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Properties

Label 12.12.111...000.1
Degree $12$
Signature $[12, 0]$
Discriminant $1.113\times 10^{24}$
Root discriminant \(100.90\)
Ramified primes $2,5,7,17$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 257*x^10 + 874*x^9 + 25914*x^8 - 71452*x^7 - 1309383*x^6 + 2723614*x^5 + 34749905*x^4 - 47923684*x^3 - 453217430*x^2 + 304232502*x + 2215713361)
 
gp: K = bnfinit(y^12 - 4*y^11 - 257*y^10 + 874*y^9 + 25914*y^8 - 71452*y^7 - 1309383*y^6 + 2723614*y^5 + 34749905*y^4 - 47923684*y^3 - 453217430*y^2 + 304232502*y + 2215713361, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 257*x^10 + 874*x^9 + 25914*x^8 - 71452*x^7 - 1309383*x^6 + 2723614*x^5 + 34749905*x^4 - 47923684*x^3 - 453217430*x^2 + 304232502*x + 2215713361);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 - 257*x^10 + 874*x^9 + 25914*x^8 - 71452*x^7 - 1309383*x^6 + 2723614*x^5 + 34749905*x^4 - 47923684*x^3 - 453217430*x^2 + 304232502*x + 2215713361)
 

\( x^{12} - 4 x^{11} - 257 x^{10} + 874 x^{9} + 25914 x^{8} - 71452 x^{7} - 1309383 x^{6} + \cdots + 2215713361 \) 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:   \(1113186255270152000000000\) \(\medspace = 2^{12}\cdot 5^{9}\cdot 7^{8}\cdot 17^{6}\) 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:  \(100.90\)
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:  $2\cdot 5^{3/4}7^{2/3}17^{1/2}\approx 100.89755746835681$
Ramified primes:   \(2\), \(5\), \(7\), \(17\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2380=2^{2}\cdot 5\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{2380}(1,·)$, $\chi_{2380}(67,·)$, $\chi_{2380}(1089,·)$, $\chi_{2380}(681,·)$, $\chi_{2380}(883,·)$, $\chi_{2380}(1429,·)$, $\chi_{2380}(407,·)$, $\chi_{2380}(543,·)$, $\chi_{2380}(2041,·)$, $\chi_{2380}(1563,·)$, $\chi_{2380}(2109,·)$, $\chi_{2380}(1087,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{17}a^{6}-\frac{2}{17}a^{5}-\frac{3}{17}a^{4}+\frac{6}{17}a^{3}+\frac{2}{17}a^{2}-\frac{4}{17}a+\frac{1}{17}$, $\frac{1}{17}a^{7}-\frac{7}{17}a^{5}-\frac{3}{17}a^{3}-\frac{7}{17}a+\frac{2}{17}$, $\frac{1}{17}a^{8}+\frac{3}{17}a^{5}-\frac{7}{17}a^{4}+\frac{8}{17}a^{3}+\frac{7}{17}a^{2}+\frac{8}{17}a+\frac{7}{17}$, $\frac{1}{17}a^{9}-\frac{1}{17}a^{5}+\frac{6}{17}a^{3}+\frac{2}{17}a^{2}+\frac{2}{17}a-\frac{3}{17}$, $\frac{1}{912020063}a^{10}+\frac{8315308}{912020063}a^{9}+\frac{21073590}{912020063}a^{8}-\frac{24871043}{912020063}a^{7}+\frac{6391077}{912020063}a^{6}-\frac{380586680}{912020063}a^{5}-\frac{410483573}{912020063}a^{4}+\frac{222616503}{912020063}a^{3}+\frac{83602970}{912020063}a^{2}-\frac{39359955}{912020063}a-\frac{4367432}{53648239}$, $\frac{1}{91\!\cdots\!83}a^{11}-\frac{121082189585}{31\!\cdots\!27}a^{10}+\frac{14\!\cdots\!16}{91\!\cdots\!83}a^{9}-\frac{17\!\cdots\!29}{91\!\cdots\!83}a^{8}+\frac{22\!\cdots\!88}{91\!\cdots\!83}a^{7}+\frac{78\!\cdots\!68}{91\!\cdots\!83}a^{6}-\frac{86\!\cdots\!65}{91\!\cdots\!83}a^{5}-\frac{28\!\cdots\!35}{91\!\cdots\!83}a^{4}-\frac{29\!\cdots\!23}{91\!\cdots\!83}a^{3}+\frac{33\!\cdots\!17}{91\!\cdots\!83}a^{2}+\frac{33\!\cdots\!85}{91\!\cdots\!83}a-\frac{14\!\cdots\!05}{31\!\cdots\!27}$ Copy content Toggle raw display

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 $2$) 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:   $\frac{28}{912020063}a^{11}+\frac{6220}{912020063}a^{10}-\frac{27920}{912020063}a^{9}-\frac{1364815}{912020063}a^{8}+\frac{4376300}{912020063}a^{7}+\frac{109388482}{912020063}a^{6}-\frac{252423766}{912020063}a^{5}-\frac{3951246695}{912020063}a^{4}+\frac{5890920830}{912020063}a^{3}+\frac{63182516921}{912020063}a^{2}-\frac{44747914208}{912020063}a-\frac{20537102366}{53648239}$, $\frac{354706423164300}{53\!\cdots\!99}a^{11}-\frac{40248957255294}{18\!\cdots\!31}a^{10}-\frac{77\!\cdots\!60}{53\!\cdots\!99}a^{9}+\frac{20\!\cdots\!55}{53\!\cdots\!99}a^{8}+\frac{62\!\cdots\!80}{53\!\cdots\!99}a^{7}-\frac{11\!\cdots\!80}{53\!\cdots\!99}a^{6}-\frac{22\!\cdots\!90}{53\!\cdots\!99}a^{5}+\frac{23\!\cdots\!20}{53\!\cdots\!99}a^{4}+\frac{36\!\cdots\!90}{53\!\cdots\!99}a^{3}-\frac{88\!\cdots\!45}{53\!\cdots\!99}a^{2}-\frac{20\!\cdots\!80}{53\!\cdots\!99}a-\frac{36\!\cdots\!17}{18\!\cdots\!31}$, $\frac{433041744480640}{53\!\cdots\!99}a^{11}-\frac{56116321717860}{18\!\cdots\!31}a^{10}-\frac{94\!\cdots\!80}{53\!\cdots\!99}a^{9}+\frac{29\!\cdots\!55}{53\!\cdots\!99}a^{8}+\frac{74\!\cdots\!00}{53\!\cdots\!99}a^{7}-\frac{19\!\cdots\!50}{53\!\cdots\!99}a^{6}-\frac{26\!\cdots\!24}{53\!\cdots\!99}a^{5}+\frac{51\!\cdots\!75}{53\!\cdots\!99}a^{4}+\frac{42\!\cdots\!00}{53\!\cdots\!99}a^{3}-\frac{53\!\cdots\!00}{53\!\cdots\!99}a^{2}-\frac{22\!\cdots\!40}{53\!\cdots\!99}a+\frac{50\!\cdots\!10}{18\!\cdots\!31}$, $\frac{76\!\cdots\!28}{91\!\cdots\!83}a^{11}+\frac{11\!\cdots\!60}{31\!\cdots\!27}a^{10}-\frac{18\!\cdots\!80}{91\!\cdots\!83}a^{9}-\frac{85\!\cdots\!80}{91\!\cdots\!83}a^{8}+\frac{17\!\cdots\!00}{91\!\cdots\!83}a^{7}+\frac{76\!\cdots\!12}{91\!\cdots\!83}a^{6}-\frac{70\!\cdots\!14}{91\!\cdots\!83}a^{5}-\frac{30\!\cdots\!20}{91\!\cdots\!83}a^{4}+\frac{13\!\cdots\!30}{91\!\cdots\!83}a^{3}+\frac{54\!\cdots\!61}{91\!\cdots\!83}a^{2}-\frac{83\!\cdots\!08}{91\!\cdots\!83}a-\frac{65\!\cdots\!73}{18\!\cdots\!31}$, $\frac{57\!\cdots\!52}{91\!\cdots\!83}a^{11}-\frac{28\!\cdots\!78}{31\!\cdots\!27}a^{10}-\frac{10\!\cdots\!00}{91\!\cdots\!83}a^{9}+\frac{17\!\cdots\!50}{91\!\cdots\!83}a^{8}+\frac{62\!\cdots\!60}{91\!\cdots\!83}a^{7}-\frac{12\!\cdots\!22}{91\!\cdots\!83}a^{6}-\frac{13\!\cdots\!24}{91\!\cdots\!83}a^{5}+\frac{43\!\cdots\!35}{91\!\cdots\!83}a^{4}+\frac{37\!\cdots\!00}{91\!\cdots\!83}a^{3}-\frac{64\!\cdots\!26}{91\!\cdots\!83}a^{2}+\frac{91\!\cdots\!68}{91\!\cdots\!83}a+\frac{67\!\cdots\!97}{18\!\cdots\!31}$, $\frac{16\!\cdots\!88}{91\!\cdots\!83}a^{11}-\frac{57\!\cdots\!34}{31\!\cdots\!27}a^{10}-\frac{32\!\cdots\!99}{91\!\cdots\!83}a^{9}+\frac{47\!\cdots\!76}{12\!\cdots\!73}a^{8}+\frac{22\!\cdots\!16}{91\!\cdots\!83}a^{7}-\frac{25\!\cdots\!25}{91\!\cdots\!83}a^{6}-\frac{63\!\cdots\!81}{91\!\cdots\!83}a^{5}+\frac{83\!\cdots\!80}{91\!\cdots\!83}a^{4}+\frac{68\!\cdots\!07}{91\!\cdots\!83}a^{3}-\frac{12\!\cdots\!38}{91\!\cdots\!83}a^{2}-\frac{15\!\cdots\!81}{91\!\cdots\!83}a+\frac{20\!\cdots\!16}{31\!\cdots\!27}$, $\frac{20\!\cdots\!72}{53\!\cdots\!99}a^{11}-\frac{24\!\cdots\!42}{18\!\cdots\!31}a^{10}-\frac{23\!\cdots\!00}{53\!\cdots\!99}a^{9}+\frac{15\!\cdots\!45}{53\!\cdots\!99}a^{8}-\frac{24\!\cdots\!60}{53\!\cdots\!99}a^{7}-\frac{11\!\cdots\!52}{53\!\cdots\!99}a^{6}+\frac{10\!\cdots\!64}{53\!\cdots\!99}a^{5}+\frac{41\!\cdots\!80}{53\!\cdots\!99}a^{4}-\frac{35\!\cdots\!60}{53\!\cdots\!99}a^{3}-\frac{64\!\cdots\!96}{53\!\cdots\!99}a^{2}+\frac{31\!\cdots\!71}{53\!\cdots\!99}a+\frac{12\!\cdots\!86}{18\!\cdots\!31}$, $\frac{10\!\cdots\!19}{91\!\cdots\!83}a^{11}-\frac{15\!\cdots\!40}{31\!\cdots\!27}a^{10}-\frac{22\!\cdots\!34}{91\!\cdots\!83}a^{9}+\frac{80\!\cdots\!21}{91\!\cdots\!83}a^{8}+\frac{19\!\cdots\!90}{91\!\cdots\!83}a^{7}-\frac{50\!\cdots\!72}{91\!\cdots\!83}a^{6}-\frac{45\!\cdots\!39}{53\!\cdots\!99}a^{5}+\frac{18\!\cdots\!55}{12\!\cdots\!73}a^{4}+\frac{14\!\cdots\!69}{91\!\cdots\!83}a^{3}-\frac{19\!\cdots\!86}{12\!\cdots\!73}a^{2}-\frac{96\!\cdots\!80}{91\!\cdots\!83}a+\frac{19\!\cdots\!22}{31\!\cdots\!27}$, $\frac{88\!\cdots\!11}{91\!\cdots\!83}a^{11}+\frac{37\!\cdots\!60}{31\!\cdots\!27}a^{10}-\frac{25\!\cdots\!58}{91\!\cdots\!83}a^{9}-\frac{28\!\cdots\!43}{91\!\cdots\!83}a^{8}+\frac{27\!\cdots\!73}{91\!\cdots\!83}a^{7}+\frac{27\!\cdots\!75}{91\!\cdots\!83}a^{6}-\frac{14\!\cdots\!75}{91\!\cdots\!83}a^{5}-\frac{12\!\cdots\!40}{91\!\cdots\!83}a^{4}+\frac{33\!\cdots\!23}{91\!\cdots\!83}a^{3}+\frac{28\!\cdots\!77}{91\!\cdots\!83}a^{2}-\frac{31\!\cdots\!80}{91\!\cdots\!83}a-\frac{82\!\cdots\!27}{31\!\cdots\!27}$, $\frac{17\!\cdots\!12}{91\!\cdots\!83}a^{11}-\frac{61\!\cdots\!13}{31\!\cdots\!27}a^{10}-\frac{35\!\cdots\!45}{91\!\cdots\!83}a^{9}+\frac{36\!\cdots\!72}{91\!\cdots\!83}a^{8}+\frac{23\!\cdots\!64}{91\!\cdots\!83}a^{7}-\frac{27\!\cdots\!40}{91\!\cdots\!83}a^{6}-\frac{24\!\cdots\!02}{32\!\cdots\!43}a^{5}+\frac{89\!\cdots\!28}{91\!\cdots\!83}a^{4}+\frac{73\!\cdots\!32}{91\!\cdots\!83}a^{3}-\frac{12\!\cdots\!16}{91\!\cdots\!83}a^{2}-\frac{18\!\cdots\!59}{91\!\cdots\!83}a+\frac{22\!\cdots\!71}{31\!\cdots\!27}$, $\frac{46\!\cdots\!74}{91\!\cdots\!83}a^{11}-\frac{16\!\cdots\!07}{31\!\cdots\!27}a^{10}-\frac{90\!\cdots\!25}{91\!\cdots\!83}a^{9}+\frac{54\!\cdots\!20}{53\!\cdots\!99}a^{8}+\frac{62\!\cdots\!96}{91\!\cdots\!83}a^{7}-\frac{65\!\cdots\!02}{91\!\cdots\!83}a^{6}-\frac{19\!\cdots\!79}{91\!\cdots\!83}a^{5}+\frac{21\!\cdots\!51}{91\!\cdots\!83}a^{4}+\frac{25\!\cdots\!23}{91\!\cdots\!83}a^{3}-\frac{32\!\cdots\!40}{91\!\cdots\!83}a^{2}-\frac{97\!\cdots\!14}{91\!\cdots\!83}a+\frac{56\!\cdots\!93}{31\!\cdots\!27}$ 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:  \( 54495459.0657 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 54495459.0657 \cdot 2}{2\cdot\sqrt{1113186255270152000000000}}\cr\approx \mathstrut & 0.211561357811 \end{aligned}\] (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 - 257*x^10 + 874*x^9 + 25914*x^8 - 71452*x^7 - 1309383*x^6 + 2723614*x^5 + 34749905*x^4 - 47923684*x^3 - 453217430*x^2 + 304232502*x + 2215713361)
 
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 - 257*x^10 + 874*x^9 + 25914*x^8 - 71452*x^7 - 1309383*x^6 + 2723614*x^5 + 34749905*x^4 - 47923684*x^3 - 453217430*x^2 + 304232502*x + 2215713361, 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 - 257*x^10 + 874*x^9 + 25914*x^8 - 71452*x^7 - 1309383*x^6 + 2723614*x^5 + 34749905*x^4 - 47923684*x^3 - 453217430*x^2 + 304232502*x + 2215713361);
 
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 - 257*x^10 + 874*x^9 + 25914*x^8 - 71452*x^7 - 1309383*x^6 + 2723614*x^5 + 34749905*x^4 - 47923684*x^3 - 453217430*x^2 + 304232502*x + 2215713361);
 
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.4.578000.1, 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 R ${\href{/padicField/3.12.0.1}{12} }$ R R ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{3}$ R ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.1.0.1}{1} }^{12}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 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}$
\(5\) Copy content Toggle raw display 5.12.9.1$x^{12} - 30 x^{8} + 225 x^{4} + 1125$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
\(7\) Copy content Toggle raw display 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}$
\(17\) Copy content Toggle raw display 17.12.6.2$x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$