Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 180*x^14 - 1120*x^13 + 14026*x^12 - 69960*x^11 + 627024*x^10 - 2519560*x^9 + 17702027*x^8 - 56416264*x^7 + 324191712*x^6 - 784977032*x^5 + 3766659562*x^4 - 6286165360*x^3 + 25396479484*x^2 - 22375524712*x + 76059908671)
gp: K = bnfinit(y^16 - 8*y^15 + 180*y^14 - 1120*y^13 + 14026*y^12 - 69960*y^11 + 627024*y^10 - 2519560*y^9 + 17702027*y^8 - 56416264*y^7 + 324191712*y^6 - 784977032*y^5 + 3766659562*y^4 - 6286165360*y^3 + 25396479484*y^2 - 22375524712*y + 76059908671, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 180*x^14 - 1120*x^13 + 14026*x^12 - 69960*x^11 + 627024*x^10 - 2519560*x^9 + 17702027*x^8 - 56416264*x^7 + 324191712*x^6 - 784977032*x^5 + 3766659562*x^4 - 6286165360*x^3 + 25396479484*x^2 - 22375524712*x + 76059908671);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 180*x^14 - 1120*x^13 + 14026*x^12 - 69960*x^11 + 627024*x^10 - 2519560*x^9 + 17702027*x^8 - 56416264*x^7 + 324191712*x^6 - 784977032*x^5 + 3766659562*x^4 - 6286165360*x^3 + 25396479484*x^2 - 22375524712*x + 76059908671)
\( x^{16} - 8 x^{15} + 180 x^{14} - 1120 x^{13} + 14026 x^{12} - 69960 x^{11} + 627024 x^{10} + \cdots + 76059908671 \)
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: |
\(10386864596368228067918124169560064\)
\(\medspace = 2^{62}\cdot 83^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(133.67\) | 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: | $2^{31/8}83^{1/2}\approx 133.66887083876682$ | ||
| Ramified primes: |
\(2\), \(83\)
| 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: | \(2656=2^{5}\cdot 83\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2656}(1,·)$, $\chi_{2656}(1161,·)$, $\chi_{2656}(333,·)$, $\chi_{2656}(1493,·)$, $\chi_{2656}(665,·)$, $\chi_{2656}(165,·)$, $\chi_{2656}(1825,·)$, $\chi_{2656}(997,·)$, $\chi_{2656}(497,·)$, $\chi_{2656}(2157,·)$, $\chi_{2656}(829,·)$, $\chi_{2656}(1329,·)$, $\chi_{2656}(1993,·)$, $\chi_{2656}(2489,·)$, $\chi_{2656}(1661,·)$, $\chi_{2656}(2325,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{18\!\cdots\!21}a^{14}-\frac{7}{18\!\cdots\!21}a^{13}+\frac{73\!\cdots\!63}{18\!\cdots\!21}a^{12}-\frac{67\!\cdots\!45}{18\!\cdots\!21}a^{11}-\frac{48\!\cdots\!01}{18\!\cdots\!21}a^{10}-\frac{11\!\cdots\!14}{18\!\cdots\!21}a^{9}+\frac{69\!\cdots\!72}{18\!\cdots\!21}a^{8}+\frac{43\!\cdots\!71}{18\!\cdots\!21}a^{7}+\frac{57\!\cdots\!91}{18\!\cdots\!21}a^{6}-\frac{79\!\cdots\!36}{18\!\cdots\!21}a^{5}-\frac{52\!\cdots\!78}{18\!\cdots\!21}a^{4}+\frac{28\!\cdots\!38}{18\!\cdots\!21}a^{3}+\frac{61\!\cdots\!19}{18\!\cdots\!21}a^{2}+\frac{12\!\cdots\!26}{18\!\cdots\!21}a+\frac{23\!\cdots\!07}{59\!\cdots\!91}$, $\frac{1}{12\!\cdots\!69}a^{15}+\frac{34022737}{12\!\cdots\!69}a^{14}+\frac{48\!\cdots\!43}{12\!\cdots\!69}a^{13}-\frac{40\!\cdots\!49}{12\!\cdots\!69}a^{12}-\frac{30\!\cdots\!75}{12\!\cdots\!69}a^{11}+\frac{49\!\cdots\!02}{12\!\cdots\!69}a^{10}-\frac{91\!\cdots\!09}{12\!\cdots\!69}a^{9}+\frac{60\!\cdots\!92}{12\!\cdots\!69}a^{8}+\frac{47\!\cdots\!09}{12\!\cdots\!69}a^{7}+\frac{21\!\cdots\!84}{12\!\cdots\!69}a^{6}-\frac{16\!\cdots\!39}{40\!\cdots\!99}a^{5}-\frac{66\!\cdots\!00}{12\!\cdots\!69}a^{4}+\frac{53\!\cdots\!43}{12\!\cdots\!69}a^{3}+\frac{35\!\cdots\!51}{12\!\cdots\!69}a^{2}-\frac{92\!\cdots\!49}{12\!\cdots\!69}a-\frac{19\!\cdots\!59}{40\!\cdots\!99}$
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
$C_{5}\times C_{5}\times C_{65535}$, which has order $1638375$ (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: |
$\frac{28332880}{32\!\cdots\!89}a^{14}-\frac{198330160}{32\!\cdots\!89}a^{13}+\frac{5008159726}{32\!\cdots\!89}a^{12}-\frac{27470666276}{32\!\cdots\!89}a^{11}+\frac{388305378410}{32\!\cdots\!89}a^{10}-\frac{1694439320000}{32\!\cdots\!89}a^{9}+\frac{17396227385793}{32\!\cdots\!89}a^{8}-\frac{59712347748528}{32\!\cdots\!89}a^{7}+\frac{484826786428284}{32\!\cdots\!89}a^{6}-\frac{12\!\cdots\!32}{32\!\cdots\!89}a^{5}+\frac{83\!\cdots\!39}{32\!\cdots\!89}a^{4}-\frac{14\!\cdots\!92}{32\!\cdots\!89}a^{3}+\frac{83\!\cdots\!72}{32\!\cdots\!89}a^{2}-\frac{76\!\cdots\!16}{32\!\cdots\!89}a+\frac{12\!\cdots\!36}{104798748388319}$, $\frac{17\!\cdots\!14}{18\!\cdots\!21}a^{14}-\frac{12\!\cdots\!98}{18\!\cdots\!21}a^{13}+\frac{28\!\cdots\!85}{18\!\cdots\!21}a^{12}-\frac{15\!\cdots\!36}{18\!\cdots\!21}a^{11}+\frac{19\!\cdots\!57}{18\!\cdots\!21}a^{10}-\frac{84\!\cdots\!24}{18\!\cdots\!21}a^{9}+\frac{76\!\cdots\!91}{18\!\cdots\!21}a^{8}-\frac{25\!\cdots\!12}{18\!\cdots\!21}a^{7}+\frac{18\!\cdots\!21}{18\!\cdots\!21}a^{6}-\frac{45\!\cdots\!50}{18\!\cdots\!21}a^{5}+\frac{25\!\cdots\!62}{18\!\cdots\!21}a^{4}-\frac{44\!\cdots\!48}{18\!\cdots\!21}a^{3}+\frac{21\!\cdots\!52}{18\!\cdots\!21}a^{2}-\frac{18\!\cdots\!14}{18\!\cdots\!21}a+\frac{24\!\cdots\!02}{59\!\cdots\!91}$, $\frac{50464632405476}{18\!\cdots\!21}a^{14}-\frac{353252426838332}{18\!\cdots\!21}a^{13}+\frac{90\!\cdots\!98}{18\!\cdots\!21}a^{12}-\frac{49\!\cdots\!72}{18\!\cdots\!21}a^{11}+\frac{71\!\cdots\!52}{18\!\cdots\!21}a^{10}-\frac{31\!\cdots\!46}{18\!\cdots\!21}a^{9}+\frac{33\!\cdots\!53}{18\!\cdots\!21}a^{8}-\frac{11\!\cdots\!92}{18\!\cdots\!21}a^{7}+\frac{10\!\cdots\!64}{18\!\cdots\!21}a^{6}-\frac{27\!\cdots\!24}{18\!\cdots\!21}a^{5}+\frac{23\!\cdots\!90}{18\!\cdots\!21}a^{4}-\frac{41\!\cdots\!08}{18\!\cdots\!21}a^{3}+\frac{33\!\cdots\!83}{18\!\cdots\!21}a^{2}-\frac{31\!\cdots\!42}{18\!\cdots\!21}a+\frac{74\!\cdots\!25}{59\!\cdots\!91}$, $\frac{32\!\cdots\!98}{12\!\cdots\!69}a^{15}-\frac{24\!\cdots\!35}{12\!\cdots\!69}a^{14}+\frac{47\!\cdots\!73}{12\!\cdots\!69}a^{13}-\frac{27\!\cdots\!77}{12\!\cdots\!69}a^{12}+\frac{29\!\cdots\!41}{12\!\cdots\!69}a^{11}-\frac{13\!\cdots\!03}{12\!\cdots\!69}a^{10}+\frac{10\!\cdots\!79}{12\!\cdots\!69}a^{9}-\frac{38\!\cdots\!51}{12\!\cdots\!69}a^{8}+\frac{23\!\cdots\!75}{12\!\cdots\!69}a^{7}-\frac{64\!\cdots\!75}{12\!\cdots\!69}a^{6}+\frac{30\!\cdots\!37}{12\!\cdots\!69}a^{5}-\frac{62\!\cdots\!42}{12\!\cdots\!69}a^{4}+\frac{23\!\cdots\!21}{12\!\cdots\!69}a^{3}-\frac{29\!\cdots\!92}{12\!\cdots\!69}a^{2}+\frac{77\!\cdots\!75}{12\!\cdots\!69}a-\frac{11\!\cdots\!51}{40\!\cdots\!99}$, $\frac{14\!\cdots\!24}{12\!\cdots\!69}a^{15}-\frac{11\!\cdots\!30}{12\!\cdots\!69}a^{14}+\frac{25\!\cdots\!48}{12\!\cdots\!69}a^{13}-\frac{14\!\cdots\!57}{12\!\cdots\!69}a^{12}+\frac{18\!\cdots\!42}{12\!\cdots\!69}a^{11}-\frac{84\!\cdots\!85}{12\!\cdots\!69}a^{10}+\frac{74\!\cdots\!12}{12\!\cdots\!69}a^{9}-\frac{27\!\cdots\!12}{12\!\cdots\!69}a^{8}+\frac{18\!\cdots\!56}{12\!\cdots\!69}a^{7}-\frac{53\!\cdots\!74}{12\!\cdots\!69}a^{6}+\frac{28\!\cdots\!19}{12\!\cdots\!69}a^{5}-\frac{58\!\cdots\!81}{12\!\cdots\!69}a^{4}+\frac{24\!\cdots\!85}{12\!\cdots\!69}a^{3}-\frac{31\!\cdots\!39}{12\!\cdots\!69}a^{2}+\frac{94\!\cdots\!36}{12\!\cdots\!69}a-\frac{14\!\cdots\!11}{40\!\cdots\!99}$, $\frac{16\!\cdots\!00}{12\!\cdots\!69}a^{15}-\frac{13\!\cdots\!70}{12\!\cdots\!69}a^{14}+\frac{21\!\cdots\!58}{12\!\cdots\!69}a^{13}+\frac{24\!\cdots\!54}{12\!\cdots\!69}a^{12}+\frac{11\!\cdots\!64}{12\!\cdots\!69}a^{11}+\frac{46\!\cdots\!17}{12\!\cdots\!69}a^{10}+\frac{32\!\cdots\!77}{12\!\cdots\!69}a^{9}+\frac{30\!\cdots\!61}{12\!\cdots\!69}a^{8}+\frac{42\!\cdots\!36}{12\!\cdots\!69}a^{7}+\frac{10\!\cdots\!74}{12\!\cdots\!69}a^{6}-\frac{39\!\cdots\!82}{12\!\cdots\!69}a^{5}+\frac{22\!\cdots\!43}{12\!\cdots\!69}a^{4}-\frac{69\!\cdots\!31}{12\!\cdots\!69}a^{3}+\frac{25\!\cdots\!15}{12\!\cdots\!69}a^{2}-\frac{52\!\cdots\!14}{12\!\cdots\!69}a+\frac{42\!\cdots\!37}{40\!\cdots\!99}$, $\frac{15\!\cdots\!84}{12\!\cdots\!69}a^{15}-\frac{11\!\cdots\!30}{12\!\cdots\!69}a^{14}+\frac{27\!\cdots\!66}{12\!\cdots\!69}a^{13}-\frac{16\!\cdots\!74}{12\!\cdots\!69}a^{12}+\frac{21\!\cdots\!92}{12\!\cdots\!69}a^{11}-\frac{99\!\cdots\!13}{12\!\cdots\!69}a^{10}+\frac{93\!\cdots\!69}{12\!\cdots\!69}a^{9}-\frac{34\!\cdots\!39}{12\!\cdots\!69}a^{8}+\frac{25\!\cdots\!60}{12\!\cdots\!69}a^{7}-\frac{75\!\cdots\!54}{12\!\cdots\!69}a^{6}+\frac{45\!\cdots\!18}{12\!\cdots\!69}a^{5}-\frac{95\!\cdots\!82}{12\!\cdots\!69}a^{4}+\frac{46\!\cdots\!01}{12\!\cdots\!69}a^{3}-\frac{60\!\cdots\!81}{12\!\cdots\!69}a^{2}+\frac{22\!\cdots\!27}{12\!\cdots\!69}a-\frac{28\!\cdots\!63}{40\!\cdots\!99}$
| 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: | \( 15753.94986242651 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 15753.94986242651 \cdot 1638375}{2\cdot\sqrt{10386864596368228067918124169560064}}\cr\approx \mathstrut & 0.307588070325566 \end{aligned}\] (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 + 180*x^14 - 1120*x^13 + 14026*x^12 - 69960*x^11 + 627024*x^10 - 2519560*x^9 + 17702027*x^8 - 56416264*x^7 + 324191712*x^6 - 784977032*x^5 + 3766659562*x^4 - 6286165360*x^3 + 25396479484*x^2 - 22375524712*x + 76059908671)
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 + 180*x^14 - 1120*x^13 + 14026*x^12 - 69960*x^11 + 627024*x^10 - 2519560*x^9 + 17702027*x^8 - 56416264*x^7 + 324191712*x^6 - 784977032*x^5 + 3766659562*x^4 - 6286165360*x^3 + 25396479484*x^2 - 22375524712*x + 76059908671, 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 + 180*x^14 - 1120*x^13 + 14026*x^12 - 69960*x^11 + 627024*x^10 - 2519560*x^9 + 17702027*x^8 - 56416264*x^7 + 324191712*x^6 - 784977032*x^5 + 3766659562*x^4 - 6286165360*x^3 + 25396479484*x^2 - 22375524712*x + 76059908671);
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 + 180*x^14 - 1120*x^13 + 14026*x^12 - 69960*x^11 + 627024*x^10 - 2519560*x^9 + 17702027*x^8 - 56416264*x^7 + 324191712*x^6 - 784977032*x^5 + 3766659562*x^4 - 6286165360*x^3 + 25396479484*x^2 - 22375524712*x + 76059908671);
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
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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.1.0.1}{1} }^{16}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.62.2 | $x^{16} + 8 x^{14} + 436 x^{12} + 32 x^{11} + 952 x^{10} + 144 x^{9} + 920 x^{8} + 448 x^{7} + 1232 x^{6} + 448 x^{5} + 584 x^{4} + 768 x^{3} + 240 x^{2} + 608 x + 508$ | $8$ | $2$ | $62$ | $C_8\times C_2$ | $[3, 4, 5]^{2}$ |
|
\(83\)
| 83.16.8.1 | $x^{16} + 664 x^{14} + 192894 x^{12} + 130 x^{11} + 32019786 x^{10} - 75446 x^{9} + 3321798615 x^{8} - 19841850 x^{7} + 220549814727 x^{6} - 1032515432 x^{5} + 9153088134336 x^{4} + 32195922972 x^{3} + 217101057166265 x^{2} + 2532564215378 x + 2252767510002356$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |