Normalized defining polynomial
\( x^{18} - 5 x^{16} - 16 x^{15} - 20 x^{14} + 152 x^{13} + 370 x^{12} + 32 x^{11} - 2786 x^{10} - 3960 x^{9} + 13300 x^{8} + 14472 x^{7} - 12096 x^{6} - 79632 x^{5} + \cdots + 106677 \)
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: | \(-37621770755235979566165000192\) \(\medspace = -\,2^{20}\cdot 3^{9}\cdot 67^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(38.68\) | 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: | $2^{4/3}3^{1/2}67^{5/6}\approx 145.0985114774686$ | ||
Ramified primes: | \(2\), \(3\), \(67\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}+\frac{1}{6}a^{4}+\frac{1}{6}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{7}+\frac{1}{6}a^{5}+\frac{1}{6}a^{3}-\frac{1}{2}a$, $\frac{1}{18}a^{8}+\frac{1}{18}a^{7}+\frac{1}{18}a^{6}+\frac{7}{18}a^{5}-\frac{5}{18}a^{4}+\frac{1}{18}a^{3}-\frac{1}{6}a^{2}+\frac{1}{6}a$, $\frac{1}{18}a^{9}+\frac{1}{3}a^{5}-\frac{2}{9}a^{3}-\frac{1}{6}a$, $\frac{1}{18}a^{10}+\frac{4}{9}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{18}a^{11}+\frac{4}{9}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{108}a^{12}-\frac{1}{54}a^{10}-\frac{1}{54}a^{9}+\frac{1}{108}a^{8}-\frac{1}{54}a^{7}-\frac{1}{18}a^{6}+\frac{23}{54}a^{5}+\frac{5}{12}a^{4}+\frac{7}{18}a^{3}+\frac{1}{3}a-\frac{1}{4}$, $\frac{1}{324}a^{13}+\frac{1}{81}a^{11}-\frac{1}{162}a^{10}+\frac{1}{324}a^{9}-\frac{1}{162}a^{8}+\frac{1}{27}a^{7}-\frac{2}{81}a^{6}+\frac{37}{108}a^{5}-\frac{1}{27}a^{4}+\frac{2}{9}a^{3}+\frac{5}{18}a^{2}+\frac{1}{12}a-\frac{1}{2}$, $\frac{1}{324}a^{14}+\frac{1}{324}a^{12}-\frac{1}{162}a^{11}+\frac{7}{324}a^{10}+\frac{1}{81}a^{9}-\frac{1}{36}a^{8}-\frac{5}{81}a^{7}+\frac{1}{108}a^{6}+\frac{4}{27}a^{5}-\frac{1}{4}a^{4}-\frac{1}{6}a^{3}-\frac{1}{12}a^{2}+\frac{1}{4}$, $\frac{1}{648}a^{15}-\frac{1}{648}a^{14}-\frac{1}{648}a^{13}-\frac{1}{216}a^{12}+\frac{1}{648}a^{11}-\frac{17}{648}a^{10}-\frac{5}{216}a^{9}+\frac{11}{648}a^{8}-\frac{37}{648}a^{7}+\frac{25}{648}a^{6}+\frac{5}{72}a^{5}-\frac{79}{216}a^{4}-\frac{17}{72}a^{3}-\frac{11}{72}a^{2}-\frac{1}{8}a+\frac{1}{8}$, $\frac{1}{1944}a^{16}-\frac{1}{1944}a^{15}-\frac{1}{1944}a^{14}-\frac{1}{648}a^{13}+\frac{1}{1944}a^{12}-\frac{53}{1944}a^{11}+\frac{7}{648}a^{10}+\frac{47}{1944}a^{9}-\frac{1}{1944}a^{8}-\frac{155}{1944}a^{7}-\frac{5}{72}a^{6}+\frac{125}{648}a^{5}-\frac{101}{216}a^{4}+\frac{61}{216}a^{3}+\frac{7}{24}a^{2}+\frac{3}{8}a$, $\frac{1}{27\!\cdots\!72}a^{17}+\frac{32\!\cdots\!93}{30\!\cdots\!08}a^{16}-\frac{88\!\cdots\!89}{27\!\cdots\!72}a^{15}+\frac{33\!\cdots\!39}{27\!\cdots\!72}a^{14}+\frac{22\!\cdots\!93}{27\!\cdots\!72}a^{13}-\frac{43\!\cdots\!69}{27\!\cdots\!72}a^{12}+\frac{92\!\cdots\!29}{27\!\cdots\!72}a^{11}+\frac{91\!\cdots\!73}{27\!\cdots\!72}a^{10}+\frac{54\!\cdots\!95}{27\!\cdots\!72}a^{9}+\frac{11\!\cdots\!07}{92\!\cdots\!24}a^{8}-\frac{19\!\cdots\!67}{27\!\cdots\!72}a^{7}+\frac{39\!\cdots\!39}{92\!\cdots\!24}a^{6}-\frac{32\!\cdots\!67}{92\!\cdots\!24}a^{5}+\frac{10\!\cdots\!21}{30\!\cdots\!08}a^{4}-\frac{14\!\cdots\!17}{30\!\cdots\!08}a^{3}-\frac{60\!\cdots\!37}{34\!\cdots\!12}a^{2}-\frac{45\!\cdots\!93}{28\!\cdots\!26}a+\frac{23\!\cdots\!69}{56\!\cdots\!52}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
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: | \( \frac{13149390871}{101161545473124} a^{17} + \frac{17105954305}{202323090946248} a^{16} - \frac{108972610469}{202323090946248} a^{15} - \frac{170418584869}{67441030315416} a^{14} - \frac{925569130669}{202323090946248} a^{13} + \frac{3251323147535}{202323090946248} a^{12} + \frac{3976104782173}{67441030315416} a^{11} + \frac{11085754317583}{202323090946248} a^{10} - \frac{63407050411535}{202323090946248} a^{9} - \frac{151288649895223}{202323090946248} a^{8} + \frac{69728492846111}{67441030315416} a^{7} + \frac{166325193411949}{67441030315416} a^{6} + \frac{25138580073269}{22480343438472} a^{5} - \frac{208906423498651}{22480343438472} a^{4} - \frac{3450147519897}{832605312536} a^{3} + \frac{41748836071829}{2497815937608} a^{2} + \frac{3757648791533}{2497815937608} a - \frac{3608743361031}{416302656268} \) (order $6$) | 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: | $\frac{68\!\cdots\!48}{75\!\cdots\!87}a^{17}+\frac{28\!\cdots\!27}{30\!\cdots\!48}a^{16}-\frac{55\!\cdots\!47}{15\!\cdots\!74}a^{15}-\frac{62\!\cdots\!79}{33\!\cdots\!72}a^{14}-\frac{28\!\cdots\!89}{75\!\cdots\!87}a^{13}+\frac{30\!\cdots\!51}{30\!\cdots\!48}a^{12}+\frac{22\!\cdots\!29}{50\!\cdots\!58}a^{11}+\frac{15\!\cdots\!73}{30\!\cdots\!48}a^{10}-\frac{31\!\cdots\!57}{15\!\cdots\!74}a^{9}-\frac{17\!\cdots\!07}{30\!\cdots\!48}a^{8}+\frac{10\!\cdots\!17}{16\!\cdots\!86}a^{7}+\frac{20\!\cdots\!19}{10\!\cdots\!16}a^{6}+\frac{87\!\cdots\!93}{83\!\cdots\!43}a^{5}-\frac{21\!\cdots\!73}{33\!\cdots\!72}a^{4}-\frac{30\!\cdots\!25}{62\!\cdots\!18}a^{3}+\frac{14\!\cdots\!03}{11\!\cdots\!24}a^{2}+\frac{23\!\cdots\!93}{62\!\cdots\!18}a-\frac{54\!\cdots\!13}{62\!\cdots\!18}$, $\frac{72\!\cdots\!97}{11\!\cdots\!53}a^{17}+\frac{54\!\cdots\!73}{92\!\cdots\!24}a^{16}-\frac{22\!\cdots\!75}{92\!\cdots\!24}a^{15}-\frac{11\!\cdots\!35}{92\!\cdots\!24}a^{14}-\frac{23\!\cdots\!15}{92\!\cdots\!24}a^{13}+\frac{64\!\cdots\!31}{92\!\cdots\!24}a^{12}+\frac{27\!\cdots\!61}{92\!\cdots\!24}a^{11}+\frac{10\!\cdots\!77}{30\!\cdots\!08}a^{10}-\frac{43\!\cdots\!15}{30\!\cdots\!08}a^{9}-\frac{35\!\cdots\!19}{92\!\cdots\!24}a^{8}+\frac{39\!\cdots\!07}{92\!\cdots\!24}a^{7}+\frac{43\!\cdots\!21}{34\!\cdots\!12}a^{6}+\frac{72\!\cdots\!11}{10\!\cdots\!36}a^{5}-\frac{43\!\cdots\!39}{10\!\cdots\!36}a^{4}-\frac{93\!\cdots\!59}{34\!\cdots\!12}a^{3}+\frac{27\!\cdots\!99}{34\!\cdots\!12}a^{2}+\frac{68\!\cdots\!55}{34\!\cdots\!12}a-\frac{28\!\cdots\!09}{56\!\cdots\!52}$, $\frac{57\!\cdots\!07}{27\!\cdots\!72}a^{17}+\frac{53\!\cdots\!05}{17\!\cdots\!56}a^{16}-\frac{36\!\cdots\!25}{27\!\cdots\!72}a^{15}-\frac{11\!\cdots\!35}{27\!\cdots\!72}a^{14}-\frac{13\!\cdots\!85}{27\!\cdots\!72}a^{13}+\frac{10\!\cdots\!15}{27\!\cdots\!72}a^{12}+\frac{29\!\cdots\!89}{27\!\cdots\!72}a^{11}+\frac{60\!\cdots\!97}{27\!\cdots\!72}a^{10}-\frac{20\!\cdots\!95}{27\!\cdots\!72}a^{9}-\frac{12\!\cdots\!73}{92\!\cdots\!24}a^{8}+\frac{74\!\cdots\!49}{27\!\cdots\!72}a^{7}+\frac{51\!\cdots\!37}{92\!\cdots\!24}a^{6}-\frac{37\!\cdots\!57}{92\!\cdots\!24}a^{5}-\frac{66\!\cdots\!91}{30\!\cdots\!08}a^{4}-\frac{28\!\cdots\!45}{30\!\cdots\!08}a^{3}+\frac{15\!\cdots\!43}{34\!\cdots\!12}a^{2}+\frac{28\!\cdots\!01}{42\!\cdots\!39}a-\frac{36\!\cdots\!49}{11\!\cdots\!04}$, $\frac{49\!\cdots\!23}{13\!\cdots\!36}a^{17}+\frac{72\!\cdots\!91}{69\!\cdots\!18}a^{16}-\frac{32\!\cdots\!95}{27\!\cdots\!72}a^{15}-\frac{17\!\cdots\!33}{34\!\cdots\!12}a^{14}-\frac{30\!\cdots\!77}{27\!\cdots\!72}a^{13}+\frac{10\!\cdots\!59}{27\!\cdots\!72}a^{12}+\frac{11\!\cdots\!35}{92\!\cdots\!24}a^{11}+\frac{30\!\cdots\!99}{27\!\cdots\!72}a^{10}-\frac{18\!\cdots\!91}{27\!\cdots\!72}a^{9}-\frac{33\!\cdots\!23}{27\!\cdots\!72}a^{8}+\frac{25\!\cdots\!29}{92\!\cdots\!24}a^{7}+\frac{19\!\cdots\!19}{92\!\cdots\!24}a^{6}+\frac{27\!\cdots\!09}{30\!\cdots\!08}a^{5}-\frac{38\!\cdots\!19}{30\!\cdots\!08}a^{4}+\frac{23\!\cdots\!03}{34\!\cdots\!12}a^{3}+\frac{77\!\cdots\!87}{10\!\cdots\!36}a^{2}-\frac{39\!\cdots\!51}{34\!\cdots\!12}a+\frac{16\!\cdots\!29}{11\!\cdots\!04}$, $\frac{59\!\cdots\!89}{27\!\cdots\!72}a^{17}+\frac{36\!\cdots\!67}{13\!\cdots\!36}a^{16}-\frac{21\!\cdots\!11}{27\!\cdots\!72}a^{15}-\frac{13\!\cdots\!71}{30\!\cdots\!08}a^{14}-\frac{27\!\cdots\!51}{27\!\cdots\!72}a^{13}+\frac{59\!\cdots\!99}{27\!\cdots\!72}a^{12}+\frac{36\!\cdots\!63}{34\!\cdots\!12}a^{11}+\frac{38\!\cdots\!05}{27\!\cdots\!72}a^{10}-\frac{12\!\cdots\!49}{27\!\cdots\!72}a^{9}-\frac{39\!\cdots\!35}{27\!\cdots\!72}a^{8}+\frac{10\!\cdots\!33}{92\!\cdots\!24}a^{7}+\frac{42\!\cdots\!09}{92\!\cdots\!24}a^{6}+\frac{94\!\cdots\!91}{30\!\cdots\!08}a^{5}-\frac{42\!\cdots\!59}{30\!\cdots\!08}a^{4}-\frac{12\!\cdots\!53}{10\!\cdots\!36}a^{3}+\frac{27\!\cdots\!13}{10\!\cdots\!36}a^{2}+\frac{84\!\cdots\!51}{85\!\cdots\!78}a-\frac{21\!\cdots\!25}{11\!\cdots\!04}$, $\frac{24\!\cdots\!51}{69\!\cdots\!18}a^{17}+\frac{88\!\cdots\!31}{13\!\cdots\!36}a^{16}-\frac{20\!\cdots\!09}{27\!\cdots\!72}a^{15}-\frac{22\!\cdots\!39}{30\!\cdots\!08}a^{14}-\frac{57\!\cdots\!35}{27\!\cdots\!72}a^{13}+\frac{50\!\cdots\!25}{27\!\cdots\!72}a^{12}+\frac{16\!\cdots\!09}{92\!\cdots\!24}a^{11}+\frac{93\!\cdots\!01}{27\!\cdots\!72}a^{10}-\frac{12\!\cdots\!21}{27\!\cdots\!72}a^{9}-\frac{67\!\cdots\!53}{27\!\cdots\!72}a^{8}+\frac{13\!\cdots\!31}{92\!\cdots\!24}a^{7}+\frac{60\!\cdots\!09}{92\!\cdots\!24}a^{6}+\frac{27\!\cdots\!03}{30\!\cdots\!08}a^{5}-\frac{43\!\cdots\!33}{30\!\cdots\!08}a^{4}-\frac{82\!\cdots\!87}{34\!\cdots\!12}a^{3}+\frac{80\!\cdots\!63}{34\!\cdots\!12}a^{2}+\frac{59\!\cdots\!73}{34\!\cdots\!12}a-\frac{17\!\cdots\!43}{11\!\cdots\!04}$, $\frac{15\!\cdots\!35}{69\!\cdots\!18}a^{17}+\frac{39\!\cdots\!59}{13\!\cdots\!36}a^{16}-\frac{78\!\cdots\!15}{92\!\cdots\!24}a^{15}-\frac{12\!\cdots\!83}{27\!\cdots\!72}a^{14}-\frac{27\!\cdots\!29}{27\!\cdots\!72}a^{13}+\frac{20\!\cdots\!43}{92\!\cdots\!24}a^{12}+\frac{30\!\cdots\!15}{27\!\cdots\!72}a^{11}+\frac{38\!\cdots\!89}{27\!\cdots\!72}a^{10}-\frac{42\!\cdots\!21}{92\!\cdots\!24}a^{9}-\frac{40\!\cdots\!09}{27\!\cdots\!72}a^{8}+\frac{35\!\cdots\!57}{27\!\cdots\!72}a^{7}+\frac{45\!\cdots\!89}{92\!\cdots\!24}a^{6}+\frac{27\!\cdots\!75}{92\!\cdots\!24}a^{5}-\frac{16\!\cdots\!75}{11\!\cdots\!04}a^{4}-\frac{41\!\cdots\!95}{30\!\cdots\!08}a^{3}+\frac{34\!\cdots\!69}{11\!\cdots\!04}a^{2}+\frac{13\!\cdots\!09}{11\!\cdots\!04}a-\frac{24\!\cdots\!11}{11\!\cdots\!04}$, $\frac{53\!\cdots\!18}{11\!\cdots\!53}a^{17}+\frac{27\!\cdots\!73}{27\!\cdots\!72}a^{16}-\frac{26\!\cdots\!91}{27\!\cdots\!72}a^{15}-\frac{28\!\cdots\!49}{27\!\cdots\!72}a^{14}-\frac{27\!\cdots\!81}{92\!\cdots\!24}a^{13}+\frac{56\!\cdots\!43}{27\!\cdots\!72}a^{12}+\frac{69\!\cdots\!05}{27\!\cdots\!72}a^{11}+\frac{46\!\cdots\!43}{92\!\cdots\!24}a^{10}-\frac{14\!\cdots\!49}{27\!\cdots\!72}a^{9}-\frac{97\!\cdots\!03}{27\!\cdots\!72}a^{8}-\frac{12\!\cdots\!09}{27\!\cdots\!72}a^{7}+\frac{10\!\cdots\!59}{10\!\cdots\!36}a^{6}+\frac{12\!\cdots\!33}{92\!\cdots\!24}a^{5}-\frac{52\!\cdots\!03}{30\!\cdots\!08}a^{4}-\frac{12\!\cdots\!45}{30\!\cdots\!08}a^{3}+\frac{10\!\cdots\!03}{34\!\cdots\!12}a^{2}+\frac{40\!\cdots\!75}{11\!\cdots\!04}a-\frac{15\!\cdots\!41}{56\!\cdots\!52}$ (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: | \( 306761663.9033038 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 306761663.9033038 \cdot 3}{6\cdot\sqrt{37621770755235979566165000192}}\cr\approx \mathstrut & 12.0689722279917 \end{aligned}\] (assuming GRH)
Galois group
$C_3\times S_3^2$ (as 18T46):
A solvable group of order 108 |
The 27 conjugacy class representatives for $C_3\times S_3^2$ |
Character table for $C_3\times S_3^2$ is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \), 3.1.804.1, 6.0.1939248.2, 6.0.1939248.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 27 sibling: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | 12.0.19400185409620625915904.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.4.2 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
2.12.16.3 | $x^{12} + 4 x^{11} + 8 x^{10} + 10 x^{9} + 8 x^{8} + 4 x^{7} + 2 x^{6} - 4 x^{5} - 4 x^{4} + 4 x^{3} + 4$ | $6$ | $2$ | $16$ | $C_6\times S_3$ | $[2]_{3}^{6}$ | |
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(67\) | 67.3.2.2 | $x^{3} + 268$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
67.3.0.1 | $x^{3} + 6 x + 65$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
67.6.5.3 | $x^{6} + 469$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
67.6.3.1 | $x^{6} + 26934 x^{2} - 19549595$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |