Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 17*x^21 + 218*x^18 - 1277*x^15 + 5609*x^12 + 3403*x^9 - 692*x^6 + 945*x^3 + 729)
gp: K = bnfinit(y^24 - 17*y^21 + 218*y^18 - 1277*y^15 + 5609*y^12 + 3403*y^9 - 692*y^6 + 945*y^3 + 729, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 17*x^21 + 218*x^18 - 1277*x^15 + 5609*x^12 + 3403*x^9 - 692*x^6 + 945*x^3 + 729);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 17*x^21 + 218*x^18 - 1277*x^15 + 5609*x^12 + 3403*x^9 - 692*x^6 + 945*x^3 + 729)
\( x^{24} - 17x^{21} + 218x^{18} - 1277x^{15} + 5609x^{12} + 3403x^{9} - 692x^{6} + 945x^{3} + 729 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(16878953293629664473677397903764439609\)
\(\medspace = 3^{36}\cdot 13^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(35.57\) | 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: | $3^{3/2}13^{3/4}\approx 35.57454845374513$ | ||
| Ramified primes: |
\(3\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $24$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(117=3^{2}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{117}(64,·)$, $\chi_{117}(1,·)$, $\chi_{117}(5,·)$, $\chi_{117}(70,·)$, $\chi_{117}(8,·)$, $\chi_{117}(73,·)$, $\chi_{117}(77,·)$, $\chi_{117}(14,·)$, $\chi_{117}(79,·)$, $\chi_{117}(83,·)$, $\chi_{117}(86,·)$, $\chi_{117}(25,·)$, $\chi_{117}(92,·)$, $\chi_{117}(31,·)$, $\chi_{117}(34,·)$, $\chi_{117}(38,·)$, $\chi_{117}(103,·)$, $\chi_{117}(40,·)$, $\chi_{117}(44,·)$, $\chi_{117}(109,·)$, $\chi_{117}(47,·)$, $\chi_{117}(112,·)$, $\chi_{117}(116,·)$, $\chi_{117}(53,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2048}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{7}+\frac{1}{3}a$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{8}+\frac{1}{9}a^{2}$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{9}+\frac{1}{9}a^{3}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{10}+\frac{1}{9}a^{4}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{11}+\frac{1}{9}a^{5}$, $\frac{1}{1647}a^{18}-\frac{37}{1647}a^{15}+\frac{253}{1647}a^{12}-\frac{748}{1647}a^{9}+\frac{316}{1647}a^{6}-\frac{55}{1647}a^{3}-\frac{1}{61}$, $\frac{1}{1647}a^{19}-\frac{37}{1647}a^{16}+\frac{253}{1647}a^{13}-\frac{748}{1647}a^{10}+\frac{316}{1647}a^{7}-\frac{55}{1647}a^{4}-\frac{1}{61}a$, $\frac{1}{1647}a^{20}-\frac{37}{1647}a^{17}+\frac{70}{1647}a^{14}-\frac{748}{1647}a^{11}+\frac{133}{1647}a^{8}-\frac{55}{1647}a^{5}-\frac{70}{549}a^{2}$, $\frac{1}{333878023359}a^{21}+\frac{21560387}{111292674453}a^{18}+\frac{1698524242}{37097558151}a^{15}-\frac{14794673017}{111292674453}a^{12}+\frac{8631920026}{37097558151}a^{9}+\frac{29103961883}{111292674453}a^{6}+\frac{63648023030}{333878023359}a^{3}-\frac{1722997796}{12365852717}$, $\frac{1}{1001634070077}a^{22}-\frac{138037736}{1001634070077}a^{19}-\frac{51407798935}{1001634070077}a^{16}-\frac{95671899992}{1001634070077}a^{13}-\frac{178752124471}{1001634070077}a^{10}+\frac{23252714197}{1001634070077}a^{7}-\frac{333275577296}{1001634070077}a^{4}-\frac{13886131616}{37097558151}a$, $\frac{1}{3004902210231}a^{23}-\frac{746194427}{3004902210231}a^{20}-\frac{140198675821}{3004902210231}a^{17}+\frac{84342480544}{3004902210231}a^{14}-\frac{836777664133}{3004902210231}a^{11}+\frac{2704151200}{49260691971}a^{8}+\frac{590514436333}{3004902210231}a^{5}-\frac{12669893453}{37097558151}a^{2}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
$C_{3}\times C_{6}$, which has order $18$ (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: |
\( \frac{186604117}{37097558151} a^{23} - \frac{9794458525}{111292674453} a^{20} + \frac{42314454700}{37097558151} a^{17} - \frac{259553185612}{37097558151} a^{14} + \frac{1178954688616}{37097558151} a^{11} + \frac{28434787913}{37097558151} a^{8} - \frac{14712814746}{12365852717} a^{5} + \frac{638512165729}{111292674453} a^{2} \)
(order $18$)
| 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: |
$\frac{186604117}{37097558151}a^{23}-\frac{299630650}{111292674453}a^{22}-\frac{9794458525}{111292674453}a^{20}+\frac{5242373503}{111292674453}a^{19}+\frac{42314454700}{37097558151}a^{17}-\frac{67944415000}{111292674453}a^{16}-\frac{259553185612}{37097558151}a^{14}+\frac{416765133400}{111292674453}a^{13}+\frac{1178954688616}{37097558151}a^{11}-\frac{1892849688535}{111292674453}a^{10}+\frac{28434787913}{37097558151}a^{8}-\frac{45657802850}{111292674453}a^{7}-\frac{14712814746}{12365852717}a^{5}+\frac{7874799900}{12365852717}a^{4}+\frac{638512165729}{111292674453}a^{2}-\frac{85429695577}{37097558151}a$, $\frac{1551393077}{1001634070077}a^{22}-\frac{27484928536}{1001634070077}a^{19}+\frac{357436634932}{1001634070077}a^{16}-\frac{2229261265579}{1001634070077}a^{13}+\frac{10198322502472}{1001634070077}a^{10}-\frac{1428951155944}{1001634070077}a^{7}-\frac{2675002242640}{1001634070077}a^{4}+\frac{28236124332}{12365852717}a+1$, $\frac{1381362398}{1001634070077}a^{22}-\frac{22806934291}{1001634070077}a^{19}+\frac{289774517074}{1001634070077}a^{16}-\frac{1618984036558}{1001634070077}a^{13}+\frac{6918000056845}{1001634070077}a^{10}+\frac{8271432308393}{1001634070077}a^{7}+\frac{2445268758083}{1001634070077}a^{4}-\frac{5201105780}{37097558151}a$, $\frac{16817650193}{3004902210231}a^{23}-\frac{295653552577}{3004902210231}a^{20}+\frac{3838665582202}{3004902210231}a^{17}-\frac{23718684405661}{3004902210231}a^{14}+\frac{108289482678337}{3004902210231}a^{11}-\frac{6779445072334}{3004902210231}a^{8}-\frac{2668839211753}{3004902210231}a^{5}+\frac{243751158679}{37097558151}a^{2}$, $\frac{23359}{19639883727}a^{21}-\frac{548824}{19639883727}a^{18}+\frac{5296900}{19639883727}a^{15}-\frac{32490724}{19639883727}a^{12}-\frac{18708521}{19639883727}a^{9}+\frac{3559451}{19639883727}a^{6}-\frac{204638}{727403101}a^{3}-\frac{220390588}{727403101}$, $\frac{299630650}{111292674453}a^{23}-\frac{2786863444}{1001634070077}a^{22}+\frac{30026572}{5473410219}a^{21}-\frac{5242373503}{111292674453}a^{20}+\frac{48684592481}{1001634070077}a^{19}-\frac{531027524}{5473410219}a^{18}+\frac{67944415000}{111292674453}a^{17}-\frac{630386077364}{1001634070077}a^{16}+\frac{6905919038}{5473410219}a^{15}-\frac{416765133400}{111292674453}a^{14}+\frac{3855407918288}{1001634070077}a^{13}-\frac{43009712444}{5473410219}a^{12}+\frac{1892849688535}{111292674453}a^{11}-\frac{17450733231584}{1001634070077}a^{10}+\frac{197038531148}{5473410219}a^{9}+\frac{45657802850}{111292674453}a^{8}-\frac{1199888650714}{1001634070077}a^{7}-\frac{27608308796}{5473410219}a^{6}-\frac{7874799900}{12365852717}a^{5}+\frac{2127809796785}{1001634070077}a^{4}-\frac{23913734024}{5473410219}a^{3}+\frac{85429695577}{37097558151}a^{2}-\frac{108998283446}{37097558151}a+\frac{1433904517}{202718897}$, $\frac{8478268010}{3004902210231}a^{23}+\frac{362309767}{37097558151}a^{22}+\frac{91689767}{111292674453}a^{21}-\frac{148038050728}{3004902210231}a^{20}-\frac{57313300330}{333878023359}a^{19}-\frac{4900792736}{333878023359}a^{18}+\frac{1916109784843}{3004902210231}a^{17}+\frac{743738013091}{333878023359}a^{16}+\frac{63733942832}{333878023359}a^{15}-\frac{11705104887670}{3004902210231}a^{14}-\frac{4590314691031}{333878023359}a^{13}-\frac{399421984673}{333878023359}a^{12}+\frac{52891441999792}{3004902210231}a^{11}+\frac{20904576176971}{333878023359}a^{10}+\frac{1818446235872}{333878023359}a^{9}+\frac{4627791656747}{3004902210231}a^{8}-\frac{781434178138}{333878023359}a^{7}-\frac{254793947744}{333878023359}a^{6}-\frac{8324964954811}{3004902210231}a^{5}-\frac{2542925831507}{333878023359}a^{4}-\frac{1323950780791}{333878023359}a^{3}+\frac{38442155321}{111292674453}a^{2}+\frac{130596612083}{12365852717}a+\frac{2738358979}{12365852717}$, $\frac{27103867487}{3004902210231}a^{23}+\frac{4954001639}{1001634070077}a^{22}-\frac{1834200149}{333878023359}a^{21}-\frac{473899593664}{3004902210231}a^{20}-\frac{86098330945}{1001634070077}a^{19}+\frac{3599573090}{37097558151}a^{18}+\frac{6139661215543}{3004902210231}a^{17}+\frac{1112867937547}{1001634070077}a^{16}-\frac{46811811455}{37097558151}a^{15}-\frac{37612002098242}{3004902210231}a^{14}-\frac{6752638679203}{1001634070077}a^{13}+\frac{291227451500}{37097558151}a^{12}+\frac{170582397073270}{3004902210231}a^{11}+\frac{30402364364503}{1001634070077}a^{10}-\frac{1335626803430}{37097558151}a^{9}+\frac{7465965751700}{3004902210231}a^{8}+\frac{4980890210327}{1001634070077}a^{7}+\frac{187143078110}{37097558151}a^{6}-\frac{12730575062089}{3004902210231}a^{5}-\frac{3714690247486}{1001634070077}a^{4}+\frac{355569163160}{333878023359}a^{3}+\frac{1107511089437}{111292674453}a^{2}+\frac{189862571689}{37097558151}a-\frac{99844750035}{12365852717}$, $\frac{18240932944}{3004902210231}a^{23}+\frac{7523846611}{1001634070077}a^{22}+\frac{91689767}{111292674453}a^{21}-\frac{319451546861}{3004902210231}a^{20}-\frac{132994941548}{1001634070077}a^{19}-\frac{4900792736}{333878023359}a^{18}+\frac{4142751276557}{3004902210231}a^{17}+\frac{1729575694826}{1001634070077}a^{16}+\frac{63733942832}{333878023359}a^{15}-\frac{25459422005474}{3004902210231}a^{14}-\frac{10767534567566}{1001634070077}a^{13}-\frac{399421984673}{333878023359}a^{12}+\frac{115903592260418}{3004902210231}a^{11}+\frac{49347965497796}{1001634070077}a^{10}+\frac{1818446235872}{333878023359}a^{9}-\frac{556312288841}{3004902210231}a^{8}-\frac{6914454051092}{1001634070077}a^{7}-\frac{254793947744}{333878023359}a^{6}+\frac{2004933112255}{3004902210231}a^{5}-\frac{4551664252409}{1001634070077}a^{4}-\frac{1323950780791}{333878023359}a^{3}+\frac{808025407750}{111292674453}a^{2}+\frac{136629851526}{12365852717}a+\frac{15104211696}{12365852717}$, $\frac{2577931900}{333878023359}a^{23}+\frac{6944744777}{1001634070077}a^{22}-\frac{26145656}{37097558151}a^{21}-\frac{45101166076}{333878023359}a^{20}-\frac{121847651590}{1001634070077}a^{19}+\frac{3810680294}{333878023359}a^{18}+\frac{584573290000}{333878023359}a^{17}+\frac{1580436104932}{1001634070077}a^{16}-\frac{47977189205}{333878023359}a^{15}-\frac{3585721728400}{333878023359}a^{14}-\frac{9731033666779}{1001634070077}a^{13}+\frac{257497071008}{333878023359}a^{12}+\frac{16289459308006}{333878023359}a^{11}+\frac{44269616896102}{1001634070077}a^{10}-\frac{1056131555072}{333878023359}a^{9}+\frac{392825989100}{333878023359}a^{8}-\frac{607110704644}{1001634070077}a^{7}-\frac{1999640380207}{333878023359}a^{6}-\frac{22584135800}{12365852717}a^{5}-\frac{3950719826440}{1001634070077}a^{4}+\frac{133943070982}{333878023359}a^{3}+\frac{1262399035783}{111292674453}a^{2}+\frac{255567764150}{37097558151}a-\frac{6574834372}{12365852717}$, $\frac{1288965950}{333878023359}a^{23}+\frac{1145282773}{1001634070077}a^{22}-\frac{277648558}{111292674453}a^{21}-\frac{22550583038}{333878023359}a^{20}-\frac{19696432991}{1001634070077}a^{19}+\frac{1634757194}{37097558151}a^{18}+\frac{292286645000}{333878023359}a^{17}+\frac{254063100068}{1001634070077}a^{16}-\frac{21259728203}{37097558151}a^{15}-\frac{1792860864200}{333878023359}a^{14}-\frac{1521624935021}{1001634070077}a^{13}+\frac{132292196363}{37097558151}a^{12}+\frac{8144729654003}{333878023359}a^{11}+\frac{6837324694343}{1001634070077}a^{10}-\frac{606579022238}{37097558151}a^{9}+\frac{196412994550}{333878023359}a^{8}+\frac{1839871381594}{1001634070077}a^{7}+\frac{84991604726}{37097558151}a^{6}-\frac{11292067900}{12365852717}a^{5}+\frac{2037143450740}{1001634070077}a^{4}+\frac{109489494034}{111292674453}a^{3}+\frac{575553180665}{111292674453}a^{2}+\frac{721322581}{37097558151}a-\frac{45344800431}{12365852717}$
| 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: | \( 533054958.55350393 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 533054958.55350393 \cdot 18}{18\cdot\sqrt{16878953293629664473677397903764439609}}\cr\approx \mathstrut & 0.491199201520010 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 17*x^21 + 218*x^18 - 1277*x^15 + 5609*x^12 + 3403*x^9 - 692*x^6 + 945*x^3 + 729)
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^24 - 17*x^21 + 218*x^18 - 1277*x^15 + 5609*x^12 + 3403*x^9 - 692*x^6 + 945*x^3 + 729, 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^24 - 17*x^21 + 218*x^18 - 1277*x^15 + 5609*x^12 + 3403*x^9 - 692*x^6 + 945*x^3 + 729);
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^24 - 17*x^21 + 218*x^18 - 1277*x^15 + 5609*x^12 + 3403*x^9 - 692*x^6 + 945*x^3 + 729);
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
$C_2\times C_{12}$ (as 24T2):
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)
| An abelian group of order 24 |
| The 24 conjugacy class representatives for $C_2\times C_{12}$ |
| Character table for $C_2\times C_{12}$ |
Intermediate fields
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 | ${\href{/padicField/2.12.0.1}{12} }^{2}$ | R | ${\href{/padicField/5.12.0.1}{12} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{12}$ | ${\href{/padicField/19.4.0.1}{4} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{4}$ | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{6}$ | ${\href{/padicField/41.12.0.1}{12} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{4}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{12}$ | ${\href{/padicField/59.12.0.1}{12} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
| 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
| 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
| 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
|
\(13\)
| 13.12.9.2 | $x^{12} + 8 x^{10} + 44 x^{9} + 63 x^{8} + 264 x^{7} + 550 x^{6} - 6336 x^{5} + 3843 x^{4} + 4532 x^{3} + 46454 x^{2} + 30668 x + 30982$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
| 13.12.9.2 | $x^{12} + 8 x^{10} + 44 x^{9} + 63 x^{8} + 264 x^{7} + 550 x^{6} - 6336 x^{5} + 3843 x^{4} + 4532 x^{3} + 46454 x^{2} + 30668 x + 30982$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |