Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 34*x^16 - 53*x^15 + 77*x^14 - 97*x^13 + 114*x^12 - 121*x^11 + 116*x^10 - 104*x^9 + 79*x^8 - 56*x^7 + 33*x^6 - 19*x^5 + 12*x^4 - 7*x^3 + 5*x^2 - 2*x + 1)
gp: K = bnfinit(y^20 - 3*y^19 + 9*y^18 - 18*y^17 + 34*y^16 - 53*y^15 + 77*y^14 - 97*y^13 + 114*y^12 - 121*y^11 + 116*y^10 - 104*y^9 + 79*y^8 - 56*y^7 + 33*y^6 - 19*y^5 + 12*y^4 - 7*y^3 + 5*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 34*x^16 - 53*x^15 + 77*x^14 - 97*x^13 + 114*x^12 - 121*x^11 + 116*x^10 - 104*x^9 + 79*x^8 - 56*x^7 + 33*x^6 - 19*x^5 + 12*x^4 - 7*x^3 + 5*x^2 - 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 34*x^16 - 53*x^15 + 77*x^14 - 97*x^13 + 114*x^12 - 121*x^11 + 116*x^10 - 104*x^9 + 79*x^8 - 56*x^7 + 33*x^6 - 19*x^5 + 12*x^4 - 7*x^3 + 5*x^2 - 2*x + 1)
\( x^{20} - 3 x^{19} + 9 x^{18} - 18 x^{17} + 34 x^{16} - 53 x^{15} + 77 x^{14} - 97 x^{13} + 114 x^{12} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2392595214874267578125\)
\(\medspace = 5^{15}\cdot 280001^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.72\) | 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: | $5^{3/4}280001^{1/2}\approx 1769.3236981512437$ | ||
| Ramified primes: |
\(5\), \(280001\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7}a^{18}+\frac{3}{7}a^{17}-\frac{1}{7}a^{15}+\frac{2}{7}a^{13}-\frac{2}{7}a^{12}-\frac{2}{7}a^{11}+\frac{2}{7}a^{10}+\frac{1}{7}a^{9}-\frac{2}{7}a^{8}-\frac{3}{7}a^{7}+\frac{3}{7}a^{6}+\frac{1}{7}a^{5}+\frac{3}{7}a^{3}+\frac{2}{7}a^{2}+\frac{1}{7}a-\frac{1}{7}$, $\frac{1}{45787}a^{19}+\frac{1453}{45787}a^{18}-\frac{3707}{45787}a^{17}+\frac{387}{1477}a^{16}+\frac{22819}{45787}a^{15}-\frac{3869}{45787}a^{14}-\frac{198}{6541}a^{13}+\frac{16138}{45787}a^{12}-\frac{1616}{6541}a^{11}-\frac{216}{1477}a^{10}-\frac{22793}{45787}a^{9}-\frac{4219}{45787}a^{8}+\frac{2691}{6541}a^{7}-\frac{6338}{45787}a^{6}-\frac{18367}{45787}a^{5}-\frac{15845}{45787}a^{4}+\frac{6340}{45787}a^{3}-\frac{11400}{45787}a^{2}-\frac{554}{6541}a+\frac{18136}{45787}$
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: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{19365}{45787} a^{19} + \frac{80529}{45787} a^{18} - \frac{197650}{45787} a^{17} + \frac{13336}{1477} a^{16} - \frac{699485}{45787} a^{15} + \frac{1068754}{45787} a^{14} - \frac{1430312}{45787} a^{13} + \frac{242521}{6541} a^{12} - \frac{1799985}{45787} a^{11} + \frac{56946}{1477} a^{10} - \frac{1543911}{45787} a^{9} + \frac{1089651}{45787} a^{8} - \frac{765314}{45787} a^{7} + \frac{248604}{45787} a^{6} - \frac{74953}{45787} a^{5} + \frac{65525}{45787} a^{4} - \frac{25694}{45787} a^{3} + \frac{93824}{45787} a^{2} + \frac{19872}{45787} a + \frac{15355}{45787} \)
(order $10$)
| 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: |
$\frac{22434}{45787}a^{19}-\frac{23365}{45787}a^{18}+\frac{64846}{45787}a^{17}-\frac{1325}{1477}a^{16}+\frac{42409}{45787}a^{15}+\frac{60793}{45787}a^{14}-\frac{180758}{45787}a^{13}+\frac{407625}{45787}a^{12}-\frac{577839}{45787}a^{11}+\frac{3237}{211}a^{10}-\frac{924309}{45787}a^{9}+\frac{764521}{45787}a^{8}-\frac{835836}{45787}a^{7}+\frac{564092}{45787}a^{6}-\frac{302410}{45787}a^{5}+\frac{206686}{45787}a^{4}-\frac{41731}{45787}a^{3}+\frac{16671}{6541}a^{2}-\frac{23375}{45787}a+\frac{19365}{45787}$, $\frac{6862}{45787}a^{19}-\frac{63408}{45787}a^{18}+\frac{137876}{45787}a^{17}-\frac{11868}{1477}a^{16}+\frac{594210}{45787}a^{15}-\frac{1045719}{45787}a^{14}+\frac{1373492}{45787}a^{13}-\frac{1838195}{45787}a^{12}+\frac{1922157}{45787}a^{11}-\frac{9664}{211}a^{10}+\frac{1827765}{45787}a^{9}-\frac{1465496}{45787}a^{8}+\frac{1121304}{45787}a^{7}-\frac{471199}{45787}a^{6}+\frac{285438}{45787}a^{5}-\frac{75839}{45787}a^{4}+\frac{79381}{45787}a^{3}-\frac{18180}{6541}a^{2}+\frac{76444}{45787}a-\frac{39080}{45787}$, $\frac{72505}{45787}a^{19}-\frac{34514}{6541}a^{18}+\frac{660837}{45787}a^{17}-\frac{43444}{1477}a^{16}+\frac{2411602}{45787}a^{15}-\frac{3739430}{45787}a^{14}+\frac{5217171}{45787}a^{13}-\frac{6443980}{45787}a^{12}+\frac{7253399}{45787}a^{11}-\frac{243099}{1477}a^{10}+\frac{6936809}{45787}a^{9}-\frac{5889089}{45787}a^{8}+\frac{4365650}{45787}a^{7}-\frac{2693627}{45787}a^{6}+\frac{1565777}{45787}a^{5}-\frac{870061}{45787}a^{4}+\frac{85945}{6541}a^{3}-\frac{435241}{45787}a^{2}+\frac{180184}{45787}a-\frac{91206}{45787}$, $\frac{68686}{45787}a^{19}-\frac{224214}{45787}a^{18}+\frac{610818}{45787}a^{17}-\frac{39966}{1477}a^{16}+\frac{2189190}{45787}a^{15}-\frac{3386624}{45787}a^{14}+\frac{4656110}{45787}a^{13}-\frac{5719649}{45787}a^{12}+\frac{6308718}{45787}a^{11}-\frac{209656}{1477}a^{10}+\frac{825413}{6541}a^{9}-\frac{692456}{6541}a^{8}+\frac{3364292}{45787}a^{7}-\frac{271292}{6541}a^{6}+\frac{1040386}{45787}a^{5}-\frac{476337}{45787}a^{4}+\frac{414248}{45787}a^{3}-\frac{298176}{45787}a^{2}+\frac{135655}{45787}a-\frac{57236}{45787}$, $\frac{24053}{45787}a^{19}-\frac{10215}{6541}a^{18}+\frac{185589}{45787}a^{17}-\frac{11359}{1477}a^{16}+\frac{605328}{45787}a^{15}-\frac{891826}{45787}a^{14}+\frac{1199022}{45787}a^{13}-\frac{1401564}{45787}a^{12}+\frac{1522494}{45787}a^{11}-\frac{49158}{1477}a^{10}+\frac{1347873}{45787}a^{9}-\frac{1081798}{45787}a^{8}+\frac{782752}{45787}a^{7}-\frac{415451}{45787}a^{6}+\frac{298575}{45787}a^{5}-\frac{171945}{45787}a^{4}+\frac{26042}{6541}a^{3}-\frac{64349}{45787}a^{2}-\frac{2874}{45787}a+\frac{5918}{45787}$, $\frac{50161}{45787}a^{19}-\frac{100545}{45787}a^{18}+\frac{314689}{45787}a^{17}-\frac{16151}{1477}a^{16}+\frac{956173}{45787}a^{15}-\frac{1263852}{45787}a^{14}+\frac{259000}{6541}a^{13}-\frac{1938996}{45787}a^{12}+\frac{322946}{6541}a^{11}-\frac{67446}{1477}a^{10}+\frac{1858984}{45787}a^{9}-\frac{1558503}{45787}a^{8}+\frac{133995}{6541}a^{7}-\frac{708082}{45787}a^{6}+\frac{339436}{45787}a^{5}-\frac{305021}{45787}a^{4}+\frac{213173}{45787}a^{3}-\frac{93131}{45787}a^{2}+\frac{10056}{6541}a-\frac{22007}{45787}$, $\frac{41851}{45787}a^{19}-\frac{139535}{45787}a^{18}+\frac{377159}{45787}a^{17}-\frac{25554}{1477}a^{16}+\frac{1398661}{45787}a^{15}-\frac{2216463}{45787}a^{14}+\frac{3061290}{45787}a^{13}-\frac{549375}{6541}a^{12}+\frac{4290341}{45787}a^{11}-\frac{145744}{1477}a^{10}+\frac{4085017}{45787}a^{9}-\frac{3527214}{45787}a^{8}+\frac{2616203}{45787}a^{7}-\frac{1538141}{45787}a^{6}+\frac{995612}{45787}a^{5}-\frac{453844}{45787}a^{4}+\frac{346348}{45787}a^{3}-\frac{288664}{45787}a^{2}+\frac{147557}{45787}a-\frac{86396}{45787}$, $\frac{1507}{6541}a^{19}-\frac{37112}{45787}a^{18}+\frac{147398}{45787}a^{17}-\frac{1472}{211}a^{16}+\frac{682554}{45787}a^{15}-\frac{159536}{6541}a^{14}+\frac{1764291}{45787}a^{13}-\frac{2324900}{45787}a^{12}+\frac{2927543}{45787}a^{11}-\frac{101697}{1477}a^{10}+\frac{3254680}{45787}a^{9}-\frac{2970881}{45787}a^{8}+\frac{2409562}{45787}a^{7}-\frac{1783153}{45787}a^{6}+\frac{1044038}{45787}a^{5}-\frac{82257}{6541}a^{4}+\frac{273657}{45787}a^{3}-\frac{257414}{45787}a^{2}+\frac{135753}{45787}a-\frac{92619}{45787}$, $\frac{9505}{45787}a^{19}+\frac{3191}{6541}a^{18}-\frac{44455}{45787}a^{17}+\frac{6613}{1477}a^{16}-\frac{358179}{45787}a^{15}+\frac{724708}{45787}a^{14}-\frac{1053457}{45787}a^{13}+\frac{1529293}{45787}a^{12}-\frac{1785295}{45787}a^{11}+\frac{64516}{1477}a^{10}-\frac{2050337}{45787}a^{9}+\frac{1715018}{45787}a^{8}-\frac{1564407}{45787}a^{7}+\frac{863332}{45787}a^{6}-\frac{594276}{45787}a^{5}+\frac{261440}{45787}a^{4}-\frac{15027}{6541}a^{3}+\frac{236682}{45787}a^{2}-\frac{8396}{45787}a+\frac{92740}{45787}$
| 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: | \( 858.376938924 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 858.376938924 \cdot 1}{10\cdot\sqrt{2392595214874267578125}}\cr\approx \mathstrut & 0.168283722445 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 34*x^16 - 53*x^15 + 77*x^14 - 97*x^13 + 114*x^12 - 121*x^11 + 116*x^10 - 104*x^9 + 79*x^8 - 56*x^7 + 33*x^6 - 19*x^5 + 12*x^4 - 7*x^3 + 5*x^2 - 2*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^20 - 3*x^19 + 9*x^18 - 18*x^17 + 34*x^16 - 53*x^15 + 77*x^14 - 97*x^13 + 114*x^12 - 121*x^11 + 116*x^10 - 104*x^9 + 79*x^8 - 56*x^7 + 33*x^6 - 19*x^5 + 12*x^4 - 7*x^3 + 5*x^2 - 2*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^20 - 3*x^19 + 9*x^18 - 18*x^17 + 34*x^16 - 53*x^15 + 77*x^14 - 97*x^13 + 114*x^12 - 121*x^11 + 116*x^10 - 104*x^9 + 79*x^8 - 56*x^7 + 33*x^6 - 19*x^5 + 12*x^4 - 7*x^3 + 5*x^2 - 2*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^20 - 3*x^19 + 9*x^18 - 18*x^17 + 34*x^16 - 53*x^15 + 77*x^14 - 97*x^13 + 114*x^12 - 121*x^11 + 116*x^10 - 104*x^9 + 79*x^8 - 56*x^7 + 33*x^6 - 19*x^5 + 12*x^4 - 7*x^3 + 5*x^2 - 2*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
$S_5^2:C_4$ (as 20T654):
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 57600 |
| The 70 conjugacy class representatives for $S_5^2:C_4$ |
| Character table for $S_5^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.2.875003125.1 |
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 20 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 40 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 | $20$ | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $20$ | $4$ | $5$ | $15$ | |||
|
\(280001\)
| $\Q_{280001}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{280001}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{280001}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{280001}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |