Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 4*x - 5)
gp: K = bnfinit(y^25 - 4*y - 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 4*x - 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 4*x - 5)
\( x^{25} - 4x - 5 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(108009270762295279786326375965420843490676304895049\)
\(\medspace = 1663\cdot 6971\cdot 492251\cdot 268235921\cdot 264590126291\cdot 266683732590664933\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(100.31\) | 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: | $1663^{1/2}6971^{1/2}492251^{1/2}268235921^{1/2}264590126291^{1/2}266683732590664933^{1/2}\approx 1.0392750875600515e+25$ | ||
| Ramified primes: |
\(1663\), \(6971\), \(492251\), \(268235921\), \(264590126291\), \(266683732590664933\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{10800\!\cdots\!95049}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{7}a^{24}+\frac{3}{7}a^{23}+\frac{2}{7}a^{22}-\frac{1}{7}a^{21}-\frac{3}{7}a^{20}-\frac{2}{7}a^{19}+\frac{1}{7}a^{18}+\frac{3}{7}a^{17}+\frac{2}{7}a^{16}-\frac{1}{7}a^{15}-\frac{3}{7}a^{14}-\frac{2}{7}a^{13}+\frac{1}{7}a^{12}+\frac{3}{7}a^{11}+\frac{2}{7}a^{10}-\frac{1}{7}a^{9}-\frac{3}{7}a^{8}-\frac{2}{7}a^{7}+\frac{1}{7}a^{6}+\frac{3}{7}a^{5}+\frac{2}{7}a^{4}-\frac{1}{7}a^{3}-\frac{3}{7}a^{2}-\frac{2}{7}a-\frac{3}{7}$
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: | $12$ | 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: |
$a^{12}-a^{11}+a^{10}-a^{9}+a^{8}-a^{7}+a^{6}-a^{5}+a^{4}-a^{3}+a^{2}-a-1$, $\frac{3}{7}a^{24}+\frac{2}{7}a^{23}+\frac{6}{7}a^{22}+\frac{4}{7}a^{21}+\frac{5}{7}a^{20}+\frac{8}{7}a^{19}+\frac{3}{7}a^{18}+\frac{9}{7}a^{17}+\frac{6}{7}a^{16}+\frac{4}{7}a^{15}+\frac{5}{7}a^{14}+\frac{1}{7}a^{13}-\frac{4}{7}a^{12}-\frac{5}{7}a^{11}-\frac{15}{7}a^{10}-\frac{17}{7}a^{9}-\frac{16}{7}a^{8}-\frac{27}{7}a^{7}-\frac{18}{7}a^{6}-\frac{19}{7}a^{5}-\frac{15}{7}a^{4}-\frac{3}{7}a^{3}-\frac{2}{7}a^{2}+\frac{8}{7}a+\frac{12}{7}$, $a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}-2a^{13}-3a^{12}-3a^{11}-3a^{10}-3a^{9}-3a^{8}-3a^{7}-3a^{6}-3a^{5}-3a^{4}-3a^{3}-a^{2}+4a+4$, $\frac{9}{7}a^{24}-\frac{15}{7}a^{23}+\frac{18}{7}a^{22}-\frac{9}{7}a^{21}+\frac{8}{7}a^{20}-\frac{11}{7}a^{19}-\frac{5}{7}a^{18}+\frac{13}{7}a^{17}-\frac{10}{7}a^{16}+\frac{19}{7}a^{15}-\frac{34}{7}a^{14}+\frac{31}{7}a^{13}-\frac{26}{7}a^{12}+\frac{41}{7}a^{11}-\frac{38}{7}a^{10}+\frac{12}{7}a^{9}-\frac{20}{7}a^{8}+\frac{24}{7}a^{7}+\frac{9}{7}a^{6}-\frac{15}{7}a^{5}-\frac{3}{7}a^{4}-\frac{16}{7}a^{3}+\frac{36}{7}a^{2}+\frac{3}{7}a-\frac{48}{7}$, $\frac{117}{7}a^{24}-\frac{90}{7}a^{23}+\frac{150}{7}a^{22}-\frac{152}{7}a^{21}+\frac{223}{7}a^{20}-\frac{108}{7}a^{19}+\frac{82}{7}a^{18}-\frac{62}{7}a^{17}+\frac{17}{7}a^{16}+\frac{86}{7}a^{15}-\frac{211}{7}a^{14}+\frac{109}{7}a^{13}-\frac{212}{7}a^{12}+\frac{183}{7}a^{11}-\frac{312}{7}a^{10}+\frac{9}{7}a^{9}-\frac{92}{7}a^{8}-\frac{10}{7}a^{7}-\frac{30}{7}a^{6}-\frac{258}{7}a^{5}+\frac{297}{7}a^{4}-\frac{110}{7}a^{3}+\frac{265}{7}a^{2}-\frac{241}{7}a-\frac{57}{7}$, $\frac{102}{7}a^{24}+\frac{124}{7}a^{23}-\frac{118}{7}a^{22}-\frac{151}{7}a^{21}+\frac{121}{7}a^{20}+\frac{188}{7}a^{19}-\frac{115}{7}a^{18}-\frac{226}{7}a^{17}+\frac{92}{7}a^{16}+\frac{276}{7}a^{15}-\frac{68}{7}a^{14}-\frac{316}{7}a^{13}+\frac{32}{7}a^{12}+\frac{362}{7}a^{11}+\frac{1}{7}a^{10}-\frac{410}{7}a^{9}-\frac{33}{7}a^{8}+\frac{482}{7}a^{7}+\frac{39}{7}a^{6}-\frac{569}{7}a^{5}-\frac{41}{7}a^{4}+\frac{682}{7}a^{3}+\frac{44}{7}a^{2}-\frac{827}{7}a-\frac{488}{7}$, $\frac{37}{7}a^{24}-\frac{50}{7}a^{23}-\frac{31}{7}a^{22}+\frac{75}{7}a^{21}+\frac{22}{7}a^{20}-\frac{116}{7}a^{19}+\frac{37}{7}a^{18}+\frac{76}{7}a^{17}-\frac{45}{7}a^{16}-\frac{93}{7}a^{15}+\frac{106}{7}a^{14}+\frac{52}{7}a^{13}-\frac{124}{7}a^{12}+\frac{13}{7}a^{11}+\frac{88}{7}a^{10}-\frac{58}{7}a^{9}-\frac{104}{7}a^{8}+\frac{150}{7}a^{7}+\frac{2}{7}a^{6}-\frac{148}{7}a^{5}+\frac{95}{7}a^{4}+\frac{145}{7}a^{3}-\frac{209}{7}a^{2}-\frac{158}{7}a+\frac{197}{7}$, $\frac{46}{7}a^{24}-\frac{51}{7}a^{23}+\frac{29}{7}a^{22}-\frac{11}{7}a^{21}-\frac{12}{7}a^{20}+\frac{27}{7}a^{19}-\frac{52}{7}a^{18}+\frac{89}{7}a^{17}-\frac{20}{7}a^{16}-\frac{39}{7}a^{15}-\frac{19}{7}a^{14}-\frac{22}{7}a^{13}+\frac{116}{7}a^{12}-\frac{65}{7}a^{11}-\frac{48}{7}a^{10}+\frac{45}{7}a^{9}+\frac{37}{7}a^{8}+\frac{76}{7}a^{7}-\frac{171}{7}a^{6}-\frac{37}{7}a^{5}+\frac{85}{7}a^{4}+\frac{17}{7}a^{3}+\frac{65}{7}a^{2}-\frac{99}{7}a-\frac{117}{7}$, $\frac{485}{7}a^{24}-\frac{540}{7}a^{23}+\frac{578}{7}a^{22}-\frac{590}{7}a^{21}+\frac{449}{7}a^{20}-\frac{249}{7}a^{19}+\frac{93}{7}a^{18}+\frac{125}{7}a^{17}-\frac{409}{7}a^{16}+\frac{551}{7}a^{15}-\frac{608}{7}a^{14}+\frac{752}{7}a^{13}-\frac{740}{7}a^{12}+\frac{531}{7}a^{11}-\frac{388}{7}a^{10}+\frac{236}{7}a^{9}+\frac{155}{7}a^{8}-\frac{494}{7}a^{7}+\frac{562}{7}a^{6}-\frac{785}{7}a^{5}+\frac{1019}{7}a^{4}-\frac{828}{7}a^{3}+\frac{624}{7}a^{2}-\frac{634}{7}a-\frac{1602}{7}$, $\frac{76}{7}a^{24}-\frac{136}{7}a^{23}-\frac{2}{7}a^{22}+\frac{127}{7}a^{21}-\frac{18}{7}a^{20}-\frac{124}{7}a^{19}+\frac{153}{7}a^{18}+\frac{172}{7}a^{17}-\frac{205}{7}a^{16}-\frac{13}{7}a^{15}+\frac{227}{7}a^{14}-\frac{82}{7}a^{13}-\frac{204}{7}a^{12}+\frac{221}{7}a^{11}+\frac{299}{7}a^{10}-\frac{265}{7}a^{9}-\frac{25}{7}a^{8}+\frac{436}{7}a^{7}-\frac{162}{7}a^{6}-\frac{381}{7}a^{5}+\frac{320}{7}a^{4}+\frac{435}{7}a^{3}-\frac{347}{7}a^{2}-\frac{19}{7}a+\frac{528}{7}$, $\frac{54}{7}a^{24}-\frac{34}{7}a^{23}-\frac{46}{7}a^{22}-\frac{159}{7}a^{21}-\frac{113}{7}a^{20}-\frac{178}{7}a^{19}-\frac{37}{7}a^{18}-\frac{48}{7}a^{17}+\frac{171}{7}a^{16}+\frac{135}{7}a^{15}+\frac{321}{7}a^{14}+\frac{172}{7}a^{13}+\frac{222}{7}a^{12}-\frac{48}{7}a^{11}-\frac{95}{7}a^{10}-\frac{397}{7}a^{9}-\frac{365}{7}a^{8}-\frac{507}{7}a^{7}-\frac{247}{7}a^{6}-\frac{118}{7}a^{5}+\frac{283}{7}a^{4}+\frac{499}{7}a^{3}+\frac{769}{7}a^{2}+\frac{718}{7}a+\frac{377}{7}$, $\frac{66}{7}a^{24}+\frac{114}{7}a^{23}-\frac{85}{7}a^{22}-\frac{136}{7}a^{21}-\frac{72}{7}a^{20}+\frac{15}{7}a^{19}+\frac{227}{7}a^{18}+\frac{107}{7}a^{17}-\frac{64}{7}a^{16}-\frac{185}{7}a^{15}-\frac{261}{7}a^{14}+\frac{134}{7}a^{13}+\frac{227}{7}a^{12}+\frac{275}{7}a^{11}-\frac{1}{7}a^{10}-\frac{451}{7}a^{9}-\frac{233}{7}a^{8}-\frac{90}{7}a^{7}+\frac{514}{7}a^{6}+\frac{471}{7}a^{5}-\frac{64}{7}a^{4}-\frac{423}{7}a^{3}-\frac{765}{7}a^{2}+\frac{57}{7}a+\frac{362}{7}$
| 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: | \( 1770854208067528.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 1770854208067528.0 \cdot 1}{2\cdot\sqrt{108009270762295279786326375965420843490676304895049}}\cr\approx \mathstrut & 0.645075742820078 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 - 4*x - 5)
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^25 - 4*x - 5, 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^25 - 4*x - 5);
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^25 - 4*x - 5);
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 non-solvable group of order 15511210043330985984000000 |
| The 1958 conjugacy class representatives for $S_{25}$ are not computed |
| Character table for $S_{25}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 | $20{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.4.0.1}{4} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/23.6.0.1}{6} }$ | $24{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }^{2}{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $20{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(1663\)
| $\Q_{1663}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1663}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(6971\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
|
\(492251\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(268235921\)
| $\Q_{268235921}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
|
\(264590126291\)
| $\Q_{264590126291}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
|
\(266683732590664933\)
| $\Q_{266683732590664933}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{266683732590664933}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |