Normalized defining polynomial
\( x^{20} - 5 x^{19} + 8 x^{18} - 10 x^{17} + 16 x^{16} - 4 x^{15} - x^{14} + x^{13} - 28 x^{12} + 24 x^{11} + \cdots + 1 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[8, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(18546881403289244170576896\) \(\medspace = 2^{10}\cdot 3^{6}\cdot 17^{4}\cdot 4153^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.34\) | 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: | $2^{15/8}3^{3/4}17^{1/2}4153^{1/2}\approx 2221.6600682148605$ | ||
Ramified primes: | \(2\), \(3\), \(17\), \(4153\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{699548070579241}a^{19}+\frac{315603094273751}{699548070579241}a^{18}+\frac{319173254885409}{699548070579241}a^{17}-\frac{76394833161040}{699548070579241}a^{16}-\frac{301310174574639}{699548070579241}a^{15}+\frac{227508140670429}{699548070579241}a^{14}+\frac{240518770147705}{699548070579241}a^{13}-\frac{31581240214051}{699548070579241}a^{12}+\frac{19846202288570}{53811390044557}a^{11}-\frac{85642092171950}{699548070579241}a^{10}+\frac{199360621575740}{699548070579241}a^{9}+\frac{224247818164635}{699548070579241}a^{8}+\frac{277070986015333}{699548070579241}a^{7}+\frac{209227698166538}{699548070579241}a^{6}-\frac{320575656442971}{699548070579241}a^{5}-\frac{29738277183596}{699548070579241}a^{4}+\frac{332598915773461}{699548070579241}a^{3}-\frac{236994140005113}{699548070579241}a^{2}-\frac{207369454645997}{699548070579241}a+\frac{347737109696318}{699548070579241}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $13$ | 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: | $\frac{85603206305892}{699548070579241}a^{19}-\frac{376164794171261}{699548070579241}a^{18}+\frac{516516606622202}{699548070579241}a^{17}-\frac{834687608932107}{699548070579241}a^{16}+\frac{12\!\cdots\!62}{699548070579241}a^{15}-\frac{10746435290289}{699548070579241}a^{14}+\frac{619305793019583}{699548070579241}a^{13}+\frac{615471709493144}{699548070579241}a^{12}-\frac{199121707601126}{53811390044557}a^{11}+\frac{862431996051817}{699548070579241}a^{10}-\frac{42\!\cdots\!52}{699548070579241}a^{9}+\frac{229988580797713}{699548070579241}a^{8}+\frac{10\!\cdots\!19}{699548070579241}a^{7}+\frac{211647855590377}{699548070579241}a^{6}+\frac{11\!\cdots\!53}{699548070579241}a^{5}-\frac{33\!\cdots\!97}{699548070579241}a^{4}-\frac{408569714827796}{699548070579241}a^{3}-\frac{906809283898057}{699548070579241}a^{2}-\frac{199766888341905}{699548070579241}a+\frac{14\!\cdots\!39}{699548070579241}$, $\frac{495202044538554}{699548070579241}a^{19}-\frac{24\!\cdots\!72}{699548070579241}a^{18}+\frac{38\!\cdots\!98}{699548070579241}a^{17}-\frac{51\!\cdots\!15}{699548070579241}a^{16}+\frac{85\!\cdots\!04}{699548070579241}a^{15}-\frac{29\!\cdots\!47}{699548070579241}a^{14}+\frac{22\!\cdots\!37}{699548070579241}a^{13}-\frac{18\!\cdots\!30}{699548070579241}a^{12}-\frac{860981397915296}{53811390044557}a^{11}+\frac{82\!\cdots\!55}{699548070579241}a^{10}-\frac{15\!\cdots\!68}{699548070579241}a^{9}+\frac{68\!\cdots\!31}{699548070579241}a^{8}+\frac{97\!\cdots\!19}{699548070579241}a^{7}-\frac{422074884506784}{699548070579241}a^{6}-\frac{94\!\cdots\!89}{699548070579241}a^{5}-\frac{17\!\cdots\!21}{699548070579241}a^{4}+\frac{11\!\cdots\!44}{699548070579241}a^{3}+\frac{83\!\cdots\!89}{699548070579241}a^{2}-\frac{47\!\cdots\!37}{699548070579241}a-\frac{337793491115254}{699548070579241}$, $\frac{212977297695516}{699548070579241}a^{19}-\frac{10\!\cdots\!09}{699548070579241}a^{18}+\frac{12\!\cdots\!40}{699548070579241}a^{17}-\frac{10\!\cdots\!98}{699548070579241}a^{16}+\frac{17\!\cdots\!48}{699548070579241}a^{15}+\frac{19\!\cdots\!69}{699548070579241}a^{14}-\frac{34\!\cdots\!17}{699548070579241}a^{13}+\frac{27\!\cdots\!61}{699548070579241}a^{12}-\frac{701547296898628}{53811390044557}a^{11}+\frac{58\!\cdots\!40}{699548070579241}a^{10}-\frac{52\!\cdots\!26}{699548070579241}a^{9}+\frac{644002765400035}{699548070579241}a^{8}+\frac{10\!\cdots\!45}{699548070579241}a^{7}-\frac{35\!\cdots\!38}{699548070579241}a^{6}-\frac{19\!\cdots\!29}{699548070579241}a^{5}-\frac{11\!\cdots\!63}{699548070579241}a^{4}+\frac{70\!\cdots\!96}{699548070579241}a^{3}+\frac{78\!\cdots\!05}{699548070579241}a^{2}-\frac{30\!\cdots\!91}{699548070579241}a-\frac{641528112102411}{699548070579241}$, $\frac{750472340806506}{699548070579241}a^{19}-\frac{36\!\cdots\!08}{699548070579241}a^{18}+\frac{58\!\cdots\!84}{699548070579241}a^{17}-\frac{80\!\cdots\!60}{699548070579241}a^{16}+\frac{13\!\cdots\!70}{699548070579241}a^{15}-\frac{50\!\cdots\!79}{699548070579241}a^{14}+\frac{42\!\cdots\!70}{699548070579241}a^{13}-\frac{31\!\cdots\!14}{699548070579241}a^{12}-\frac{13\!\cdots\!02}{53811390044557}a^{11}+\frac{12\!\cdots\!84}{699548070579241}a^{10}-\frac{25\!\cdots\!07}{699548070579241}a^{9}+\frac{12\!\cdots\!36}{699548070579241}a^{8}+\frac{12\!\cdots\!67}{699548070579241}a^{7}+\frac{12\!\cdots\!23}{699548070579241}a^{6}-\frac{13\!\cdots\!65}{699548070579241}a^{5}-\frac{27\!\cdots\!77}{699548070579241}a^{4}+\frac{17\!\cdots\!99}{699548070579241}a^{3}+\frac{11\!\cdots\!42}{699548070579241}a^{2}-\frac{51\!\cdots\!23}{699548070579241}a-\frac{15\!\cdots\!85}{699548070579241}$, $\frac{507149942401422}{699548070579241}a^{19}-\frac{21\!\cdots\!92}{699548070579241}a^{18}+\frac{22\!\cdots\!34}{699548070579241}a^{17}-\frac{24\!\cdots\!75}{699548070579241}a^{16}+\frac{43\!\cdots\!66}{699548070579241}a^{15}+\frac{44\!\cdots\!50}{699548070579241}a^{14}-\frac{19\!\cdots\!18}{699548070579241}a^{13}+\frac{35\!\cdots\!48}{699548070579241}a^{12}-\frac{12\!\cdots\!49}{53811390044557}a^{11}+\frac{33\!\cdots\!96}{699548070579241}a^{10}-\frac{12\!\cdots\!44}{699548070579241}a^{9}-\frac{54\!\cdots\!59}{699548070579241}a^{8}+\frac{17\!\cdots\!72}{699548070579241}a^{7}+\frac{676815399410935}{699548070579241}a^{6}-\frac{45\!\cdots\!29}{699548070579241}a^{5}-\frac{25\!\cdots\!06}{699548070579241}a^{4}+\frac{919825484079000}{699548070579241}a^{3}+\frac{12\!\cdots\!46}{699548070579241}a^{2}-\frac{28\!\cdots\!28}{699548070579241}a-\frac{729365454201352}{699548070579241}$, $\frac{2739270678912}{6417872207149}a^{19}-\frac{12634244846030}{6417872207149}a^{18}+\frac{16511514122638}{6417872207149}a^{17}-\frac{17721743197045}{6417872207149}a^{16}+\frac{27971685743590}{6417872207149}a^{15}+\frac{17766885748620}{6417872207149}a^{14}-\frac{26193658823426}{6417872207149}a^{13}+\frac{31276387223645}{6417872207149}a^{12}-\frac{8263018389906}{493682477473}a^{11}+\frac{61761893842673}{6417872207149}a^{10}-\frac{86906033540442}{6417872207149}a^{9}+\frac{8018269231172}{6417872207149}a^{8}+\frac{110084346496296}{6417872207149}a^{7}-\frac{30709242437329}{6417872207149}a^{6}-\frac{7928638756864}{6417872207149}a^{5}-\frac{134872358876340}{6417872207149}a^{4}+\frac{61367191278600}{6417872207149}a^{3}+\frac{63591536019872}{6417872207149}a^{2}-\frac{30206892287644}{6417872207149}a+\frac{876070756643}{6417872207149}$, $\frac{50078302048393}{699548070579241}a^{19}-\frac{516195449598091}{699548070579241}a^{18}+\frac{16\!\cdots\!74}{699548070579241}a^{17}-\frac{22\!\cdots\!47}{699548070579241}a^{16}+\frac{31\!\cdots\!21}{699548070579241}a^{15}-\frac{41\!\cdots\!88}{699548070579241}a^{14}+\frac{354568010533869}{699548070579241}a^{13}-\frac{863992016698082}{699548070579241}a^{12}-\frac{117606487226646}{53811390044557}a^{11}+\frac{86\!\cdots\!10}{699548070579241}a^{10}-\frac{54\!\cdots\!59}{699548070579241}a^{9}+\frac{10\!\cdots\!13}{699548070579241}a^{8}-\frac{16\!\cdots\!26}{699548070579241}a^{7}-\frac{55\!\cdots\!09}{699548070579241}a^{6}-\frac{23\!\cdots\!01}{699548070579241}a^{5}+\frac{895015076905400}{699548070579241}a^{4}+\frac{13\!\cdots\!05}{699548070579241}a^{3}-\frac{22\!\cdots\!29}{699548070579241}a^{2}-\frac{37\!\cdots\!24}{699548070579241}a+\frac{533939103835741}{699548070579241}$, $\frac{783898828221634}{699548070579241}a^{19}-\frac{36\!\cdots\!32}{699548070579241}a^{18}+\frac{53\!\cdots\!31}{699548070579241}a^{17}-\frac{77\!\cdots\!70}{699548070579241}a^{16}+\frac{13\!\cdots\!42}{699548070579241}a^{15}-\frac{41\!\cdots\!41}{699548070579241}a^{14}+\frac{67\!\cdots\!30}{699548070579241}a^{13}-\frac{46\!\cdots\!77}{699548070579241}a^{12}-\frac{12\!\cdots\!18}{53811390044557}a^{11}+\frac{73\!\cdots\!37}{699548070579241}a^{10}-\frac{25\!\cdots\!86}{699548070579241}a^{9}+\frac{87\!\cdots\!06}{699548070579241}a^{8}+\frac{10\!\cdots\!04}{699548070579241}a^{7}+\frac{81\!\cdots\!92}{699548070579241}a^{6}-\frac{14\!\cdots\!93}{699548070579241}a^{5}-\frac{29\!\cdots\!55}{699548070579241}a^{4}+\frac{86\!\cdots\!45}{699548070579241}a^{3}+\frac{11\!\cdots\!58}{699548070579241}a^{2}-\frac{16\!\cdots\!67}{699548070579241}a-\frac{18\!\cdots\!33}{699548070579241}$, $\frac{446168130462497}{699548070579241}a^{19}-\frac{19\!\cdots\!71}{699548070579241}a^{18}+\frac{20\!\cdots\!95}{699548070579241}a^{17}-\frac{20\!\cdots\!17}{699548070579241}a^{16}+\frac{38\!\cdots\!12}{699548070579241}a^{15}+\frac{36\!\cdots\!70}{699548070579241}a^{14}-\frac{26\!\cdots\!53}{699548070579241}a^{13}+\frac{17\!\cdots\!17}{699548070579241}a^{12}-\frac{10\!\cdots\!89}{53811390044557}a^{11}+\frac{30\!\cdots\!57}{699548070579241}a^{10}-\frac{76\!\cdots\!66}{699548070579241}a^{9}-\frac{39\!\cdots\!67}{699548070579241}a^{8}+\frac{18\!\cdots\!13}{699548070579241}a^{7}+\frac{517624343488888}{699548070579241}a^{6}-\frac{77\!\cdots\!77}{699548070579241}a^{5}-\frac{22\!\cdots\!31}{699548070579241}a^{4}+\frac{30\!\cdots\!05}{699548070579241}a^{3}+\frac{16\!\cdots\!89}{699548070579241}a^{2}-\frac{32\!\cdots\!92}{699548070579241}a-\frac{24\!\cdots\!07}{699548070579241}$, $\frac{180078036511550}{699548070579241}a^{19}-\frac{772193171903494}{699548070579241}a^{18}+\frac{779728890935270}{699548070579241}a^{17}-\frac{720495126840352}{699548070579241}a^{16}+\frac{16\!\cdots\!57}{699548070579241}a^{15}+\frac{14\!\cdots\!87}{699548070579241}a^{14}-\frac{844438239159448}{699548070579241}a^{13}-\frac{196212669947670}{699548070579241}a^{12}-\frac{413300900893042}{53811390044557}a^{11}+\frac{349576338667670}{699548070579241}a^{10}-\frac{22\!\cdots\!29}{699548070579241}a^{9}-\frac{975618268874014}{699548070579241}a^{8}+\frac{81\!\cdots\!35}{699548070579241}a^{7}+\frac{26\!\cdots\!79}{699548070579241}a^{6}-\frac{37\!\cdots\!82}{699548070579241}a^{5}-\frac{93\!\cdots\!91}{699548070579241}a^{4}+\frac{316921790482965}{699548070579241}a^{3}+\frac{88\!\cdots\!92}{699548070579241}a^{2}+\frac{567269281692330}{699548070579241}a-\frac{11\!\cdots\!76}{699548070579241}$, $\frac{255270296267952}{699548070579241}a^{19}-\frac{12\!\cdots\!36}{699548070579241}a^{18}+\frac{19\!\cdots\!86}{699548070579241}a^{17}-\frac{28\!\cdots\!45}{699548070579241}a^{16}+\frac{48\!\cdots\!66}{699548070579241}a^{15}-\frac{20\!\cdots\!32}{699548070579241}a^{14}+\frac{20\!\cdots\!33}{699548070579241}a^{13}-\frac{13\!\cdots\!84}{699548070579241}a^{12}-\frac{469851875968206}{53811390044557}a^{11}+\frac{46\!\cdots\!29}{699548070579241}a^{10}-\frac{96\!\cdots\!39}{699548070579241}a^{9}+\frac{56\!\cdots\!05}{699548070579241}a^{8}+\frac{29\!\cdots\!48}{699548070579241}a^{7}+\frac{16\!\cdots\!07}{699548070579241}a^{6}-\frac{41\!\cdots\!76}{699548070579241}a^{5}-\frac{10\!\cdots\!56}{699548070579241}a^{4}+\frac{53\!\cdots\!55}{699548070579241}a^{3}+\frac{33\!\cdots\!53}{699548070579241}a^{2}-\frac{456383367491586}{699548070579241}a-\frac{479920099410190}{699548070579241}$, $\frac{11947897862868}{699548070579241}a^{19}+\frac{293331388525280}{699548070579241}a^{18}-\frac{16\!\cdots\!64}{699548070579241}a^{17}+\frac{27\!\cdots\!40}{699548070579241}a^{16}-\frac{41\!\cdots\!38}{699548070579241}a^{15}+\frac{73\!\cdots\!97}{699548070579241}a^{14}-\frac{42\!\cdots\!55}{699548070579241}a^{13}+\frac{54\!\cdots\!78}{699548070579241}a^{12}-\frac{351445755143053}{53811390044557}a^{11}-\frac{49\!\cdots\!59}{699548070579241}a^{10}+\frac{27\!\cdots\!24}{699548070579241}a^{9}-\frac{12\!\cdots\!90}{699548070579241}a^{8}+\frac{74\!\cdots\!53}{699548070579241}a^{7}+\frac{10\!\cdots\!19}{699548070579241}a^{6}+\frac{48\!\cdots\!60}{699548070579241}a^{5}-\frac{81\!\cdots\!85}{699548070579241}a^{4}-\frac{10\!\cdots\!44}{699548070579241}a^{3}+\frac{36\!\cdots\!57}{699548070579241}a^{2}+\frac{18\!\cdots\!09}{699548070579241}a+\frac{307976107493143}{699548070579241}$, $\frac{25385207767199}{699548070579241}a^{19}-\frac{75205724698976}{699548070579241}a^{18}+\frac{37968539403016}{699548070579241}a^{17}-\frac{349459462625132}{699548070579241}a^{16}+\frac{913089525104317}{699548070579241}a^{15}-\frac{910082683535151}{699548070579241}a^{14}+\frac{24\!\cdots\!31}{699548070579241}a^{13}-\frac{22\!\cdots\!22}{699548070579241}a^{12}+\frac{107958336097899}{53811390044557}a^{11}-\frac{25\!\cdots\!75}{699548070579241}a^{10}-\frac{433201939063445}{699548070579241}a^{9}+\frac{499303015393620}{699548070579241}a^{8}-\frac{26\!\cdots\!68}{699548070579241}a^{7}+\frac{49\!\cdots\!35}{699548070579241}a^{6}-\frac{19\!\cdots\!57}{699548070579241}a^{5}-\frac{795925756942872}{699548070579241}a^{4}-\frac{37\!\cdots\!62}{699548070579241}a^{3}+\frac{530316995580913}{699548070579241}a^{2}+\frac{24\!\cdots\!72}{699548070579241}a-\frac{14\!\cdots\!75}{699548070579241}$ (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: | \( 96380.8387012 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{6}\cdot 96380.8387012 \cdot 1}{2\cdot\sqrt{18546881403289244170576896}}\cr\approx \mathstrut & 0.176256212511 \end{aligned}\] (assuming GRH)
Galois group
$C_2^9.C_2^4:S_5$ (as 20T964):
A non-solvable group of order 983040 |
The 155 conjugacy class representatives for $C_2^9.C_2^4:S_5$ |
Character table for $C_2^9.C_2^4:S_5$ |
Intermediate fields
5.5.70601.1, 10.6.44860510809.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 20 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | $16{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.5.0.1}{5} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.10.6 | $x^{10} - 6 x^{9} + 42 x^{8} - 104 x^{7} - 256 x^{6} - 112 x^{5} - 1568 x^{4} - 2016 x^{3} - 2832 x^{2} - 4960 x - 3616$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2]^{10}$ |
2.10.0.1 | $x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(3\) | 3.8.6.3 | $x^{8} - 6 x^{4} + 18$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ |
3.12.0.1 | $x^{12} + x^{6} + x^{5} + x^{4} + x^{2} + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(17\) | 17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.3.0.1 | $x^{3} + x + 14$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
17.3.0.1 | $x^{3} + x + 14$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
17.3.0.1 | $x^{3} + x + 14$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
17.3.0.1 | $x^{3} + x + 14$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(4153\) | Deg $4$ | $2$ | $2$ | $2$ | |||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |