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Properties

Label 23.15.124...500.1
Degree $23$
Signature $[15, 4]$
Discriminant $1.240\times 10^{40}$
Root discriminant \(55.36\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{23}$ (as 23T7)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 2*x^22 - 20*x^21 + 36*x^20 + 175*x^19 - 273*x^18 - 876*x^17 + 1134*x^16 + 2749*x^15 - 2812*x^14 - 5571*x^13 + 4267*x^12 + 7255*x^11 - 3939*x^10 - 5887*x^9 + 2181*x^8 + 2825*x^7 - 734*x^6 - 749*x^5 + 154*x^4 + 99*x^3 - 19*x^2 - 5*x + 1)
 
gp: K = bnfinit(y^23 - 2*y^22 - 20*y^21 + 36*y^20 + 175*y^19 - 273*y^18 - 876*y^17 + 1134*y^16 + 2749*y^15 - 2812*y^14 - 5571*y^13 + 4267*y^12 + 7255*y^11 - 3939*y^10 - 5887*y^9 + 2181*y^8 + 2825*y^7 - 734*y^6 - 749*y^5 + 154*y^4 + 99*y^3 - 19*y^2 - 5*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 2*x^22 - 20*x^21 + 36*x^20 + 175*x^19 - 273*x^18 - 876*x^17 + 1134*x^16 + 2749*x^15 - 2812*x^14 - 5571*x^13 + 4267*x^12 + 7255*x^11 - 3939*x^10 - 5887*x^9 + 2181*x^8 + 2825*x^7 - 734*x^6 - 749*x^5 + 154*x^4 + 99*x^3 - 19*x^2 - 5*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 2*x^22 - 20*x^21 + 36*x^20 + 175*x^19 - 273*x^18 - 876*x^17 + 1134*x^16 + 2749*x^15 - 2812*x^14 - 5571*x^13 + 4267*x^12 + 7255*x^11 - 3939*x^10 - 5887*x^9 + 2181*x^8 + 2825*x^7 - 734*x^6 - 749*x^5 + 154*x^4 + 99*x^3 - 19*x^2 - 5*x + 1)
 

\( x^{23} - 2 x^{22} - 20 x^{21} + 36 x^{20} + 175 x^{19} - 273 x^{18} - 876 x^{17} + 1134 x^{16} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $23$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[15, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(12400700046148013345324171691662893757500\) \(\medspace = 2^{2}\cdot 5^{4}\cdot 7499\cdot 275729\cdot 49942556731\cdot 48034096151655703\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(55.36\)
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:  $2\cdot 5^{2/3}7499^{1/2}275729^{1/2}49942556731^{1/2}48034096151655703^{1/2}\approx 1.3024561044692664e+19$
Ramified primes:   \(2\), \(5\), \(7499\), \(275729\), \(49942556731\), \(48034096151655703\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  $\Q(\sqrt{49602\!\cdots\!57503}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Yes
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:  $18$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $a$, $3a^{22}-9a^{21}-53a^{20}+165a^{19}+399a^{18}-1288a^{17}-1671a^{16}+5585a^{15}+4257a^{14}-14720a^{13}-6755a^{12}+24257a^{11}+6514a^{10}-24795a^{9}-3451a^{8}+15114a^{7}+649a^{6}-5084a^{5}+201a^{4}+815a^{3}-91a^{2}-49a+8$, $8a^{22}-21a^{21}-147a^{20}+380a^{19}+1165a^{18}-2914a^{17}-5208a^{16}+12338a^{15}+14380a^{14}-31515a^{13}-25147a^{12}+49897a^{11}+27319a^{10}-48611a^{9}-17181a^{8}+28187a^{7}+5253a^{6}-9157a^{5}-354a^{4}+1458a^{3}-106a^{2}-87a+14$, $a^{22}-2a^{21}-20a^{20}+36a^{19}+175a^{18}-273a^{17}-876a^{16}+1134a^{15}+2749a^{14}-2812a^{13}-5571a^{12}+4267a^{11}+7255a^{10}-3939a^{9}-5887a^{8}+2181a^{7}+2825a^{6}-734a^{5}-748a^{4}+153a^{3}+95a^{2}-15a-3$, $3a^{22}-5a^{21}-62a^{20}+88a^{19}+561a^{18}-644a^{17}-2901a^{16}+2526a^{15}+9381a^{14}-5687a^{13}-19525a^{12}+7230a^{11}+26032a^{10}-4562a^{9}-21600a^{8}+656a^{7}+10656a^{6}+622a^{5}-2980a^{4}-281a^{3}+447a^{2}+33a-29$, $9a^{22}-26a^{21}-160a^{20}+474a^{19}+1212a^{18}-3675a^{17}-5092a^{16}+15809a^{15}+12884a^{14}-41303a^{13}-19758a^{12}+67496a^{11}+17023a^{10}-68662a^{9}-5735a^{8}+42112a^{7}-2177a^{6}-14624a^{5}+2356a^{4}+2526a^{3}-606a^{2}-162a+46$, $a^{22}-2a^{21}-20a^{20}+36a^{19}+175a^{18}-273a^{17}-876a^{16}+1134a^{15}+2749a^{14}-2812a^{13}-5571a^{12}+4267a^{11}+7255a^{10}-3939a^{9}-5887a^{8}+2181a^{7}+2825a^{6}-734a^{5}-749a^{4}+153a^{3}+99a^{2}-15a-4$, $15a^{22}-34a^{21}-290a^{20}+614a^{19}+2445a^{18}-4684a^{17}-11769a^{16}+19639a^{15}+35476a^{14}-49343a^{13}-68970a^{12}+76086a^{11}+85819a^{10}-71148a^{9}-65729a^{8}+38838a^{7}+28807a^{6}-11689a^{5}-6436a^{4}+1745a^{3}+627a^{2}-98a-20$, $11a^{22}-10a^{21}-243a^{20}+156a^{19}+2333a^{18}-907a^{17}-12665a^{16}+2038a^{15}+42427a^{14}+1455a^{13}-90038a^{12}-17353a^{11}+119998a^{10}+37357a^{9}-96830a^{8}-37288a^{7}+44567a^{6}+17971a^{5}-10988a^{4}-3822a^{3}+1368a^{2}+275a-75$, $4a^{21}-9a^{20}-77a^{19}+162a^{18}+644a^{17}-1230a^{16}-3059a^{15}+5124a^{14}+9033a^{13}-12770a^{12}-17027a^{11}+19518a^{10}+20232a^{9}-18147a^{8}-14449a^{7}+9991a^{6}+5678a^{5}-3140a^{4}-1062a^{3}+501a^{2}+70a-30$, $16a^{22}-43a^{21}-292a^{20}+781a^{19}+2292a^{18}-6022a^{17}-10108a^{16}+25706a^{15}+27371a^{14}-66471a^{13}-46479a^{12}+107238a^{11}+48061a^{10}-107577a^{9}-27266a^{8}+65289a^{7}+5857a^{6}-22703a^{5}+1024a^{4}+3967a^{3}-532a^{2}-253a+47$, $2a^{22}+4a^{21}-56a^{20}-86a^{19}+634a^{18}+815a^{17}-3866a^{16}-4409a^{15}+14058a^{14}+14773a^{13}-31611a^{12}-31271a^{11}+43943a^{10}+41144a^{9}-36801a^{8}-32091a^{7}+17897a^{6}+13701a^{5}-4998a^{4}-2866a^{3}+775a^{2}+226a-54$, $2a^{22}-3a^{21}-42a^{20}+52a^{19}+386a^{18}-371a^{17}-2025a^{16}+1392a^{15}+6632a^{14}-2875a^{13}-13954a^{12}+2963a^{11}+18777a^{10}-623a^{9}-15712a^{8}-1525a^{7}+7821a^{6}+1355a^{5}-2199a^{4}-430a^{3}+312a^{2}+45a-17$, $a^{22}+3a^{21}-31a^{20}-61a^{19}+372a^{18}+548a^{17}-2362a^{16}-2831a^{15}+8884a^{14}+9150a^{13}-20661a^{12}-18878a^{11}+29859a^{10}+24418a^{9}-26243a^{8}-18798a^{7}+13485a^{6}+7852a^{5}-3882a^{4}-1530a^{3}+574a^{2}+101a-33$, $14a^{22}-22a^{21}-290a^{20}+381a^{19}+2624a^{18}-2720a^{17}-13517a^{16}+10250a^{15}+43284a^{14}-21496a^{13}-88386a^{12}+23511a^{11}+113877a^{10}-8827a^{9}-88940a^{8}-5282a^{7}+39389a^{6}+5318a^{5}-9124a^{4}-1336a^{3}+995a^{2}+96a-40$, $16a^{22}-29a^{21}-323a^{20}+510a^{19}+2849a^{18}-3737a^{17}-14329a^{16}+14731a^{15}+44915a^{14}-33585a^{13}-90059a^{12}+44060a^{11}+114281a^{10}-30611a^{9}-88071a^{8}+8440a^{7}+38385a^{6}+682a^{5}-8599a^{4}-657a^{3}+876a^{2}+69a-33$, $28a^{22}-46a^{21}-578a^{20}+805a^{19}+5218a^{18}-5842a^{17}-26872a^{16}+22621a^{15}+86287a^{14}-49853a^{13}-177478a^{12}+60877a^{11}+231882a^{10}-34699a^{9}-185583a^{8}+1336a^{7}+85545a^{6}+6570a^{5}-20961a^{4}-2184a^{3}+2425a^{2}+191a-100$, $20a^{21}-40a^{20}-394a^{19}+708a^{18}+3382a^{17}-5248a^{16}-16509a^{15}+21112a^{14}+50055a^{13}-49948a^{12}-96522a^{11}+70497a^{10}+116449a^{9}-57902a^{8}-83285a^{7}+26549a^{6}+31955a^{5}-6909a^{4}-5611a^{3}+1081a^{2}+348a-71$ Copy content Toggle raw display (assuming GRH)
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:  \( 2323935089500 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{4}\cdot 2323935089500 \cdot 1}{2\cdot\sqrt{12400700046148013345324171691662893757500}}\cr\approx \mathstrut & 0.532893355106987 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^23 - 2*x^22 - 20*x^21 + 36*x^20 + 175*x^19 - 273*x^18 - 876*x^17 + 1134*x^16 + 2749*x^15 - 2812*x^14 - 5571*x^13 + 4267*x^12 + 7255*x^11 - 3939*x^10 - 5887*x^9 + 2181*x^8 + 2825*x^7 - 734*x^6 - 749*x^5 + 154*x^4 + 99*x^3 - 19*x^2 - 5*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^23 - 2*x^22 - 20*x^21 + 36*x^20 + 175*x^19 - 273*x^18 - 876*x^17 + 1134*x^16 + 2749*x^15 - 2812*x^14 - 5571*x^13 + 4267*x^12 + 7255*x^11 - 3939*x^10 - 5887*x^9 + 2181*x^8 + 2825*x^7 - 734*x^6 - 749*x^5 + 154*x^4 + 99*x^3 - 19*x^2 - 5*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^23 - 2*x^22 - 20*x^21 + 36*x^20 + 175*x^19 - 273*x^18 - 876*x^17 + 1134*x^16 + 2749*x^15 - 2812*x^14 - 5571*x^13 + 4267*x^12 + 7255*x^11 - 3939*x^10 - 5887*x^9 + 2181*x^8 + 2825*x^7 - 734*x^6 - 749*x^5 + 154*x^4 + 99*x^3 - 19*x^2 - 5*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^23 - 2*x^22 - 20*x^21 + 36*x^20 + 175*x^19 - 273*x^18 - 876*x^17 + 1134*x^16 + 2749*x^15 - 2812*x^14 - 5571*x^13 + 4267*x^12 + 7255*x^11 - 3939*x^10 - 5887*x^9 + 2181*x^8 + 2825*x^7 - 734*x^6 - 749*x^5 + 154*x^4 + 99*x^3 - 19*x^2 - 5*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

$S_{23}$ (as 23T7):

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 25852016738884976640000
The 1255 conjugacy class representatives for $S_{23}$ are not computed
Character table for $S_{23}$ 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]
 

Sibling fields

Degree 46 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 $17{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ R ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ $22{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ $17{,}\,{\href{/padicField/19.6.0.1}{6} }$ ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ $18{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ $18{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ $15{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ $15{,}\,{\href{/padicField/47.8.0.1}{8} }$ ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
2.9.0.1$x^{9} + x^{4} + 1$$1$$9$$0$$C_9$$[\ ]^{9}$
2.12.0.1$x^{12} + x^{7} + x^{6} + x^{5} + x^{3} + x + 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.5.0.1$x^{5} + 4 x + 3$$1$$5$$0$$C_5$$[\ ]^{5}$
5.6.4.2$x^{6} + 10 x^{3} - 25$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.11.0.1$x^{11} + 3 x + 3$$1$$11$$0$$C_{11}$$[\ ]^{11}$
\(7499\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $21$$1$$21$$0$$C_{21}$$[\ ]^{21}$
\(275729\) Copy content Toggle raw display $\Q_{275729}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{275729}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $11$$1$$11$$0$$C_{11}$$[\ ]^{11}$
\(49942556731\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $16$$1$$16$$0$$C_{16}$$[\ ]^{16}$
\(48034096151655703\) Copy content Toggle raw display $\Q_{48034096151655703}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $11$$1$$11$$0$$C_{11}$$[\ ]^{11}$