Normalized defining polynomial
\( x^{18} - 6 x^{17} + 53 x^{16} - 224 x^{15} + 1059 x^{14} - 3090 x^{13} + 9543 x^{12} - 18332 x^{11} + \cdots + 271568 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-5627055140868788987624861696\) \(\medspace = -\,2^{12}\cdot 11^{9}\cdot 17^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(34.81\) | 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: | $2^{2/3}11^{1/2}17^{2/3}\approx 34.80825783744108$ | ||
Ramified primes: | \(2\), \(11\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{12}a^{10}-\frac{1}{12}a^{9}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{3}a^{4}+\frac{1}{6}a^{3}-\frac{1}{6}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{12}a^{11}-\frac{1}{12}a^{9}-\frac{1}{4}a^{7}-\frac{5}{12}a^{5}-\frac{1}{2}a^{4}+\frac{1}{6}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{120}a^{12}-\frac{1}{30}a^{11}-\frac{1}{40}a^{10}+\frac{1}{30}a^{9}+\frac{7}{40}a^{8}-\frac{1}{5}a^{7}-\frac{29}{120}a^{6}-\frac{1}{3}a^{5}+\frac{11}{60}a^{4}+\frac{1}{15}a^{3}-\frac{4}{15}a^{2}+\frac{13}{30}a-\frac{7}{30}$, $\frac{1}{120}a^{13}+\frac{1}{120}a^{11}+\frac{1}{60}a^{10}+\frac{7}{120}a^{9}-\frac{1}{24}a^{7}-\frac{1}{20}a^{6}-\frac{7}{30}a^{5}+\frac{2}{15}a^{4}+\frac{1}{6}a^{3}-\frac{7}{15}a^{2}+\frac{1}{6}a+\frac{1}{15}$, $\frac{1}{120}a^{14}-\frac{1}{30}a^{11}-\frac{1}{30}a^{9}-\frac{13}{60}a^{8}-\frac{1}{10}a^{7}-\frac{29}{120}a^{6}-\frac{11}{30}a^{5}-\frac{7}{20}a^{4}-\frac{11}{30}a^{3}+\frac{13}{30}a^{2}-\frac{1}{30}a+\frac{7}{30}$, $\frac{1}{720}a^{15}-\frac{1}{360}a^{14}-\frac{1}{720}a^{13}-\frac{1}{360}a^{12}-\frac{11}{720}a^{11}+\frac{7}{180}a^{10}-\frac{7}{720}a^{9}+\frac{11}{360}a^{8}+\frac{17}{120}a^{7}+\frac{71}{360}a^{6}+\frac{43}{90}a^{5}+\frac{2}{5}a^{4}-\frac{41}{180}a^{3}+\frac{1}{5}a^{2}-\frac{1}{2}a+\frac{5}{18}$, $\frac{1}{4601520}a^{16}-\frac{439}{1533840}a^{15}+\frac{10111}{4601520}a^{14}-\frac{25}{83664}a^{13}-\frac{4891}{1533840}a^{12}+\frac{547}{306768}a^{11}-\frac{4547}{4601520}a^{10}-\frac{10181}{920304}a^{9}-\frac{249313}{1150380}a^{8}+\frac{148279}{1150380}a^{7}-\frac{153287}{1150380}a^{6}+\frac{33821}{104580}a^{5}-\frac{22102}{57519}a^{4}-\frac{1369}{2772}a^{3}+\frac{27029}{191730}a^{2}-\frac{12349}{52290}a-\frac{2302}{26145}$, $\frac{1}{33\!\cdots\!80}a^{17}-\frac{82\!\cdots\!31}{16\!\cdots\!24}a^{16}+\frac{37\!\cdots\!97}{33\!\cdots\!80}a^{15}-\frac{80\!\cdots\!59}{18\!\cdots\!36}a^{14}+\frac{95\!\cdots\!47}{33\!\cdots\!80}a^{13}-\frac{49\!\cdots\!07}{50\!\cdots\!80}a^{12}+\frac{47\!\cdots\!31}{66\!\cdots\!96}a^{11}-\frac{38\!\cdots\!73}{13\!\cdots\!40}a^{10}-\frac{51\!\cdots\!77}{76\!\cdots\!20}a^{9}-\frac{88\!\cdots\!51}{79\!\cdots\!40}a^{8}+\frac{19\!\cdots\!13}{83\!\cdots\!20}a^{7}-\frac{39\!\cdots\!97}{16\!\cdots\!24}a^{6}+\frac{40\!\cdots\!89}{83\!\cdots\!20}a^{5}+\frac{89\!\cdots\!89}{19\!\cdots\!60}a^{4}-\frac{13\!\cdots\!87}{29\!\cdots\!90}a^{3}+\frac{16\!\cdots\!11}{41\!\cdots\!60}a^{2}+\frac{58\!\cdots\!17}{12\!\cdots\!82}a-\frac{14\!\cdots\!97}{19\!\cdots\!30}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
Rank: | $8$ | 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: | $\frac{33\!\cdots\!07}{83\!\cdots\!20}a^{17}-\frac{11\!\cdots\!07}{23\!\cdots\!20}a^{16}+\frac{68\!\cdots\!27}{23\!\cdots\!67}a^{15}-\frac{43\!\cdots\!59}{23\!\cdots\!20}a^{14}+\frac{17\!\cdots\!09}{27\!\cdots\!40}a^{13}-\frac{58\!\cdots\!97}{23\!\cdots\!20}a^{12}+\frac{12\!\cdots\!31}{25\!\cdots\!64}a^{11}-\frac{19\!\cdots\!19}{16\!\cdots\!40}a^{10}+\frac{65\!\cdots\!93}{20\!\cdots\!30}a^{9}-\frac{17\!\cdots\!59}{41\!\cdots\!60}a^{8}-\frac{31\!\cdots\!39}{41\!\cdots\!60}a^{7}+\frac{10\!\cdots\!51}{41\!\cdots\!60}a^{6}-\frac{11\!\cdots\!43}{69\!\cdots\!10}a^{5}-\frac{36\!\cdots\!59}{13\!\cdots\!20}a^{4}-\frac{27\!\cdots\!11}{20\!\cdots\!30}a^{3}-\frac{65\!\cdots\!58}{10\!\cdots\!15}a^{2}-\frac{94\!\cdots\!12}{10\!\cdots\!85}a+\frac{98\!\cdots\!42}{95\!\cdots\!65}$, $\frac{34\!\cdots\!97}{46\!\cdots\!40}a^{17}-\frac{41\!\cdots\!17}{16\!\cdots\!40}a^{16}+\frac{35\!\cdots\!43}{83\!\cdots\!20}a^{15}-\frac{80\!\cdots\!63}{55\!\cdots\!80}a^{14}+\frac{23\!\cdots\!27}{23\!\cdots\!70}a^{13}-\frac{96\!\cdots\!63}{33\!\cdots\!48}a^{12}+\frac{10\!\cdots\!79}{83\!\cdots\!20}a^{11}-\frac{37\!\cdots\!55}{15\!\cdots\!88}a^{10}+\frac{47\!\cdots\!01}{69\!\cdots\!10}a^{9}-\frac{90\!\cdots\!27}{14\!\cdots\!45}a^{8}+\frac{57\!\cdots\!11}{41\!\cdots\!60}a^{7}+\frac{64\!\cdots\!43}{69\!\cdots\!10}a^{6}+\frac{22\!\cdots\!97}{20\!\cdots\!30}a^{5}+\frac{23\!\cdots\!17}{59\!\cdots\!80}a^{4}+\frac{20\!\cdots\!41}{29\!\cdots\!90}a^{3}-\frac{89\!\cdots\!09}{34\!\cdots\!05}a^{2}+\frac{14\!\cdots\!86}{95\!\cdots\!65}a-\frac{54\!\cdots\!46}{95\!\cdots\!65}$, $\frac{95\!\cdots\!31}{33\!\cdots\!80}a^{17}-\frac{25\!\cdots\!61}{92\!\cdots\!80}a^{16}+\frac{69\!\cdots\!93}{33\!\cdots\!80}a^{15}-\frac{47\!\cdots\!91}{41\!\cdots\!60}a^{14}+\frac{55\!\cdots\!21}{11\!\cdots\!60}a^{13}-\frac{32\!\cdots\!51}{18\!\cdots\!60}a^{12}+\frac{14\!\cdots\!61}{30\!\cdots\!80}a^{11}-\frac{91\!\cdots\!21}{83\!\cdots\!20}a^{10}+\frac{30\!\cdots\!43}{16\!\cdots\!40}a^{9}-\frac{37\!\cdots\!87}{16\!\cdots\!40}a^{8}+\frac{76\!\cdots\!63}{10\!\cdots\!15}a^{7}+\frac{64\!\cdots\!27}{41\!\cdots\!60}a^{6}-\frac{38\!\cdots\!19}{83\!\cdots\!20}a^{5}-\frac{25\!\cdots\!09}{83\!\cdots\!12}a^{4}+\frac{19\!\cdots\!17}{13\!\cdots\!20}a^{3}-\frac{14\!\cdots\!01}{41\!\cdots\!60}a^{2}+\frac{23\!\cdots\!33}{95\!\cdots\!65}a-\frac{48\!\cdots\!02}{45\!\cdots\!65}$, $\frac{43\!\cdots\!53}{11\!\cdots\!60}a^{17}-\frac{51\!\cdots\!87}{16\!\cdots\!40}a^{16}+\frac{74\!\cdots\!91}{33\!\cdots\!80}a^{15}-\frac{18\!\cdots\!13}{16\!\cdots\!40}a^{14}+\frac{14\!\cdots\!57}{33\!\cdots\!80}a^{13}-\frac{55\!\cdots\!33}{38\!\cdots\!60}a^{12}+\frac{17\!\cdots\!27}{47\!\cdots\!40}a^{11}-\frac{13\!\cdots\!97}{16\!\cdots\!40}a^{10}+\frac{48\!\cdots\!99}{38\!\cdots\!60}a^{9}-\frac{10\!\cdots\!11}{55\!\cdots\!80}a^{8}+\frac{22\!\cdots\!55}{20\!\cdots\!03}a^{7}-\frac{16\!\cdots\!11}{10\!\cdots\!15}a^{6}-\frac{47\!\cdots\!11}{92\!\cdots\!80}a^{5}-\frac{52\!\cdots\!03}{83\!\cdots\!12}a^{4}+\frac{33\!\cdots\!51}{69\!\cdots\!10}a^{3}-\frac{25\!\cdots\!41}{13\!\cdots\!20}a^{2}+\frac{51\!\cdots\!15}{38\!\cdots\!46}a-\frac{45\!\cdots\!71}{63\!\cdots\!91}$, $\frac{44\!\cdots\!01}{79\!\cdots\!40}a^{17}-\frac{32\!\cdots\!93}{16\!\cdots\!24}a^{16}+\frac{38\!\cdots\!03}{18\!\cdots\!60}a^{15}-\frac{23\!\cdots\!33}{41\!\cdots\!60}a^{14}+\frac{48\!\cdots\!27}{16\!\cdots\!40}a^{13}-\frac{16\!\cdots\!21}{34\!\cdots\!05}a^{12}+\frac{94\!\cdots\!69}{55\!\cdots\!80}a^{11}-\frac{57\!\cdots\!19}{83\!\cdots\!20}a^{10}+\frac{17\!\cdots\!19}{41\!\cdots\!60}a^{9}+\frac{47\!\cdots\!47}{83\!\cdots\!20}a^{8}+\frac{11\!\cdots\!39}{16\!\cdots\!24}a^{7}+\frac{17\!\cdots\!51}{83\!\cdots\!20}a^{6}+\frac{12\!\cdots\!61}{41\!\cdots\!60}a^{5}+\frac{91\!\cdots\!41}{41\!\cdots\!60}a^{4}+\frac{59\!\cdots\!91}{10\!\cdots\!15}a^{3}-\frac{47\!\cdots\!61}{11\!\cdots\!35}a^{2}-\frac{98\!\cdots\!54}{95\!\cdots\!65}a+\frac{47\!\cdots\!99}{38\!\cdots\!46}$, $\frac{24\!\cdots\!89}{53\!\cdots\!60}a^{17}-\frac{25\!\cdots\!61}{69\!\cdots\!10}a^{16}+\frac{17\!\cdots\!27}{66\!\cdots\!96}a^{15}-\frac{55\!\cdots\!03}{41\!\cdots\!60}a^{14}+\frac{17\!\cdots\!01}{33\!\cdots\!80}a^{13}-\frac{29\!\cdots\!91}{16\!\cdots\!40}a^{12}+\frac{15\!\cdots\!69}{33\!\cdots\!80}a^{11}-\frac{21\!\cdots\!41}{20\!\cdots\!30}a^{10}+\frac{13\!\cdots\!01}{83\!\cdots\!20}a^{9}-\frac{34\!\cdots\!37}{16\!\cdots\!40}a^{8}+\frac{15\!\cdots\!67}{11\!\cdots\!35}a^{7}+\frac{52\!\cdots\!41}{83\!\cdots\!20}a^{6}-\frac{16\!\cdots\!23}{83\!\cdots\!20}a^{5}+\frac{16\!\cdots\!61}{13\!\cdots\!20}a^{4}+\frac{41\!\cdots\!17}{41\!\cdots\!06}a^{3}-\frac{19\!\cdots\!03}{13\!\cdots\!20}a^{2}+\frac{10\!\cdots\!16}{31\!\cdots\!55}a-\frac{22\!\cdots\!49}{19\!\cdots\!30}$, $\frac{68\!\cdots\!37}{47\!\cdots\!40}a^{17}-\frac{15\!\cdots\!87}{16\!\cdots\!40}a^{16}+\frac{20\!\cdots\!43}{33\!\cdots\!80}a^{15}-\frac{42\!\cdots\!91}{16\!\cdots\!40}a^{14}+\frac{54\!\cdots\!89}{66\!\cdots\!96}a^{13}-\frac{14\!\cdots\!07}{83\!\cdots\!20}a^{12}+\frac{83\!\cdots\!27}{60\!\cdots\!36}a^{11}+\frac{89\!\cdots\!41}{16\!\cdots\!40}a^{10}-\frac{84\!\cdots\!07}{27\!\cdots\!40}a^{9}+\frac{23\!\cdots\!49}{33\!\cdots\!48}a^{8}-\frac{10\!\cdots\!93}{83\!\cdots\!20}a^{7}+\frac{84\!\cdots\!81}{92\!\cdots\!68}a^{6}+\frac{30\!\cdots\!77}{16\!\cdots\!24}a^{5}-\frac{16\!\cdots\!65}{92\!\cdots\!68}a^{4}+\frac{15\!\cdots\!91}{69\!\cdots\!10}a^{3}+\frac{26\!\cdots\!83}{83\!\cdots\!12}a^{2}-\frac{23\!\cdots\!54}{10\!\cdots\!85}a+\frac{15\!\cdots\!23}{10\!\cdots\!85}$, $\frac{12\!\cdots\!11}{13\!\cdots\!20}a^{17}-\frac{55\!\cdots\!49}{12\!\cdots\!80}a^{16}+\frac{28\!\cdots\!16}{78\!\cdots\!55}a^{15}-\frac{50\!\cdots\!39}{41\!\cdots\!60}a^{14}+\frac{10\!\cdots\!85}{20\!\cdots\!88}a^{13}-\frac{98\!\cdots\!33}{11\!\cdots\!80}a^{12}+\frac{29\!\cdots\!89}{15\!\cdots\!10}a^{11}+\frac{96\!\cdots\!01}{13\!\cdots\!20}a^{10}-\frac{75\!\cdots\!63}{38\!\cdots\!60}a^{9}+\frac{50\!\cdots\!39}{31\!\cdots\!20}a^{8}-\frac{12\!\cdots\!03}{31\!\cdots\!20}a^{7}+\frac{70\!\cdots\!53}{34\!\cdots\!98}a^{6}+\frac{74\!\cdots\!33}{15\!\cdots\!10}a^{5}+\frac{66\!\cdots\!82}{78\!\cdots\!55}a^{4}+\frac{50\!\cdots\!69}{62\!\cdots\!64}a^{3}+\frac{53\!\cdots\!51}{52\!\cdots\!70}a^{2}-\frac{12\!\cdots\!03}{14\!\cdots\!10}a+\frac{46\!\cdots\!38}{71\!\cdots\!05}$ (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: | \( 3932278.1508 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 3932278.1508 \cdot 3}{2\cdot\sqrt{5627055140868788987624861696}}\cr\approx \mathstrut & 1.2000897294 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 18 |
The 6 conjugacy class representatives for $C_3^2 : C_2$ |
Character table for $C_3^2 : C_2$ |
Intermediate fields
\(\Q(\sqrt{-11}) \), 3.1.12716.2 x3, 3.1.44.1 x3, 3.1.12716.1 x3, 3.1.3179.1 x3, 6.0.1778663216.2, 6.0.21296.1, 6.0.1778663216.1, 6.0.111166451.1, 9.1.22617481454656.1 x9 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{6}$ | ${\href{/padicField/5.3.0.1}{3} }^{6}$ | ${\href{/padicField/7.2.0.1}{2} }^{9}$ | R | ${\href{/padicField/13.2.0.1}{2} }^{9}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{9}$ | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.2.0.1}{2} }^{9}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.1.0.1}{1} }^{18}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(11\) | 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(17\) | 17.6.4.1 | $x^{6} + 48 x^{5} + 879 x^{4} + 7682 x^{3} + 44916 x^{2} + 265428 x + 127425$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
17.6.4.1 | $x^{6} + 48 x^{5} + 879 x^{4} + 7682 x^{3} + 44916 x^{2} + 265428 x + 127425$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
17.6.4.1 | $x^{6} + 48 x^{5} + 879 x^{4} + 7682 x^{3} + 44916 x^{2} + 265428 x + 127425$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |