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Properties

Label 21.1.162...568.1
Degree $21$
Signature $[1, 10]$
Discriminant $1.624\times 10^{80}$
Root discriminant \(6600.15\)
Ramified primes $2,7,11,13,239$
Class number not computed
Class group not computed
Galois group $S_3\times F_7$ (as 21T15)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 343*x^19 - 1162*x^18 + 50421*x^17 + 341628*x^16 - 3539039*x^15 - 41773926*x^14 + 59992415*x^13 + 2617225520*x^12 + 8927122659*x^11 - 63601829382*x^10 - 500020233713*x^9 + 343600855164*x^8 + 14413854343811*x^7 + 58121867075622*x^6 + 69534025707132*x^5 - 474189524383064*x^4 - 3007387918123568*x^3 - 7993067522148464*x^2 - 4353580806135904*x - 862855657735952)
 
gp: K = bnfinit(y^21 - 343*y^19 - 1162*y^18 + 50421*y^17 + 341628*y^16 - 3539039*y^15 - 41773926*y^14 + 59992415*y^13 + 2617225520*y^12 + 8927122659*y^11 - 63601829382*y^10 - 500020233713*y^9 + 343600855164*y^8 + 14413854343811*y^7 + 58121867075622*y^6 + 69534025707132*y^5 - 474189524383064*y^4 - 3007387918123568*y^3 - 7993067522148464*y^2 - 4353580806135904*y - 862855657735952, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 343*x^19 - 1162*x^18 + 50421*x^17 + 341628*x^16 - 3539039*x^15 - 41773926*x^14 + 59992415*x^13 + 2617225520*x^12 + 8927122659*x^11 - 63601829382*x^10 - 500020233713*x^9 + 343600855164*x^8 + 14413854343811*x^7 + 58121867075622*x^6 + 69534025707132*x^5 - 474189524383064*x^4 - 3007387918123568*x^3 - 7993067522148464*x^2 - 4353580806135904*x - 862855657735952);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 343*x^19 - 1162*x^18 + 50421*x^17 + 341628*x^16 - 3539039*x^15 - 41773926*x^14 + 59992415*x^13 + 2617225520*x^12 + 8927122659*x^11 - 63601829382*x^10 - 500020233713*x^9 + 343600855164*x^8 + 14413854343811*x^7 + 58121867075622*x^6 + 69534025707132*x^5 - 474189524383064*x^4 - 3007387918123568*x^3 - 7993067522148464*x^2 - 4353580806135904*x - 862855657735952)
 

\( x^{21} - 343 x^{19} - 1162 x^{18} + 50421 x^{17} + 341628 x^{16} - 3539039 x^{15} + \cdots - 862855657735952 \) Copy content Toggle raw display

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(162\!\cdots\!568\) \(\medspace = 2^{33}\cdot 7^{15}\cdot 11^{19}\cdot 13^{19}\cdot 239^{7}\) 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:  \(6600.15\)
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:  $2^{27/14}7^{5/6}11^{13/14}13^{13/14}239^{1/2}\approx 29880.55290223505$
Ramified primes:   \(2\), \(7\), \(11\), \(13\), \(239\) 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:  \(\Q(\sqrt{478478}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{44}a^{8}-\frac{5}{22}a^{7}+\frac{1}{11}a^{6}+\frac{5}{22}a^{5}-\frac{7}{44}a^{4}-\frac{1}{11}a^{3}-\frac{1}{22}a^{2}+\frac{5}{11}a-\frac{2}{11}$, $\frac{1}{44}a^{9}-\frac{2}{11}a^{7}+\frac{3}{22}a^{6}+\frac{5}{44}a^{5}-\frac{2}{11}a^{4}+\frac{1}{22}a^{3}-\frac{1}{2}a^{2}+\frac{4}{11}a+\frac{2}{11}$, $\frac{1}{44}a^{10}-\frac{2}{11}a^{7}-\frac{7}{44}a^{6}+\frac{3}{22}a^{5}-\frac{5}{22}a^{4}-\frac{5}{22}a^{3}-\frac{2}{11}a-\frac{5}{11}$, $\frac{1}{44}a^{11}+\frac{1}{44}a^{7}-\frac{3}{22}a^{6}+\frac{1}{11}a^{5}-\frac{5}{22}a^{3}-\frac{1}{22}a^{2}+\frac{2}{11}a-\frac{5}{11}$, $\frac{1}{88}a^{12}-\frac{1}{88}a^{10}-\frac{1}{88}a^{8}-\frac{19}{88}a^{6}-\frac{1}{22}a^{5}-\frac{1}{11}a^{4}-\frac{7}{22}a^{3}+\frac{3}{22}a^{2}-\frac{1}{11}a-\frac{1}{11}$, $\frac{1}{88}a^{13}-\frac{1}{88}a^{11}-\frac{1}{88}a^{9}-\frac{19}{88}a^{7}-\frac{1}{22}a^{6}-\frac{1}{11}a^{5}+\frac{2}{11}a^{4}+\frac{3}{22}a^{3}+\frac{9}{22}a^{2}-\frac{1}{11}a$, $\frac{1}{12584}a^{14}-\frac{7}{1573}a^{13}+\frac{19}{12584}a^{12}-\frac{10}{1573}a^{11}+\frac{5}{968}a^{10}-\frac{23}{3146}a^{9}+\frac{59}{12584}a^{8}+\frac{25}{286}a^{7}+\frac{85}{6292}a^{6}+\frac{58}{1573}a^{5}-\frac{71}{484}a^{4}+\frac{685}{1573}a^{3}-\frac{537}{3146}a^{2}-\frac{149}{1573}a-\frac{504}{1573}$, $\frac{1}{25168}a^{15}-\frac{57}{12584}a^{13}+\frac{63}{12584}a^{12}-\frac{67}{6292}a^{11}-\frac{85}{12584}a^{10}+\frac{9}{1144}a^{9}+\frac{57}{12584}a^{8}+\frac{2139}{25168}a^{7}-\frac{199}{968}a^{6}+\frac{243}{6292}a^{5}+\frac{245}{1573}a^{4}+\frac{706}{1573}a^{3}+\frac{630}{1573}a^{2}-\frac{134}{1573}a+\frac{45}{1573}$, $\frac{1}{50336}a^{16}-\frac{1}{50336}a^{15}-\frac{1}{25168}a^{14}+\frac{5}{968}a^{13}+\frac{9}{25168}a^{12}-\frac{141}{25168}a^{11}+\frac{53}{12584}a^{10}+\frac{15}{1573}a^{9}+\frac{53}{50336}a^{8}+\frac{12355}{50336}a^{7}+\frac{3155}{25168}a^{6}+\frac{2861}{12584}a^{5}+\frac{133}{1573}a^{4}-\frac{20}{1573}a^{3}-\frac{178}{1573}a^{2}-\frac{443}{3146}a+\frac{1189}{3146}$, $\frac{1}{553696}a^{17}+\frac{1}{553696}a^{16}-\frac{1}{69212}a^{14}-\frac{289}{276848}a^{13}-\frac{117}{21296}a^{12}+\frac{491}{138424}a^{11}-\frac{1231}{138424}a^{10}-\frac{4223}{553696}a^{9}+\frac{201}{553696}a^{8}+\frac{23523}{138424}a^{7}+\frac{15219}{138424}a^{6}+\frac{2295}{69212}a^{5}+\frac{593}{17303}a^{4}-\frac{15531}{34606}a^{3}+\frac{5033}{17303}a^{2}-\frac{441}{2662}a+\frac{652}{17303}$, $\frac{1}{7751744}a^{18}+\frac{5}{7751744}a^{17}-\frac{51}{7751744}a^{16}-\frac{41}{7751744}a^{15}-\frac{1}{484484}a^{14}-\frac{1275}{553696}a^{13}-\frac{6543}{3875872}a^{12}-\frac{1493}{3875872}a^{11}-\frac{77247}{7751744}a^{10}-\frac{86167}{7751744}a^{9}-\frac{65671}{7751744}a^{8}-\frac{848205}{7751744}a^{7}+\frac{912031}{3875872}a^{6}-\frac{373193}{1937936}a^{5}+\frac{24021}{968968}a^{4}-\frac{56477}{242242}a^{3}-\frac{20689}{484484}a^{2}+\frac{148355}{484484}a+\frac{153661}{484484}$, $\frac{1}{100772672}a^{19}-\frac{3}{100772672}a^{18}-\frac{3}{14396096}a^{17}-\frac{179}{100772672}a^{16}-\frac{115}{12596584}a^{15}+\frac{37}{4580576}a^{14}-\frac{3083}{4580576}a^{13}+\frac{212341}{50386336}a^{12}+\frac{997649}{100772672}a^{11}-\frac{141815}{100772672}a^{10}-\frac{144007}{14396096}a^{9}-\frac{643255}{100772672}a^{8}+\frac{12566419}{50386336}a^{7}-\frac{359701}{25193168}a^{6}+\frac{1123351}{12596584}a^{5}+\frac{389497}{6298292}a^{4}-\frac{1487971}{6298292}a^{3}+\frac{209103}{484484}a^{2}+\frac{50537}{899756}a+\frac{676458}{1574573}$, $\frac{1}{28\!\cdots\!56}a^{20}+\frac{10\!\cdots\!69}{28\!\cdots\!56}a^{19}+\frac{41\!\cdots\!85}{70\!\cdots\!64}a^{18}+\frac{21\!\cdots\!39}{14\!\cdots\!28}a^{17}-\frac{23\!\cdots\!65}{28\!\cdots\!56}a^{16}-\frac{63\!\cdots\!91}{40\!\cdots\!08}a^{15}+\frac{68\!\cdots\!47}{20\!\cdots\!04}a^{14}+\frac{11\!\cdots\!21}{50\!\cdots\!76}a^{13}-\frac{63\!\cdots\!13}{28\!\cdots\!56}a^{12}+\frac{16\!\cdots\!59}{36\!\cdots\!28}a^{11}-\frac{33\!\cdots\!29}{88\!\cdots\!08}a^{10}-\frac{61\!\cdots\!33}{14\!\cdots\!28}a^{9}+\frac{13\!\cdots\!99}{40\!\cdots\!08}a^{8}-\frac{43\!\cdots\!23}{28\!\cdots\!56}a^{7}-\frac{47\!\cdots\!85}{20\!\cdots\!04}a^{6}-\frac{43\!\cdots\!75}{17\!\cdots\!04}a^{5}+\frac{31\!\cdots\!99}{35\!\cdots\!32}a^{4}-\frac{40\!\cdots\!35}{17\!\cdots\!16}a^{3}+\frac{16\!\cdots\!99}{63\!\cdots\!22}a^{2}-\frac{31\!\cdots\!11}{17\!\cdots\!16}a-\frac{82\!\cdots\!01}{17\!\cdots\!16}$ Copy content Toggle raw display

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

not computed

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:  $10$
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^21 - 343*x^19 - 1162*x^18 + 50421*x^17 + 341628*x^16 - 3539039*x^15 - 41773926*x^14 + 59992415*x^13 + 2617225520*x^12 + 8927122659*x^11 - 63601829382*x^10 - 500020233713*x^9 + 343600855164*x^8 + 14413854343811*x^7 + 58121867075622*x^6 + 69534025707132*x^5 - 474189524383064*x^4 - 3007387918123568*x^3 - 7993067522148464*x^2 - 4353580806135904*x - 862855657735952)
 
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^21 - 343*x^19 - 1162*x^18 + 50421*x^17 + 341628*x^16 - 3539039*x^15 - 41773926*x^14 + 59992415*x^13 + 2617225520*x^12 + 8927122659*x^11 - 63601829382*x^10 - 500020233713*x^9 + 343600855164*x^8 + 14413854343811*x^7 + 58121867075622*x^6 + 69534025707132*x^5 - 474189524383064*x^4 - 3007387918123568*x^3 - 7993067522148464*x^2 - 4353580806135904*x - 862855657735952, 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^21 - 343*x^19 - 1162*x^18 + 50421*x^17 + 341628*x^16 - 3539039*x^15 - 41773926*x^14 + 59992415*x^13 + 2617225520*x^12 + 8927122659*x^11 - 63601829382*x^10 - 500020233713*x^9 + 343600855164*x^8 + 14413854343811*x^7 + 58121867075622*x^6 + 69534025707132*x^5 - 474189524383064*x^4 - 3007387918123568*x^3 - 7993067522148464*x^2 - 4353580806135904*x - 862855657735952);
 
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^21 - 343*x^19 - 1162*x^18 + 50421*x^17 + 341628*x^16 - 3539039*x^15 - 41773926*x^14 + 59992415*x^13 + 2617225520*x^12 + 8927122659*x^11 - 63601829382*x^10 - 500020233713*x^9 + 343600855164*x^8 + 14413854343811*x^7 + 58121867075622*x^6 + 69534025707132*x^5 - 474189524383064*x^4 - 3007387918123568*x^3 - 7993067522148464*x^2 - 4353580806135904*x - 862855657735952);
 
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

$S_3\times F_7$ (as 21T15):

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 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$

Intermediate fields

3.1.273416.1, 7.1.9197851611565769152.1

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 42 siblings: 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 ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ R R R ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.3.0.1}{3} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ $21$ ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.2.0.1}{2} }^{10}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.7.0.1}{7} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.3.0.1}{3} }^{7}$ ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

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 2.7.6.1$x^{7} + 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.27.121$x^{14} + 6 x^{12} + 4 x^{9} + 4 x^{7} + 4 x^{4} + 6$$14$$1$$27$$(C_7:C_3) \times C_2$$[3]_{7}^{3}$
\(7\) Copy content Toggle raw display 7.3.0.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
7.18.15.5$x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$$6$$3$$15$$C_6 \times C_3$$[\ ]_{6}^{3}$
\(11\) Copy content Toggle raw display 11.7.6.1$x^{7} + 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
11.14.13.1$x^{14} + 22$$14$$1$$13$$(C_7:C_3) \times C_2$$[\ ]_{14}^{3}$
\(13\) Copy content Toggle raw display 13.7.6.1$x^{7} + 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
13.14.13.2$x^{14} + 26$$14$$1$$13$$D_{14}$$[\ ]_{14}^{2}$
\(239\) Copy content Toggle raw display $\Q_{239}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{239}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{239}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{239}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{239}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{239}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{239}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$