Properties

Label 19.1.495...343.1
Degree $19$
Signature $[1, 9]$
Discriminant $-4.958\times 10^{28}$
Root discriminant \(32.38\)
Ramified prime $1543$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{19}$ (as 19T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 9*x^18 + 24*x^17 - 6*x^16 - 42*x^15 + 4*x^14 + 34*x^13 + 74*x^12 + 46*x^11 - 238*x^10 - 49*x^9 + 181*x^8 + 195*x^7 + 116*x^6 - 193*x^5 - 275*x^4 + 106*x^3 - 8*x^2 + 121*x - 1)
 
gp: K = bnfinit(y^19 - 9*y^18 + 24*y^17 - 6*y^16 - 42*y^15 + 4*y^14 + 34*y^13 + 74*y^12 + 46*y^11 - 238*y^10 - 49*y^9 + 181*y^8 + 195*y^7 + 116*y^6 - 193*y^5 - 275*y^4 + 106*y^3 - 8*y^2 + 121*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 9*x^18 + 24*x^17 - 6*x^16 - 42*x^15 + 4*x^14 + 34*x^13 + 74*x^12 + 46*x^11 - 238*x^10 - 49*x^9 + 181*x^8 + 195*x^7 + 116*x^6 - 193*x^5 - 275*x^4 + 106*x^3 - 8*x^2 + 121*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 - 9*x^18 + 24*x^17 - 6*x^16 - 42*x^15 + 4*x^14 + 34*x^13 + 74*x^12 + 46*x^11 - 238*x^10 - 49*x^9 + 181*x^8 + 195*x^7 + 116*x^6 - 193*x^5 - 275*x^4 + 106*x^3 - 8*x^2 + 121*x - 1)
 

\( x^{19} - 9 x^{18} + 24 x^{17} - 6 x^{16} - 42 x^{15} + 4 x^{14} + 34 x^{13} + 74 x^{12} + 46 x^{11} - 238 x^{10} - 49 x^{9} + 181 x^{8} + 195 x^{7} + 116 x^{6} - 193 x^{5} - 275 x^{4} + 106 x^{3} + \cdots - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $19$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-49578494467761916312526550343\) \(\medspace = -\,1543^{9}\) 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:  \(32.38\)
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:  $1543^{1/2}\approx 39.28103868280471$
Ramified primes:   \(1543\) 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:  \(\Q(\sqrt{-1543}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{13}-\frac{1}{9}a^{12}-\frac{1}{9}a^{10}-\frac{1}{9}a^{9}+\frac{1}{9}a^{8}+\frac{2}{9}a^{6}+\frac{2}{9}a^{5}-\frac{2}{9}a^{4}-\frac{2}{9}a^{2}-\frac{2}{9}a+\frac{2}{9}$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}-\frac{1}{9}a^{11}-\frac{1}{9}a^{9}-\frac{1}{9}a^{8}+\frac{2}{9}a^{7}+\frac{2}{9}a^{5}+\frac{2}{9}a^{4}-\frac{2}{9}a^{3}-\frac{2}{9}a-\frac{2}{9}$, $\frac{1}{855}a^{16}-\frac{11}{285}a^{15}+\frac{41}{855}a^{14}+\frac{89}{855}a^{13}+\frac{14}{171}a^{12}+\frac{11}{285}a^{11}+\frac{103}{855}a^{10}+\frac{82}{855}a^{9}-\frac{4}{57}a^{8}-\frac{67}{285}a^{7}-\frac{17}{855}a^{6}-\frac{371}{855}a^{5}+\frac{73}{171}a^{4}-\frac{29}{285}a^{3}+\frac{251}{855}a^{2}-\frac{169}{855}a-\frac{61}{855}$, $\frac{1}{2565}a^{17}-\frac{98}{2565}a^{15}-\frac{26}{855}a^{14}-\frac{413}{2565}a^{13}-\frac{32}{2565}a^{12}-\frac{43}{2565}a^{11}-\frac{43}{855}a^{10}+\frac{122}{855}a^{9}+\frac{194}{2565}a^{8}-\frac{5}{27}a^{7}-\frac{184}{855}a^{6}+\frac{92}{2565}a^{5}-\frac{487}{2565}a^{4}+\frac{179}{513}a^{3}+\frac{298}{855}a^{2}+\frac{917}{2565}a+\frac{1027}{2565}$, $\frac{1}{4772442701295}a^{18}+\frac{662887919}{4772442701295}a^{17}+\frac{1752360862}{4772442701295}a^{16}+\frac{33487901806}{954488540259}a^{15}+\frac{13895980616}{954488540259}a^{14}+\frac{58546782047}{1590814233765}a^{13}+\frac{736040120899}{4772442701295}a^{12}-\frac{723266576861}{4772442701295}a^{11}-\frac{43584455216}{318162846753}a^{10}+\frac{474576464078}{4772442701295}a^{9}+\frac{935269937}{38800347165}a^{8}+\frac{647391070363}{4772442701295}a^{7}-\frac{1487278745026}{4772442701295}a^{6}-\frac{660860588378}{1590814233765}a^{5}-\frac{1990918487548}{4772442701295}a^{4}+\frac{326430003014}{4772442701295}a^{3}-\frac{26896025713}{60410667105}a^{2}+\frac{185213667625}{954488540259}a-\frac{51519525688}{251181194805}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

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:  $9$
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:   $\frac{17959875136}{530271411255}a^{18}-\frac{170742520601}{530271411255}a^{17}+\frac{168526829131}{176757137085}a^{16}-\frac{249098257613}{530271411255}a^{15}-\frac{107705448476}{58919045695}a^{14}+\frac{141861836681}{106054282251}a^{13}+\frac{1026407780539}{530271411255}a^{12}+\frac{196483523537}{530271411255}a^{11}-\frac{213101128832}{530271411255}a^{10}-\frac{256272399431}{35351427417}a^{9}+\frac{32404310506}{12933449055}a^{8}+\frac{4995260337812}{530271411255}a^{7}-\frac{8235336552}{58919045695}a^{6}-\frac{419460757742}{106054282251}a^{5}-\frac{1360146353143}{530271411255}a^{4}-\frac{1658717815748}{530271411255}a^{3}+\frac{1831912804}{353278755}a^{2}-\frac{623739678769}{530271411255}a+\frac{507780115297}{530271411255}$, $\frac{13156927406}{4772442701295}a^{18}-\frac{29591640715}{954488540259}a^{17}+\frac{109483527277}{954488540259}a^{16}-\frac{461920476377}{4772442701295}a^{15}-\frac{258869780084}{954488540259}a^{14}+\frac{4827003128}{16745412987}a^{13}+\frac{3067488956732}{4772442701295}a^{12}-\frac{277717320521}{954488540259}a^{11}-\frac{258386622394}{318162846753}a^{10}-\frac{1551479184340}{954488540259}a^{9}+\frac{11331086483}{7760069433}a^{8}+\frac{3106736399332}{954488540259}a^{7}+\frac{1419688731886}{4772442701295}a^{6}-\frac{1024009356113}{318162846753}a^{5}-\frac{2603687341768}{954488540259}a^{4}-\frac{3058554546107}{4772442701295}a^{3}+\frac{2342384036}{635901759}a^{2}+\frac{991610363252}{954488540259}a+\frac{1824113337292}{4772442701295}$, $\frac{42669179771}{4772442701295}a^{18}-\frac{334653386162}{4772442701295}a^{17}+\frac{596805388544}{4772442701295}a^{16}+\frac{848855610022}{4772442701295}a^{15}-\frac{25025270768}{50236238961}a^{14}+\frac{3735872006}{83727064935}a^{13}-\frac{1561202891824}{4772442701295}a^{12}+\frac{295109856557}{251181194805}a^{11}+\frac{330932984561}{318162846753}a^{10}-\frac{5562234879449}{4772442701295}a^{9}-\frac{84517697401}{38800347165}a^{8}+\frac{5383654096436}{4772442701295}a^{7}+\frac{7442914958137}{4772442701295}a^{6}+\frac{7209378381409}{1590814233765}a^{5}-\frac{4074757663706}{4772442701295}a^{4}-\frac{1972104993328}{954488540259}a^{3}+\frac{286539964}{60410667105}a^{2}+\frac{472190295728}{954488540259}a+\frac{3527831449339}{4772442701295}$, $\frac{179374741}{110987039565}a^{18}-\frac{3701085019}{110987039565}a^{17}+\frac{25590312451}{110987039565}a^{16}-\frac{71663645878}{110987039565}a^{15}+\frac{44860567228}{110987039565}a^{14}+\frac{47022721987}{36995679855}a^{13}-\frac{35277665647}{22197407913}a^{12}-\frac{24021131401}{22197407913}a^{11}+\frac{41117508368}{36995679855}a^{10}+\frac{781793777}{5841423135}a^{9}+\frac{3609172273}{902333655}a^{8}-\frac{292017712856}{110987039565}a^{7}-\frac{36868288612}{5841423135}a^{6}+\frac{162142107617}{36995679855}a^{5}+\frac{409488479663}{110987039565}a^{4}-\frac{99131109319}{110987039565}a^{3}+\frac{809404279}{280979847}a^{2}-\frac{180538537792}{110987039565}a+\frac{108776696858}{110987039565}$, $\frac{1213365566}{78236765595}a^{18}-\frac{11519989592}{78236765595}a^{17}+\frac{34715751704}{78236765595}a^{16}-\frac{23264664908}{78236765595}a^{15}-\frac{9002950439}{15647353119}a^{14}+\frac{13450168954}{26078921865}a^{13}+\frac{5510932091}{78236765595}a^{12}+\frac{95301531908}{78236765595}a^{11}-\frac{344972290}{5215784373}a^{10}-\frac{252866301329}{78236765595}a^{9}+\frac{394527154}{636071265}a^{8}+\frac{221715247346}{78236765595}a^{7}+\frac{35067046522}{78236765595}a^{6}+\frac{50389294454}{26078921865}a^{5}-\frac{316847425331}{78236765595}a^{4}-\frac{13898712415}{15647353119}a^{3}+\frac{1318508029}{990338805}a^{2}-\frac{6794209435}{15647353119}a+\frac{75077899054}{78236765595}$, $\frac{21587547676}{4772442701295}a^{18}-\frac{204093079054}{4772442701295}a^{17}+\frac{125740279940}{954488540259}a^{16}-\frac{123915700130}{954488540259}a^{15}+\frac{133182683302}{4772442701295}a^{14}-\frac{58359328667}{530271411255}a^{13}-\frac{15377793235}{50236238961}a^{12}+\frac{6248593061482}{4772442701295}a^{11}-\frac{321221377918}{1590814233765}a^{10}-\frac{6071394919324}{4772442701295}a^{9}-\frac{1926935149}{12933449055}a^{8}-\frac{792817402987}{954488540259}a^{7}+\frac{14579637031844}{4772442701295}a^{6}+\frac{241896317311}{176757137085}a^{5}-\frac{14963876953072}{4772442701295}a^{4}-\frac{6374647569502}{4772442701295}a^{3}-\frac{52486314469}{60410667105}a^{2}+\frac{1224540768557}{4772442701295}a+\frac{3227493096854}{4772442701295}$, $\frac{47936257897}{4772442701295}a^{18}-\frac{448044733822}{4772442701295}a^{17}+\frac{1202302108546}{4772442701295}a^{16}+\frac{294510683534}{4772442701295}a^{15}-\frac{5068119670088}{4772442701295}a^{14}+\frac{186958326428}{318162846753}a^{13}+\frac{9100301884528}{4772442701295}a^{12}-\frac{6106874069861}{4772442701295}a^{11}-\frac{2721805230883}{1590814233765}a^{10}-\frac{82914318008}{954488540259}a^{9}+\frac{53670820609}{38800347165}a^{8}+\frac{819778705741}{251181194805}a^{7}-\frac{292468243294}{251181194805}a^{6}-\frac{1305481878095}{318162846753}a^{5}+\frac{3873241031444}{4772442701295}a^{4}+\frac{6368708899649}{4772442701295}a^{3}+\frac{60637748177}{60410667105}a^{2}+\frac{3555906193187}{4772442701295}a-\frac{3267898875646}{4772442701295}$, $\frac{36486838}{60410667105}a^{18}-\frac{904109332}{60410667105}a^{17}+\frac{5679276124}{60410667105}a^{16}-\frac{10611709492}{60410667105}a^{15}-\frac{1560990148}{12082133421}a^{14}+\frac{3401553518}{6712296345}a^{13}+\frac{1864371938}{12082133421}a^{12}-\frac{32509293452}{60410667105}a^{11}-\frac{1785516857}{4027377807}a^{10}-\frac{41513809339}{60410667105}a^{9}+\frac{265006868}{163714545}a^{8}+\frac{79548617746}{60410667105}a^{7}-\frac{125138019436}{60410667105}a^{6}-\frac{13972983007}{6712296345}a^{5}-\frac{26084257816}{60410667105}a^{4}+\frac{7178994551}{60410667105}a^{3}+\frac{136723395311}{60410667105}a^{2}-\frac{6787950947}{12082133421}a-\frac{6261296407}{60410667105}$, $a$ 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:  \( 17317158.8896 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 17317158.8896 \cdot 1}{2\cdot\sqrt{49578494467761916312526550343}}\cr\approx \mathstrut & 1.18699472696 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^19 - 9*x^18 + 24*x^17 - 6*x^16 - 42*x^15 + 4*x^14 + 34*x^13 + 74*x^12 + 46*x^11 - 238*x^10 - 49*x^9 + 181*x^8 + 195*x^7 + 116*x^6 - 193*x^5 - 275*x^4 + 106*x^3 - 8*x^2 + 121*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^19 - 9*x^18 + 24*x^17 - 6*x^16 - 42*x^15 + 4*x^14 + 34*x^13 + 74*x^12 + 46*x^11 - 238*x^10 - 49*x^9 + 181*x^8 + 195*x^7 + 116*x^6 - 193*x^5 - 275*x^4 + 106*x^3 - 8*x^2 + 121*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^19 - 9*x^18 + 24*x^17 - 6*x^16 - 42*x^15 + 4*x^14 + 34*x^13 + 74*x^12 + 46*x^11 - 238*x^10 - 49*x^9 + 181*x^8 + 195*x^7 + 116*x^6 - 193*x^5 - 275*x^4 + 106*x^3 - 8*x^2 + 121*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^19 - 9*x^18 + 24*x^17 - 6*x^16 - 42*x^15 + 4*x^14 + 34*x^13 + 74*x^12 + 46*x^11 - 238*x^10 - 49*x^9 + 181*x^8 + 195*x^7 + 116*x^6 - 193*x^5 - 275*x^4 + 106*x^3 - 8*x^2 + 121*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

$D_{19}$ (as 19T2):

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 38
The 11 conjugacy class representatives for $D_{19}$
Character table for $D_{19}$

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

Galois closure: 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 $19$ ${\href{/padicField/3.2.0.1}{2} }^{9}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.2.0.1}{2} }^{9}{,}\,{\href{/padicField/5.1.0.1}{1} }$ $19$ ${\href{/padicField/11.2.0.1}{2} }^{9}{,}\,{\href{/padicField/11.1.0.1}{1} }$ $19$ $19$ ${\href{/padicField/19.2.0.1}{2} }^{9}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.2.0.1}{2} }^{9}{,}\,{\href{/padicField/23.1.0.1}{1} }$ $19$ $19$ $19$ ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.2.0.1}{2} }^{9}{,}\,{\href{/padicField/43.1.0.1}{1} }$ $19$ $19$ ${\href{/padicField/59.2.0.1}{2} }^{9}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(1543\) Copy content Toggle raw display $\Q_{1543}$$x$$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$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$