Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 3*x^14 + 139*x^13 - 825*x^12 + 2322*x^11 + 1617*x^10 - 46309*x^9 + 182354*x^8 - 452696*x^7 + 1527356*x^6 - 2049201*x^5 - 1662839*x^4 + 8808112*x^3 - 11939112*x^2 - 7607472*x + 27618832)
gp: K = bnfinit(y^16 - 5*y^15 - 3*y^14 + 139*y^13 - 825*y^12 + 2322*y^11 + 1617*y^10 - 46309*y^9 + 182354*y^8 - 452696*y^7 + 1527356*y^6 - 2049201*y^5 - 1662839*y^4 + 8808112*y^3 - 11939112*y^2 - 7607472*y + 27618832, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 - 3*x^14 + 139*x^13 - 825*x^12 + 2322*x^11 + 1617*x^10 - 46309*x^9 + 182354*x^8 - 452696*x^7 + 1527356*x^6 - 2049201*x^5 - 1662839*x^4 + 8808112*x^3 - 11939112*x^2 - 7607472*x + 27618832);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 - 3*x^14 + 139*x^13 - 825*x^12 + 2322*x^11 + 1617*x^10 - 46309*x^9 + 182354*x^8 - 452696*x^7 + 1527356*x^6 - 2049201*x^5 - 1662839*x^4 + 8808112*x^3 - 11939112*x^2 - 7607472*x + 27618832)
\( x^{16} - 5 x^{15} - 3 x^{14} + 139 x^{13} - 825 x^{12} + 2322 x^{11} + 1617 x^{10} - 46309 x^{9} + \cdots + 27618832 \)
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: |
\(42832588164190465585875603485409\)
\(\medspace = 3^{8}\cdot 97^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(94.84\) | 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: | $3^{1/2}97^{7/8}\approx 94.8387660758292$ | ||
| Ramified primes: |
\(3\), \(97\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{1648}a^{14}+\frac{11}{1648}a^{13}-\frac{295}{1648}a^{12}+\frac{159}{1648}a^{11}-\frac{301}{1648}a^{10}-\frac{137}{824}a^{9}-\frac{331}{1648}a^{8}+\frac{819}{1648}a^{7}+\frac{83}{824}a^{6}-\frac{67}{412}a^{5}-\frac{185}{412}a^{4}+\frac{783}{1648}a^{3}-\frac{423}{1648}a^{2}-\frac{97}{412}a+\frac{1}{4}$, $\frac{1}{10\!\cdots\!16}a^{15}+\frac{11\!\cdots\!09}{10\!\cdots\!16}a^{14}+\frac{10\!\cdots\!07}{10\!\cdots\!16}a^{13}+\frac{23\!\cdots\!25}{10\!\cdots\!16}a^{12}-\frac{17\!\cdots\!91}{10\!\cdots\!16}a^{11}+\frac{49\!\cdots\!93}{25\!\cdots\!04}a^{10}-\frac{15\!\cdots\!47}{10\!\cdots\!16}a^{9}+\frac{16\!\cdots\!57}{10\!\cdots\!16}a^{8}+\frac{63\!\cdots\!51}{25\!\cdots\!04}a^{7}-\frac{71\!\cdots\!39}{62\!\cdots\!92}a^{6}+\frac{10\!\cdots\!83}{25\!\cdots\!04}a^{5}-\frac{11\!\cdots\!93}{10\!\cdots\!16}a^{4}+\frac{26\!\cdots\!75}{10\!\cdots\!16}a^{3}+\frac{21\!\cdots\!87}{51\!\cdots\!08}a^{2}+\frac{69\!\cdots\!79}{54\!\cdots\!32}a-\frac{34\!\cdots\!11}{12\!\cdots\!84}$
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
$C_{2}\times C_{34}$, which has order $68$ (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: |
$\frac{11\!\cdots\!43}{51\!\cdots\!08}a^{15}-\frac{61\!\cdots\!63}{51\!\cdots\!08}a^{14}-\frac{47\!\cdots\!37}{51\!\cdots\!08}a^{13}+\frac{17\!\cdots\!69}{51\!\cdots\!08}a^{12}-\frac{94\!\cdots\!59}{51\!\cdots\!08}a^{11}+\frac{12\!\cdots\!63}{25\!\cdots\!04}a^{10}+\frac{30\!\cdots\!83}{51\!\cdots\!08}a^{9}-\frac{56\!\cdots\!71}{51\!\cdots\!08}a^{8}+\frac{98\!\cdots\!03}{25\!\cdots\!04}a^{7}-\frac{10\!\cdots\!83}{12\!\cdots\!52}a^{6}+\frac{41\!\cdots\!27}{12\!\cdots\!52}a^{5}-\frac{28\!\cdots\!67}{51\!\cdots\!08}a^{4}-\frac{18\!\cdots\!65}{51\!\cdots\!08}a^{3}+\frac{40\!\cdots\!57}{16\!\cdots\!69}a^{2}-\frac{12\!\cdots\!21}{27\!\cdots\!16}a+\frac{88\!\cdots\!13}{31\!\cdots\!46}$, $\frac{26\!\cdots\!75}{51\!\cdots\!08}a^{15}-\frac{15\!\cdots\!51}{51\!\cdots\!08}a^{14}-\frac{65\!\cdots\!01}{51\!\cdots\!08}a^{13}+\frac{40\!\cdots\!13}{51\!\cdots\!08}a^{12}-\frac{23\!\cdots\!59}{51\!\cdots\!08}a^{11}+\frac{33\!\cdots\!27}{25\!\cdots\!04}a^{10}+\frac{53\!\cdots\!19}{51\!\cdots\!08}a^{9}-\frac{14\!\cdots\!63}{51\!\cdots\!08}a^{8}+\frac{27\!\cdots\!31}{25\!\cdots\!04}a^{7}-\frac{29\!\cdots\!55}{12\!\cdots\!52}a^{6}+\frac{91\!\cdots\!03}{12\!\cdots\!52}a^{5}-\frac{63\!\cdots\!19}{51\!\cdots\!08}a^{4}-\frac{10\!\cdots\!69}{51\!\cdots\!08}a^{3}+\frac{11\!\cdots\!09}{16\!\cdots\!69}a^{2}-\frac{58\!\cdots\!73}{27\!\cdots\!16}a-\frac{42\!\cdots\!01}{31\!\cdots\!46}$, $\frac{33\!\cdots\!61}{51\!\cdots\!08}a^{15}-\frac{16\!\cdots\!97}{51\!\cdots\!08}a^{14}-\frac{16\!\cdots\!47}{51\!\cdots\!08}a^{13}+\frac{48\!\cdots\!31}{51\!\cdots\!08}a^{12}-\frac{26\!\cdots\!97}{51\!\cdots\!08}a^{11}+\frac{34\!\cdots\!97}{25\!\cdots\!04}a^{10}+\frac{95\!\cdots\!09}{51\!\cdots\!08}a^{9}-\frac{16\!\cdots\!25}{51\!\cdots\!08}a^{8}+\frac{29\!\cdots\!65}{25\!\cdots\!04}a^{7}-\frac{29\!\cdots\!55}{12\!\cdots\!84}a^{6}+\frac{10\!\cdots\!49}{12\!\cdots\!52}a^{5}-\frac{59\!\cdots\!65}{51\!\cdots\!08}a^{4}-\frac{13\!\cdots\!67}{51\!\cdots\!08}a^{3}+\frac{10\!\cdots\!43}{16\!\cdots\!69}a^{2}-\frac{25\!\cdots\!83}{27\!\cdots\!16}a-\frac{12\!\cdots\!27}{31\!\cdots\!46}$, $\frac{97\!\cdots\!85}{92\!\cdots\!44}a^{15}+\frac{21\!\cdots\!55}{46\!\cdots\!72}a^{14}-\frac{18\!\cdots\!65}{46\!\cdots\!72}a^{13}+\frac{53\!\cdots\!27}{46\!\cdots\!72}a^{12}-\frac{58\!\cdots\!11}{23\!\cdots\!36}a^{11}+\frac{63\!\cdots\!75}{92\!\cdots\!44}a^{10}+\frac{48\!\cdots\!19}{92\!\cdots\!44}a^{9}-\frac{10\!\cdots\!23}{46\!\cdots\!72}a^{8}+\frac{42\!\cdots\!75}{92\!\cdots\!44}a^{7}-\frac{99\!\cdots\!95}{46\!\cdots\!72}a^{6}+\frac{20\!\cdots\!48}{57\!\cdots\!59}a^{5}+\frac{10\!\cdots\!59}{92\!\cdots\!44}a^{4}-\frac{56\!\cdots\!93}{46\!\cdots\!72}a^{3}+\frac{19\!\cdots\!35}{92\!\cdots\!44}a^{2}+\frac{42\!\cdots\!67}{49\!\cdots\!88}a-\frac{98\!\cdots\!95}{22\!\cdots\!12}$, $\frac{15\!\cdots\!91}{36\!\cdots\!76}a^{15}+\frac{33\!\cdots\!23}{36\!\cdots\!76}a^{14}-\frac{39\!\cdots\!11}{36\!\cdots\!76}a^{13}+\frac{14\!\cdots\!03}{36\!\cdots\!76}a^{12}+\frac{12\!\cdots\!35}{36\!\cdots\!76}a^{11}-\frac{79\!\cdots\!19}{92\!\cdots\!44}a^{10}+\frac{19\!\cdots\!07}{36\!\cdots\!76}a^{9}-\frac{45\!\cdots\!09}{36\!\cdots\!76}a^{8}-\frac{25\!\cdots\!53}{92\!\cdots\!44}a^{7}+\frac{33\!\cdots\!11}{23\!\cdots\!36}a^{6}-\frac{40\!\cdots\!19}{92\!\cdots\!44}a^{5}+\frac{12\!\cdots\!49}{36\!\cdots\!76}a^{4}-\frac{13\!\cdots\!63}{36\!\cdots\!76}a^{3}+\frac{21\!\cdots\!97}{18\!\cdots\!88}a^{2}-\frac{16\!\cdots\!51}{19\!\cdots\!52}a+\frac{40\!\cdots\!63}{44\!\cdots\!24}$, $\frac{14\!\cdots\!89}{92\!\cdots\!44}a^{15}-\frac{68\!\cdots\!67}{92\!\cdots\!44}a^{14}+\frac{54\!\cdots\!07}{92\!\cdots\!44}a^{13}+\frac{11\!\cdots\!25}{92\!\cdots\!44}a^{12}-\frac{76\!\cdots\!47}{92\!\cdots\!44}a^{11}+\frac{33\!\cdots\!77}{11\!\cdots\!18}a^{10}-\frac{33\!\cdots\!63}{92\!\cdots\!44}a^{9}-\frac{12\!\cdots\!95}{92\!\cdots\!44}a^{8}+\frac{67\!\cdots\!87}{11\!\cdots\!18}a^{7}+\frac{17\!\cdots\!45}{23\!\cdots\!36}a^{6}+\frac{20\!\cdots\!49}{23\!\cdots\!36}a^{5}+\frac{38\!\cdots\!31}{92\!\cdots\!44}a^{4}-\frac{47\!\cdots\!61}{92\!\cdots\!44}a^{3}-\frac{22\!\cdots\!35}{46\!\cdots\!72}a^{2}+\frac{11\!\cdots\!77}{12\!\cdots\!97}a+\frac{57\!\cdots\!56}{56\!\cdots\!53}$, $\frac{24\!\cdots\!09}{12\!\cdots\!52}a^{15}-\frac{93\!\cdots\!17}{12\!\cdots\!52}a^{14}-\frac{21\!\cdots\!35}{12\!\cdots\!52}a^{13}+\frac{31\!\cdots\!91}{12\!\cdots\!52}a^{12}-\frac{15\!\cdots\!49}{12\!\cdots\!52}a^{11}+\frac{17\!\cdots\!07}{64\!\cdots\!76}a^{10}+\frac{91\!\cdots\!17}{12\!\cdots\!52}a^{9}-\frac{10\!\cdots\!13}{12\!\cdots\!52}a^{8}+\frac{15\!\cdots\!17}{64\!\cdots\!76}a^{7}-\frac{16\!\cdots\!25}{32\!\cdots\!38}a^{6}+\frac{35\!\cdots\!34}{16\!\cdots\!69}a^{5}-\frac{13\!\cdots\!49}{12\!\cdots\!52}a^{4}-\frac{79\!\cdots\!55}{12\!\cdots\!52}a^{3}+\frac{83\!\cdots\!04}{16\!\cdots\!69}a^{2}-\frac{11\!\cdots\!47}{68\!\cdots\!54}a-\frac{59\!\cdots\!89}{15\!\cdots\!23}$
| 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: | \( 22316334.7724 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 22316334.7724 \cdot 68}{2\cdot\sqrt{42832588164190465585875603485409}}\cr\approx \mathstrut & 0.281613566598 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 3*x^14 + 139*x^13 - 825*x^12 + 2322*x^11 + 1617*x^10 - 46309*x^9 + 182354*x^8 - 452696*x^7 + 1527356*x^6 - 2049201*x^5 - 1662839*x^4 + 8808112*x^3 - 11939112*x^2 - 7607472*x + 27618832)
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 - 5*x^15 - 3*x^14 + 139*x^13 - 825*x^12 + 2322*x^11 + 1617*x^10 - 46309*x^9 + 182354*x^8 - 452696*x^7 + 1527356*x^6 - 2049201*x^5 - 1662839*x^4 + 8808112*x^3 - 11939112*x^2 - 7607472*x + 27618832, 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 - 5*x^15 - 3*x^14 + 139*x^13 - 825*x^12 + 2322*x^11 + 1617*x^10 - 46309*x^9 + 182354*x^8 - 452696*x^7 + 1527356*x^6 - 2049201*x^5 - 1662839*x^4 + 8808112*x^3 - 11939112*x^2 - 7607472*x + 27618832);
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 - 5*x^15 - 3*x^14 + 139*x^13 - 825*x^12 + 2322*x^11 + 1617*x^10 - 46309*x^9 + 182354*x^8 - 452696*x^7 + 1527356*x^6 - 2049201*x^5 - 1662839*x^4 + 8808112*x^3 - 11939112*x^2 - 7607472*x + 27618832);
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
$C_2^2:C_8$ (as 16T24):
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 20 conjugacy class representatives for $C_2^2 : C_8$ |
| Character table for $C_2^2 : C_8$ |
Intermediate fields
| \(\Q(\sqrt{97}) \), 4.2.2738019.1, 4.4.912673.1, 4.2.28227.1, 8.4.727184560303017.1, 8.0.6544661042727153.1, 8.4.7496748044361.1 |
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
| Galois closure: | deg 32 |
| Degree 16 sibling: | 16.8.528797384743092167726859302289.1 |
| Minimal sibling: | 16.8.528797384743092167726859302289.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(97\)
| 97.16.14.1 | $x^{16} + 768 x^{15} + 258088 x^{14} + 49572096 x^{13} + 5953168060 x^{12} + 457847744256 x^{11} + 22036223404888 x^{10} + 608019502810368 x^{9} + 7433960430005160 x^{8} + 3040097514126336 x^{7} + 550905610125696 x^{6} + 57235766095872 x^{5} + 4296185158440 x^{4} + 44325875768832 x^{3} + 2119059344887056 x^{2} + 58091109365638656 x + 696714908933731396$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |