Normalized defining polynomial
\( x^{18} - 3 x^{17} - 2 x^{16} + 27 x^{15} - 63 x^{14} - 182 x^{13} + 326 x^{12} + 24 x^{11} + 1072 x^{10} + 3160 x^{9} + 1576 x^{8} + 3344 x^{7} + 7792 x^{6} + 7808 x^{5} + 30080 x^{4} + \cdots + 4096 \)
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: | \(-5632074928630950758566400360448\) \(\medspace = -\,2^{18}\cdot 3^{9}\cdot 127^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(51.09\) | 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: | $2^{3/2}3^{1/2}127^{5/6}\approx 277.50792385820444$ | ||
Ramified primes: | \(2\), \(3\), \(127\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{8}a^{9}+\frac{1}{16}a^{8}-\frac{1}{16}a^{7}+\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a$, $\frac{1}{32}a^{12}-\frac{1}{32}a^{11}-\frac{1}{16}a^{10}+\frac{1}{32}a^{9}+\frac{3}{32}a^{8}+\frac{1}{16}a^{7}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{13}-\frac{1}{32}a^{11}+\frac{1}{32}a^{10}-\frac{1}{8}a^{9}-\frac{1}{32}a^{8}+\frac{1}{8}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{64}a^{14}-\frac{1}{64}a^{13}-\frac{1}{64}a^{11}-\frac{1}{64}a^{10}-\frac{1}{16}a^{9}+\frac{1}{32}a^{8}-\frac{1}{16}a^{7}+\frac{1}{8}a^{6}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{256}a^{15}+\frac{1}{256}a^{14}-\frac{1}{128}a^{13}-\frac{1}{256}a^{12}+\frac{5}{256}a^{11}-\frac{7}{128}a^{10}-\frac{11}{128}a^{9}-\frac{1}{32}a^{8}-\frac{1}{32}a^{7}-\frac{5}{32}a^{6}-\frac{3}{32}a^{5}-\frac{3}{16}a^{4}-\frac{7}{16}a^{3}-\frac{1}{4}a$, $\frac{1}{512}a^{16}-\frac{1}{512}a^{15}-\frac{1}{128}a^{14}+\frac{3}{512}a^{13}+\frac{7}{512}a^{12}+\frac{1}{64}a^{11}-\frac{13}{256}a^{10}+\frac{9}{128}a^{9}+\frac{5}{64}a^{8}-\frac{7}{64}a^{7}-\frac{9}{64}a^{6}-\frac{1}{32}a^{4}+\frac{3}{16}a^{3}-\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{11\!\cdots\!32}a^{17}+\frac{98\!\cdots\!17}{11\!\cdots\!32}a^{16}-\frac{53\!\cdots\!39}{55\!\cdots\!16}a^{15}-\frac{56\!\cdots\!45}{11\!\cdots\!32}a^{14}+\frac{55\!\cdots\!29}{11\!\cdots\!32}a^{13}+\frac{34\!\cdots\!99}{55\!\cdots\!16}a^{12}+\frac{40\!\cdots\!99}{55\!\cdots\!16}a^{11}+\frac{25\!\cdots\!43}{69\!\cdots\!52}a^{10}+\frac{60\!\cdots\!55}{69\!\cdots\!52}a^{9}+\frac{53\!\cdots\!87}{13\!\cdots\!04}a^{8}+\frac{10\!\cdots\!89}{13\!\cdots\!04}a^{7}-\frac{11\!\cdots\!37}{69\!\cdots\!52}a^{6}-\frac{95\!\cdots\!51}{54\!\cdots\!76}a^{5}+\frac{34\!\cdots\!61}{17\!\cdots\!88}a^{4}-\frac{19\!\cdots\!39}{87\!\cdots\!44}a^{3}-\frac{14\!\cdots\!85}{43\!\cdots\!72}a^{2}+\frac{18\!\cdots\!85}{43\!\cdots\!72}a-\frac{22\!\cdots\!91}{10\!\cdots\!68}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{693570908493955}{432739011821809664} a^{17} + \frac{2340308272355869}{432739011821809664} a^{16} + \frac{262576486421757}{216369505910904832} a^{15} - \frac{18976322657208169}{432739011821809664} a^{14} + \frac{50802472029220505}{432739011821809664} a^{13} + \frac{53823693559736835}{216369505910904832} a^{12} - \frac{133874644475789629}{216369505910904832} a^{11} + \frac{5121641378728171}{27046188238863104} a^{10} - \frac{47915558313508693}{27046188238863104} a^{9} - \frac{238981647733541057}{54092376477726208} a^{8} - \frac{45313337058619219}{54092376477726208} a^{7} - \frac{133093303032384041}{27046188238863104} a^{6} - \frac{290897290481288057}{27046188238863104} a^{5} - \frac{56579125697596667}{6761547059715776} a^{4} - \frac{152010996061114863}{3380773529857888} a^{3} - \frac{134790296096409437}{1690386764928944} a^{2} - \frac{112464181087348891}{1690386764928944} a - \frac{6773562299056895}{422596691232236} \) (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: | $\frac{37\!\cdots\!83}{11\!\cdots\!32}a^{17}-\frac{12\!\cdots\!21}{11\!\cdots\!32}a^{16}-\frac{16\!\cdots\!37}{55\!\cdots\!16}a^{15}+\frac{10\!\cdots\!77}{11\!\cdots\!32}a^{14}-\frac{27\!\cdots\!73}{11\!\cdots\!32}a^{13}-\frac{29\!\cdots\!31}{55\!\cdots\!16}a^{12}+\frac{71\!\cdots\!61}{55\!\cdots\!16}a^{11}-\frac{11\!\cdots\!19}{34\!\cdots\!76}a^{10}+\frac{25\!\cdots\!61}{69\!\cdots\!52}a^{9}+\frac{13\!\cdots\!37}{13\!\cdots\!04}a^{8}+\frac{29\!\cdots\!07}{13\!\cdots\!04}a^{7}+\frac{72\!\cdots\!49}{69\!\cdots\!52}a^{6}+\frac{12\!\cdots\!99}{54\!\cdots\!76}a^{5}+\frac{32\!\cdots\!65}{17\!\cdots\!88}a^{4}+\frac{82\!\cdots\!83}{87\!\cdots\!44}a^{3}+\frac{74\!\cdots\!47}{43\!\cdots\!72}a^{2}+\frac{63\!\cdots\!23}{43\!\cdots\!72}a+\frac{43\!\cdots\!59}{10\!\cdots\!68}$, $\frac{18\!\cdots\!57}{11\!\cdots\!32}a^{17}+\frac{55\!\cdots\!77}{11\!\cdots\!32}a^{16}-\frac{24\!\cdots\!79}{55\!\cdots\!16}a^{15}+\frac{71\!\cdots\!55}{11\!\cdots\!32}a^{14}+\frac{19\!\cdots\!53}{11\!\cdots\!32}a^{13}-\frac{70\!\cdots\!25}{55\!\cdots\!16}a^{12}-\frac{24\!\cdots\!17}{55\!\cdots\!16}a^{11}+\frac{36\!\cdots\!95}{69\!\cdots\!52}a^{10}-\frac{21\!\cdots\!61}{69\!\cdots\!52}a^{9}+\frac{21\!\cdots\!03}{13\!\cdots\!04}a^{8}+\frac{35\!\cdots\!73}{13\!\cdots\!04}a^{7}-\frac{17\!\cdots\!45}{69\!\cdots\!52}a^{6}+\frac{21\!\cdots\!45}{54\!\cdots\!76}a^{5}+\frac{10\!\cdots\!37}{17\!\cdots\!88}a^{4}+\frac{51\!\cdots\!37}{87\!\cdots\!44}a^{3}+\frac{14\!\cdots\!99}{43\!\cdots\!72}a^{2}+\frac{19\!\cdots\!57}{43\!\cdots\!72}a+\frac{16\!\cdots\!05}{10\!\cdots\!68}$, $\frac{22\!\cdots\!69}{87\!\cdots\!16}a^{17}-\frac{76\!\cdots\!75}{87\!\cdots\!16}a^{16}-\frac{78\!\cdots\!23}{43\!\cdots\!08}a^{15}+\frac{61\!\cdots\!75}{87\!\cdots\!16}a^{14}-\frac{16\!\cdots\!71}{87\!\cdots\!16}a^{13}-\frac{17\!\cdots\!05}{43\!\cdots\!08}a^{12}+\frac{43\!\cdots\!43}{43\!\cdots\!08}a^{11}-\frac{83\!\cdots\!41}{27\!\cdots\!88}a^{10}+\frac{16\!\cdots\!93}{54\!\cdots\!76}a^{9}+\frac{76\!\cdots\!27}{10\!\cdots\!52}a^{8}+\frac{17\!\cdots\!81}{10\!\cdots\!52}a^{7}+\frac{42\!\cdots\!91}{54\!\cdots\!76}a^{6}+\frac{94\!\cdots\!11}{54\!\cdots\!76}a^{5}+\frac{19\!\cdots\!99}{13\!\cdots\!44}a^{4}+\frac{50\!\cdots\!51}{68\!\cdots\!72}a^{3}+\frac{44\!\cdots\!73}{34\!\cdots\!36}a^{2}+\frac{37\!\cdots\!49}{34\!\cdots\!36}a+\frac{24\!\cdots\!33}{85\!\cdots\!84}$, $\frac{15\!\cdots\!55}{21\!\cdots\!04}a^{17}-\frac{66\!\cdots\!67}{21\!\cdots\!04}a^{16}+\frac{10\!\cdots\!09}{54\!\cdots\!76}a^{15}+\frac{41\!\cdots\!13}{21\!\cdots\!04}a^{14}-\frac{15\!\cdots\!15}{21\!\cdots\!04}a^{13}-\frac{15\!\cdots\!27}{27\!\cdots\!88}a^{12}+\frac{37\!\cdots\!93}{10\!\cdots\!52}a^{11}-\frac{19\!\cdots\!33}{54\!\cdots\!76}a^{10}+\frac{27\!\cdots\!51}{27\!\cdots\!88}a^{9}+\frac{31\!\cdots\!07}{27\!\cdots\!88}a^{8}-\frac{25\!\cdots\!27}{27\!\cdots\!88}a^{7}+\frac{41\!\cdots\!69}{17\!\cdots\!68}a^{6}+\frac{36\!\cdots\!57}{13\!\cdots\!44}a^{5}+\frac{50\!\cdots\!41}{68\!\cdots\!72}a^{4}+\frac{61\!\cdots\!81}{34\!\cdots\!36}a^{3}+\frac{32\!\cdots\!03}{17\!\cdots\!68}a^{2}+\frac{14\!\cdots\!38}{21\!\cdots\!71}a+\frac{16\!\cdots\!61}{21\!\cdots\!71}$, $\frac{62\!\cdots\!69}{55\!\cdots\!16}a^{17}-\frac{17\!\cdots\!15}{55\!\cdots\!16}a^{16}-\frac{28\!\cdots\!75}{27\!\cdots\!08}a^{15}+\frac{12\!\cdots\!03}{55\!\cdots\!16}a^{14}-\frac{34\!\cdots\!83}{55\!\cdots\!16}a^{13}-\frac{43\!\cdots\!69}{27\!\cdots\!08}a^{12}+\frac{42\!\cdots\!03}{27\!\cdots\!08}a^{11}-\frac{26\!\cdots\!23}{34\!\cdots\!76}a^{10}+\frac{41\!\cdots\!43}{17\!\cdots\!88}a^{9}+\frac{21\!\cdots\!55}{69\!\cdots\!52}a^{8}+\frac{21\!\cdots\!29}{69\!\cdots\!52}a^{7}+\frac{12\!\cdots\!11}{34\!\cdots\!76}a^{6}+\frac{82\!\cdots\!53}{27\!\cdots\!88}a^{5}+\frac{15\!\cdots\!65}{87\!\cdots\!44}a^{4}+\frac{51\!\cdots\!69}{10\!\cdots\!68}a^{3}+\frac{13\!\cdots\!85}{21\!\cdots\!36}a^{2}+\frac{16\!\cdots\!95}{21\!\cdots\!36}a+\frac{11\!\cdots\!25}{54\!\cdots\!34}$, $\frac{51\!\cdots\!99}{13\!\cdots\!04}a^{17}-\frac{44\!\cdots\!15}{34\!\cdots\!76}a^{16}-\frac{40\!\cdots\!33}{13\!\cdots\!04}a^{15}+\frac{14\!\cdots\!95}{13\!\cdots\!04}a^{14}-\frac{18\!\cdots\!13}{69\!\cdots\!52}a^{13}-\frac{81\!\cdots\!77}{13\!\cdots\!04}a^{12}+\frac{25\!\cdots\!13}{17\!\cdots\!88}a^{11}-\frac{17\!\cdots\!43}{69\!\cdots\!52}a^{10}+\frac{36\!\cdots\!37}{87\!\cdots\!44}a^{9}+\frac{40\!\cdots\!09}{43\!\cdots\!72}a^{8}+\frac{10\!\cdots\!27}{87\!\cdots\!44}a^{7}+\frac{16\!\cdots\!71}{17\!\cdots\!88}a^{6}+\frac{28\!\cdots\!93}{17\!\cdots\!68}a^{5}+\frac{98\!\cdots\!49}{87\!\cdots\!44}a^{4}+\frac{21\!\cdots\!81}{21\!\cdots\!36}a^{3}+\frac{36\!\cdots\!07}{21\!\cdots\!36}a^{2}+\frac{71\!\cdots\!71}{54\!\cdots\!34}a+\frac{68\!\cdots\!71}{27\!\cdots\!17}$, $\frac{14\!\cdots\!53}{55\!\cdots\!16}a^{17}-\frac{49\!\cdots\!75}{55\!\cdots\!16}a^{16}-\frac{61\!\cdots\!45}{27\!\cdots\!08}a^{15}+\frac{40\!\cdots\!47}{55\!\cdots\!16}a^{14}-\frac{10\!\cdots\!99}{55\!\cdots\!16}a^{13}-\frac{11\!\cdots\!27}{27\!\cdots\!08}a^{12}+\frac{28\!\cdots\!67}{27\!\cdots\!08}a^{11}-\frac{20\!\cdots\!23}{69\!\cdots\!52}a^{10}+\frac{10\!\cdots\!19}{34\!\cdots\!76}a^{9}+\frac{51\!\cdots\!87}{69\!\cdots\!52}a^{8}+\frac{10\!\cdots\!05}{69\!\cdots\!52}a^{7}+\frac{29\!\cdots\!65}{34\!\cdots\!76}a^{6}+\frac{48\!\cdots\!41}{27\!\cdots\!88}a^{5}+\frac{15\!\cdots\!25}{10\!\cdots\!68}a^{4}+\frac{32\!\cdots\!75}{43\!\cdots\!72}a^{3}+\frac{36\!\cdots\!99}{27\!\cdots\!17}a^{2}+\frac{24\!\cdots\!85}{21\!\cdots\!36}a+\frac{16\!\cdots\!87}{54\!\cdots\!34}$, $\frac{11\!\cdots\!07}{55\!\cdots\!16}a^{17}-\frac{38\!\cdots\!73}{55\!\cdots\!16}a^{16}-\frac{20\!\cdots\!43}{27\!\cdots\!08}a^{15}+\frac{30\!\cdots\!25}{55\!\cdots\!16}a^{14}-\frac{85\!\cdots\!17}{55\!\cdots\!16}a^{13}-\frac{81\!\cdots\!93}{27\!\cdots\!08}a^{12}+\frac{22\!\cdots\!21}{27\!\cdots\!08}a^{11}-\frac{22\!\cdots\!27}{69\!\cdots\!52}a^{10}+\frac{80\!\cdots\!29}{34\!\cdots\!76}a^{9}+\frac{36\!\cdots\!69}{69\!\cdots\!52}a^{8}+\frac{45\!\cdots\!43}{69\!\cdots\!52}a^{7}+\frac{22\!\cdots\!71}{34\!\cdots\!76}a^{6}+\frac{34\!\cdots\!75}{27\!\cdots\!88}a^{5}+\frac{42\!\cdots\!93}{43\!\cdots\!72}a^{4}+\frac{24\!\cdots\!85}{43\!\cdots\!72}a^{3}+\frac{10\!\cdots\!41}{10\!\cdots\!68}a^{2}+\frac{16\!\cdots\!27}{21\!\cdots\!36}a+\frac{86\!\cdots\!49}{54\!\cdots\!34}$ (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: | \( 621058194.3189892 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 621058194.3189892 \cdot 2}{6\cdot\sqrt{5632074928630950758566400360448}}\cr\approx \mathstrut & 1.33136042959378 \end{aligned}\] (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
\(\Q(\sqrt{-3}) \), 3.3.1016.1, 6.0.435483.1, 6.0.27870912.1 |
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: | 12.0.12932938664955809431289856.3 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
\(3\) | 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(127\) | $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
127.2.1.1 | $x^{2} + 381$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
127.2.1.1 | $x^{2} + 381$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
127.2.1.1 | $x^{2} + 381$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
127.3.2.1 | $x^{3} + 127$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
127.6.5.1 | $x^{6} + 635$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |