↑

Properties

Label 32.0.50636421157...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 11^{16}$
Root discriminant $38.42$
Ramified primes $2, 3, 5, 11$
Class number Not computed
Class group Not computed
Galois group $C_2^3\times C_4$ (as 32T34)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43046721, 0, -23914845, 0, 4782969, 0, -590490, 0, 328050, 0, -411885, 0, 141426, 0, -10170, 0, -911, 0, -1130, 0, 1746, 0, -565, 0, 50, 0, -10, 0, 9, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 5*x^30 + 9*x^28 - 10*x^26 + 50*x^24 - 565*x^22 + 1746*x^20 - 1130*x^18 - 911*x^16 - 10170*x^14 + 141426*x^12 - 411885*x^10 + 328050*x^8 - 590490*x^6 + 4782969*x^4 - 23914845*x^2 + 43046721)
 
gp: K = bnfinit(x^32 - 5*x^30 + 9*x^28 - 10*x^26 + 50*x^24 - 565*x^22 + 1746*x^20 - 1130*x^18 - 911*x^16 - 10170*x^14 + 141426*x^12 - 411885*x^10 + 328050*x^8 - 590490*x^6 + 4782969*x^4 - 23914845*x^2 + 43046721, 1)
 

Normalized defining polynomial

\( x^{32} - 5 x^{30} + 9 x^{28} - 10 x^{26} + 50 x^{24} - 565 x^{22} + 1746 x^{20} - 1130 x^{18} - 911 x^{16} - 10170 x^{14} + 141426 x^{12} - 411885 x^{10} + 328050 x^{8} - 590490 x^{6} + 4782969 x^{4} - 23914845 x^{2} + 43046721 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(506364211574415072751124736000000000000000000000000=2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.42$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(660=2^{2}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(131,·)$, $\chi_{660}(133,·)$, $\chi_{660}(263,·)$, $\chi_{660}(397,·)$, $\chi_{660}(527,·)$, $\chi_{660}(529,·)$, $\chi_{660}(659,·)$, $\chi_{660}(23,·)$, $\chi_{660}(287,·)$, $\chi_{660}(419,·)$, $\chi_{660}(551,·)$, $\chi_{660}(43,·)$, $\chi_{660}(307,·)$, $\chi_{660}(439,·)$, $\chi_{660}(571,·)$, $\chi_{660}(67,·)$, $\chi_{660}(197,·)$, $\chi_{660}(199,·)$, $\chi_{660}(329,·)$, $\chi_{660}(331,·)$, $\chi_{660}(461,·)$, $\chi_{660}(463,·)$, $\chi_{660}(593,·)$, $\chi_{660}(89,·)$, $\chi_{660}(221,·)$, $\chi_{660}(353,·)$, $\chi_{660}(617,·)$, $\chi_{660}(109,·)$, $\chi_{660}(241,·)$, $\chi_{660}(373,·)$, $\chi_{660}(637,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{15} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{18} + \frac{4}{9} a^{16} - \frac{1}{9} a^{12} - \frac{4}{9} a^{10} + \frac{2}{9} a^{8} + \frac{4}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{19} + \frac{4}{27} a^{17} - \frac{1}{3} a^{15} - \frac{10}{27} a^{13} - \frac{13}{27} a^{11} - \frac{7}{27} a^{9} + \frac{1}{3} a^{7} + \frac{4}{27} a^{5} - \frac{11}{27} a^{3} - \frac{1}{3} a$, $\frac{1}{2511} a^{20} + \frac{1}{81} a^{18} + \frac{2}{9} a^{16} - \frac{16}{81} a^{14} - \frac{10}{81} a^{12} + \frac{1235}{2511} a^{10} + \frac{1}{9} a^{8} - \frac{26}{81} a^{6} + \frac{4}{81} a^{4} - \frac{1}{9} a^{2} - \frac{15}{31}$, $\frac{1}{7533} a^{21} + \frac{1}{243} a^{19} + \frac{2}{27} a^{17} + \frac{65}{243} a^{15} - \frac{10}{243} a^{13} + \frac{1235}{7533} a^{11} + \frac{10}{27} a^{9} + \frac{55}{243} a^{7} + \frac{85}{243} a^{5} + \frac{8}{27} a^{3} + \frac{16}{93} a$, $\frac{1}{22599} a^{22} + \frac{4}{22599} a^{20} - \frac{1}{81} a^{18} + \frac{65}{729} a^{16} - \frac{307}{729} a^{14} + \frac{9605}{22599} a^{12} + \frac{790}{2511} a^{10} + \frac{55}{729} a^{8} + \frac{301}{729} a^{6} - \frac{4}{81} a^{4} + \frac{109}{279} a^{2} + \frac{14}{31}$, $\frac{1}{67797} a^{23} + \frac{4}{67797} a^{21} - \frac{1}{243} a^{19} + \frac{65}{2187} a^{17} - \frac{1036}{2187} a^{15} - \frac{12994}{67797} a^{13} + \frac{3301}{7533} a^{11} + \frac{784}{2187} a^{9} + \frac{1030}{2187} a^{7} - \frac{85}{243} a^{5} + \frac{388}{837} a^{3} + \frac{15}{31} a$, $\frac{1}{3254256} a^{24} - \frac{2}{203391} a^{22} + \frac{1}{22599} a^{20} - \frac{4147}{104976} a^{18} - \frac{1669}{6561} a^{16} + \frac{96245}{203391} a^{14} + \frac{121121}{361584} a^{12} + \frac{81547}{203391} a^{10} + \frac{1672}{6561} a^{8} + \frac{1933}{11664} a^{6} + \frac{83}{837} a^{4} - \frac{23}{93} a^{2} - \frac{223}{496}$, $\frac{1}{9762768} a^{25} - \frac{2}{610173} a^{23} + \frac{1}{67797} a^{21} - \frac{4147}{314928} a^{19} - \frac{1669}{19683} a^{17} + \frac{299636}{610173} a^{15} - \frac{240463}{1084752} a^{13} + \frac{81547}{610173} a^{11} + \frac{1672}{19683} a^{9} - \frac{9731}{34992} a^{7} - \frac{754}{2511} a^{5} - \frac{116}{279} a^{3} - \frac{719}{1488} a$, $\frac{1}{304920532944} a^{26} - \frac{9383}{304920532944} a^{24} - \frac{253}{235278189} a^{22} + \frac{1789523}{304920532944} a^{20} - \frac{423165835}{9836146224} a^{18} + \frac{4032593450}{19057533309} a^{16} - \frac{1768513607}{3764451024} a^{14} - \frac{47548964975}{304920532944} a^{12} + \frac{4937470084}{19057533309} a^{10} + \frac{9999317}{121433904} a^{8} + \frac{961445827}{3764451024} a^{6} + \frac{10586818}{26142021} a^{4} - \frac{2109925}{15491568} a^{2} + \frac{198913}{5163856}$, $\frac{1}{914761598832} a^{27} - \frac{9383}{914761598832} a^{25} - \frac{253}{705834567} a^{23} + \frac{1789523}{914761598832} a^{21} - \frac{423165835}{29508438672} a^{19} + \frac{4032593450}{57172599927} a^{17} - \frac{5532964631}{11293353072} a^{15} - \frac{47548964975}{914761598832} a^{13} + \frac{4937470084}{57172599927} a^{11} + \frac{9999317}{364301712} a^{9} - \frac{2803005197}{11293353072} a^{7} - \frac{15555203}{78426063} a^{5} - \frac{2109925}{46474704} a^{3} + \frac{5362769}{15491568} a$, $\frac{1}{2744284796496} a^{28} + \frac{1}{686071199124} a^{26} + \frac{7747}{152460266472} a^{24} + \frac{40708403}{2744284796496} a^{22} - \frac{1659463}{22131329004} a^{20} + \frac{73428679369}{1372142398248} a^{18} + \frac{25715117473}{304920532944} a^{16} + \frac{10100215297}{686071199124} a^{14} - \frac{220294413469}{1372142398248} a^{12} + \frac{4303727933}{9836146224} a^{10} - \frac{2531248421}{8470014804} a^{8} + \frac{928864601}{1882225512} a^{6} + \frac{43914283}{139424112} a^{4} - \frac{2419447}{11618676} a^{2} + \frac{101}{83288}$, $\frac{1}{8232854389488} a^{29} + \frac{1}{2058213597372} a^{27} + \frac{7747}{457380799416} a^{25} + \frac{40708403}{8232854389488} a^{23} - \frac{1659463}{66393987012} a^{21} + \frac{73428679369}{4116427194744} a^{19} + \frac{25715117473}{914761598832} a^{17} + \frac{10100215297}{2058213597372} a^{15} - \frac{1592436811717}{4116427194744} a^{13} + \frac{4303727933}{29508438672} a^{11} - \frac{2531248421}{25410044412} a^{9} + \frac{2811090113}{5646676536} a^{7} + \frac{43914283}{418272336} a^{5} - \frac{14038123}{34856028} a^{3} + \frac{101}{249864} a$, $\frac{1}{24698563168464} a^{30} + \frac{1}{6174640792116} a^{28} - \frac{1}{686071199124} a^{26} - \frac{1484173}{24698563168464} a^{24} + \frac{58463603}{6174640792116} a^{22} - \frac{550403959}{6174640792116} a^{20} + \frac{4740163973}{94630510224} a^{18} + \frac{2841730302313}{6174640792116} a^{16} + \frac{2258696642215}{6174640792116} a^{14} - \frac{189386401741}{2744284796496} a^{12} + \frac{24882265963}{76230133236} a^{10} + \frac{338306375}{2823338268} a^{8} + \frac{397013297}{3764451024} a^{6} + \frac{23123105}{104568084} a^{4} + \frac{4117319}{11618676} a^{2} + \frac{58169}{322741}$, $\frac{1}{74095689505392} a^{31} + \frac{1}{18523922376348} a^{29} - \frac{1}{2058213597372} a^{27} - \frac{1484173}{74095689505392} a^{25} + \frac{58463603}{18523922376348} a^{23} - \frac{550403959}{18523922376348} a^{21} + \frac{4740163973}{283891530672} a^{19} + \frac{2841730302313}{18523922376348} a^{17} + \frac{2258696642215}{18523922376348} a^{15} - \frac{189386401741}{8232854389488} a^{13} + \frac{24882265963}{228690399708} a^{11} - \frac{2485031893}{8470014804} a^{9} + \frac{397013297}{11293353072} a^{7} - \frac{81444979}{313704252} a^{5} + \frac{4117319}{34856028} a^{3} - \frac{264572}{968223} a$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{832135}{8232854389488} a^{31} + \frac{166427}{914761598832} a^{29} - \frac{832135}{4116427194744} a^{27} + \frac{4160675}{4116427194744} a^{25} + \frac{638641}{914761598832} a^{23} + \frac{16143419}{457380799416} a^{21} - \frac{94031255}{4116427194744} a^{19} - \frac{151614997}{8232854389488} a^{17} - \frac{94031255}{457380799416} a^{15} - \frac{2092819525}{4116427194744} a^{13} - \frac{94031255}{11293353072} a^{11} + \frac{4160675}{627408504} a^{9} - \frac{832135}{69712056} a^{7} + \frac{499281}{5163856} a^{5} + \frac{5824945}{69712056} a^{3} + \frac{4493529}{5163856} a \) (order $60$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3\times C_4$ (as 32T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{-165}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-15}) \), \(\Q(i, \sqrt{33})\), \(\Q(i, \sqrt{11})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{3}, \sqrt{11})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{-3}, \sqrt{11})\), \(\Q(\sqrt{3}, \sqrt{-11})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{165})\), \(\Q(i, \sqrt{55})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{33})\), \(\Q(\sqrt{-5}, \sqrt{33})\), \(\Q(\sqrt{15}, \sqrt{33})\), \(\Q(\sqrt{-15}, \sqrt{33})\), \(\Q(\sqrt{5}, \sqrt{-33})\), \(\Q(\sqrt{-5}, \sqrt{-33})\), \(\Q(\sqrt{-15}, \sqrt{-33})\), \(\Q(\sqrt{15}, \sqrt{-33})\), \(\Q(\sqrt{5}, \sqrt{11})\), \(\Q(\sqrt{-5}, \sqrt{11})\), \(\Q(\sqrt{11}, \sqrt{15})\), \(\Q(\sqrt{11}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{-5}, \sqrt{-11})\), \(\Q(\sqrt{-11}, \sqrt{-15})\), \(\Q(\sqrt{-11}, \sqrt{15})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{55})\), \(\Q(\sqrt{3}, \sqrt{-55})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{-55})\), \(\Q(\sqrt{-3}, \sqrt{55})\), 4.0.242000.2, 4.4.15125.1, 4.0.18000.1, \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 4.0.136125.2, 4.4.2178000.1, 8.0.303595776.1, 8.0.189747360000.8, 8.0.189747360000.5, 8.0.2342560000.1, 8.0.189747360000.7, 8.0.12960000.1, 8.0.189747360000.10, 8.8.189747360000.1, 8.0.189747360000.4, 8.0.741200625.1, 8.0.189747360000.2, 8.0.189747360000.9, 8.0.189747360000.3, 8.0.189747360000.6, 8.0.189747360000.1, 8.0.58564000000.2, 8.0.324000000.1, \(\Q(\zeta_{20})\), 8.0.4743684000000.8, 8.0.4743684000000.3, 8.8.18530015625.1, 8.0.18530015625.3, 8.8.4743684000000.2, 8.0.4743684000000.7, 8.0.4743684000000.1, 8.0.4743684000000.10, 8.0.4743684000000.6, 8.0.58564000000.3, 8.8.58564000000.1, 8.0.4743684000000.4, 8.8.4743684000000.3, 8.0.58564000000.1, 8.0.228765625.1, 8.0.4743684000000.2, 8.0.18530015625.1, 8.0.4743684000000.5, 8.8.4743684000000.1, 8.0.324000000.3, \(\Q(\zeta_{60})^+\), 8.0.4743684000000.9, 8.0.18530015625.2, 8.0.324000000.2, \(\Q(\zeta_{15})\), 16.0.36004060626969600000000.1, 16.0.22502537891856000000000000.2, 16.0.22502537891856000000000000.5, 16.0.3429742096000000000000.1, 16.0.22502537891856000000000000.8, 16.0.22502537891856000000000000.7, \(\Q(\zeta_{60})\), 16.0.22502537891856000000000000.3, 16.16.22502537891856000000000000.1, 16.0.22502537891856000000000000.10, 16.0.343361479062744140625.1, 16.0.22502537891856000000000000.4, 16.0.22502537891856000000000000.1, 16.0.22502537891856000000000000.9, 16.0.22502537891856000000000000.6

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$