Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 4*x^14 + 112*x^13 - 254*x^12 - 296*x^11 + 1457*x^10 - 949*x^9 - 1463*x^8 + 1998*x^7 - 97*x^6 - 807*x^5 + 256*x^4 + 86*x^3 - 39*x^2 - x + 1)
gp: K = bnfinit(y^16 - 8*y^15 + 4*y^14 + 112*y^13 - 254*y^12 - 296*y^11 + 1457*y^10 - 949*y^9 - 1463*y^8 + 1998*y^7 - 97*y^6 - 807*y^5 + 256*y^4 + 86*y^3 - 39*y^2 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 4*x^14 + 112*x^13 - 254*x^12 - 296*x^11 + 1457*x^10 - 949*x^9 - 1463*x^8 + 1998*x^7 - 97*x^6 - 807*x^5 + 256*x^4 + 86*x^3 - 39*x^2 - x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 4*x^14 + 112*x^13 - 254*x^12 - 296*x^11 + 1457*x^10 - 949*x^9 - 1463*x^8 + 1998*x^7 - 97*x^6 - 807*x^5 + 256*x^4 + 86*x^3 - 39*x^2 - x + 1)
\( x^{16} - 8 x^{15} + 4 x^{14} + 112 x^{13} - 254 x^{12} - 296 x^{11} + 1457 x^{10} - 949 x^{9} - 1463 x^{8} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(11981707062330062500000000\)
\(\medspace = 2^{8}\cdot 5^{12}\cdot 61^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(36.93\) | 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: | $2\cdot 5^{3/4}61^{1/2}\approx 52.23028750207817$ | ||
| Ramified primes: |
\(2\), \(5\), \(61\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{436}a^{14}-\frac{7}{436}a^{13}-\frac{7}{218}a^{12}+\frac{175}{436}a^{11}-\frac{143}{436}a^{10}-\frac{46}{109}a^{9}+\frac{3}{109}a^{8}+\frac{215}{436}a^{7}-\frac{18}{109}a^{6}-\frac{3}{436}a^{5}-\frac{45}{109}a^{4}+\frac{34}{109}a^{3}-\frac{13}{218}a^{2}+\frac{45}{218}a-\frac{99}{436}$, $\frac{1}{21364}a^{15}+\frac{17}{21364}a^{14}+\frac{1017}{21364}a^{13}+\frac{57}{21364}a^{12}+\frac{4481}{10682}a^{11}-\frac{1122}{5341}a^{10}+\frac{1395}{10682}a^{9}+\frac{3337}{21364}a^{8}-\frac{1780}{5341}a^{7}+\frac{957}{5341}a^{6}+\frac{10539}{21364}a^{5}-\frac{115}{3052}a^{4}-\frac{1433}{10682}a^{3}+\frac{3063}{21364}a^{2}-\frac{711}{5341}a+\frac{2131}{5341}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $15$ | 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: |
$\frac{15166}{5341}a^{15}-\frac{113745}{5341}a^{14}+\frac{4257}{5341}a^{13}+\frac{1697462}{5341}a^{12}-\frac{3004609}{5341}a^{11}-\frac{5942024}{5341}a^{10}+\frac{19056613}{5341}a^{9}-\frac{5048688}{5341}a^{8}-\frac{24252631}{5341}a^{7}+\frac{18120546}{5341}a^{6}+\frac{7064283}{5341}a^{5}-\frac{1199777}{763}a^{4}-\frac{216249}{5341}a^{3}+\frac{1128605}{5341}a^{2}-\frac{36604}{5341}a-\frac{34301}{5341}$, $\frac{2112}{5341}a^{15}-\frac{15840}{5341}a^{14}-\frac{158}{5341}a^{13}+\frac{241267}{5341}a^{12}-\frac{414222}{5341}a^{11}-\frac{904244}{5341}a^{10}+\frac{2728501}{5341}a^{9}-\frac{382959}{5341}a^{8}-\frac{3954522}{5341}a^{7}+\frac{2404803}{5341}a^{6}+\frac{1701202}{5341}a^{5}-\frac{207894}{763}a^{4}-\frac{180979}{5341}a^{3}+\frac{256885}{5341}a^{2}-\frac{18875}{5341}a-\frac{1186}{5341}$, $\frac{34301}{5341}a^{15}-\frac{259242}{5341}a^{14}+\frac{23459}{5341}a^{13}+\frac{3845969}{5341}a^{12}-\frac{7014992}{5341}a^{11}-\frac{13157705}{5341}a^{10}+\frac{44034533}{5341}a^{9}-\frac{13495036}{5341}a^{8}-\frac{55231051}{5341}a^{7}+\frac{44280767}{5341}a^{6}+\frac{14793349}{5341}a^{5}-\frac{2945232}{763}a^{4}+\frac{382617}{5341}a^{3}+\frac{2733637}{5341}a^{2}-\frac{209134}{5341}a-\frac{76246}{5341}$, $a-1$, $\frac{25869}{21364}a^{15}-\frac{192033}{21364}a^{14}-\frac{11381}{21364}a^{13}+\frac{730633}{5341}a^{12}-\frac{2446587}{10682}a^{11}-\frac{10941221}{21364}a^{10}+\frac{16149351}{10682}a^{9}-\frac{4458917}{21364}a^{8}-\frac{11494715}{5341}a^{7}+\frac{6876484}{5341}a^{6}+\frac{4966147}{5341}a^{5}-\frac{1106027}{1526}a^{4}-\frac{3537483}{21364}a^{3}+\frac{2730303}{21364}a^{2}+\frac{297681}{21364}a-\frac{129011}{21364}$, $\frac{16901}{21364}a^{15}-\frac{136239}{21364}a^{14}+\frac{71049}{21364}a^{13}+\frac{1926011}{21364}a^{12}-\frac{2207783}{10682}a^{11}-\frac{2647755}{10682}a^{10}+\frac{12990249}{10682}a^{9}-\frac{15596083}{21364}a^{8}-\frac{7568917}{5341}a^{7}+\frac{9201312}{5341}a^{6}+\frac{4707561}{21364}a^{5}-\frac{2669613}{3052}a^{4}+\frac{576411}{5341}a^{3}+\frac{3025111}{21364}a^{2}-\frac{210353}{10682}a-\frac{58159}{10682}$, $\frac{5249}{21364}a^{15}-\frac{56199}{21364}a^{14}+\frac{117969}{21364}a^{13}+\frac{647485}{21364}a^{12}-\frac{13169}{98}a^{11}+\frac{57556}{5341}a^{10}+\frac{7114595}{10682}a^{9}-\frac{18155303}{21364}a^{8}-\frac{2782218}{5341}a^{7}+\frac{7320609}{5341}a^{6}-\frac{4619157}{21364}a^{5}-\frac{1725791}{3052}a^{4}+\frac{1410491}{10682}a^{3}+\frac{1784975}{21364}a^{2}-\frac{39595}{5341}a-\frac{14850}{5341}$, $\frac{43003}{21364}a^{15}-\frac{305691}{21364}a^{14}-\frac{100761}{21364}a^{13}+\frac{1182332}{5341}a^{12}-\frac{1676842}{5341}a^{11}-\frac{18770075}{21364}a^{10}+\frac{23008787}{10682}a^{9}+\frac{686297}{21364}a^{8}-\frac{15807047}{5341}a^{7}+\frac{14097635}{10682}a^{6}+\frac{12070573}{10682}a^{5}-\frac{439990}{763}a^{4}-\frac{3469185}{21364}a^{3}+\frac{932387}{21364}a^{2}+\frac{238521}{21364}a+\frac{25567}{21364}$, $\frac{7583}{5341}a^{15}-\frac{114137}{10682}a^{14}+\frac{7001}{10682}a^{13}+\frac{3400559}{21364}a^{12}-\frac{1528593}{5341}a^{11}-\frac{11798641}{21364}a^{10}+\frac{9695225}{5341}a^{9}-\frac{5528741}{10682}a^{8}-\frac{12512950}{5341}a^{7}+\frac{9774056}{5341}a^{6}+\frac{15172413}{21364}a^{5}-\frac{2793703}{3052}a^{4}-\frac{491053}{21364}a^{3}+\frac{742981}{5341}a^{2}-\frac{31607}{21364}a-\frac{103147}{21364}$, $\frac{48411}{5341}a^{15}-\frac{1471587}{21364}a^{14}+\frac{190413}{21364}a^{13}+\frac{21726813}{21364}a^{12}-\frac{40493727}{21364}a^{11}-\frac{36483235}{10682}a^{10}+\frac{63019344}{5341}a^{9}-\frac{42476591}{10682}a^{8}-\frac{311786303}{21364}a^{7}+\frac{65352323}{5341}a^{6}+\frac{19785755}{5341}a^{5}-\frac{17216831}{3052}a^{4}+\frac{2954355}{21364}a^{3}+\frac{8196915}{10682}a^{2}-\frac{1050601}{21364}a-\frac{135581}{5341}$, $\frac{7583}{5341}a^{15}-\frac{113353}{10682}a^{14}+\frac{1513}{10682}a^{13}+\frac{3389289}{21364}a^{12}-\frac{1476016}{5341}a^{11}-\frac{11969455}{21364}a^{10}+\frac{9361388}{5341}a^{9}-\frac{4568635}{10682}a^{8}-\frac{11739681}{5341}a^{7}+\frac{8346490}{5341}a^{6}+\frac{13084719}{21364}a^{5}-\frac{2005405}{3052}a^{4}-\frac{373943}{21364}a^{3}+\frac{385624}{5341}a^{2}-\frac{136173}{21364}a-\frac{34057}{21364}$, $\frac{197623}{21364}a^{15}-\frac{373869}{5341}a^{14}+\frac{72827}{10682}a^{13}+\frac{11088773}{10682}a^{12}-\frac{40581683}{21364}a^{11}-\frac{18950426}{5341}a^{10}+\frac{127213301}{10682}a^{9}-\frac{77922367}{21364}a^{8}-\frac{319076379}{21364}a^{7}+\frac{126719555}{10682}a^{6}+\frac{87533085}{21364}a^{5}-\frac{4149881}{763}a^{4}-\frac{740877}{21364}a^{3}+\frac{15248117}{21364}a^{2}-\frac{565325}{21364}a-\frac{204581}{10682}$, $\frac{5051}{763}a^{15}-\frac{148779}{3052}a^{14}-\frac{13835}{3052}a^{13}+\frac{2254205}{3052}a^{12}-\frac{3698831}{3052}a^{11}-\frac{4166681}{1526}a^{10}+\frac{6037873}{763}a^{9}-\frac{1920109}{1526}a^{8}-\frac{32048991}{3052}a^{7}+\frac{5040913}{763}a^{6}+\frac{2694505}{763}a^{5}-\frac{1374787}{436}a^{4}-\frac{770433}{3052}a^{3}+\frac{615395}{1526}a^{2}-\frac{36957}{3052}a-\frac{6224}{763}$, $\frac{167291}{21364}a^{15}-\frac{308936}{5341}a^{14}-\frac{44277}{10682}a^{13}+\frac{18678399}{21364}a^{12}-\frac{31013105}{21364}a^{11}-\frac{68522921}{21364}a^{10}+\frac{100666597}{10682}a^{9}-\frac{34761849}{21364}a^{8}-\frac{264568421}{21364}a^{7}+\frac{84151537}{10682}a^{6}+\frac{43663959}{10682}a^{5}-\frac{10988965}{3052}a^{4}-\frac{1759522}{5341}a^{3}+\frac{9022789}{21364}a^{2}-\frac{143}{98}a-\frac{215757}{21364}$, $\frac{106181}{21364}a^{15}-\frac{202501}{5341}a^{14}+\frac{121201}{21364}a^{13}+\frac{11952701}{21364}a^{12}-\frac{11251865}{10682}a^{11}-\frac{40051317}{21364}a^{10}+\frac{34976863}{5341}a^{9}-\frac{47453325}{21364}a^{8}-\frac{174208935}{21364}a^{7}+\frac{145403029}{21364}a^{6}+\frac{45809651}{21364}a^{5}-\frac{2403144}{763}a^{4}+\frac{539563}{10682}a^{3}+\frac{2225543}{5341}a^{2}-\frac{581577}{21364}a-\frac{208589}{21364}$
| 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: | \( 31643195.2071 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 31643195.2071 \cdot 1}{2\cdot\sqrt{11981707062330062500000000}}\cr\approx \mathstrut & 0.299551098571 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 4*x^14 + 112*x^13 - 254*x^12 - 296*x^11 + 1457*x^10 - 949*x^9 - 1463*x^8 + 1998*x^7 - 97*x^6 - 807*x^5 + 256*x^4 + 86*x^3 - 39*x^2 - x + 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^16 - 8*x^15 + 4*x^14 + 112*x^13 - 254*x^12 - 296*x^11 + 1457*x^10 - 949*x^9 - 1463*x^8 + 1998*x^7 - 97*x^6 - 807*x^5 + 256*x^4 + 86*x^3 - 39*x^2 - x + 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 8*x^15 + 4*x^14 + 112*x^13 - 254*x^12 - 296*x^11 + 1457*x^10 - 949*x^9 - 1463*x^8 + 1998*x^7 - 97*x^6 - 807*x^5 + 256*x^4 + 86*x^3 - 39*x^2 - x + 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 4*x^14 + 112*x^13 - 254*x^12 - 296*x^11 + 1457*x^10 - 949*x^9 - 1463*x^8 + 1998*x^7 - 97*x^6 - 807*x^5 + 256*x^4 + 86*x^3 - 39*x^2 - x + 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_2^3:C_4$ (as 16T33):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $C_2^3:C_4$ |
| Character table for $C_2^3:C_4$ |
Intermediate fields
| \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{305}) \), 4.4.465125.1 x2, \(\Q(\sqrt{5}, \sqrt{61})\), 4.4.7625.1 x2, 8.8.3461460250000.1 x2, 8.8.216341265625.1, 8.8.692292050000.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
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 | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(5\)
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(61\)
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |