Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*x + 1)
gp: K = bnfinit(y^20 - 3*y^19 + 5*y^18 - 9*y^17 + 18*y^16 - 29*y^15 + 37*y^14 - 24*y^13 + 26*y^12 - 13*y^11 + 31*y^10 - 13*y^9 + 26*y^8 - 24*y^7 + 37*y^6 - 29*y^5 + 18*y^4 - 9*y^3 + 5*y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 3*x^19 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*x + 1)
\( x^{20} - 3 x^{19} + 5 x^{18} - 9 x^{17} + 18 x^{16} - 29 x^{15} + 37 x^{14} - 24 x^{13} + 26 x^{12} + \cdots + 1 \)
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: |
\(591319123885120791015625\)
\(\medspace = 5^{10}\cdot 7^{10}\cdot 11^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.44\) | 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: | $5^{1/2}7^{1/2}11^{4/5}\approx 40.28544546409263$ | ||
| Ramified primes: |
\(5\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $10$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{15}-\frac{2}{5}a^{14}+\frac{2}{5}a^{13}+\frac{1}{5}a^{12}+\frac{2}{5}a^{10}-\frac{1}{5}a^{9}+\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{17}-\frac{1}{5}a^{15}+\frac{1}{5}a^{14}+\frac{2}{5}a^{13}-\frac{2}{5}a^{12}+\frac{2}{5}a^{11}-\frac{2}{5}a^{9}+\frac{2}{5}a^{8}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{10955}a^{18}-\frac{456}{10955}a^{17}+\frac{618}{10955}a^{16}-\frac{250}{2191}a^{15}+\frac{2563}{10955}a^{14}-\frac{5161}{10955}a^{13}+\frac{4183}{10955}a^{12}+\frac{779}{1565}a^{11}-\frac{5109}{10955}a^{10}-\frac{957}{2191}a^{9}-\frac{5109}{10955}a^{8}+\frac{779}{1565}a^{7}+\frac{4183}{10955}a^{6}-\frac{5161}{10955}a^{5}+\frac{2563}{10955}a^{4}-\frac{250}{2191}a^{3}+\frac{618}{10955}a^{2}-\frac{456}{10955}a+\frac{1}{10955}$, $\frac{1}{208145}a^{19}-\frac{4}{208145}a^{18}+\frac{9224}{208145}a^{17}-\frac{6744}{208145}a^{16}-\frac{8634}{41629}a^{15}+\frac{97253}{208145}a^{14}+\frac{83702}{208145}a^{13}+\frac{64493}{208145}a^{12}-\frac{1313}{10955}a^{11}-\frac{1929}{29735}a^{10}-\frac{13352}{41629}a^{9}-\frac{3516}{10955}a^{8}-\frac{13464}{208145}a^{7}-\frac{3571}{29735}a^{6}+\frac{9222}{29735}a^{5}+\frac{11948}{29735}a^{4}+\frac{2780}{5947}a^{3}-\frac{6171}{29735}a^{2}-\frac{1346}{41629}a+\frac{9216}{208145}$
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 $1$
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 $2$)
| 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: |
$\frac{262}{208145}a^{19}-\frac{128671}{208145}a^{18}+\frac{29871}{29735}a^{17}-\frac{4831}{5947}a^{16}+\frac{74074}{29735}a^{15}-\frac{162763}{29735}a^{14}+\frac{166474}{29735}a^{13}-\frac{958388}{208145}a^{12}-\frac{57839}{10955}a^{11}-\frac{384675}{41629}a^{10}-\frac{128932}{29735}a^{9}-\frac{135103}{10955}a^{8}-\frac{409030}{41629}a^{7}-\frac{2341398}{208145}a^{6}+\frac{519587}{208145}a^{5}-\frac{233706}{41629}a^{4}-\frac{144442}{208145}a^{3}+\frac{271189}{208145}a^{2}+\frac{192049}{208145}a-\frac{2626}{208145}$, $\frac{14941}{29735}a^{19}-\frac{61946}{208145}a^{18}-\frac{124541}{208145}a^{17}-\frac{100442}{208145}a^{16}+\frac{321382}{208145}a^{15}+\frac{77453}{41629}a^{14}-\frac{1117047}{208145}a^{13}+\frac{2825111}{208145}a^{12}+\frac{12026}{1565}a^{11}+\frac{1893511}{208145}a^{10}+\frac{1859254}{208145}a^{9}+\frac{40853}{2191}a^{8}+\frac{197}{19}a^{7}+\frac{148546}{41629}a^{6}-\frac{852846}{208145}a^{5}+\frac{1936078}{208145}a^{4}-\frac{881988}{208145}a^{3}-\frac{70284}{41629}a^{2}-\frac{131961}{208145}a+\frac{352328}{208145}$, $a$, $\frac{91002}{208145}a^{19}-\frac{43116}{41629}a^{18}+\frac{292281}{208145}a^{17}-\frac{585759}{208145}a^{16}+\frac{1185084}{208145}a^{15}-\frac{1674634}{208145}a^{14}+\frac{1918013}{208145}a^{13}-\frac{427822}{208145}a^{12}+\frac{67703}{10955}a^{11}+\frac{35686}{29735}a^{10}+\frac{1753918}{208145}a^{9}+\frac{3783}{10955}a^{8}+\frac{258233}{41629}a^{7}-\frac{211592}{29735}a^{6}+\frac{202693}{29735}a^{5}-\frac{215987}{29735}a^{4}-\frac{4268}{29735}a^{3}-\frac{13142}{29735}a^{2}-\frac{200171}{208145}a+\frac{81768}{208145}$, $\frac{146}{10955}a^{19}-\frac{13168}{10955}a^{18}+\frac{4597}{1565}a^{17}-\frac{5908}{1565}a^{16}+\frac{11482}{1565}a^{15}-\frac{24729}{1565}a^{14}+\frac{34984}{1565}a^{13}-\frac{261766}{10955}a^{12}+\frac{55523}{10955}a^{11}-\frac{178057}{10955}a^{10}+\frac{8449}{1565}a^{9}-\frac{254256}{10955}a^{8}+\frac{16488}{10955}a^{7}-\frac{174733}{10955}a^{6}+\frac{229659}{10955}a^{5}-\frac{246251}{10955}a^{4}+\frac{136009}{10955}a^{3}-\frac{48128}{10955}a^{2}+\frac{27546}{10955}a-\frac{3779}{2191}$, $\frac{246333}{208145}a^{19}-\frac{495151}{208145}a^{18}+\frac{109577}{41629}a^{17}-\frac{1232494}{208145}a^{16}+\frac{2649326}{208145}a^{15}-\frac{3351679}{208145}a^{14}+\frac{3312978}{208145}a^{13}+\frac{827094}{208145}a^{12}+\frac{35538}{2191}a^{11}+\frac{1029237}{208145}a^{10}+\frac{5119367}{208145}a^{9}+\frac{134783}{10955}a^{8}+\frac{4243237}{208145}a^{7}-\frac{2081559}{208145}a^{6}+\frac{693811}{41629}a^{5}-\frac{747209}{208145}a^{4}-\frac{101139}{208145}a^{3}-\frac{143544}{208145}a^{2}-\frac{45858}{208145}a+\frac{166191}{208145}$, $\frac{29662}{41629}a^{19}-\frac{343713}{208145}a^{18}+\frac{75452}{29735}a^{17}-\frac{168004}{29735}a^{16}+\frac{335153}{29735}a^{15}-\frac{97259}{5947}a^{14}+\frac{638966}{29735}a^{13}-\frac{617327}{41629}a^{12}+\frac{286912}{10955}a^{11}-\frac{1968766}{208145}a^{10}+\frac{673001}{29735}a^{9}-\frac{5113}{2191}a^{8}+\frac{4578459}{208145}a^{7}-\frac{3357271}{208145}a^{6}+\frac{4480004}{208145}a^{5}-\frac{3601803}{208145}a^{4}+\frac{3467591}{208145}a^{3}-\frac{507197}{41629}a^{2}+\frac{1290477}{208145}a-\frac{188389}{208145}$, $\frac{2626}{208145}a^{19}-\frac{1088}{29735}a^{18}-\frac{115541}{208145}a^{17}+\frac{185463}{208145}a^{16}-\frac{121817}{208145}a^{15}+\frac{442364}{208145}a^{14}-\frac{1042179}{208145}a^{13}+\frac{1102294}{208145}a^{12}-\frac{46848}{10955}a^{11}-\frac{1133079}{208145}a^{10}-\frac{1841969}{208145}a^{9}-\frac{49298}{10955}a^{8}-\frac{2498681}{208145}a^{7}-\frac{2108174}{208145}a^{6}-\frac{2244236}{208145}a^{5}+\frac{443433}{208145}a^{4}-\frac{1121262}{208145}a^{3}-\frac{168076}{208145}a^{2}+\frac{10882}{29735}a+\frac{184171}{208145}$, $\frac{445027}{208145}a^{19}-\frac{956724}{208145}a^{18}+\frac{1047817}{208145}a^{17}-\frac{2330512}{208145}a^{16}+\frac{1043712}{41629}a^{15}-\frac{6713057}{208145}a^{14}+\frac{6726519}{208145}a^{13}+\frac{135476}{208145}a^{12}+\frac{368521}{10955}a^{11}-\frac{1277449}{208145}a^{10}+\frac{1721711}{41629}a^{9}+\frac{94194}{10955}a^{8}+\frac{6787642}{208145}a^{7}-\frac{6692706}{208145}a^{6}+\frac{6431162}{208145}a^{5}-\frac{2507657}{208145}a^{4}+\frac{320471}{41629}a^{3}-\frac{1703782}{208145}a^{2}+\frac{215905}{41629}a-\frac{54331}{208145}$
| 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: | \( 4090.28443721 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 4090.28443721 \cdot 1}{2\cdot\sqrt{591319123885120791015625}}\cr\approx \mathstrut & 0.255041723934 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*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 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*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 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*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 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*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
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 100 |
| The 16 conjugacy class representatives for $D_5^2$ |
| Character table for $D_5^2$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}, \sqrt{-7})\), 10.2.109853253125.1 x5 |
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 10 siblings: | data not computed |
| Degree 20 sibling: | data not computed |
| Degree 25 sibling: | data not computed |
| Minimal sibling: | 10.2.109853253125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | R | R | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{10}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{10}$ | ${\href{/padicField/31.2.0.1}{2} }^{10}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(11\)
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.0.1 | $x^{5} + 10 x^{2} + 9$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 11.5.0.1 | $x^{5} + 10 x^{2} + 9$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |