Normalized defining polynomial
\( x^{15} - 60 x^{13} - 63 x^{12} + 1341 x^{11} + 2727 x^{10} - 12674 x^{9} - 40869 x^{8} + 31254 x^{7} + 243271 x^{6} + 196857 x^{5} - 393162 x^{4} - 847458 x^{3} + \cdots + 12167 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[15, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(138622929316925604547946697\) \(\medspace = 3^{15}\cdot 11^{13}\cdot 23^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(55.31\) | 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: | $3^{241/162}11^{9/10}23^{2/3}\approx 358.80911756371285$ | ||
Ramified primes: | \(3\), \(11\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{33}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{46}a^{11}+\frac{9}{46}a^{9}-\frac{17}{46}a^{8}-\frac{8}{23}a^{7}+\frac{13}{46}a^{6}+\frac{11}{23}a^{5}-\frac{21}{46}a^{4}-\frac{3}{46}a^{3}-\frac{1}{2}a$, $\frac{1}{46}a^{12}+\frac{9}{46}a^{10}-\frac{17}{46}a^{9}-\frac{8}{23}a^{8}+\frac{13}{46}a^{7}+\frac{11}{23}a^{6}-\frac{21}{46}a^{5}-\frac{3}{46}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{46}a^{13}+\frac{3}{23}a^{10}-\frac{5}{46}a^{9}+\frac{5}{46}a^{8}+\frac{5}{46}a^{7}+\frac{3}{23}a^{5}+\frac{5}{46}a^{4}-\frac{19}{46}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{12\!\cdots\!06}a^{14}+\frac{1370582028003}{28\!\cdots\!61}a^{13}+\frac{11\!\cdots\!01}{12\!\cdots\!06}a^{12}-\frac{476042798426446}{64\!\cdots\!03}a^{11}+\frac{11\!\cdots\!80}{64\!\cdots\!03}a^{10}-\frac{30\!\cdots\!15}{64\!\cdots\!03}a^{9}-\frac{48\!\cdots\!31}{12\!\cdots\!06}a^{8}+\frac{45\!\cdots\!15}{12\!\cdots\!06}a^{7}-\frac{15\!\cdots\!50}{64\!\cdots\!03}a^{6}+\frac{11\!\cdots\!65}{28\!\cdots\!61}a^{5}-\frac{829197257205122}{28\!\cdots\!61}a^{4}+\frac{24\!\cdots\!87}{56\!\cdots\!22}a^{3}-\frac{12653543406775}{122541950361907}a^{2}-\frac{122129403538203}{245083900723814}a+\frac{13370030390824}{122541950361907}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $14$ | 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: | $\frac{39\!\cdots\!85}{12\!\cdots\!06}a^{14}-\frac{402746262548505}{56\!\cdots\!22}a^{13}-\frac{21\!\cdots\!41}{12\!\cdots\!06}a^{12}+\frac{12\!\cdots\!67}{64\!\cdots\!03}a^{11}+\frac{23\!\cdots\!36}{64\!\cdots\!03}a^{10}-\frac{28\!\cdots\!25}{12\!\cdots\!06}a^{9}-\frac{24\!\cdots\!26}{64\!\cdots\!03}a^{8}-\frac{22\!\cdots\!96}{64\!\cdots\!03}a^{7}+\frac{11\!\cdots\!82}{64\!\cdots\!03}a^{6}+\frac{90\!\cdots\!37}{28\!\cdots\!61}a^{5}-\frac{90\!\cdots\!79}{56\!\cdots\!22}a^{4}-\frac{22\!\cdots\!90}{28\!\cdots\!61}a^{3}-\frac{16\!\cdots\!25}{245083900723814}a^{2}-\frac{18\!\cdots\!37}{122541950361907}a+\frac{39\!\cdots\!01}{245083900723814}$, $\frac{368748726176169}{64\!\cdots\!03}a^{14}-\frac{47850023790447}{56\!\cdots\!22}a^{13}-\frac{21\!\cdots\!66}{64\!\cdots\!03}a^{12}+\frac{88\!\cdots\!58}{64\!\cdots\!03}a^{11}+\frac{48\!\cdots\!20}{64\!\cdots\!03}a^{10}+\frac{55\!\cdots\!51}{12\!\cdots\!06}a^{9}-\frac{10\!\cdots\!87}{12\!\cdots\!06}a^{8}-\frac{14\!\cdots\!15}{12\!\cdots\!06}a^{7}+\frac{22\!\cdots\!67}{64\!\cdots\!03}a^{6}+\frac{23\!\cdots\!21}{28\!\cdots\!61}a^{5}-\frac{10\!\cdots\!35}{56\!\cdots\!22}a^{4}-\frac{11\!\cdots\!83}{56\!\cdots\!22}a^{3}-\frac{44\!\cdots\!57}{245083900723814}a^{2}-\frac{10\!\cdots\!69}{245083900723814}a+\frac{11\!\cdots\!71}{245083900723814}$, $\frac{97681176961776}{64\!\cdots\!03}a^{14}+\frac{13628846444295}{28\!\cdots\!61}a^{13}-\frac{77\!\cdots\!22}{64\!\cdots\!03}a^{12}-\frac{36\!\cdots\!67}{12\!\cdots\!06}a^{11}+\frac{20\!\cdots\!67}{64\!\cdots\!03}a^{10}+\frac{90\!\cdots\!41}{12\!\cdots\!06}a^{9}-\frac{44\!\cdots\!91}{12\!\cdots\!06}a^{8}-\frac{57\!\cdots\!83}{64\!\cdots\!03}a^{7}+\frac{18\!\cdots\!49}{12\!\cdots\!06}a^{6}+\frac{14\!\cdots\!22}{28\!\cdots\!61}a^{5}+\frac{18\!\cdots\!67}{56\!\cdots\!22}a^{4}-\frac{61\!\cdots\!19}{56\!\cdots\!22}a^{3}-\frac{13\!\cdots\!77}{122541950361907}a^{2}-\frac{69\!\cdots\!73}{245083900723814}a+\frac{30\!\cdots\!02}{122541950361907}$, $\frac{466429903137945}{64\!\cdots\!03}a^{14}-\frac{20592330901857}{56\!\cdots\!22}a^{13}-\frac{29\!\cdots\!88}{64\!\cdots\!03}a^{12}-\frac{18\!\cdots\!51}{12\!\cdots\!06}a^{11}+\frac{68\!\cdots\!87}{64\!\cdots\!03}a^{10}+\frac{72\!\cdots\!96}{64\!\cdots\!03}a^{9}-\frac{73\!\cdots\!39}{64\!\cdots\!03}a^{8}-\frac{26\!\cdots\!81}{12\!\cdots\!06}a^{7}+\frac{64\!\cdots\!83}{12\!\cdots\!06}a^{6}+\frac{38\!\cdots\!43}{28\!\cdots\!61}a^{5}-\frac{44\!\cdots\!34}{28\!\cdots\!61}a^{4}-\frac{37\!\cdots\!87}{122541950361907}a^{3}-\frac{71\!\cdots\!11}{245083900723814}a^{2}-\frac{88\!\cdots\!71}{122541950361907}a+\frac{17\!\cdots\!75}{245083900723814}$, $\frac{432580983011511}{64\!\cdots\!03}a^{14}-\frac{39165327120835}{56\!\cdots\!22}a^{13}-\frac{52\!\cdots\!49}{12\!\cdots\!06}a^{12}+\frac{51\!\cdots\!99}{12\!\cdots\!06}a^{11}+\frac{12\!\cdots\!53}{12\!\cdots\!06}a^{10}+\frac{90\!\cdots\!11}{12\!\cdots\!06}a^{9}-\frac{64\!\cdots\!93}{64\!\cdots\!03}a^{8}-\frac{98\!\cdots\!48}{64\!\cdots\!03}a^{7}+\frac{57\!\cdots\!09}{12\!\cdots\!06}a^{6}+\frac{61\!\cdots\!61}{56\!\cdots\!22}a^{5}-\frac{13\!\cdots\!21}{56\!\cdots\!22}a^{4}-\frac{30\!\cdots\!47}{122541950361907}a^{3}-\frac{27\!\cdots\!27}{122541950361907}a^{2}-\frac{67\!\cdots\!80}{122541950361907}a+\frac{13\!\cdots\!25}{245083900723814}$, $\frac{25\!\cdots\!89}{64\!\cdots\!03}a^{14}-\frac{481488339031855}{56\!\cdots\!22}a^{13}-\frac{14\!\cdots\!09}{64\!\cdots\!03}a^{12}+\frac{14\!\cdots\!05}{64\!\cdots\!03}a^{11}+\frac{31\!\cdots\!98}{64\!\cdots\!03}a^{10}+\frac{62\!\cdots\!59}{12\!\cdots\!06}a^{9}-\frac{67\!\cdots\!47}{12\!\cdots\!06}a^{8}-\frac{69\!\cdots\!87}{12\!\cdots\!06}a^{7}+\frac{15\!\cdots\!80}{64\!\cdots\!03}a^{6}+\frac{13\!\cdots\!97}{28\!\cdots\!61}a^{5}-\frac{11\!\cdots\!55}{56\!\cdots\!22}a^{4}-\frac{64\!\cdots\!29}{56\!\cdots\!22}a^{3}-\frac{23\!\cdots\!01}{245083900723814}a^{2}-\frac{54\!\cdots\!15}{245083900723814}a+\frac{56\!\cdots\!53}{245083900723814}$, $\frac{334899806049735}{64\!\cdots\!03}a^{14}-\frac{66423020009425}{56\!\cdots\!22}a^{13}-\frac{36\!\cdots\!05}{12\!\cdots\!06}a^{12}+\frac{20\!\cdots\!33}{64\!\cdots\!03}a^{11}+\frac{80\!\cdots\!19}{12\!\cdots\!06}a^{10}+\frac{11\!\cdots\!85}{64\!\cdots\!03}a^{9}-\frac{84\!\cdots\!95}{12\!\cdots\!06}a^{8}-\frac{40\!\cdots\!65}{64\!\cdots\!03}a^{7}+\frac{19\!\cdots\!30}{64\!\cdots\!03}a^{6}+\frac{32\!\cdots\!17}{56\!\cdots\!22}a^{5}-\frac{75\!\cdots\!94}{28\!\cdots\!61}a^{4}-\frac{79\!\cdots\!43}{56\!\cdots\!22}a^{3}-\frac{14\!\cdots\!50}{122541950361907}a^{2}-\frac{65\!\cdots\!87}{245083900723814}a+\frac{72\!\cdots\!21}{245083900723814}$, $\frac{70\!\cdots\!29}{12\!\cdots\!06}a^{14}-\frac{314702514561712}{28\!\cdots\!61}a^{13}-\frac{39\!\cdots\!79}{12\!\cdots\!06}a^{12}+\frac{18\!\cdots\!15}{64\!\cdots\!03}a^{11}+\frac{86\!\cdots\!75}{12\!\cdots\!06}a^{10}+\frac{70\!\cdots\!97}{64\!\cdots\!03}a^{9}-\frac{45\!\cdots\!82}{64\!\cdots\!03}a^{8}-\frac{49\!\cdots\!09}{64\!\cdots\!03}a^{7}+\frac{21\!\cdots\!88}{64\!\cdots\!03}a^{6}+\frac{36\!\cdots\!83}{56\!\cdots\!22}a^{5}-\frac{73\!\cdots\!06}{28\!\cdots\!61}a^{4}-\frac{44\!\cdots\!61}{28\!\cdots\!61}a^{3}-\frac{33\!\cdots\!99}{245083900723814}a^{2}-\frac{76\!\cdots\!39}{245083900723814}a+\frac{79\!\cdots\!99}{245083900723814}$, $\frac{98\!\cdots\!41}{12\!\cdots\!06}a^{14}-\frac{651807138082563}{56\!\cdots\!22}a^{13}-\frac{57\!\cdots\!49}{12\!\cdots\!06}a^{12}+\frac{12\!\cdots\!88}{64\!\cdots\!03}a^{11}+\frac{64\!\cdots\!41}{64\!\cdots\!03}a^{10}+\frac{71\!\cdots\!75}{12\!\cdots\!06}a^{9}-\frac{68\!\cdots\!98}{64\!\cdots\!03}a^{8}-\frac{96\!\cdots\!95}{64\!\cdots\!03}a^{7}+\frac{30\!\cdots\!70}{64\!\cdots\!03}a^{6}+\frac{31\!\cdots\!13}{28\!\cdots\!61}a^{5}-\frac{15\!\cdots\!03}{56\!\cdots\!22}a^{4}-\frac{73\!\cdots\!27}{28\!\cdots\!61}a^{3}-\frac{58\!\cdots\!41}{245083900723814}a^{2}-\frac{69\!\cdots\!32}{122541950361907}a+\frac{14\!\cdots\!53}{245083900723814}$, $\frac{43\!\cdots\!76}{64\!\cdots\!03}a^{14}-\frac{701901406216755}{56\!\cdots\!22}a^{13}-\frac{49\!\cdots\!97}{12\!\cdots\!06}a^{12}+\frac{36\!\cdots\!31}{12\!\cdots\!06}a^{11}+\frac{10\!\cdots\!47}{12\!\cdots\!06}a^{10}+\frac{32\!\cdots\!45}{12\!\cdots\!06}a^{9}-\frac{58\!\cdots\!67}{64\!\cdots\!03}a^{8}-\frac{69\!\cdots\!31}{64\!\cdots\!03}a^{7}+\frac{53\!\cdots\!05}{12\!\cdots\!06}a^{6}+\frac{48\!\cdots\!03}{56\!\cdots\!22}a^{5}-\frac{17\!\cdots\!35}{56\!\cdots\!22}a^{4}-\frac{58\!\cdots\!60}{28\!\cdots\!61}a^{3}-\frac{21\!\cdots\!70}{122541950361907}a^{2}-\frac{49\!\cdots\!66}{122541950361907}a+\frac{10\!\cdots\!09}{245083900723814}$, $\frac{71\!\cdots\!29}{64\!\cdots\!03}a^{14}-\frac{12\!\cdots\!57}{56\!\cdots\!22}a^{13}-\frac{80\!\cdots\!71}{12\!\cdots\!06}a^{12}+\frac{36\!\cdots\!62}{64\!\cdots\!03}a^{11}+\frac{89\!\cdots\!18}{64\!\cdots\!03}a^{10}+\frac{14\!\cdots\!43}{64\!\cdots\!03}a^{9}-\frac{94\!\cdots\!62}{64\!\cdots\!03}a^{8}-\frac{20\!\cdots\!39}{12\!\cdots\!06}a^{7}+\frac{43\!\cdots\!82}{64\!\cdots\!03}a^{6}+\frac{37\!\cdots\!00}{28\!\cdots\!61}a^{5}-\frac{15\!\cdots\!90}{28\!\cdots\!61}a^{4}-\frac{91\!\cdots\!66}{28\!\cdots\!61}a^{3}-\frac{67\!\cdots\!23}{245083900723814}a^{2}-\frac{15\!\cdots\!51}{245083900723814}a+\frac{79\!\cdots\!98}{122541950361907}$, $\frac{58\!\cdots\!83}{12\!\cdots\!06}a^{14}-\frac{217268964363708}{28\!\cdots\!61}a^{13}-\frac{33\!\cdots\!45}{12\!\cdots\!06}a^{12}+\frac{20\!\cdots\!99}{12\!\cdots\!06}a^{11}+\frac{37\!\cdots\!80}{64\!\cdots\!03}a^{10}+\frac{31\!\cdots\!59}{12\!\cdots\!06}a^{9}-\frac{39\!\cdots\!61}{64\!\cdots\!03}a^{8}-\frac{10\!\cdots\!01}{12\!\cdots\!06}a^{7}+\frac{36\!\cdots\!79}{12\!\cdots\!06}a^{6}+\frac{17\!\cdots\!20}{28\!\cdots\!61}a^{5}-\frac{44\!\cdots\!31}{245083900723814}a^{4}-\frac{41\!\cdots\!54}{28\!\cdots\!61}a^{3}-\frac{16\!\cdots\!13}{122541950361907}a^{2}-\frac{38\!\cdots\!53}{122541950361907}a+\frac{38\!\cdots\!39}{122541950361907}$, $\frac{10\!\cdots\!27}{12\!\cdots\!06}a^{14}-\frac{926761243064371}{56\!\cdots\!22}a^{13}-\frac{55\!\cdots\!47}{12\!\cdots\!06}a^{12}+\frac{54\!\cdots\!13}{12\!\cdots\!06}a^{11}+\frac{61\!\cdots\!61}{64\!\cdots\!03}a^{10}+\frac{68\!\cdots\!15}{64\!\cdots\!03}a^{9}-\frac{13\!\cdots\!27}{12\!\cdots\!06}a^{8}-\frac{67\!\cdots\!19}{64\!\cdots\!03}a^{7}+\frac{60\!\cdots\!21}{12\!\cdots\!06}a^{6}+\frac{25\!\cdots\!72}{28\!\cdots\!61}a^{5}-\frac{11\!\cdots\!86}{28\!\cdots\!61}a^{4}-\frac{12\!\cdots\!21}{56\!\cdots\!22}a^{3}-\frac{45\!\cdots\!93}{245083900723814}a^{2}-\frac{10\!\cdots\!35}{245083900723814}a+\frac{10\!\cdots\!91}{245083900723814}$, $\frac{10\!\cdots\!09}{12\!\cdots\!06}a^{14}-\frac{872665231557377}{56\!\cdots\!22}a^{13}-\frac{56\!\cdots\!89}{12\!\cdots\!06}a^{12}+\frac{49\!\cdots\!49}{12\!\cdots\!06}a^{11}+\frac{12\!\cdots\!69}{12\!\cdots\!06}a^{10}+\frac{11\!\cdots\!00}{64\!\cdots\!03}a^{9}-\frac{66\!\cdots\!30}{64\!\cdots\!03}a^{8}-\frac{14\!\cdots\!85}{12\!\cdots\!06}a^{7}+\frac{61\!\cdots\!67}{12\!\cdots\!06}a^{6}+\frac{52\!\cdots\!23}{56\!\cdots\!22}a^{5}-\frac{11\!\cdots\!07}{28\!\cdots\!61}a^{4}-\frac{64\!\cdots\!80}{28\!\cdots\!61}a^{3}-\frac{23\!\cdots\!93}{122541950361907}a^{2}-\frac{10\!\cdots\!83}{245083900723814}a+\frac{54\!\cdots\!70}{122541950361907}$ (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: | \( 181222525.173 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 181222525.173 \cdot 1}{2\cdot\sqrt{138622929316925604547946697}}\cr\approx \mathstrut & 0.252182288159 \end{aligned}\] (assuming GRH)
Galois group
$C_3^4:C_{10}$ (as 15T33):
A solvable group of order 810 |
The 18 conjugacy class representatives for $C_3^4:C_{10}$ |
Character table for $C_3^4:C_{10}$ |
Intermediate fields
\(\Q(\zeta_{11})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 15 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 45 siblings: | data not computed |
Minimal sibling: | 15.15.138622929316925604547946697.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{3}$ | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ | ${\href{/padicField/17.5.0.1}{5} }^{3}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | R | ${\href{/padicField/29.5.0.1}{5} }^{3}$ | ${\href{/padicField/31.5.0.1}{5} }^{3}$ | ${\href{/padicField/37.5.0.1}{5} }^{3}$ | ${\href{/padicField/41.5.0.1}{5} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.15.15.31 | $x^{15} - 9 x^{14} + 111 x^{13} + 4596 x^{12} + 29925 x^{11} + 55764 x^{10} + 33786 x^{9} + 2916 x^{8} - 2025 x^{7} + 40851 x^{6} + 21384 x^{5} - 20331 x^{4} - 3483 x^{3} + 10449 x^{2} - 2673 x + 243$ | $3$ | $5$ | $15$ | 15T33 | $[3/2, 3/2, 3/2, 3/2]_{2}^{5}$ |
\(11\) | 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
\(23\) | $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.3.2.1 | $x^{3} + 23$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
23.3.2.1 | $x^{3} + 23$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |