↑

Properties

Label 12.12.129...217.2
Degree $12$
Signature $[12, 0]$
Discriminant $1.298\times 10^{28}$
Root discriminant \(220.17\)
Ramified primes $3,7,17$
Class number $6$ (GRH)
Class group [6] (GRH)
Galois group $C_{12}$ (as 12T1)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 273*x^10 - 203*x^9 + 22995*x^8 + 16842*x^7 - 733096*x^6 - 283563*x^5 + 8467347*x^4 - 1448881*x^3 - 20439468*x^2 + 18954180*x - 4504472)
 
gp: K = bnfinit(y^12 - 273*y^10 - 203*y^9 + 22995*y^8 + 16842*y^7 - 733096*y^6 - 283563*y^5 + 8467347*y^4 - 1448881*y^3 - 20439468*y^2 + 18954180*y - 4504472, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 273*x^10 - 203*x^9 + 22995*x^8 + 16842*x^7 - 733096*x^6 - 283563*x^5 + 8467347*x^4 - 1448881*x^3 - 20439468*x^2 + 18954180*x - 4504472);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 273*x^10 - 203*x^9 + 22995*x^8 + 16842*x^7 - 733096*x^6 - 283563*x^5 + 8467347*x^4 - 1448881*x^3 - 20439468*x^2 + 18954180*x - 4504472)
 

\( x^{12} - 273 x^{10} - 203 x^{9} + 22995 x^{8} + 16842 x^{7} - 733096 x^{6} - 283563 x^{5} + \cdots - 4504472 \) Copy content Toggle raw display

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:   \(12977865756870219436044086217\) \(\medspace = 3^{18}\cdot 7^{10}\cdot 17^{9}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(220.17\)
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:  $3^{3/2}7^{5/6}17^{3/4}\approx 220.1744599625188$
Ramified primes:   \(3\), \(7\), \(17\) Copy content Toggle raw display
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:  \(1071=3^{2}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1071}(256,·)$, $\chi_{1071}(1,·)$, $\chi_{1071}(67,·)$, $\chi_{1071}(47,·)$, $\chi_{1071}(1067,·)$, $\chi_{1071}(205,·)$, $\chi_{1071}(1007,·)$, $\chi_{1071}(16,·)$, $\chi_{1071}(883,·)$, $\chi_{1071}(803,·)$, $\chi_{1071}(251,·)$, $\chi_{1071}(752,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{14}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{14}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{28}a^{8}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{532}a^{9}-\frac{3}{266}a^{7}+\frac{1}{38}a^{6}-\frac{11}{76}a^{5}+\frac{9}{38}a^{4}+\frac{5}{76}a^{3}-\frac{7}{38}a^{2}-\frac{9}{19}a-\frac{7}{19}$, $\frac{1}{70657580}a^{10}-\frac{50643}{70657580}a^{9}+\frac{222973}{17664395}a^{8}-\frac{468553}{17664395}a^{7}-\frac{1957607}{70657580}a^{6}-\frac{728867}{10093940}a^{5}+\frac{670637}{10093940}a^{4}+\frac{2090573}{10093940}a^{3}+\frac{62683}{265630}a^{2}-\frac{518}{24985}a-\frac{935656}{2523485}$, $\frac{1}{15\!\cdots\!60}a^{11}-\frac{189883382218087}{38\!\cdots\!65}a^{10}-\frac{45\!\cdots\!43}{15\!\cdots\!60}a^{9}+\frac{25\!\cdots\!92}{38\!\cdots\!65}a^{8}+\frac{87\!\cdots\!13}{15\!\cdots\!60}a^{7}-\frac{15\!\cdots\!67}{77\!\cdots\!30}a^{6}+\frac{58\!\cdots\!23}{55\!\cdots\!95}a^{5}-\frac{57\!\cdots\!53}{55\!\cdots\!95}a^{4}+\frac{84\!\cdots\!09}{22\!\cdots\!80}a^{3}+\frac{18\!\cdots\!37}{55\!\cdots\!95}a^{2}+\frac{36\!\cdots\!59}{55\!\cdots\!95}a+\frac{39\!\cdots\!76}{11\!\cdots\!59}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{6}$, which has order $6$ (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 $2$) Copy content Toggle raw display
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:   $\frac{16\!\cdots\!65}{31\!\cdots\!52}a^{11}+\frac{42\!\cdots\!20}{77\!\cdots\!13}a^{10}-\frac{46\!\cdots\!09}{31\!\cdots\!52}a^{9}-\frac{80\!\cdots\!55}{31\!\cdots\!52}a^{8}+\frac{54\!\cdots\!13}{44\!\cdots\!36}a^{7}+\frac{47\!\cdots\!09}{22\!\cdots\!18}a^{6}-\frac{42\!\cdots\!59}{11\!\cdots\!59}a^{5}-\frac{23\!\cdots\!45}{44\!\cdots\!36}a^{4}+\frac{18\!\cdots\!33}{44\!\cdots\!36}a^{3}+\frac{14\!\cdots\!97}{44\!\cdots\!36}a^{2}-\frac{92\!\cdots\!08}{11\!\cdots\!59}a+\frac{28\!\cdots\!30}{11\!\cdots\!59}$, $\frac{61\!\cdots\!86}{77\!\cdots\!13}a^{11}+\frac{14\!\cdots\!03}{22\!\cdots\!18}a^{10}-\frac{16\!\cdots\!98}{77\!\cdots\!13}a^{9}-\frac{53\!\cdots\!11}{15\!\cdots\!26}a^{8}+\frac{39\!\cdots\!67}{22\!\cdots\!18}a^{7}+\frac{63\!\cdots\!09}{22\!\cdots\!18}a^{6}-\frac{59\!\cdots\!79}{11\!\cdots\!59}a^{5}-\frac{78\!\cdots\!63}{11\!\cdots\!59}a^{4}+\frac{12\!\cdots\!55}{22\!\cdots\!18}a^{3}+\frac{10\!\cdots\!69}{22\!\cdots\!18}a^{2}-\frac{25\!\cdots\!03}{22\!\cdots\!18}a+\frac{26\!\cdots\!94}{11\!\cdots\!59}$, $\frac{12843819}{5520409600699}a^{11}+\frac{387810}{290547873721}a^{10}-\frac{7013543141}{11040819201398}a^{9}-\frac{4586297166}{5520409600699}a^{8}+\frac{293911702830}{5520409600699}a^{7}+\frac{377927136210}{5520409600699}a^{6}-\frac{139801770315}{83013678206}a^{5}-\frac{1217200289094}{788629942957}a^{4}+\frac{30478161093111}{1577259885914}a^{3}+\frac{4812231533580}{788629942957}a^{2}-\frac{37964851213698}{788629942957}a+\frac{20434263715293}{788629942957}$, $\frac{34\!\cdots\!77}{11\!\cdots\!59}a^{11}+\frac{31\!\cdots\!81}{15\!\cdots\!26}a^{10}-\frac{69\!\cdots\!77}{82\!\cdots\!54}a^{9}-\frac{18\!\cdots\!95}{15\!\cdots\!26}a^{8}+\frac{11\!\cdots\!89}{15\!\cdots\!26}a^{7}+\frac{15\!\cdots\!03}{15\!\cdots\!26}a^{6}-\frac{49\!\cdots\!53}{22\!\cdots\!18}a^{5}-\frac{25\!\cdots\!41}{11\!\cdots\!59}a^{4}+\frac{27\!\cdots\!56}{11\!\cdots\!59}a^{3}+\frac{23\!\cdots\!89}{22\!\cdots\!18}a^{2}-\frac{13\!\cdots\!55}{22\!\cdots\!18}a+\frac{23\!\cdots\!72}{11\!\cdots\!59}$, $\frac{12\!\cdots\!97}{31\!\cdots\!52}a^{11}+\frac{75\!\cdots\!01}{15\!\cdots\!26}a^{10}-\frac{33\!\cdots\!67}{31\!\cdots\!52}a^{9}-\frac{65\!\cdots\!09}{31\!\cdots\!52}a^{8}+\frac{27\!\cdots\!89}{31\!\cdots\!52}a^{7}+\frac{13\!\cdots\!43}{77\!\cdots\!13}a^{6}-\frac{59\!\cdots\!81}{22\!\cdots\!18}a^{5}-\frac{20\!\cdots\!85}{44\!\cdots\!36}a^{4}+\frac{12\!\cdots\!29}{44\!\cdots\!36}a^{3}+\frac{14\!\cdots\!95}{44\!\cdots\!36}a^{2}-\frac{10\!\cdots\!67}{22\!\cdots\!18}a+\frac{37\!\cdots\!92}{11\!\cdots\!59}$, $\frac{46\!\cdots\!29}{15\!\cdots\!60}a^{11}+\frac{11\!\cdots\!48}{38\!\cdots\!65}a^{10}-\frac{63\!\cdots\!87}{77\!\cdots\!30}a^{9}-\frac{42\!\cdots\!17}{31\!\cdots\!52}a^{8}+\frac{10\!\cdots\!79}{15\!\cdots\!60}a^{7}+\frac{89\!\cdots\!53}{77\!\cdots\!30}a^{6}-\frac{92\!\cdots\!93}{44\!\cdots\!36}a^{5}-\frac{12\!\cdots\!99}{44\!\cdots\!36}a^{4}+\frac{65\!\cdots\!08}{29\!\cdots\!05}a^{3}+\frac{20\!\cdots\!27}{11\!\cdots\!20}a^{2}-\frac{48\!\cdots\!47}{11\!\cdots\!90}a+\frac{77\!\cdots\!06}{55\!\cdots\!95}$, $\frac{16\!\cdots\!26}{38\!\cdots\!65}a^{11}+\frac{25\!\cdots\!49}{77\!\cdots\!30}a^{10}-\frac{87\!\cdots\!51}{77\!\cdots\!30}a^{9}-\frac{13\!\cdots\!39}{77\!\cdots\!30}a^{8}+\frac{72\!\cdots\!71}{77\!\cdots\!30}a^{7}+\frac{11\!\cdots\!87}{77\!\cdots\!30}a^{6}-\frac{16\!\cdots\!58}{55\!\cdots\!95}a^{5}-\frac{38\!\cdots\!39}{11\!\cdots\!90}a^{4}+\frac{35\!\cdots\!73}{11\!\cdots\!90}a^{3}+\frac{22\!\cdots\!71}{11\!\cdots\!90}a^{2}-\frac{37\!\cdots\!49}{55\!\cdots\!95}a+\frac{24\!\cdots\!21}{11\!\cdots\!59}$, $\frac{68\!\cdots\!39}{15\!\cdots\!60}a^{11}+\frac{59\!\cdots\!31}{15\!\cdots\!60}a^{10}-\frac{18\!\cdots\!11}{15\!\cdots\!60}a^{9}-\frac{10\!\cdots\!04}{55\!\cdots\!95}a^{8}+\frac{15\!\cdots\!11}{15\!\cdots\!60}a^{7}+\frac{12\!\cdots\!37}{82\!\cdots\!40}a^{6}-\frac{16\!\cdots\!47}{55\!\cdots\!95}a^{5}-\frac{83\!\cdots\!17}{22\!\cdots\!80}a^{4}+\frac{13\!\cdots\!29}{44\!\cdots\!36}a^{3}+\frac{11\!\cdots\!11}{55\!\cdots\!95}a^{2}-\frac{36\!\cdots\!68}{55\!\cdots\!95}a+\frac{63\!\cdots\!23}{29\!\cdots\!05}$, $\frac{17\!\cdots\!11}{22\!\cdots\!80}a^{11}+\frac{29\!\cdots\!88}{55\!\cdots\!95}a^{10}-\frac{46\!\cdots\!43}{22\!\cdots\!80}a^{9}-\frac{46\!\cdots\!99}{15\!\cdots\!60}a^{8}+\frac{27\!\cdots\!11}{15\!\cdots\!60}a^{7}+\frac{96\!\cdots\!13}{38\!\cdots\!65}a^{6}-\frac{60\!\cdots\!03}{11\!\cdots\!90}a^{5}-\frac{13\!\cdots\!09}{22\!\cdots\!80}a^{4}+\frac{13\!\cdots\!93}{22\!\cdots\!80}a^{3}+\frac{66\!\cdots\!01}{22\!\cdots\!80}a^{2}-\frac{14\!\cdots\!89}{11\!\cdots\!90}a+\frac{63\!\cdots\!25}{11\!\cdots\!59}$, $\frac{60\!\cdots\!89}{82\!\cdots\!40}a^{11}+\frac{38\!\cdots\!93}{77\!\cdots\!30}a^{10}-\frac{15\!\cdots\!75}{77\!\cdots\!13}a^{9}-\frac{44\!\cdots\!69}{15\!\cdots\!60}a^{8}+\frac{74\!\cdots\!89}{44\!\cdots\!36}a^{7}+\frac{18\!\cdots\!09}{77\!\cdots\!30}a^{6}-\frac{11\!\cdots\!01}{22\!\cdots\!80}a^{5}-\frac{68\!\cdots\!81}{11\!\cdots\!20}a^{4}+\frac{63\!\cdots\!13}{11\!\cdots\!90}a^{3}+\frac{72\!\cdots\!29}{22\!\cdots\!80}a^{2}-\frac{67\!\cdots\!63}{55\!\cdots\!95}a+\frac{25\!\cdots\!31}{55\!\cdots\!95}$, $\frac{29\!\cdots\!81}{77\!\cdots\!30}a^{11}+\frac{55\!\cdots\!61}{15\!\cdots\!60}a^{10}-\frac{80\!\cdots\!61}{77\!\cdots\!30}a^{9}-\frac{54\!\cdots\!19}{31\!\cdots\!52}a^{8}+\frac{66\!\cdots\!61}{77\!\cdots\!30}a^{7}+\frac{22\!\cdots\!53}{15\!\cdots\!60}a^{6}-\frac{29\!\cdots\!39}{11\!\cdots\!59}a^{5}-\frac{39\!\cdots\!93}{11\!\cdots\!59}a^{4}+\frac{15\!\cdots\!46}{55\!\cdots\!95}a^{3}+\frac{48\!\cdots\!39}{22\!\cdots\!80}a^{2}-\frac{30\!\cdots\!23}{55\!\cdots\!95}a+\frac{96\!\cdots\!53}{55\!\cdots\!95}$ Copy content Toggle raw display (assuming GRH)
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:  \( 2515816493.4522147 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 2515816493.4522147 \cdot 6}{2\cdot\sqrt{12977865756870219436044086217}}\cr\approx \mathstrut & 0.271367931295293 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 273*x^10 - 203*x^9 + 22995*x^8 + 16842*x^7 - 733096*x^6 - 283563*x^5 + 8467347*x^4 - 1448881*x^3 - 20439468*x^2 + 18954180*x - 4504472)
 
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 - 273*x^10 - 203*x^9 + 22995*x^8 + 16842*x^7 - 733096*x^6 - 283563*x^5 + 8467347*x^4 - 1448881*x^3 - 20439468*x^2 + 18954180*x - 4504472, 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 - 273*x^10 - 203*x^9 + 22995*x^8 + 16842*x^7 - 733096*x^6 - 283563*x^5 + 8467347*x^4 - 1448881*x^3 - 20439468*x^2 + 18954180*x - 4504472);
 
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 - 273*x^10 - 203*x^9 + 22995*x^8 + 16842*x^7 - 733096*x^6 - 283563*x^5 + 8467347*x^4 - 1448881*x^3 - 20439468*x^2 + 18954180*x - 4504472);
 
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

$C_{12}$ (as 12T1):

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

\(\Q(\sqrt{17}) \), 3.3.3969.2, 4.4.2166633.1, 6.6.77394297393.2

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 ${\href{/padicField/2.3.0.1}{3} }^{4}$ R ${\href{/padicField/5.4.0.1}{4} }^{3}$ R ${\href{/padicField/11.4.0.1}{4} }^{3}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ R ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{3}$ ${\href{/padicField/29.12.0.1}{12} }$ ${\href{/padicField/31.12.0.1}{12} }$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.3.0.1}{3} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.12.18.61$x^{12} + 12 x^{11} + 42 x^{10} + 42 x^{9} + 54 x^{8} + 18 x^{7} + 21 x^{6} + 72 x^{5} + 108 x^{4} + 36 x^{3} + 180$$6$$2$$18$$C_{12}$$[2]_{2}^{2}$
\(7\) Copy content Toggle raw display 7.12.10.4$x^{12} - 42 x^{6} + 147$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
\(17\) Copy content Toggle raw display 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}$