Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 338*x^10 + 334*x^9 + 43730*x^8 - 42394*x^7 - 2726655*x^6 + 2288070*x^5 + 83876760*x^4 - 44603207*x^3 - 1136572641*x^2 + 122837871*x + 4028583637)
gp: K = bnfinit(y^12 - y^11 - 338*y^10 + 334*y^9 + 43730*y^8 - 42394*y^7 - 2726655*y^6 + 2288070*y^5 + 83876760*y^4 - 44603207*y^3 - 1136572641*y^2 + 122837871*y + 4028583637, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 338*x^10 + 334*x^9 + 43730*x^8 - 42394*x^7 - 2726655*x^6 + 2288070*x^5 + 83876760*x^4 - 44603207*x^3 - 1136572641*x^2 + 122837871*x + 4028583637);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 338*x^10 + 334*x^9 + 43730*x^8 - 42394*x^7 - 2726655*x^6 + 2288070*x^5 + 83876760*x^4 - 44603207*x^3 - 1136572641*x^2 + 122837871*x + 4028583637)
\( x^{12} - x^{11} - 338 x^{10} + 334 x^{9} + 43730 x^{8} - 42394 x^{7} - 2726655 x^{6} + 2288070 x^{5} + \cdots + 4028583637 \)
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: |
\(59343998293561502407627433\)
\(\medspace = 7^{10}\cdot 11^{6}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(140.53\) | 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: | $7^{5/6}11^{1/2}17^{3/4}\approx 140.53399760248536$ | ||
| Ramified primes: |
\(7\), \(11\), \(17\)
| 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: | \(1309=7\cdot 11\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1309}(1,·)$, $\chi_{1309}(67,·)$, $\chi_{1309}(769,·)$, $\chi_{1309}(1002,·)$, $\chi_{1309}(395,·)$, $\chi_{1309}(846,·)$, $\chi_{1309}(208,·)$, $\chi_{1309}(562,·)$, $\chi_{1309}(375,·)$, $\chi_{1309}(472,·)$, $\chi_{1309}(285,·)$, $\chi_{1309}(254,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{4472}a^{9}-\frac{205}{1118}a^{8}+\frac{293}{2236}a^{7}+\frac{905}{4472}a^{6}+\frac{2131}{4472}a^{5}-\frac{2177}{4472}a^{4}+\frac{1901}{4472}a^{3}+\frac{1033}{2236}a^{2}-\frac{1863}{4472}a-\frac{1}{344}$, $\frac{1}{340529384}a^{10}-\frac{17081}{170264692}a^{9}+\frac{33472529}{170264692}a^{8}+\frac{74390145}{340529384}a^{7}+\frac{66033757}{340529384}a^{6}+\frac{163386149}{340529384}a^{5}-\frac{44766789}{340529384}a^{4}-\frac{25245413}{85132346}a^{3}-\frac{103750147}{340529384}a^{2}+\frac{161537637}{340529384}a+\frac{2601057}{13097284}$, $\frac{1}{11\!\cdots\!44}a^{11}+\frac{868540635078691}{56\!\cdots\!72}a^{10}+\frac{18\!\cdots\!43}{11\!\cdots\!44}a^{9}+\frac{14\!\cdots\!21}{11\!\cdots\!44}a^{8}+\frac{17\!\cdots\!79}{11\!\cdots\!44}a^{7}-\frac{14\!\cdots\!97}{56\!\cdots\!72}a^{6}-\frac{28\!\cdots\!57}{56\!\cdots\!72}a^{5}-\frac{63\!\cdots\!41}{11\!\cdots\!44}a^{4}+\frac{81\!\cdots\!25}{43\!\cdots\!44}a^{3}+\frac{96\!\cdots\!63}{11\!\cdots\!44}a^{2}-\frac{13\!\cdots\!09}{11\!\cdots\!44}a-\frac{42\!\cdots\!69}{87\!\cdots\!88}$
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
$C_{2}$, 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: |
$\frac{84\!\cdots\!35}{40\!\cdots\!24}a^{11}+\frac{68\!\cdots\!87}{50\!\cdots\!53}a^{10}-\frac{24\!\cdots\!47}{40\!\cdots\!24}a^{9}-\frac{15\!\cdots\!25}{40\!\cdots\!24}a^{8}+\frac{26\!\cdots\!67}{40\!\cdots\!24}a^{7}+\frac{19\!\cdots\!51}{50\!\cdots\!53}a^{6}-\frac{15\!\cdots\!94}{50\!\cdots\!53}a^{5}-\frac{52\!\cdots\!13}{31\!\cdots\!48}a^{4}+\frac{11\!\cdots\!39}{20\!\cdots\!12}a^{3}+\frac{12\!\cdots\!63}{40\!\cdots\!24}a^{2}-\frac{81\!\cdots\!61}{40\!\cdots\!24}a-\frac{36\!\cdots\!47}{31\!\cdots\!48}$, $\frac{84\!\cdots\!35}{40\!\cdots\!24}a^{11}+\frac{68\!\cdots\!87}{50\!\cdots\!53}a^{10}-\frac{24\!\cdots\!47}{40\!\cdots\!24}a^{9}-\frac{15\!\cdots\!25}{40\!\cdots\!24}a^{8}+\frac{26\!\cdots\!67}{40\!\cdots\!24}a^{7}+\frac{19\!\cdots\!51}{50\!\cdots\!53}a^{6}-\frac{15\!\cdots\!94}{50\!\cdots\!53}a^{5}-\frac{52\!\cdots\!13}{31\!\cdots\!48}a^{4}+\frac{11\!\cdots\!39}{20\!\cdots\!12}a^{3}+\frac{12\!\cdots\!63}{40\!\cdots\!24}a^{2}-\frac{81\!\cdots\!61}{40\!\cdots\!24}a-\frac{36\!\cdots\!95}{31\!\cdots\!48}$, $\frac{98\!\cdots\!97}{28\!\cdots\!86}a^{11}+\frac{27\!\cdots\!71}{11\!\cdots\!44}a^{10}-\frac{57\!\cdots\!91}{56\!\cdots\!72}a^{9}-\frac{38\!\cdots\!43}{56\!\cdots\!72}a^{8}+\frac{11\!\cdots\!87}{11\!\cdots\!44}a^{7}+\frac{77\!\cdots\!99}{11\!\cdots\!44}a^{6}-\frac{54\!\cdots\!97}{11\!\cdots\!44}a^{5}-\frac{34\!\cdots\!43}{11\!\cdots\!44}a^{4}+\frac{24\!\cdots\!87}{28\!\cdots\!86}a^{3}+\frac{61\!\cdots\!23}{11\!\cdots\!44}a^{2}-\frac{33\!\cdots\!81}{11\!\cdots\!44}a-\frac{90\!\cdots\!63}{43\!\cdots\!44}$, $\frac{19\!\cdots\!25}{11\!\cdots\!44}a^{11}+\frac{15\!\cdots\!73}{11\!\cdots\!44}a^{10}-\frac{57\!\cdots\!41}{11\!\cdots\!44}a^{9}-\frac{42\!\cdots\!49}{11\!\cdots\!44}a^{8}+\frac{29\!\cdots\!53}{56\!\cdots\!72}a^{7}+\frac{41\!\cdots\!47}{11\!\cdots\!44}a^{6}-\frac{26\!\cdots\!33}{11\!\cdots\!44}a^{5}-\frac{88\!\cdots\!03}{56\!\cdots\!72}a^{4}+\frac{58\!\cdots\!88}{14\!\cdots\!93}a^{3}+\frac{15\!\cdots\!79}{56\!\cdots\!72}a^{2}-\frac{38\!\cdots\!13}{28\!\cdots\!86}a-\frac{88\!\cdots\!77}{87\!\cdots\!88}$, $\frac{35\!\cdots\!57}{87\!\cdots\!88}a^{11}+\frac{12\!\cdots\!37}{56\!\cdots\!72}a^{10}-\frac{33\!\cdots\!61}{28\!\cdots\!86}a^{9}-\frac{71\!\cdots\!63}{11\!\cdots\!44}a^{8}+\frac{14\!\cdots\!37}{11\!\cdots\!44}a^{7}+\frac{74\!\cdots\!11}{11\!\cdots\!44}a^{6}-\frac{69\!\cdots\!75}{11\!\cdots\!44}a^{5}-\frac{16\!\cdots\!35}{56\!\cdots\!72}a^{4}+\frac{10\!\cdots\!71}{87\!\cdots\!88}a^{3}+\frac{48\!\cdots\!93}{87\!\cdots\!88}a^{2}-\frac{12\!\cdots\!77}{28\!\cdots\!86}a-\frac{95\!\cdots\!53}{43\!\cdots\!44}$, $\frac{59\!\cdots\!45}{87\!\cdots\!88}a^{11}+\frac{45\!\cdots\!61}{11\!\cdots\!44}a^{10}-\frac{23\!\cdots\!11}{11\!\cdots\!44}a^{9}-\frac{13\!\cdots\!25}{11\!\cdots\!44}a^{8}+\frac{12\!\cdots\!51}{56\!\cdots\!72}a^{7}+\frac{13\!\cdots\!37}{11\!\cdots\!44}a^{6}-\frac{91\!\cdots\!71}{87\!\cdots\!88}a^{5}-\frac{12\!\cdots\!81}{21\!\cdots\!22}a^{4}+\frac{11\!\cdots\!61}{56\!\cdots\!72}a^{3}+\frac{59\!\cdots\!07}{56\!\cdots\!72}a^{2}-\frac{41\!\cdots\!27}{56\!\cdots\!72}a-\frac{35\!\cdots\!91}{87\!\cdots\!88}$, $\frac{13\!\cdots\!41}{11\!\cdots\!44}a^{11}+\frac{39\!\cdots\!85}{56\!\cdots\!72}a^{10}-\frac{38\!\cdots\!17}{11\!\cdots\!44}a^{9}-\frac{22\!\cdots\!99}{11\!\cdots\!44}a^{8}+\frac{41\!\cdots\!19}{11\!\cdots\!44}a^{7}+\frac{11\!\cdots\!13}{56\!\cdots\!72}a^{6}-\frac{96\!\cdots\!59}{56\!\cdots\!72}a^{5}-\frac{10\!\cdots\!85}{11\!\cdots\!44}a^{4}+\frac{18\!\cdots\!73}{56\!\cdots\!72}a^{3}+\frac{19\!\cdots\!23}{11\!\cdots\!44}a^{2}-\frac{13\!\cdots\!97}{11\!\cdots\!44}a-\frac{58\!\cdots\!97}{87\!\cdots\!88}$, $\frac{12\!\cdots\!15}{56\!\cdots\!72}a^{11}-\frac{18\!\cdots\!11}{11\!\cdots\!44}a^{10}-\frac{46\!\cdots\!55}{56\!\cdots\!72}a^{9}+\frac{39\!\cdots\!09}{14\!\cdots\!93}a^{8}+\frac{13\!\cdots\!75}{11\!\cdots\!44}a^{7}-\frac{48\!\cdots\!41}{11\!\cdots\!44}a^{6}-\frac{76\!\cdots\!53}{11\!\cdots\!44}a^{5}-\frac{91\!\cdots\!49}{87\!\cdots\!88}a^{4}+\frac{85\!\cdots\!91}{56\!\cdots\!72}a^{3}+\frac{47\!\cdots\!63}{11\!\cdots\!44}a^{2}-\frac{67\!\cdots\!47}{11\!\cdots\!44}a-\frac{82\!\cdots\!07}{43\!\cdots\!44}$, $\frac{41\!\cdots\!13}{11\!\cdots\!44}a^{11}-\frac{33\!\cdots\!21}{11\!\cdots\!44}a^{10}-\frac{15\!\cdots\!70}{14\!\cdots\!93}a^{9}+\frac{99\!\cdots\!11}{11\!\cdots\!44}a^{8}+\frac{67\!\cdots\!59}{56\!\cdots\!72}a^{7}-\frac{54\!\cdots\!81}{56\!\cdots\!72}a^{6}-\frac{15\!\cdots\!99}{28\!\cdots\!86}a^{5}+\frac{51\!\cdots\!31}{11\!\cdots\!44}a^{4}+\frac{92\!\cdots\!49}{87\!\cdots\!88}a^{3}-\frac{47\!\cdots\!93}{56\!\cdots\!72}a^{2}-\frac{59\!\cdots\!73}{11\!\cdots\!44}a+\frac{15\!\cdots\!27}{43\!\cdots\!44}$, $\frac{29\!\cdots\!27}{87\!\cdots\!88}a^{11}+\frac{24\!\cdots\!01}{11\!\cdots\!44}a^{10}-\frac{57\!\cdots\!47}{56\!\cdots\!72}a^{9}-\frac{70\!\cdots\!27}{11\!\cdots\!44}a^{8}+\frac{60\!\cdots\!81}{56\!\cdots\!72}a^{7}+\frac{18\!\cdots\!05}{28\!\cdots\!86}a^{6}-\frac{28\!\cdots\!67}{56\!\cdots\!72}a^{5}-\frac{33\!\cdots\!81}{11\!\cdots\!44}a^{4}+\frac{10\!\cdots\!37}{11\!\cdots\!44}a^{3}+\frac{30\!\cdots\!35}{56\!\cdots\!72}a^{2}-\frac{38\!\cdots\!77}{11\!\cdots\!44}a-\frac{46\!\cdots\!85}{21\!\cdots\!22}$, $\frac{16\!\cdots\!17}{28\!\cdots\!86}a^{11}+\frac{81\!\cdots\!31}{11\!\cdots\!44}a^{10}-\frac{25\!\cdots\!27}{14\!\cdots\!93}a^{9}-\frac{11\!\cdots\!63}{56\!\cdots\!72}a^{8}+\frac{23\!\cdots\!07}{11\!\cdots\!44}a^{7}+\frac{21\!\cdots\!05}{11\!\cdots\!44}a^{6}-\frac{14\!\cdots\!27}{11\!\cdots\!44}a^{5}-\frac{82\!\cdots\!89}{11\!\cdots\!44}a^{4}+\frac{20\!\cdots\!75}{56\!\cdots\!72}a^{3}+\frac{74\!\cdots\!39}{87\!\cdots\!88}a^{2}-\frac{43\!\cdots\!27}{11\!\cdots\!44}a+\frac{26\!\cdots\!12}{10\!\cdots\!61}$
| 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: | \( 1377319421.77 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 1377319421.77 \cdot 2}{2\cdot\sqrt{59343998293561502407627433}}\cr\approx \mathstrut & 0.732328975366 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 338*x^10 + 334*x^9 + 43730*x^8 - 42394*x^7 - 2726655*x^6 + 2288070*x^5 + 83876760*x^4 - 44603207*x^3 - 1136572641*x^2 + 122837871*x + 4028583637)
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 - 338*x^10 + 334*x^9 + 43730*x^8 - 42394*x^7 - 2726655*x^6 + 2288070*x^5 + 83876760*x^4 - 44603207*x^3 - 1136572641*x^2 + 122837871*x + 4028583637, 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 - 338*x^10 + 334*x^9 + 43730*x^8 - 42394*x^7 - 2726655*x^6 + 2288070*x^5 + 83876760*x^4 - 44603207*x^3 - 1136572641*x^2 + 122837871*x + 4028583637);
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 - 338*x^10 + 334*x^9 + 43730*x^8 - 42394*x^7 - 2726655*x^6 + 2288070*x^5 + 83876760*x^4 - 44603207*x^3 - 1136572641*x^2 + 122837871*x + 4028583637);
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
| \(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.4.29129177.2, 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 | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.12.0.1}{12} }$ | R | R | ${\href{/padicField/13.1.0.1}{1} }^{12}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.1.0.1}{1} }^{12}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.12.10.5 | $x^{12} - 154 x^{6} - 1421$ | $6$ | $2$ | $10$ | $C_{12}$ | $[\ ]_{6}^{2}$ |
|
\(11\)
| 11.12.6.2 | $x^{12} + 363 x^{8} - 5324 x^{6} + 87846 x^{4} - 1127357 x^{2} + 3543122$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |