Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 163*x^14 - 282*x^13 + 12233*x^12 - 18086*x^11 + 549386*x^10 - 679232*x^9 + 16099839*x^8 - 16070158*x^7 + 314665218*x^6 - 238996272*x^5 + 4001369686*x^4 - 2066649212*x^3 + 30257685388*x^2 - 8015671568*x + 104203730401)
gp: K = bnfinit(y^16 - 2*y^15 + 163*y^14 - 282*y^13 + 12233*y^12 - 18086*y^11 + 549386*y^10 - 679232*y^9 + 16099839*y^8 - 16070158*y^7 + 314665218*y^6 - 238996272*y^5 + 4001369686*y^4 - 2066649212*y^3 + 30257685388*y^2 - 8015671568*y + 104203730401, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 163*x^14 - 282*x^13 + 12233*x^12 - 18086*x^11 + 549386*x^10 - 679232*x^9 + 16099839*x^8 - 16070158*x^7 + 314665218*x^6 - 238996272*x^5 + 4001369686*x^4 - 2066649212*x^3 + 30257685388*x^2 - 8015671568*x + 104203730401);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 163*x^14 - 282*x^13 + 12233*x^12 - 18086*x^11 + 549386*x^10 - 679232*x^9 + 16099839*x^8 - 16070158*x^7 + 314665218*x^6 - 238996272*x^5 + 4001369686*x^4 - 2066649212*x^3 + 30257685388*x^2 - 8015671568*x + 104203730401)
\( x^{16} - 2 x^{15} + 163 x^{14} - 282 x^{13} + 12233 x^{12} - 18086 x^{11} + 549386 x^{10} + \cdots + 104203730401 \)
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: |
\(605544796424780341033364025769984\)
\(\medspace = 2^{24}\cdot 11^{8}\cdot 17^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(111.91\) | 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: | $2^{3/2}11^{1/2}17^{7/8}\approx 111.91392883271797$ | ||
| Ramified primes: |
\(2\), \(11\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1496=2^{3}\cdot 11\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1496}(1,·)$, $\chi_{1496}(1409,·)$, $\chi_{1496}(461,·)$, $\chi_{1496}(1165,·)$, $\chi_{1496}(1233,·)$, $\chi_{1496}(21,·)$, $\chi_{1496}(89,·)$, $\chi_{1496}(285,·)$, $\chi_{1496}(353,·)$, $\chi_{1496}(529,·)$, $\chi_{1496}(637,·)$, $\chi_{1496}(373,·)$, $\chi_{1496}(441,·)$, $\chi_{1496}(705,·)$, $\chi_{1496}(1341,·)$, $\chi_{1496}(1429,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{10\!\cdots\!69}a^{15}-\frac{42\!\cdots\!99}{10\!\cdots\!69}a^{14}-\frac{39\!\cdots\!38}{10\!\cdots\!69}a^{13}-\frac{38\!\cdots\!88}{10\!\cdots\!69}a^{12}+\frac{40\!\cdots\!41}{10\!\cdots\!69}a^{11}+\frac{17\!\cdots\!05}{10\!\cdots\!69}a^{10}-\frac{39\!\cdots\!30}{10\!\cdots\!69}a^{9}+\frac{42\!\cdots\!83}{10\!\cdots\!69}a^{8}-\frac{19\!\cdots\!69}{10\!\cdots\!69}a^{7}-\frac{87\!\cdots\!57}{10\!\cdots\!69}a^{6}+\frac{22\!\cdots\!35}{10\!\cdots\!69}a^{5}+\frac{36\!\cdots\!25}{10\!\cdots\!69}a^{4}+\frac{36\!\cdots\!97}{10\!\cdots\!69}a^{3}-\frac{40\!\cdots\!35}{10\!\cdots\!69}a^{2}-\frac{30\!\cdots\!72}{10\!\cdots\!69}a-\frac{35\!\cdots\!42}{10\!\cdots\!69}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{5}\times C_{120}\times C_{1680}$, which has order $1008000$ (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: |
$\frac{94\!\cdots\!16}{10\!\cdots\!69}a^{15}-\frac{18\!\cdots\!23}{10\!\cdots\!69}a^{14}+\frac{15\!\cdots\!48}{10\!\cdots\!69}a^{13}-\frac{26\!\cdots\!64}{10\!\cdots\!69}a^{12}+\frac{11\!\cdots\!56}{10\!\cdots\!69}a^{11}-\frac{16\!\cdots\!39}{10\!\cdots\!69}a^{10}+\frac{52\!\cdots\!72}{10\!\cdots\!69}a^{9}-\frac{62\!\cdots\!76}{10\!\cdots\!69}a^{8}+\frac{15\!\cdots\!28}{10\!\cdots\!69}a^{7}-\frac{14\!\cdots\!26}{10\!\cdots\!69}a^{6}+\frac{28\!\cdots\!72}{10\!\cdots\!69}a^{5}-\frac{22\!\cdots\!12}{10\!\cdots\!69}a^{4}+\frac{32\!\cdots\!90}{10\!\cdots\!69}a^{3}-\frac{19\!\cdots\!23}{10\!\cdots\!69}a^{2}+\frac{17\!\cdots\!78}{10\!\cdots\!69}a-\frac{74\!\cdots\!96}{10\!\cdots\!69}$, $\frac{20\!\cdots\!66}{10\!\cdots\!69}a^{15}-\frac{22\!\cdots\!21}{10\!\cdots\!69}a^{14}+\frac{26\!\cdots\!46}{10\!\cdots\!69}a^{13}-\frac{18\!\cdots\!44}{10\!\cdots\!69}a^{12}+\frac{16\!\cdots\!56}{10\!\cdots\!69}a^{11}-\frac{44\!\cdots\!93}{10\!\cdots\!69}a^{10}+\frac{55\!\cdots\!62}{10\!\cdots\!69}a^{9}+\frac{10\!\cdots\!25}{10\!\cdots\!69}a^{8}+\frac{12\!\cdots\!40}{10\!\cdots\!69}a^{7}+\frac{88\!\cdots\!86}{10\!\cdots\!69}a^{6}+\frac{16\!\cdots\!36}{10\!\cdots\!69}a^{5}+\frac{23\!\cdots\!01}{10\!\cdots\!69}a^{4}+\frac{12\!\cdots\!94}{10\!\cdots\!69}a^{3}+\frac{29\!\cdots\!98}{10\!\cdots\!69}a^{2}+\frac{41\!\cdots\!80}{10\!\cdots\!69}a+\frac{15\!\cdots\!16}{10\!\cdots\!69}$, $\frac{69\!\cdots\!48}{10\!\cdots\!69}a^{15}+\frac{65\!\cdots\!96}{10\!\cdots\!69}a^{14}+\frac{95\!\cdots\!32}{10\!\cdots\!69}a^{13}+\frac{10\!\cdots\!52}{10\!\cdots\!69}a^{12}+\frac{59\!\cdots\!06}{10\!\cdots\!69}a^{11}+\frac{72\!\cdots\!40}{10\!\cdots\!69}a^{10}+\frac{21\!\cdots\!54}{10\!\cdots\!69}a^{9}+\frac{29\!\cdots\!64}{10\!\cdots\!69}a^{8}+\frac{50\!\cdots\!92}{10\!\cdots\!69}a^{7}+\frac{75\!\cdots\!25}{10\!\cdots\!69}a^{6}+\frac{73\!\cdots\!78}{10\!\cdots\!69}a^{5}+\frac{12\!\cdots\!92}{10\!\cdots\!69}a^{4}+\frac{62\!\cdots\!30}{10\!\cdots\!69}a^{3}+\frac{11\!\cdots\!85}{10\!\cdots\!69}a^{2}+\frac{23\!\cdots\!26}{10\!\cdots\!69}a+\frac{47\!\cdots\!96}{10\!\cdots\!69}$, $\frac{20\!\cdots\!18}{10\!\cdots\!69}a^{15}-\frac{80\!\cdots\!69}{10\!\cdots\!69}a^{14}+\frac{44\!\cdots\!86}{10\!\cdots\!69}a^{13}-\frac{12\!\cdots\!03}{10\!\cdots\!69}a^{12}+\frac{39\!\cdots\!78}{10\!\cdots\!69}a^{11}-\frac{83\!\cdots\!73}{10\!\cdots\!69}a^{10}+\frac{18\!\cdots\!44}{10\!\cdots\!69}a^{9}-\frac{32\!\cdots\!98}{10\!\cdots\!69}a^{8}+\frac{54\!\cdots\!84}{10\!\cdots\!69}a^{7}-\frac{80\!\cdots\!15}{10\!\cdots\!69}a^{6}+\frac{95\!\cdots\!94}{10\!\cdots\!69}a^{5}-\frac{12\!\cdots\!98}{10\!\cdots\!69}a^{4}+\frac{94\!\cdots\!28}{10\!\cdots\!69}a^{3}-\frac{11\!\cdots\!99}{10\!\cdots\!69}a^{2}+\frac{41\!\cdots\!94}{10\!\cdots\!69}a-\frac{43\!\cdots\!80}{10\!\cdots\!69}$, $\frac{69\!\cdots\!48}{10\!\cdots\!69}a^{15}+\frac{65\!\cdots\!96}{10\!\cdots\!69}a^{14}+\frac{95\!\cdots\!32}{10\!\cdots\!69}a^{13}+\frac{10\!\cdots\!52}{10\!\cdots\!69}a^{12}+\frac{59\!\cdots\!06}{10\!\cdots\!69}a^{11}+\frac{72\!\cdots\!40}{10\!\cdots\!69}a^{10}+\frac{21\!\cdots\!54}{10\!\cdots\!69}a^{9}+\frac{29\!\cdots\!64}{10\!\cdots\!69}a^{8}+\frac{50\!\cdots\!92}{10\!\cdots\!69}a^{7}+\frac{75\!\cdots\!25}{10\!\cdots\!69}a^{6}+\frac{73\!\cdots\!78}{10\!\cdots\!69}a^{5}+\frac{12\!\cdots\!92}{10\!\cdots\!69}a^{4}+\frac{62\!\cdots\!30}{10\!\cdots\!69}a^{3}+\frac{11\!\cdots\!85}{10\!\cdots\!69}a^{2}+\frac{23\!\cdots\!26}{10\!\cdots\!69}a+\frac{46\!\cdots\!27}{10\!\cdots\!69}$, $\frac{13\!\cdots\!86}{10\!\cdots\!69}a^{15}+\frac{17\!\cdots\!84}{10\!\cdots\!69}a^{14}+\frac{18\!\cdots\!50}{10\!\cdots\!69}a^{13}+\frac{31\!\cdots\!16}{10\!\cdots\!69}a^{12}+\frac{11\!\cdots\!64}{10\!\cdots\!69}a^{11}+\frac{25\!\cdots\!00}{10\!\cdots\!69}a^{10}+\frac{41\!\cdots\!22}{10\!\cdots\!69}a^{9}+\frac{12\!\cdots\!00}{10\!\cdots\!69}a^{8}+\frac{96\!\cdots\!88}{10\!\cdots\!69}a^{7}+\frac{41\!\cdots\!60}{10\!\cdots\!69}a^{6}+\frac{14\!\cdots\!32}{10\!\cdots\!69}a^{5}+\frac{10\!\cdots\!60}{10\!\cdots\!69}a^{4}+\frac{11\!\cdots\!04}{10\!\cdots\!69}a^{3}+\frac{16\!\cdots\!85}{10\!\cdots\!69}a^{2}+\frac{45\!\cdots\!50}{10\!\cdots\!69}a+\frac{12\!\cdots\!63}{10\!\cdots\!69}$, $\frac{47\!\cdots\!40}{10\!\cdots\!69}a^{15}+\frac{60\!\cdots\!52}{10\!\cdots\!69}a^{14}+\frac{64\!\cdots\!64}{10\!\cdots\!69}a^{13}+\frac{10\!\cdots\!72}{10\!\cdots\!69}a^{12}+\frac{40\!\cdots\!88}{10\!\cdots\!69}a^{11}+\frac{81\!\cdots\!28}{10\!\cdots\!69}a^{10}+\frac{14\!\cdots\!04}{10\!\cdots\!69}a^{9}+\frac{38\!\cdots\!72}{10\!\cdots\!69}a^{8}+\frac{34\!\cdots\!60}{10\!\cdots\!69}a^{7}+\frac{11\!\cdots\!92}{10\!\cdots\!69}a^{6}+\frac{50\!\cdots\!56}{10\!\cdots\!69}a^{5}+\frac{21\!\cdots\!09}{10\!\cdots\!69}a^{4}+\frac{42\!\cdots\!84}{10\!\cdots\!69}a^{3}+\frac{24\!\cdots\!16}{10\!\cdots\!69}a^{2}+\frac{16\!\cdots\!56}{10\!\cdots\!69}a+\frac{12\!\cdots\!35}{10\!\cdots\!69}$
| 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: | \( 3640.012213375973 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.012213375973 \cdot 1008000}{2\cdot\sqrt{605544796424780341033364025769984}}\cr\approx \mathstrut & 0.181091966955762 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 163*x^14 - 282*x^13 + 12233*x^12 - 18086*x^11 + 549386*x^10 - 679232*x^9 + 16099839*x^8 - 16070158*x^7 + 314665218*x^6 - 238996272*x^5 + 4001369686*x^4 - 2066649212*x^3 + 30257685388*x^2 - 8015671568*x + 104203730401)
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 - 2*x^15 + 163*x^14 - 282*x^13 + 12233*x^12 - 18086*x^11 + 549386*x^10 - 679232*x^9 + 16099839*x^8 - 16070158*x^7 + 314665218*x^6 - 238996272*x^5 + 4001369686*x^4 - 2066649212*x^3 + 30257685388*x^2 - 8015671568*x + 104203730401, 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 - 2*x^15 + 163*x^14 - 282*x^13 + 12233*x^12 - 18086*x^11 + 549386*x^10 - 679232*x^9 + 16099839*x^8 - 16070158*x^7 + 314665218*x^6 - 238996272*x^5 + 4001369686*x^4 - 2066649212*x^3 + 30257685388*x^2 - 8015671568*x + 104203730401);
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 - 2*x^15 + 163*x^14 - 282*x^13 + 12233*x^12 - 18086*x^11 + 549386*x^10 - 679232*x^9 + 16099839*x^8 - 16070158*x^7 + 314665218*x^6 - 238996272*x^5 + 4001369686*x^4 - 2066649212*x^3 + 30257685388*x^2 - 8015671568*x + 104203730401);
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
$C_2\times C_8$ (as 16T5):
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)
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_8\times C_2$ |
| Character table for $C_8\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-374}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{17}, \sqrt{-22})\), 4.4.4913.1, 4.0.38046272.6, 8.0.1447518813097984.43, 8.0.24607819822665728.4, \(\Q(\zeta_{17})^+\) |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
|
\(11\)
| 11.16.8.1 | $x^{16} + 88 x^{14} + 3402 x^{12} + 14 x^{11} + 74230 x^{10} - 1064 x^{9} + 995839 x^{8} - 40250 x^{7} + 8494703 x^{6} - 233814 x^{5} + 46227770 x^{4} + 1580390 x^{3} + 151894730 x^{2} + 9934568 x + 239653341$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(17\)
| 17.16.14.1 | $x^{16} + 128 x^{15} + 7192 x^{14} + 232064 x^{13} + 4716796 x^{12} + 62185088 x^{11} + 525781480 x^{10} + 2696730752 x^{9} + 7365142088 x^{8} + 8090194432 x^{7} + 4732152320 x^{6} + 1682759680 x^{5} + 456414056 x^{4} + 996830464 x^{3} + 7439529968 x^{2} + 33582546688 x + 66368009604$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |