834236*x^2 + 10394682*x + 366872221);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 207*x^10 - 4*x^9 + 16434*x^8 + 36*x^7 - 637037*x^6 + 16272*x^5 + 12549855*x^4 - 755976*x^3 - 115834236*x^2 + 10394682*x + 366872221)
\( x^{12} - 207 x^{10} - 4 x^{9} + 16434 x^{8} + 36 x^{7} - 637037 x^{6} + 16272 x^{5} + 12549855 x^{4} + \cdots + 366872221 \)
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: |
\(1662226402746312000000000\)
\(\medspace = 2^{12}\cdot 3^{16}\cdot 5^{9}\cdot 13^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(104.33\) | 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: | $2\cdot 3^{4/3}5^{3/4}13^{1/2}\approx 104.32558964964244$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(13\)
| 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: | \(2340=2^{2}\cdot 3^{2}\cdot 5\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2340}(1,·)$, $\chi_{2340}(1507,·)$, $\chi_{2340}(1249,·)$, $\chi_{2340}(103,·)$, $\chi_{2340}(781,·)$, $\chi_{2340}(2029,·)$, $\chi_{2340}(883,·)$, $\chi_{2340}(469,·)$, $\chi_{2340}(727,·)$, $\chi_{2340}(1561,·)$, $\chi_{2340}(2287,·)$, $\chi_{2340}(1663,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13}a^{6}-\frac{6}{13}a^{4}-\frac{2}{13}a^{3}-\frac{4}{13}a^{2}+\frac{6}{13}a+\frac{1}{13}$, $\frac{1}{13}a^{7}-\frac{6}{13}a^{5}-\frac{2}{13}a^{4}-\frac{4}{13}a^{3}+\frac{6}{13}a^{2}+\frac{1}{13}a$, $\frac{1}{13}a^{8}-\frac{2}{13}a^{5}-\frac{1}{13}a^{4}-\frac{6}{13}a^{3}+\frac{3}{13}a^{2}-\frac{3}{13}a+\frac{6}{13}$, $\frac{1}{13}a^{9}-\frac{1}{13}a^{5}-\frac{5}{13}a^{4}-\frac{1}{13}a^{3}+\frac{2}{13}a^{2}+\frac{5}{13}a+\frac{2}{13}$, $\frac{1}{1035134347}a^{10}+\frac{20309165}{1035134347}a^{9}+\frac{4602718}{1035134347}a^{8}-\frac{17375552}{1035134347}a^{7}-\frac{5739622}{1035134347}a^{6}+\frac{35866600}{1035134347}a^{5}+\frac{142825645}{1035134347}a^{4}-\frac{475907892}{1035134347}a^{3}+\frac{147700211}{1035134347}a^{2}+\frac{18327610}{79625719}a+\frac{8550691}{79625719}$, $\frac{1}{17\!\cdots\!47}a^{11}+\frac{245644662026}{17\!\cdots\!47}a^{10}+\frac{34\!\cdots\!60}{17\!\cdots\!47}a^{9}+\frac{48\!\cdots\!62}{17\!\cdots\!47}a^{8}-\frac{28\!\cdots\!07}{17\!\cdots\!47}a^{7}-\frac{54\!\cdots\!20}{17\!\cdots\!47}a^{6}+\frac{27\!\cdots\!92}{17\!\cdots\!47}a^{5}+\frac{30\!\cdots\!34}{17\!\cdots\!47}a^{4}+\frac{81\!\cdots\!56}{17\!\cdots\!47}a^{3}-\frac{67\!\cdots\!25}{17\!\cdots\!47}a^{2}-\frac{20\!\cdots\!10}{17\!\cdots\!47}a+\frac{60\!\cdots\!98}{17\!\cdots\!47}$
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: |
$\frac{108}{1035134347}a^{11}-\frac{2784}{1035134347}a^{10}-\frac{21240}{1035134347}a^{9}+\frac{493755}{1035134347}a^{8}+\frac{1525140}{1035134347}a^{7}-\frac{31313650}{1035134347}a^{6}-\frac{46943346}{1035134347}a^{5}+\frac{12270795}{14579357}a^{4}+\frac{561394430}{1035134347}a^{3}-\frac{10177592751}{1035134347}a^{2}-\frac{1211772546}{1035134347}a+\frac{34578296770}{1035134347}$, $\frac{706253967720}{15\!\cdots\!71}a^{11}-\frac{139790818660304}{13\!\cdots\!19}a^{10}-\frac{11\!\cdots\!60}{13\!\cdots\!19}a^{9}+\frac{28\!\cdots\!95}{13\!\cdots\!19}a^{8}+\frac{71\!\cdots\!40}{13\!\cdots\!19}a^{7}-\frac{21\!\cdots\!50}{13\!\cdots\!19}a^{6}-\frac{20\!\cdots\!56}{13\!\cdots\!19}a^{5}+\frac{71\!\cdots\!45}{13\!\cdots\!19}a^{4}+\frac{23\!\cdots\!00}{13\!\cdots\!19}a^{3}-\frac{98\!\cdots\!50}{13\!\cdots\!19}a^{2}-\frac{76\!\cdots\!00}{13\!\cdots\!19}a+\frac{38\!\cdots\!13}{13\!\cdots\!19}$, $\frac{706253967720}{15\!\cdots\!71}a^{11}-\frac{139790818660304}{13\!\cdots\!19}a^{10}-\frac{11\!\cdots\!60}{13\!\cdots\!19}a^{9}+\frac{28\!\cdots\!95}{13\!\cdots\!19}a^{8}+\frac{71\!\cdots\!40}{13\!\cdots\!19}a^{7}-\frac{21\!\cdots\!50}{13\!\cdots\!19}a^{6}-\frac{20\!\cdots\!56}{13\!\cdots\!19}a^{5}+\frac{71\!\cdots\!45}{13\!\cdots\!19}a^{4}+\frac{23\!\cdots\!00}{13\!\cdots\!19}a^{3}-\frac{98\!\cdots\!50}{13\!\cdots\!19}a^{2}-\frac{76\!\cdots\!00}{13\!\cdots\!19}a+\frac{37\!\cdots\!94}{13\!\cdots\!19}$, $\frac{438817762482408}{17\!\cdots\!47}a^{11}-\frac{37\!\cdots\!14}{17\!\cdots\!47}a^{10}-\frac{11\!\cdots\!20}{24\!\cdots\!57}a^{9}+\frac{63\!\cdots\!30}{17\!\cdots\!47}a^{8}+\frac{55\!\cdots\!40}{17\!\cdots\!47}a^{7}-\frac{37\!\cdots\!10}{17\!\cdots\!47}a^{6}-\frac{16\!\cdots\!48}{17\!\cdots\!47}a^{5}+\frac{97\!\cdots\!75}{17\!\cdots\!47}a^{4}+\frac{21\!\cdots\!80}{17\!\cdots\!47}a^{3}-\frac{10\!\cdots\!36}{17\!\cdots\!47}a^{2}-\frac{82\!\cdots\!56}{17\!\cdots\!47}a+\frac{30\!\cdots\!27}{17\!\cdots\!47}$, $\frac{10\!\cdots\!48}{17\!\cdots\!47}a^{11}-\frac{65\!\cdots\!36}{17\!\cdots\!47}a^{10}-\frac{18\!\cdots\!20}{17\!\cdots\!47}a^{9}+\frac{12\!\cdots\!90}{17\!\cdots\!47}a^{8}+\frac{11\!\cdots\!60}{17\!\cdots\!47}a^{7}-\frac{81\!\cdots\!00}{17\!\cdots\!47}a^{6}-\frac{38\!\cdots\!66}{19\!\cdots\!23}a^{5}+\frac{24\!\cdots\!30}{17\!\cdots\!47}a^{4}+\frac{39\!\cdots\!30}{17\!\cdots\!47}a^{3}-\frac{30\!\cdots\!01}{17\!\cdots\!47}a^{2}-\frac{12\!\cdots\!46}{17\!\cdots\!47}a+\frac{11\!\cdots\!86}{17\!\cdots\!47}$, $\frac{86\!\cdots\!72}{17\!\cdots\!47}a^{11}+\frac{53\!\cdots\!96}{17\!\cdots\!47}a^{10}-\frac{14\!\cdots\!59}{17\!\cdots\!47}a^{9}-\frac{92\!\cdots\!41}{17\!\cdots\!47}a^{8}+\frac{90\!\cdots\!76}{17\!\cdots\!47}a^{7}+\frac{56\!\cdots\!87}{17\!\cdots\!47}a^{6}-\frac{23\!\cdots\!05}{17\!\cdots\!47}a^{5}-\frac{15\!\cdots\!60}{17\!\cdots\!47}a^{4}+\frac{26\!\cdots\!97}{17\!\cdots\!47}a^{3}+\frac{16\!\cdots\!44}{17\!\cdots\!47}a^{2}-\frac{84\!\cdots\!12}{17\!\cdots\!47}a-\frac{54\!\cdots\!33}{17\!\cdots\!47}$, $\frac{96\!\cdots\!20}{17\!\cdots\!47}a^{11}+\frac{52\!\cdots\!06}{17\!\cdots\!47}a^{10}-\frac{16\!\cdots\!26}{17\!\cdots\!47}a^{9}-\frac{91\!\cdots\!89}{17\!\cdots\!47}a^{8}+\frac{10\!\cdots\!64}{17\!\cdots\!47}a^{7}+\frac{55\!\cdots\!98}{17\!\cdots\!47}a^{6}-\frac{27\!\cdots\!34}{17\!\cdots\!47}a^{5}-\frac{14\!\cdots\!70}{17\!\cdots\!47}a^{4}+\frac{31\!\cdots\!18}{17\!\cdots\!47}a^{3}+\frac{16\!\cdots\!32}{17\!\cdots\!47}a^{2}-\frac{12\!\cdots\!89}{17\!\cdots\!47}a-\frac{58\!\cdots\!54}{17\!\cdots\!47}$, $\frac{88\!\cdots\!79}{17\!\cdots\!47}a^{11}-\frac{31\!\cdots\!35}{17\!\cdots\!47}a^{10}-\frac{17\!\cdots\!69}{17\!\cdots\!47}a^{9}+\frac{48\!\cdots\!37}{13\!\cdots\!19}a^{8}+\frac{13\!\cdots\!61}{17\!\cdots\!47}a^{7}-\frac{47\!\cdots\!34}{17\!\cdots\!47}a^{6}-\frac{48\!\cdots\!72}{17\!\cdots\!47}a^{5}+\frac{17\!\cdots\!95}{17\!\cdots\!47}a^{4}+\frac{83\!\cdots\!55}{17\!\cdots\!47}a^{3}-\frac{29\!\cdots\!12}{17\!\cdots\!47}a^{2}-\frac{41\!\cdots\!57}{13\!\cdots\!19}a+\frac{19\!\cdots\!25}{17\!\cdots\!47}$, $\frac{40\!\cdots\!08}{17\!\cdots\!47}a^{11}+\frac{53\!\cdots\!38}{17\!\cdots\!47}a^{10}-\frac{75\!\cdots\!81}{17\!\cdots\!47}a^{9}-\frac{95\!\cdots\!36}{17\!\cdots\!47}a^{8}+\frac{52\!\cdots\!81}{17\!\cdots\!47}a^{7}+\frac{61\!\cdots\!41}{17\!\cdots\!47}a^{6}-\frac{12\!\cdots\!92}{13\!\cdots\!19}a^{5}-\frac{14\!\cdots\!61}{13\!\cdots\!19}a^{4}+\frac{24\!\cdots\!02}{17\!\cdots\!47}a^{3}+\frac{31\!\cdots\!83}{17\!\cdots\!47}a^{2}-\frac{13\!\cdots\!97}{17\!\cdots\!47}a-\frac{21\!\cdots\!78}{17\!\cdots\!47}$, $\frac{26\!\cdots\!62}{17\!\cdots\!47}a^{11}+\frac{35\!\cdots\!18}{17\!\cdots\!47}a^{10}-\frac{40\!\cdots\!29}{17\!\cdots\!47}a^{9}-\frac{59\!\cdots\!49}{17\!\cdots\!47}a^{8}+\frac{20\!\cdots\!34}{17\!\cdots\!47}a^{7}+\frac{35\!\cdots\!13}{17\!\cdots\!47}a^{6}-\frac{41\!\cdots\!34}{17\!\cdots\!47}a^{5}-\frac{93\!\cdots\!91}{17\!\cdots\!47}a^{4}+\frac{35\!\cdots\!89}{17\!\cdots\!47}a^{3}+\frac{10\!\cdots\!70}{17\!\cdots\!47}a^{2}-\frac{13\!\cdots\!05}{17\!\cdots\!47}a-\frac{38\!\cdots\!17}{17\!\cdots\!47}$, $\frac{72\!\cdots\!30}{17\!\cdots\!47}a^{11}+\frac{22\!\cdots\!83}{17\!\cdots\!47}a^{10}-\frac{11\!\cdots\!60}{13\!\cdots\!19}a^{9}-\frac{45\!\cdots\!16}{17\!\cdots\!47}a^{8}+\frac{10\!\cdots\!98}{17\!\cdots\!47}a^{7}+\frac{33\!\cdots\!09}{17\!\cdots\!47}a^{6}-\frac{35\!\cdots\!91}{17\!\cdots\!47}a^{5}-\frac{11\!\cdots\!06}{17\!\cdots\!47}a^{4}+\frac{50\!\cdots\!95}{17\!\cdots\!47}a^{3}+\frac{16\!\cdots\!53}{17\!\cdots\!47}a^{2}-\frac{22\!\cdots\!70}{17\!\cdots\!47}a-\frac{70\!\cdots\!11}{17\!\cdots\!47}$
| 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: | \( 60912458.8577 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 60912458.8577 \cdot 2}{2\cdot\sqrt{1662226402746312000000000}}\cr\approx \mathstrut & 0.193517832406 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 207*x^10 - 4*x^9 + 16434*x^8 + 36*x^7 - 637037*x^6 + 16272*x^5 + 12549855*x^4 - 755976*x^3 - 115834236*x^2 + 10394682*x + 366872221)
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 - 207*x^10 - 4*x^9 + 16434*x^8 + 36*x^7 - 637037*x^6 + 16272*x^5 + 12549855*x^4 - 755976*x^3 - 115834236*x^2 + 10394682*x + 366872221, 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 - 207*x^10 - 4*x^9 + 16434*x^8 + 36*x^7 - 637037*x^6 + 16272*x^5 + 12549855*x^4 - 755976*x^3 - 115834236*x^2 + 10394682*x + 366872221);
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 - 207*x^10 - 4*x^9 + 16434*x^8 + 36*x^7 - 637037*x^6 + 16272*x^5 + 12549855*x^4 - 755976*x^3 - 115834236*x^2 + 10394682*x + 366872221);
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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.4.338000.1, 6.6.820125.1 |
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 | R | R | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.12.25 | $x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704$ | $2$ | $6$ | $12$ | $C_{12}$ | $[2]^{6}$ |
|
\(3\)
| 3.12.16.14 | $x^{12} + 24 x^{11} + 216 x^{10} + 768 x^{9} - 432 x^{8} - 10368 x^{7} - 18414 x^{6} + 27864 x^{5} + 83592 x^{4} + 10800 x^{3} + 64800 x^{2} + 901125$ | $3$ | $4$ | $16$ | $C_{12}$ | $[2]^{4}$ |
|
\(5\)
| 5.12.9.1 | $x^{12} - 30 x^{8} + 225 x^{4} + 1125$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(13\)
| 13.12.6.2 | $x^{12} - 21970 x^{6} + 314171 x^{4} - 4084223 x^{2} + 9653618$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |