Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 34*x^14 - 78*x^13 + 166*x^12 - 296*x^11 + 592*x^10 - 1594*x^9 + 3932*x^8 - 9382*x^7 + 20318*x^6 - 34912*x^5 + 44991*x^4 - 39494*x^3 + 23226*x^2 - 7276*x + 1076)
gp: K = bnfinit(y^16 - 8*y^15 + 34*y^14 - 78*y^13 + 166*y^12 - 296*y^11 + 592*y^10 - 1594*y^9 + 3932*y^8 - 9382*y^7 + 20318*y^6 - 34912*y^5 + 44991*y^4 - 39494*y^3 + 23226*y^2 - 7276*y + 1076, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 34*x^14 - 78*x^13 + 166*x^12 - 296*x^11 + 592*x^10 - 1594*x^9 + 3932*x^8 - 9382*x^7 + 20318*x^6 - 34912*x^5 + 44991*x^4 - 39494*x^3 + 23226*x^2 - 7276*x + 1076);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 34*x^14 - 78*x^13 + 166*x^12 - 296*x^11 + 592*x^10 - 1594*x^9 + 3932*x^8 - 9382*x^7 + 20318*x^6 - 34912*x^5 + 44991*x^4 - 39494*x^3 + 23226*x^2 - 7276*x + 1076)
\( x^{16} - 8 x^{15} + 34 x^{14} - 78 x^{13} + 166 x^{12} - 296 x^{11} + 592 x^{10} - 1594 x^{9} + \cdots + 1076 \)
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: | $[0, 8]$ | 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 a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{12}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{8}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{336}a^{13}-\frac{5}{84}a^{12}+\frac{5}{168}a^{11}+\frac{13}{336}a^{10}-\frac{17}{168}a^{9}+\frac{1}{168}a^{8}-\frac{73}{336}a^{7}+\frac{19}{84}a^{6}+\frac{29}{168}a^{5}-\frac{137}{336}a^{4}-\frac{15}{56}a^{3}-\frac{17}{56}a^{2}-\frac{5}{12}a-\frac{11}{84}$, $\frac{1}{116482464}a^{14}+\frac{47315}{38827488}a^{13}-\frac{5227}{510888}a^{12}-\frac{72239}{1848928}a^{11}-\frac{2177543}{116482464}a^{10}+\frac{1593}{12164}a^{9}+\frac{826829}{5546784}a^{8}+\frac{2607743}{116482464}a^{7}-\frac{2085785}{9706872}a^{6}+\frac{3848287}{38827488}a^{5}+\frac{44933471}{116482464}a^{4}+\frac{1490133}{3235624}a^{3}-\frac{22018615}{58241232}a^{2}+\frac{138304}{404453}a+\frac{14425139}{29120616}$, $\frac{1}{11\!\cdots\!44}a^{15}-\frac{36334507}{11\!\cdots\!44}a^{14}-\frac{308304724247}{665138301829008}a^{13}-\frac{189166583696591}{39\!\cdots\!48}a^{12}-\frac{381128409488747}{11\!\cdots\!44}a^{11}+\frac{290731126640167}{59\!\cdots\!72}a^{10}+\frac{137310229917521}{570118544424864}a^{9}+\frac{29\!\cdots\!83}{11\!\cdots\!44}a^{8}+\frac{23632038922241}{855177816637296}a^{7}+\frac{22093938115511}{190039514808288}a^{6}+\frac{83506487206709}{17\!\cdots\!92}a^{5}-\frac{644922887957545}{59\!\cdots\!72}a^{4}+\frac{27\!\cdots\!11}{59\!\cdots\!72}a^{3}+\frac{78084529745009}{427588908318648}a^{2}-\frac{9394590642355}{22504679385192}a+\frac{271317111140951}{14\!\cdots\!68}$
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
$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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: | $7$ | 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: |
$\frac{60869315765}{78766377848172}a^{15}-\frac{947674616197}{157532755696344}a^{14}+\frac{414457265381}{17503639521816}a^{13}-\frac{1230585784735}{26255459282724}a^{12}+\frac{2015256616303}{22504679385192}a^{11}-\frac{26569918112245}{157532755696344}a^{10}+\frac{8461019079259}{26255459282724}a^{9}-\frac{159930650974849}{157532755696344}a^{8}+\frac{388183499712343}{157532755696344}a^{7}-\frac{48173866551835}{8751819760908}a^{6}+\frac{18\!\cdots\!07}{157532755696344}a^{5}-\frac{28\!\cdots\!55}{157532755696344}a^{4}+\frac{367445113758461}{19691594462043}a^{3}-\frac{128747842459433}{11252339692596}a^{2}+\frac{70963947322483}{19691594462043}a-\frac{44756618855771}{39383188924086}$, $\frac{8906926245137}{11\!\cdots\!44}a^{15}-\frac{44809701762323}{11\!\cdots\!44}a^{14}+\frac{2199282848841}{221712767276336}a^{13}-\frac{646890283417}{570118544424864}a^{12}+\frac{300694084038533}{11\!\cdots\!44}a^{11}-\frac{49652026500343}{59\!\cdots\!72}a^{10}+\frac{354146875756351}{39\!\cdots\!48}a^{9}-\frac{737656426007471}{17\!\cdots\!92}a^{8}+\frac{29\!\cdots\!53}{59\!\cdots\!72}a^{7}-\frac{20\!\cdots\!03}{13\!\cdots\!16}a^{6}+\frac{27\!\cdots\!79}{11\!\cdots\!44}a^{5}+\frac{53\!\cdots\!37}{59\!\cdots\!72}a^{4}-\frac{28\!\cdots\!33}{59\!\cdots\!72}a^{3}+\frac{31\!\cdots\!11}{427588908318648}a^{2}-\frac{34\!\cdots\!11}{29\!\cdots\!36}a+\frac{41998483117021}{14\!\cdots\!68}$, $\frac{8087360317343}{11\!\cdots\!44}a^{15}-\frac{51834858650191}{59\!\cdots\!72}a^{14}+\frac{60004694278723}{13\!\cdots\!16}a^{13}-\frac{532685520476161}{39\!\cdots\!48}a^{12}+\frac{105090408248375}{427588908318648}a^{11}-\frac{58\!\cdots\!89}{11\!\cdots\!44}a^{10}+\frac{34\!\cdots\!65}{39\!\cdots\!48}a^{9}-\frac{63\!\cdots\!19}{29\!\cdots\!36}a^{8}+\frac{72\!\cdots\!01}{11\!\cdots\!44}a^{7}-\frac{61\!\cdots\!75}{443425534552672}a^{6}+\frac{23\!\cdots\!53}{748280589557634}a^{5}-\frac{71\!\cdots\!95}{11\!\cdots\!44}a^{4}+\frac{48\!\cdots\!97}{59\!\cdots\!72}a^{3}-\frac{43\!\cdots\!01}{59\!\cdots\!72}a^{2}+\frac{93\!\cdots\!29}{29\!\cdots\!36}a-\frac{24\!\cdots\!65}{427588908318648}$, $\frac{2260971422183}{11\!\cdots\!44}a^{15}-\frac{62989472818649}{11\!\cdots\!44}a^{14}+\frac{21363408337909}{665138301829008}a^{13}-\frac{433558605857077}{39\!\cdots\!48}a^{12}+\frac{22\!\cdots\!87}{11\!\cdots\!44}a^{11}-\frac{23\!\cdots\!17}{59\!\cdots\!72}a^{10}+\frac{26\!\cdots\!09}{39\!\cdots\!48}a^{9}-\frac{137176452725927}{90018717540768}a^{8}+\frac{40\!\cdots\!73}{855177816637296}a^{7}-\frac{673087759365889}{63346504936096}a^{6}+\frac{41\!\cdots\!59}{17\!\cdots\!92}a^{5}-\frac{29\!\cdots\!09}{59\!\cdots\!72}a^{4}+\frac{40\!\cdots\!49}{59\!\cdots\!72}a^{3}-\frac{18\!\cdots\!09}{29\!\cdots\!36}a^{2}+\frac{78\!\cdots\!83}{29\!\cdots\!36}a-\frac{10\!\cdots\!43}{213794454159324}$, $\frac{1856557907867}{13\!\cdots\!16}a^{15}-\frac{7121245534961}{665138301829008}a^{14}+\frac{55692124343735}{13\!\cdots\!16}a^{13}-\frac{108776594886991}{13\!\cdots\!16}a^{12}+\frac{3778157928875}{23754939351036}a^{11}-\frac{392909338559405}{13\!\cdots\!16}a^{10}+\frac{760247534793763}{13\!\cdots\!16}a^{9}-\frac{74198678409140}{41571143864313}a^{8}+\frac{57\!\cdots\!77}{13\!\cdots\!16}a^{7}-\frac{12\!\cdots\!49}{13\!\cdots\!16}a^{6}+\frac{23\!\cdots\!75}{110856383638168}a^{5}-\frac{14\!\cdots\!33}{443425534552672}a^{4}+\frac{21\!\cdots\!81}{665138301829008}a^{3}-\frac{13\!\cdots\!57}{665138301829008}a^{2}+\frac{708427738478459}{110856383638168}a-\frac{127450146351341}{110856383638168}$, $\frac{61746651752431}{11\!\cdots\!44}a^{15}-\frac{3907493006251}{106897227079662}a^{14}+\frac{8965253175151}{63346504936096}a^{13}-\frac{10\!\cdots\!89}{39\!\cdots\!48}a^{12}+\frac{35\!\cdots\!99}{59\!\cdots\!72}a^{11}-\frac{11\!\cdots\!65}{11\!\cdots\!44}a^{10}+\frac{11\!\cdots\!23}{570118544424864}a^{9}-\frac{36\!\cdots\!61}{59\!\cdots\!72}a^{8}+\frac{17\!\cdots\!01}{11\!\cdots\!44}a^{7}-\frac{15\!\cdots\!75}{443425534552672}a^{6}+\frac{22\!\cdots\!45}{315065511392688}a^{5}-\frac{13\!\cdots\!47}{11\!\cdots\!44}a^{4}+\frac{71\!\cdots\!79}{59\!\cdots\!72}a^{3}-\frac{43\!\cdots\!05}{59\!\cdots\!72}a^{2}+\frac{66\!\cdots\!11}{29\!\cdots\!36}a+\frac{21\!\cdots\!21}{29\!\cdots\!36}$, $\frac{840441010823}{997707452743512}a^{15}-\frac{9925664445941}{39\!\cdots\!48}a^{14}-\frac{1459232664139}{13\!\cdots\!16}a^{13}+\frac{14534659961095}{332569150914504}a^{12}-\frac{190314654586555}{39\!\cdots\!48}a^{11}+\frac{589872166013551}{39\!\cdots\!48}a^{10}-\frac{8515618889389}{47509878702072}a^{9}+\frac{256700546189443}{39\!\cdots\!48}a^{8}-\frac{56\!\cdots\!11}{39\!\cdots\!48}a^{7}+\frac{836449169458265}{332569150914504}a^{6}-\frac{28\!\cdots\!13}{39\!\cdots\!48}a^{5}+\frac{12\!\cdots\!31}{570118544424864}a^{4}-\frac{33\!\cdots\!39}{997707452743512}a^{3}+\frac{66\!\cdots\!95}{19\!\cdots\!24}a^{2}-\frac{32\!\cdots\!29}{249426863185878}a+\frac{22\!\cdots\!37}{997707452743512}$
| 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: | \( 160289.989776 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 160289.989776 \cdot 16}{2\cdot\sqrt{11981707062330062500000000}}\cr\approx \mathstrut & 0.899862152977 \end{aligned}\] (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 + 34*x^14 - 78*x^13 + 166*x^12 - 296*x^11 + 592*x^10 - 1594*x^9 + 3932*x^8 - 9382*x^7 + 20318*x^6 - 34912*x^5 + 44991*x^4 - 39494*x^3 + 23226*x^2 - 7276*x + 1076)
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 + 34*x^14 - 78*x^13 + 166*x^12 - 296*x^11 + 592*x^10 - 1594*x^9 + 3932*x^8 - 9382*x^7 + 20318*x^6 - 34912*x^5 + 44991*x^4 - 39494*x^3 + 23226*x^2 - 7276*x + 1076, 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 + 34*x^14 - 78*x^13 + 166*x^12 - 296*x^11 + 592*x^10 - 1594*x^9 + 3932*x^8 - 9382*x^7 + 20318*x^6 - 34912*x^5 + 44991*x^4 - 39494*x^3 + 23226*x^2 - 7276*x + 1076);
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 + 34*x^14 - 78*x^13 + 166*x^12 - 296*x^11 + 592*x^10 - 1594*x^9 + 3932*x^8 - 9382*x^7 + 20318*x^6 - 34912*x^5 + 44991*x^4 - 39494*x^3 + 23226*x^2 - 7276*x + 1076);
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, 8.0.3461460250000.1 x2, 8.0.692292050000.2 x2, 8.8.216341265625.1 |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
|
\(5\)
| 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 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |