Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 10*x^16 - 15*x^15 + 26*x^14 - 34*x^13 - 10*x^12 + 14*x^11 - 112*x^10 + 111*x^9 - 70*x^8 + 88*x^7 + 28*x^6 + 10*x^5 + 16*x^4 - 19*x^3 - 8*x^2 - 4*x - 1)
gp: K = bnfinit(y^18 - 2*y^17 + 10*y^16 - 15*y^15 + 26*y^14 - 34*y^13 - 10*y^12 + 14*y^11 - 112*y^10 + 111*y^9 - 70*y^8 + 88*y^7 + 28*y^6 + 10*y^5 + 16*y^4 - 19*y^3 - 8*y^2 - 4*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 + 10*x^16 - 15*x^15 + 26*x^14 - 34*x^13 - 10*x^12 + 14*x^11 - 112*x^10 + 111*x^9 - 70*x^8 + 88*x^7 + 28*x^6 + 10*x^5 + 16*x^4 - 19*x^3 - 8*x^2 - 4*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 + 10*x^16 - 15*x^15 + 26*x^14 - 34*x^13 - 10*x^12 + 14*x^11 - 112*x^10 + 111*x^9 - 70*x^8 + 88*x^7 + 28*x^6 + 10*x^5 + 16*x^4 - 19*x^3 - 8*x^2 - 4*x - 1)
\( x^{18} - 2 x^{17} + 10 x^{16} - 15 x^{15} + 26 x^{14} - 34 x^{13} - 10 x^{12} + 14 x^{11} - 112 x^{10} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[6, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(666757183921928000000000\)
\(\medspace = 2^{12}\cdot 5^{9}\cdot 11^{6}\cdot 19^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.06\) | 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: | $2^{2/3}5^{1/2}11^{1/2}19^{2/3}\approx 83.82439143954362$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7}a^{16}+\frac{3}{7}a^{15}+\frac{2}{7}a^{14}+\frac{3}{7}a^{13}+\frac{2}{7}a^{12}-\frac{2}{7}a^{11}-\frac{3}{7}a^{10}+\frac{3}{7}a^{9}-\frac{3}{7}a^{5}-\frac{1}{7}a^{4}-\frac{3}{7}a^{3}+\frac{3}{7}a^{2}+\frac{2}{7}a+\frac{3}{7}$, $\frac{1}{81324051706303}a^{17}+\frac{5389805174497}{81324051706303}a^{16}-\frac{38736130503925}{81324051706303}a^{15}-\frac{364087149705}{81324051706303}a^{14}-\frac{1380065826917}{81324051706303}a^{13}-\frac{24033927240628}{81324051706303}a^{12}+\frac{22671611299546}{81324051706303}a^{11}+\frac{4760997911095}{81324051706303}a^{10}-\frac{23615721286811}{81324051706303}a^{9}+\frac{845125662232}{11617721672329}a^{8}-\frac{1393350343067}{11617721672329}a^{7}-\frac{36704428198217}{81324051706303}a^{6}+\frac{17990975066535}{81324051706303}a^{5}+\frac{37335434516280}{81324051706303}a^{4}-\frac{18290141425011}{81324051706303}a^{3}-\frac{32864190265243}{81324051706303}a^{2}-\frac{24070450732201}{81324051706303}a-\frac{36992956410784}{81324051706303}$
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
Trivial group, which has order $1$ (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: |
$\frac{30049762513220}{81324051706303}a^{17}-\frac{58652113999426}{81324051706303}a^{16}+\frac{297122121927909}{81324051706303}a^{15}-\frac{437955746478829}{81324051706303}a^{14}+\frac{757983587453769}{81324051706303}a^{13}-\frac{996987203017149}{81324051706303}a^{12}-\frac{346344682346336}{81324051706303}a^{11}+\frac{402223657457578}{81324051706303}a^{10}-\frac{33\!\cdots\!71}{81324051706303}a^{9}+\frac{469555436399100}{11617721672329}a^{8}-\frac{275556323971022}{11617721672329}a^{7}+\frac{26\!\cdots\!14}{81324051706303}a^{6}+\frac{975670083196603}{81324051706303}a^{5}+\frac{84652756249407}{81324051706303}a^{4}+\frac{442831265178244}{81324051706303}a^{3}-\frac{721915663410146}{81324051706303}a^{2}-\frac{297264051914576}{81324051706303}a-\frac{61225093303541}{81324051706303}$, $\frac{24201312}{601415843}a^{17}-\frac{3688822}{85916549}a^{16}+\frac{28812316}{85916549}a^{15}-\frac{190229248}{601415843}a^{14}+\frac{63059492}{85916549}a^{13}-\frac{758296513}{601415843}a^{12}-\frac{121055488}{601415843}a^{11}-\frac{1258218574}{601415843}a^{10}-\frac{891096518}{601415843}a^{9}+\frac{74441286}{85916549}a^{8}-\frac{155038294}{85916549}a^{7}+\frac{6059561726}{601415843}a^{6}-\frac{3299814978}{601415843}a^{5}+\frac{664545728}{85916549}a^{4}-\frac{1532377032}{601415843}a^{3}-\frac{77197730}{85916549}a^{2}-\frac{379287838}{601415843}a-\frac{508901306}{601415843}$, $a$, $\frac{16274342041828}{81324051706303}a^{17}-\frac{17637197737326}{81324051706303}a^{16}+\frac{118448309655098}{81324051706303}a^{15}-\frac{8926686499457}{11617721672329}a^{14}+\frac{50215047311838}{81324051706303}a^{13}+\frac{80047311235730}{81324051706303}a^{12}-\frac{152937916784130}{11617721672329}a^{11}+\frac{609454153065175}{81324051706303}a^{10}-\frac{15\!\cdots\!19}{81324051706303}a^{9}-\frac{29573217474492}{11617721672329}a^{8}+\frac{310379486288972}{11617721672329}a^{7}-\frac{14\!\cdots\!51}{81324051706303}a^{6}+\frac{402350240155561}{11617721672329}a^{5}-\frac{492080527534247}{81324051706303}a^{4}+\frac{95994830967873}{81324051706303}a^{3}-\frac{56050682249818}{81324051706303}a^{2}-\frac{71244221934847}{11617721672329}a+\frac{10293977126598}{81324051706303}$, $\frac{1825114582938}{81324051706303}a^{17}-\frac{116140193343}{81324051706303}a^{16}+\frac{14481520053381}{81324051706303}a^{15}-\frac{344678750551}{11617721672329}a^{14}+\frac{34984050541793}{81324051706303}a^{13}-\frac{56690555570237}{81324051706303}a^{12}+\frac{300539083522}{11617721672329}a^{11}-\frac{214771751333323}{81324051706303}a^{10}-\frac{71859378086758}{81324051706303}a^{9}-\frac{14875276026972}{11617721672329}a^{8}-\frac{25356828782196}{11617721672329}a^{7}+\frac{685103263325003}{81324051706303}a^{6}-\frac{32908487465335}{11617721672329}a^{5}+\frac{666988700234492}{81324051706303}a^{4}-\frac{56239378985106}{81324051706303}a^{3}-\frac{16205327929494}{81324051706303}a^{2}-\frac{4133940826462}{11617721672329}a-\frac{98863906011846}{81324051706303}$, $\frac{1825114582938}{81324051706303}a^{17}-\frac{116140193343}{81324051706303}a^{16}+\frac{14481520053381}{81324051706303}a^{15}-\frac{344678750551}{11617721672329}a^{14}+\frac{34984050541793}{81324051706303}a^{13}-\frac{56690555570237}{81324051706303}a^{12}+\frac{300539083522}{11617721672329}a^{11}-\frac{214771751333323}{81324051706303}a^{10}-\frac{71859378086758}{81324051706303}a^{9}-\frac{14875276026972}{11617721672329}a^{8}-\frac{25356828782196}{11617721672329}a^{7}+\frac{685103263325003}{81324051706303}a^{6}-\frac{32908487465335}{11617721672329}a^{5}+\frac{666988700234492}{81324051706303}a^{4}-\frac{56239378985106}{81324051706303}a^{3}-\frac{16205327929494}{81324051706303}a^{2}-\frac{4133940826462}{11617721672329}a-\frac{180187957718149}{81324051706303}$, $\frac{2782381613452}{11617721672329}a^{17}-\frac{7132061431659}{81324051706303}a^{16}+\frac{115673117916060}{81324051706303}a^{15}+\frac{51891191349887}{81324051706303}a^{14}-\frac{105128531232230}{81324051706303}a^{13}+\frac{330009503356569}{81324051706303}a^{12}-\frac{15\!\cdots\!77}{81324051706303}a^{11}+\frac{246700881523714}{81324051706303}a^{10}-\frac{12\!\cdots\!43}{81324051706303}a^{9}-\frac{249835222427236}{11617721672329}a^{8}+\frac{501183901720238}{11617721672329}a^{7}-\frac{208904034737200}{11617721672329}a^{6}+\frac{30\!\cdots\!54}{81324051706303}a^{5}+\frac{619825371806098}{81324051706303}a^{4}+\frac{35645683409604}{81324051706303}a^{3}+\frac{116041394522667}{81324051706303}a^{2}-\frac{727035603066646}{81324051706303}a-\frac{170195855034711}{81324051706303}$, $\frac{18035077624159}{81324051706303}a^{17}-\frac{21597214454630}{81324051706303}a^{16}+\frac{142684176007063}{81324051706303}a^{15}-\frac{121747107494056}{81324051706303}a^{14}+\frac{202503666807986}{81324051706303}a^{13}-\frac{36588475519377}{11617721672329}a^{12}-\frac{607607230485939}{81324051706303}a^{11}-\frac{4635019888856}{11617721672329}a^{10}-\frac{11\!\cdots\!38}{81324051706303}a^{9}+\frac{41064073806708}{11617721672329}a^{8}+\frac{94961432366603}{11617721672329}a^{7}+\frac{13\!\cdots\!00}{81324051706303}a^{6}-\frac{9589080799876}{81324051706303}a^{5}+\frac{980105493524825}{81324051706303}a^{4}-\frac{38530450665886}{11617721672329}a^{3}-\frac{734741452625427}{81324051706303}a^{2}-\frac{80018165355627}{81324051706303}a-\frac{848934398232}{81324051706303}$, $\frac{3473856116923}{11617721672329}a^{17}-\frac{65157164946091}{81324051706303}a^{16}+\frac{285062872626739}{81324051706303}a^{15}-\frac{544705633429008}{81324051706303}a^{14}+\frac{961943670899229}{81324051706303}a^{13}-\frac{13\!\cdots\!12}{81324051706303}a^{12}+\frac{500289390644192}{81324051706303}a^{11}+\frac{273147337128543}{81324051706303}a^{10}-\frac{31\!\cdots\!41}{81324051706303}a^{9}+\frac{661232783864003}{11617721672329}a^{8}-\frac{630770642970906}{11617721672329}a^{7}+\frac{569264890416820}{11617721672329}a^{6}-\frac{994212668986317}{81324051706303}a^{5}+\frac{446216457530362}{81324051706303}a^{4}+\frac{514481160639456}{81324051706303}a^{3}-\frac{510183947844553}{81324051706303}a^{2}+\frac{24255972140622}{81324051706303}a-\frac{71072309537582}{81324051706303}$, $\frac{10540787580336}{81324051706303}a^{17}-\frac{17853693281593}{81324051706303}a^{16}+\frac{95337945536187}{81324051706303}a^{15}-\frac{121556032404995}{81324051706303}a^{14}+\frac{198853106359079}{81324051706303}a^{13}-\frac{251620507444811}{81324051706303}a^{12}-\frac{240596164265423}{81324051706303}a^{11}+\frac{156567646282262}{81324051706303}a^{10}-\frac{981489642387514}{81324051706303}a^{9}+\frac{110794510203275}{11617721672329}a^{8}-\frac{27978721971660}{11617721672329}a^{7}+\frac{601835786362604}{81324051706303}a^{6}+\frac{263198552294379}{81324051706303}a^{5}+\frac{22130557651826}{81324051706303}a^{4}+\frac{118807601631346}{81324051706303}a^{3}-\frac{167389194310159}{81324051706303}a^{2}+\frac{20187792342565}{81324051706303}a-\frac{76088086975216}{81324051706303}$, $\frac{14674813843958}{81324051706303}a^{17}-\frac{8295429331155}{81324051706303}a^{16}+\frac{93891491849215}{81324051706303}a^{15}+\frac{11348686875889}{81324051706303}a^{14}-\frac{39319003281008}{81324051706303}a^{13}+\frac{204018530636175}{81324051706303}a^{12}-\frac{11\!\cdots\!71}{81324051706303}a^{11}+\frac{353638466810502}{81324051706303}a^{10}-\frac{177106304678978}{11617721672329}a^{9}-\frac{120276543086035}{11617721672329}a^{8}+\frac{342178696235087}{11617721672329}a^{7}-\frac{12\!\cdots\!84}{81324051706303}a^{6}+\frac{29\!\cdots\!96}{81324051706303}a^{5}-\frac{281351822168521}{81324051706303}a^{4}+\frac{362752195281097}{81324051706303}a^{3}-\frac{187058721258195}{81324051706303}a^{2}-\frac{481453366671761}{81324051706303}a-\frac{22881394253756}{11617721672329}$
| 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: | \( 80091.8197007 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 80091.8197007 \cdot 1}{2\cdot\sqrt{666757183921928000000000}}\cr\approx \mathstrut & 0.193122776977 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 10*x^16 - 15*x^15 + 26*x^14 - 34*x^13 - 10*x^12 + 14*x^11 - 112*x^10 + 111*x^9 - 70*x^8 + 88*x^7 + 28*x^6 + 10*x^5 + 16*x^4 - 19*x^3 - 8*x^2 - 4*x - 1)
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^18 - 2*x^17 + 10*x^16 - 15*x^15 + 26*x^14 - 34*x^13 - 10*x^12 + 14*x^11 - 112*x^10 + 111*x^9 - 70*x^8 + 88*x^7 + 28*x^6 + 10*x^5 + 16*x^4 - 19*x^3 - 8*x^2 - 4*x - 1, 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^18 - 2*x^17 + 10*x^16 - 15*x^15 + 26*x^14 - 34*x^13 - 10*x^12 + 14*x^11 - 112*x^10 + 111*x^9 - 70*x^8 + 88*x^7 + 28*x^6 + 10*x^5 + 16*x^4 - 19*x^3 - 8*x^2 - 4*x - 1);
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^18 - 2*x^17 + 10*x^16 - 15*x^15 + 26*x^14 - 34*x^13 - 10*x^12 + 14*x^11 - 112*x^10 + 111*x^9 - 70*x^8 + 88*x^7 + 28*x^6 + 10*x^5 + 16*x^4 - 19*x^3 - 8*x^2 - 4*x - 1);
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
$C_3\times S_3^2$ (as 18T46):
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 solvable group of order 108 |
| The 27 conjugacy class representatives for $C_3\times S_3^2$ |
| Character table for $C_3\times S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 3.1.44.1, 6.6.722000.1, 6.2.242000.1 |
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]
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.57717900270250000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{9}$ | R | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(5\)
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(11\)
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(19\)
| 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 19.6.4.2 | $x^{6} - 342 x^{3} + 722$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |