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Properties

Label 12.12.129...217.1
Degree $12$
Signature $[12, 0]$
Discriminant $1.298\times 10^{28}$
Root discriminant \(220.17\)
Ramified primes $3,7,17$
Class number $6$ (GRH)
Class group [6] (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 273*x^10 - 266*x^9 + 22995*x^8 + 26796*x^7 - 759367*x^6 - 1041642*x^5 + 10115364*x^4 + 13933640*x^3 - 50535072*x^2 - 52157952*x + 75464704)
 
gp: K = bnfinit(y^12 - 273*y^10 - 266*y^9 + 22995*y^8 + 26796*y^7 - 759367*y^6 - 1041642*y^5 + 10115364*y^4 + 13933640*y^3 - 50535072*y^2 - 52157952*y + 75464704, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 273*x^10 - 266*x^9 + 22995*x^8 + 26796*x^7 - 759367*x^6 - 1041642*x^5 + 10115364*x^4 + 13933640*x^3 - 50535072*x^2 - 52157952*x + 75464704);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 273*x^10 - 266*x^9 + 22995*x^8 + 26796*x^7 - 759367*x^6 - 1041642*x^5 + 10115364*x^4 + 13933640*x^3 - 50535072*x^2 - 52157952*x + 75464704)
 

\( x^{12} - 273 x^{10} - 266 x^{9} + 22995 x^{8} + 26796 x^{7} - 759367 x^{6} - 1041642 x^{5} + \cdots + 75464704 \) Copy content Toggle raw display

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

Invariants

Degree:  $12$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[12, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(12977865756870219436044086217\) \(\medspace = 3^{18}\cdot 7^{10}\cdot 17^{9}\) 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:  \(220.17\)
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:  $3^{3/2}7^{5/6}17^{3/4}\approx 220.1744599625188$
Ramified primes:   \(3\), \(7\), \(17\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1071=3^{2}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1071}(1,·)$, $\chi_{1071}(38,·)$, $\chi_{1071}(353,·)$, $\chi_{1071}(970,·)$, $\chi_{1071}(781,·)$, $\chi_{1071}(1007,·)$, $\chi_{1071}(562,·)$, $\chi_{1071}(883,·)$, $\chi_{1071}(373,·)$, $\chi_{1071}(761,·)$, $\chi_{1071}(251,·)$, $\chi_{1071}(446,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{112}a^{6}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}+\frac{1}{16}a^{3}+\frac{1}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{224}a^{7}-\frac{1}{16}a^{4}+\frac{1}{32}a^{3}+\frac{1}{16}a^{2}+\frac{1}{4}a$, $\frac{1}{896}a^{8}+\frac{1}{448}a^{6}-\frac{1}{16}a^{5}-\frac{1}{128}a^{4}+\frac{3}{16}a^{3}+\frac{7}{32}a^{2}+\frac{3}{8}a$, $\frac{1}{34048}a^{9}-\frac{1}{1792}a^{8}-\frac{3}{17024}a^{7}-\frac{9}{2432}a^{6}-\frac{201}{4864}a^{5}+\frac{251}{4864}a^{4}-\frac{165}{1216}a^{3}+\frac{139}{1216}a^{2}-\frac{121}{304}a+\frac{7}{19}$, $\frac{1}{136192}a^{10}+\frac{1}{136192}a^{9}-\frac{3}{68096}a^{8}-\frac{123}{68096}a^{7}+\frac{177}{136192}a^{6}-\frac{425}{19456}a^{5}-\frac{145}{4864}a^{4}+\frac{563}{4864}a^{3}+\frac{61}{1216}a^{2}-\frac{13}{152}a-\frac{3}{19}$, $\frac{1}{10\!\cdots\!12}a^{11}-\frac{325630156779}{14\!\cdots\!16}a^{10}+\frac{1674229524655}{25\!\cdots\!28}a^{9}+\frac{11812099388269}{26\!\cdots\!24}a^{8}-\frac{218081545068973}{14\!\cdots\!16}a^{7}-\frac{46719668822603}{14\!\cdots\!16}a^{6}+\frac{161488857016759}{38\!\cdots\!32}a^{5}+\frac{171381272797053}{36\!\cdots\!04}a^{4}+\frac{290515246356221}{18\!\cdots\!52}a^{3}-\frac{807574626209033}{45\!\cdots\!88}a^{2}+\frac{98572846190001}{282621895462168}a-\frac{338763569035}{751653977293}$ 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:  $2$

Class group and class number

$C_{6}$, which has order $6$ (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:  $11$
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:   $\frac{119529}{858007119872}a^{11}-\frac{5614083}{16302135277568}a^{10}-\frac{2586247}{72777389632}a^{9}+\frac{420495273}{8151067638784}a^{8}+\frac{43693692975}{16302135277568}a^{7}-\frac{55317337059}{16302135277568}a^{6}-\frac{84638493999}{1164438234112}a^{5}+\frac{30015748251}{582219117056}a^{4}+\frac{12827445867}{15321555712}a^{3}-\frac{10503243729}{72777389632}a^{2}-\frac{21089805075}{4548586852}a-\frac{4911353141}{1137146713}$, $\frac{18679350845}{75\!\cdots\!68}a^{11}-\frac{63111176189}{75\!\cdots\!68}a^{10}-\frac{615596457743}{944885143037696}a^{9}+\frac{5798352581047}{37\!\cdots\!84}a^{8}+\frac{57215055470255}{10\!\cdots\!24}a^{7}-\frac{117832426035131}{10\!\cdots\!24}a^{6}-\frac{869148372777997}{539934367450112}a^{5}+\frac{700919893219317}{269967183725056}a^{4}+\frac{25\!\cdots\!95}{134983591862528}a^{3}-\frac{775815480227419}{33745897965632}a^{2}-\frac{147375999388177}{2109118622852}a+\frac{779188099415}{11218716079}$, $\frac{59411566275}{75\!\cdots\!68}a^{11}-\frac{173173528239}{75\!\cdots\!68}a^{10}-\frac{3941623024629}{18\!\cdots\!92}a^{9}+\frac{15036294640637}{37\!\cdots\!84}a^{8}+\frac{184575088077065}{10\!\cdots\!24}a^{7}-\frac{306065462080185}{10\!\cdots\!24}a^{6}-\frac{28\!\cdots\!93}{539934367450112}a^{5}+\frac{18\!\cdots\!63}{269967183725056}a^{4}+\frac{85\!\cdots\!99}{134983591862528}a^{3}-\frac{22\!\cdots\!95}{33745897965632}a^{2}-\frac{497598069597741}{2109118622852}a+\frac{2606710422997}{11218716079}$, $\frac{528552269785}{13\!\cdots\!12}a^{11}-\frac{43756314255027}{25\!\cdots\!28}a^{10}-\frac{316924444295105}{31\!\cdots\!16}a^{9}+\frac{41\!\cdots\!81}{12\!\cdots\!64}a^{8}+\frac{19\!\cdots\!87}{25\!\cdots\!28}a^{7}-\frac{29\!\cdots\!41}{13\!\cdots\!12}a^{6}-\frac{35\!\cdots\!75}{18\!\cdots\!52}a^{5}+\frac{38\!\cdots\!25}{90\!\cdots\!76}a^{4}+\frac{86\!\cdots\!97}{45\!\cdots\!88}a^{3}-\frac{41\!\cdots\!91}{14874836603272}a^{2}-\frac{16\!\cdots\!27}{282621895462168}a+\frac{469536685572261}{751653977293}$, $\frac{385633272544817}{50\!\cdots\!56}a^{11}-\frac{11\!\cdots\!33}{50\!\cdots\!56}a^{10}-\frac{64\!\cdots\!93}{31\!\cdots\!16}a^{9}+\frac{13\!\cdots\!37}{36\!\cdots\!04}a^{8}+\frac{84\!\cdots\!57}{50\!\cdots\!56}a^{7}-\frac{13\!\cdots\!17}{50\!\cdots\!56}a^{6}-\frac{18\!\cdots\!01}{36\!\cdots\!04}a^{5}+\frac{12\!\cdots\!33}{18\!\cdots\!52}a^{4}+\frac{57\!\cdots\!11}{90\!\cdots\!76}a^{3}-\frac{15\!\cdots\!65}{22\!\cdots\!44}a^{2}-\frac{69\!\cdots\!55}{282621895462168}a+\frac{17\!\cdots\!79}{751653977293}$, $\frac{12719675252647}{50\!\cdots\!56}a^{11}-\frac{28411243591219}{50\!\cdots\!56}a^{10}-\frac{853745666894957}{12\!\cdots\!64}a^{9}+\frac{111914014513859}{13\!\cdots\!12}a^{8}+\frac{28\!\cdots\!35}{50\!\cdots\!56}a^{7}-\frac{15\!\cdots\!45}{26\!\cdots\!24}a^{6}-\frac{65\!\cdots\!93}{36\!\cdots\!04}a^{5}+\frac{28\!\cdots\!75}{18\!\cdots\!52}a^{4}+\frac{20\!\cdots\!71}{90\!\cdots\!76}a^{3}-\frac{41\!\cdots\!71}{22\!\cdots\!44}a^{2}-\frac{60\!\cdots\!81}{70655473865542}a+\frac{586702284427129}{751653977293}$, $\frac{314083892177299}{25\!\cdots\!28}a^{11}+\frac{109245529430437}{36\!\cdots\!04}a^{10}-\frac{41\!\cdots\!57}{12\!\cdots\!64}a^{9}-\frac{14\!\cdots\!97}{12\!\cdots\!64}a^{8}+\frac{65\!\cdots\!51}{25\!\cdots\!28}a^{7}+\frac{24\!\cdots\!47}{25\!\cdots\!28}a^{6}-\frac{64\!\cdots\!77}{90\!\cdots\!76}a^{5}-\frac{27\!\cdots\!33}{90\!\cdots\!76}a^{4}+\frac{11\!\cdots\!25}{22\!\cdots\!44}a^{3}+\frac{33\!\cdots\!29}{11\!\cdots\!72}a^{2}+\frac{30\!\cdots\!57}{282621895462168}a-\frac{28\!\cdots\!51}{751653977293}$, $\frac{588545191095091}{50\!\cdots\!56}a^{11}-\frac{218939216083153}{72\!\cdots\!08}a^{10}-\frac{39\!\cdots\!11}{12\!\cdots\!64}a^{9}+\frac{12\!\cdots\!17}{25\!\cdots\!28}a^{8}+\frac{18\!\cdots\!89}{72\!\cdots\!08}a^{7}-\frac{17\!\cdots\!27}{50\!\cdots\!56}a^{6}-\frac{28\!\cdots\!33}{36\!\cdots\!04}a^{5}+\frac{15\!\cdots\!51}{18\!\cdots\!52}a^{4}+\frac{84\!\cdots\!75}{90\!\cdots\!76}a^{3}-\frac{19\!\cdots\!87}{22\!\cdots\!44}a^{2}-\frac{24\!\cdots\!09}{70655473865542}a+\frac{24\!\cdots\!27}{751653977293}$, $\frac{1454839354553}{72\!\cdots\!08}a^{11}+\frac{6240631165281}{50\!\cdots\!56}a^{10}-\frac{54870289897577}{90\!\cdots\!76}a^{9}-\frac{252335675868773}{36\!\cdots\!04}a^{8}+\frac{43\!\cdots\!09}{72\!\cdots\!08}a^{7}+\frac{26\!\cdots\!89}{50\!\cdots\!56}a^{6}-\frac{86\!\cdots\!71}{36\!\cdots\!04}a^{5}-\frac{98\!\cdots\!53}{18\!\cdots\!52}a^{4}+\frac{33\!\cdots\!13}{90\!\cdots\!76}a^{3}-\frac{34\!\cdots\!21}{22\!\cdots\!44}a^{2}-\frac{21\!\cdots\!55}{141310947731084}a+\frac{931214992424795}{751653977293}$, $\frac{44150455837657}{38\!\cdots\!32}a^{11}-\frac{16\!\cdots\!73}{50\!\cdots\!56}a^{10}-\frac{12\!\cdots\!21}{39\!\cdots\!52}a^{9}+\frac{13\!\cdots\!11}{25\!\cdots\!28}a^{8}+\frac{12\!\cdots\!53}{50\!\cdots\!56}a^{7}-\frac{10\!\cdots\!11}{26\!\cdots\!24}a^{6}-\frac{28\!\cdots\!45}{36\!\cdots\!04}a^{5}+\frac{16\!\cdots\!93}{18\!\cdots\!52}a^{4}+\frac{83\!\cdots\!43}{90\!\cdots\!76}a^{3}-\frac{10\!\cdots\!93}{118998692826176}a^{2}-\frac{47\!\cdots\!33}{141310947731084}a+\frac{23\!\cdots\!89}{751653977293}$, $\frac{4096024967579}{72\!\cdots\!08}a^{11}-\frac{17482872032749}{50\!\cdots\!56}a^{10}-\frac{950892018569049}{63\!\cdots\!32}a^{9}-\frac{9557892622421}{19\!\cdots\!16}a^{8}+\frac{60\!\cdots\!45}{50\!\cdots\!56}a^{7}+\frac{24\!\cdots\!31}{50\!\cdots\!56}a^{6}-\frac{13\!\cdots\!45}{36\!\cdots\!04}a^{5}-\frac{37\!\cdots\!67}{18\!\cdots\!52}a^{4}+\frac{38\!\cdots\!23}{90\!\cdots\!76}a^{3}+\frac{63\!\cdots\!21}{22\!\cdots\!44}a^{2}-\frac{53\!\cdots\!26}{35327736932771}a-\frac{731445614378545}{751653977293}$ 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:  \( 6566400059444.978 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 6566400059444.978 \cdot 6}{2\cdot\sqrt{12977865756870219436044086217}}\cr\approx \mathstrut & 708.283137830824 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 273*x^10 - 266*x^9 + 22995*x^8 + 26796*x^7 - 759367*x^6 - 1041642*x^5 + 10115364*x^4 + 13933640*x^3 - 50535072*x^2 - 52157952*x + 75464704)
 
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^12 - 273*x^10 - 266*x^9 + 22995*x^8 + 26796*x^7 - 759367*x^6 - 1041642*x^5 + 10115364*x^4 + 13933640*x^3 - 50535072*x^2 - 52157952*x + 75464704, 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^12 - 273*x^10 - 266*x^9 + 22995*x^8 + 26796*x^7 - 759367*x^6 - 1041642*x^5 + 10115364*x^4 + 13933640*x^3 - 50535072*x^2 - 52157952*x + 75464704);
 
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^12 - 273*x^10 - 266*x^9 + 22995*x^8 + 26796*x^7 - 759367*x^6 - 1041642*x^5 + 10115364*x^4 + 13933640*x^3 - 50535072*x^2 - 52157952*x + 75464704);
 
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

$C_{12}$ (as 12T1):

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 cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{17}) \), 3.3.3969.1, 4.4.2166633.1, 6.6.77394297393.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.1.0.1}{1} }^{12}$ R ${\href{/padicField/5.12.0.1}{12} }$ R ${\href{/padicField/11.12.0.1}{12} }$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ R ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.12.0.1}{12} }$ ${\href{/padicField/31.4.0.1}{4} }^{3}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.1.0.1}{1} }^{12}$ ${\href{/padicField/53.3.0.1}{3} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{6}$

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
\(3\) Copy content Toggle raw display 3.12.18.88$x^{12} + 12 x^{11} + 42 x^{10} + 42 x^{9} + 54 x^{8} + 18 x^{7} + 21 x^{6} + 72 x^{5} + 54 x^{4} + 36 x^{3} + 45$$6$$2$$18$$C_{12}$$[2]_{2}^{2}$
\(7\) Copy content Toggle raw display 7.12.10.6$x^{12} - 28 x^{6} - 98$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
\(17\) Copy content Toggle raw display 17.12.9.1$x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$