Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 39*x^16 + 599*x^14 - 4620*x^12 + 19019*x^10 - 41678*x^8 + 46956*x^6 - 23998*x^4 + 3718*x^2 - 169)
gp: K = bnfinit(y^18 - 39*y^16 + 599*y^14 - 4620*y^12 + 19019*y^10 - 41678*y^8 + 46956*y^6 - 23998*y^4 + 3718*y^2 - 169, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 39*x^16 + 599*x^14 - 4620*x^12 + 19019*x^10 - 41678*x^8 + 46956*x^6 - 23998*x^4 + 3718*x^2 - 169);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 39*x^16 + 599*x^14 - 4620*x^12 + 19019*x^10 - 41678*x^8 + 46956*x^6 - 23998*x^4 + 3718*x^2 - 169)
\( x^{18} - 39 x^{16} + 599 x^{14} - 4620 x^{12} + 19019 x^{10} - 41678 x^{8} + 46956 x^{6} - 23998 x^{4} + \cdots - 169 \)
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: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1200997449871582137072287481856\)
\(\medspace = 2^{18}\cdot 7^{16}\cdot 13^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(46.89\) | 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: | $2^{3/2}7^{8/9}13^{5/6}\approx 135.21640283526824$ | ||
| Ramified primes: |
\(2\), \(7\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13}a^{12}+\frac{1}{13}a^{8}-\frac{5}{13}a^{6}$, $\frac{1}{13}a^{13}+\frac{1}{13}a^{9}-\frac{5}{13}a^{7}$, $\frac{1}{13}a^{14}+\frac{1}{13}a^{10}-\frac{5}{13}a^{8}$, $\frac{1}{13}a^{15}+\frac{1}{13}a^{11}-\frac{5}{13}a^{9}$, $\frac{1}{18282616703}a^{16}+\frac{405447117}{18282616703}a^{14}+\frac{280926877}{18282616703}a^{12}+\frac{7335694813}{18282616703}a^{10}+\frac{874840081}{18282616703}a^{8}+\frac{9082625767}{18282616703}a^{6}+\frac{508643706}{1406355131}a^{4}-\frac{224372932}{1406355131}a^{2}-\frac{105456372}{1406355131}$, $\frac{1}{18282616703}a^{17}+\frac{405447117}{18282616703}a^{15}+\frac{280926877}{18282616703}a^{13}+\frac{7335694813}{18282616703}a^{11}+\frac{874840081}{18282616703}a^{9}+\frac{9082625767}{18282616703}a^{7}+\frac{508643706}{1406355131}a^{5}-\frac{224372932}{1406355131}a^{3}-\frac{105456372}{1406355131}a$
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: | $17$ | 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: |
$\frac{161309}{43426643}a^{16}-\frac{5789611}{43426643}a^{14}+\frac{77752657}{43426643}a^{12}-\frac{470702266}{43426643}a^{10}+\frac{1127850759}{43426643}a^{8}+\frac{142352884}{43426643}a^{6}-\frac{292469597}{3340511}a^{4}+\frac{278376927}{3340511}a^{2}-\frac{27422653}{3340511}$, $\frac{161309}{43426643}a^{16}-\frac{5789611}{43426643}a^{14}+\frac{77752657}{43426643}a^{12}-\frac{470702266}{43426643}a^{10}+\frac{1127850759}{43426643}a^{8}+\frac{142352884}{43426643}a^{6}-\frac{292469597}{3340511}a^{4}+\frac{278376927}{3340511}a^{2}-\frac{24082142}{3340511}$, $\frac{15137617}{1406355131}a^{16}-\frac{7454100344}{18282616703}a^{14}+\frac{109630513747}{18282616703}a^{12}-\frac{790359600955}{18282616703}a^{10}+\frac{2914462315932}{18282616703}a^{8}-\frac{5323842365015}{18282616703}a^{6}+\frac{350343079876}{1406355131}a^{4}-\frac{116740285450}{1406355131}a^{2}+\frac{8114174645}{1406355131}$, $\frac{117273567}{18282616703}a^{16}-\frac{4478844711}{18282616703}a^{14}+\frac{66636518451}{18282616703}a^{12}-\frac{488477471564}{18282616703}a^{10}+\frac{1845503438600}{18282616703}a^{8}-\frac{3480172269800}{18282616703}a^{6}+\frac{233721389563}{1406355131}a^{4}-\frac{77226710431}{1406355131}a^{2}+\frac{5847033068}{1406355131}$, $\frac{185184656}{18282616703}a^{16}-\frac{6916270942}{18282616703}a^{14}+\frac{99370387048}{18282616703}a^{12}-\frac{686643125550}{18282616703}a^{10}+\frac{2320328608139}{18282616703}a^{8}-\frac{3420241705636}{18282616703}a^{6}+\frac{110591689226}{1406355131}a^{4}+\frac{39969975836}{1406355131}a^{2}-\frac{4291548714}{1406355131}$, $\frac{500313629}{18282616703}a^{16}-\frac{18915052937}{18282616703}a^{14}+\frac{277106918926}{18282616703}a^{12}-\frac{1980658739630}{18282616703}a^{10}+\frac{7152837616770}{18282616703}a^{8}-\frac{12342736956946}{18282616703}a^{6}+\frac{687564106085}{1406355131}a^{4}-\frac{136225991739}{1406355131}a^{2}+\frac{7545388801}{1406355131}$, $\frac{63197357}{18282616703}a^{16}-\frac{2369154163}{18282616703}a^{14}+\frac{34277312447}{18282616703}a^{12}-\frac{240318106505}{18282616703}a^{10}+\frac{841124003961}{18282616703}a^{8}-\frac{1377972340623}{18282616703}a^{6}+\frac{71315270989}{1406355131}a^{4}-\frac{13631200801}{1406355131}a^{2}+\frac{2496601179}{1406355131}$, $\frac{141156021}{18282616703}a^{16}-\frac{405450323}{1406355131}a^{14}+\frac{75811680769}{18282616703}a^{12}-\frac{526385945076}{18282616703}a^{10}+\frac{139151659607}{1406355131}a^{8}-\frac{218638087421}{1406355131}a^{6}+\frac{129910557003}{1406355131}a^{4}-\frac{10060265473}{1406355131}a^{2}+\frac{291115696}{1406355131}$, $\frac{14296870}{1406355131}a^{17}-\frac{751716085}{18282616703}a^{16}-\frac{6469977924}{18282616703}a^{15}+\frac{28434102784}{18282616703}a^{14}+\frac{82079816688}{18282616703}a^{13}-\frac{417037251737}{18282616703}a^{12}-\frac{434176259914}{18282616703}a^{11}+\frac{2988879879919}{18282616703}a^{10}+\frac{546372482235}{18282616703}a^{9}-\frac{10869524580205}{18282616703}a^{8}+\frac{2763364734889}{18282616703}a^{7}+\frac{19144549236925}{18282616703}a^{6}-\frac{653638413969}{1406355131}a^{5}-\frac{1141153036452}{1406355131}a^{4}+\frac{515662824902}{1406355131}a^{3}+\frac{298329285806}{1406355131}a^{2}-\frac{52571096297}{1406355131}a-\frac{21633355101}{1406355131}$, $\frac{117273567}{18282616703}a^{16}-\frac{4478844711}{18282616703}a^{14}+\frac{66636518451}{18282616703}a^{12}-\frac{488477471564}{18282616703}a^{10}+\frac{1845503438600}{18282616703}a^{8}-\frac{3480172269800}{18282616703}a^{6}+\frac{233721389563}{1406355131}a^{4}-\frac{77226710431}{1406355131}a^{2}-a+\frac{5847033068}{1406355131}$, $\frac{751716085}{18282616703}a^{17}-\frac{819627174}{18282616703}a^{16}-\frac{28434102784}{18282616703}a^{15}+\frac{30871529015}{18282616703}a^{14}+\frac{417037251737}{18282616703}a^{13}-\frac{449771120334}{18282616703}a^{12}-\frac{2988879879919}{18282616703}a^{11}+\frac{3187045533905}{18282616703}a^{10}+\frac{10869524580205}{18282616703}a^{9}-\frac{11344349749744}{18282616703}a^{8}-\frac{19144549236925}{18282616703}a^{7}+\frac{19084618672761}{18282616703}a^{6}+\frac{1141153036452}{1406355131}a^{5}-\frac{1018023336115}{1406355131}a^{4}-\frac{298329285806}{1406355131}a^{3}+\frac{181132599539}{1406355131}a^{2}+\frac{20226999970}{1406355131}a-\frac{8682063057}{1406355131}$, $\frac{708032361}{18282616703}a^{17}+\frac{182954142}{18282616703}a^{16}-\frac{27353275701}{18282616703}a^{15}-\frac{548181897}{1406355131}a^{14}+\frac{414288129732}{18282616703}a^{13}+\frac{109255851154}{18282616703}a^{12}-\frac{3127201681626}{18282616703}a^{11}-\frac{840353312028}{18282616703}a^{10}+\frac{12435389766399}{18282616703}a^{9}+\frac{3444016581032}{18282616703}a^{8}-\frac{25753786421865}{18282616703}a^{7}-\frac{7485972858356}{18282616703}a^{6}+\frac{2041888679806}{1406355131}a^{5}+\frac{635878898787}{1406355131}a^{4}-\frac{885369387541}{1406355131}a^{3}-\frac{297532481347}{1406355131}a^{2}+\frac{72125768995}{1406355131}a+\frac{23009543447}{1406355131}$, $\frac{751716085}{18282616703}a^{17}-\frac{765550964}{18282616703}a^{16}-\frac{28434102784}{18282616703}a^{15}+\frac{28761838467}{18282616703}a^{14}+\frac{417037251737}{18282616703}a^{13}-\frac{417411914330}{18282616703}a^{12}-\frac{2988879879919}{18282616703}a^{11}+\frac{2938886168846}{18282616703}a^{10}+\frac{10869524580205}{18282616703}a^{9}-\frac{10339970315105}{18282616703}a^{8}-\frac{19144549236925}{18282616703}a^{7}+\frac{16982418743584}{18282616703}a^{6}+\frac{1141153036452}{1406355131}a^{5}-\frac{855617217541}{1406355131}a^{4}-\frac{298329285806}{1406355131}a^{3}+\frac{117537089909}{1406355131}a^{2}+\frac{20226999970}{1406355131}a-\frac{5331631168}{1406355131}$, $\frac{721330015}{18282616703}a^{17}-\frac{128877932}{18282616703}a^{16}-\frac{27725752429}{18282616703}a^{15}+\frac{5016674113}{18282616703}a^{14}+\frac{416722179265}{18282616703}a^{13}-\frac{5915126550}{1406355131}a^{12}-\frac{239038907406}{1406355131}a^{11}+\frac{592193946969}{18282616703}a^{10}+\frac{12107990288435}{18282616703}a^{9}-\frac{2439637146393}{18282616703}a^{8}-\frac{24215885942737}{18282616703}a^{7}+\frac{5383772929179}{18282616703}a^{6}+\frac{1815513128978}{1406355131}a^{5}-\frac{473472780213}{1406355131}a^{4}-\frac{722809692657}{1406355131}a^{3}+\frac{233936971717}{1406355131}a^{2}+\frac{53200685296}{1406355131}a-\frac{18252756427}{1406355131}$, $\frac{600433}{43426643}a^{17}-\frac{8583922}{18282616703}a^{16}-\frac{22492380}{43426643}a^{15}+\frac{304204660}{18282616703}a^{14}+\frac{324826708}{43426643}a^{13}-\frac{3977493383}{18282616703}a^{12}-\frac{2266280833}{43426643}a^{11}+\frac{22456567171}{18282616703}a^{10}+\frac{7819933486}{43426643}a^{9}-\frac{38899356458}{18282616703}a^{8}-\frac{12209743513}{43426643}a^{7}-\frac{99997574341}{18282616703}a^{6}+\frac{515390578}{3340511}a^{5}+\frac{31930579502}{1406355131}a^{4}+\frac{15242751}{3340511}a^{3}-\frac{31731807816}{1406355131}a^{2}-\frac{2929257}{3340511}a+\frac{4883545607}{1406355131}$, $\frac{169656488}{18282616703}a^{17}-\frac{949884943}{18282616703}a^{16}-\frac{519529841}{1406355131}a^{15}+\frac{35838445261}{18282616703}a^{14}+\frac{106821801621}{18282616703}a^{13}-\frac{523489476741}{18282616703}a^{12}-\frac{860049197376}{18282616703}a^{11}+\frac{3725122571278}{18282616703}a^{10}+\frac{3771416058996}{18282616703}a^{9}-\frac{13359491930308}{18282616703}a^{8}-\frac{9023873337484}{18282616703}a^{7}+\frac{22806881340934}{18282616703}a^{6}+\frac{862254449615}{1406355131}a^{5}-\frac{1254886619299}{1406355131}a^{4}-\frac{460092176231}{1406355131}a^{3}+\frac{246549079018}{1406355131}a^{2}+\frac{39121916884}{1406355131}a-\frac{11613635987}{1406355131}$, $\frac{592989308}{18282616703}a^{17}+\frac{456598446}{18282616703}a^{16}-\frac{22664337271}{18282616703}a^{15}-\frac{17154041387}{18282616703}a^{14}+\frac{337766147175}{18282616703}a^{13}+\frac{248900424224}{18282616703}a^{12}-\frac{2485014023584}{18282616703}a^{11}-\frac{1751106107999}{18282616703}a^{10}+\frac{9466198354906}{18282616703}a^{9}+\frac{6144442363594}{18282616703}a^{8}-\frac{1400606384565}{1406355131}a^{7}-\frac{9984799510282}{18282616703}a^{6}+\frac{1282880080682}{1406355131}a^{5}+\frac{477365529413}{1406355131}a^{4}-\frac{470640219927}{1406355131}a^{3}-\frac{35506769116}{1406355131}a^{2}+\frac{40383998897}{1406355131}a-\frac{2475215219}{1406355131}$
| 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: | \( 1751416910.41 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 1751416910.41 \cdot 1}{2\cdot\sqrt{1200997449871582137072287481856}}\cr\approx \mathstrut & 0.209473178326 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 39*x^16 + 599*x^14 - 4620*x^12 + 19019*x^10 - 41678*x^8 + 46956*x^6 - 23998*x^4 + 3718*x^2 - 169)
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 - 39*x^16 + 599*x^14 - 4620*x^12 + 19019*x^10 - 41678*x^8 + 46956*x^6 - 23998*x^4 + 3718*x^2 - 169, 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 - 39*x^16 + 599*x^14 - 4620*x^12 + 19019*x^10 - 41678*x^8 + 46956*x^6 - 23998*x^4 + 3718*x^2 - 169);
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 - 39*x^16 + 599*x^14 - 4620*x^12 + 19019*x^10 - 41678*x^8 + 46956*x^6 - 23998*x^4 + 3718*x^2 - 169);
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
$C_3^2.A_4$ (as 18T47):
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 20 conjugacy class representatives for $C_3^2.A_4$ |
| Character table for $C_3^2.A_4$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 6.6.25969216.1, 9.9.164648481361.1 |
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 18 siblings: | data not computed |
| Degree 36 sibling: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/17.9.0.1}{9} }^{2}$ | ${\href{/padicField/19.9.0.1}{9} }^{2}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.9.0.1}{9} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.18.89 | $x^{18} + 18 x^{17} + 66 x^{16} - 272 x^{15} - 1608 x^{14} + 7008 x^{13} + 83536 x^{12} + 346688 x^{11} + 922880 x^{10} + 2307136 x^{9} + 7066496 x^{8} + 20902656 x^{7} + 47520384 x^{6} + 81117696 x^{5} + 108969728 x^{4} + 117408768 x^{3} + 95319808 x^{2} + 50121216 x + 12416512$ | $2$ | $9$ | $18$ | $C_2^2 : C_9$ | $[2, 2]^{9}$ |
|
\(7\)
| 7.9.8.2 | $x^{9} + 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |
| 7.9.8.2 | $x^{9} + 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ | |
|
\(13\)
| 13.6.5.6 | $x^{6} + 78$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 13.6.5.3 | $x^{6} + 39$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |