Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 5*x^14 + 28*x^13 + 30*x^12 - 611*x^11 + 468*x^10 + 5496*x^9 - 10545*x^8 - 31116*x^7 + 150416*x^6 - 219357*x^5 + 89578*x^4 - 68160*x^3 + 648725*x^2 - 1271875*x + 855625)
gp: K = bnfinit(y^16 - 5*y^15 + 5*y^14 + 28*y^13 + 30*y^12 - 611*y^11 + 468*y^10 + 5496*y^9 - 10545*y^8 - 31116*y^7 + 150416*y^6 - 219357*y^5 + 89578*y^4 - 68160*y^3 + 648725*y^2 - 1271875*y + 855625, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 5*x^14 + 28*x^13 + 30*x^12 - 611*x^11 + 468*x^10 + 5496*x^9 - 10545*x^8 - 31116*x^7 + 150416*x^6 - 219357*x^5 + 89578*x^4 - 68160*x^3 + 648725*x^2 - 1271875*x + 855625);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 + 5*x^14 + 28*x^13 + 30*x^12 - 611*x^11 + 468*x^10 + 5496*x^9 - 10545*x^8 - 31116*x^7 + 150416*x^6 - 219357*x^5 + 89578*x^4 - 68160*x^3 + 648725*x^2 - 1271875*x + 855625)
\( x^{16} - 5 x^{15} + 5 x^{14} + 28 x^{13} + 30 x^{12} - 611 x^{11} + 468 x^{10} + 5496 x^{9} + \cdots + 855625 \)
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: |
\(35028437828275611780530528881\)
\(\medspace = 31^{4}\cdot 41^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(60.82\) | 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: | $31^{1/2}41^{7/8}\approx 143.50489707977238$ | ||
| Ramified primes: |
\(31\), \(41\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{10}a^{13}+\frac{3}{10}a^{10}+\frac{2}{5}a^{8}+\frac{3}{10}a^{7}-\frac{2}{5}a^{6}-\frac{1}{10}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}+\frac{3}{10}a$, $\frac{1}{2950}a^{14}+\frac{7}{590}a^{13}-\frac{57}{295}a^{12}-\frac{447}{2950}a^{11}+\frac{7}{118}a^{10}+\frac{632}{1475}a^{9}+\frac{403}{2950}a^{8}-\frac{1209}{2950}a^{7}+\frac{67}{295}a^{6}-\frac{141}{2950}a^{5}+\frac{51}{2950}a^{4}-\frac{396}{1475}a^{3}+\frac{1423}{2950}a^{2}-\frac{43}{590}a+\frac{18}{59}$, $\frac{1}{16\!\cdots\!50}a^{15}-\frac{16\!\cdots\!33}{33\!\cdots\!55}a^{14}-\frac{98\!\cdots\!79}{33\!\cdots\!50}a^{13}+\frac{71\!\cdots\!89}{84\!\cdots\!75}a^{12}-\frac{28\!\cdots\!03}{16\!\cdots\!75}a^{11}-\frac{18\!\cdots\!61}{16\!\cdots\!50}a^{10}+\frac{22\!\cdots\!19}{84\!\cdots\!75}a^{9}+\frac{36\!\cdots\!93}{84\!\cdots\!75}a^{8}+\frac{19\!\cdots\!11}{91\!\cdots\!50}a^{7}-\frac{38\!\cdots\!33}{84\!\cdots\!75}a^{6}-\frac{36\!\cdots\!32}{84\!\cdots\!75}a^{5}-\frac{65\!\cdots\!27}{16\!\cdots\!50}a^{4}-\frac{19\!\cdots\!16}{84\!\cdots\!75}a^{3}-\frac{54\!\cdots\!47}{16\!\cdots\!75}a^{2}-\frac{82\!\cdots\!23}{67\!\cdots\!10}a+\frac{13\!\cdots\!01}{36\!\cdots\!86}$
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
$C_{8}$, which has order $8$ (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: |
$\frac{76\!\cdots\!14}{10\!\cdots\!75}a^{15}-\frac{11\!\cdots\!11}{42\!\cdots\!50}a^{14}-\frac{27\!\cdots\!77}{42\!\cdots\!50}a^{13}+\frac{23\!\cdots\!42}{10\!\cdots\!75}a^{12}+\frac{23\!\cdots\!69}{42\!\cdots\!50}a^{11}-\frac{81\!\cdots\!83}{21\!\cdots\!50}a^{10}-\frac{27\!\cdots\!78}{10\!\cdots\!75}a^{9}+\frac{85\!\cdots\!93}{21\!\cdots\!50}a^{8}-\frac{12\!\cdots\!31}{85\!\cdots\!10}a^{7}-\frac{29\!\cdots\!49}{10\!\cdots\!75}a^{6}+\frac{14\!\cdots\!63}{21\!\cdots\!50}a^{5}-\frac{83\!\cdots\!11}{21\!\cdots\!50}a^{4}-\frac{18\!\cdots\!43}{10\!\cdots\!75}a^{3}-\frac{84\!\cdots\!21}{85\!\cdots\!10}a^{2}+\frac{27\!\cdots\!51}{85\!\cdots\!10}a-\frac{16\!\cdots\!76}{85\!\cdots\!71}$, $\frac{17\!\cdots\!56}{10\!\cdots\!75}a^{15}-\frac{92\!\cdots\!69}{42\!\cdots\!50}a^{14}+\frac{35\!\cdots\!67}{42\!\cdots\!50}a^{13}+\frac{82\!\cdots\!18}{10\!\cdots\!75}a^{12}-\frac{85\!\cdots\!49}{42\!\cdots\!50}a^{11}-\frac{60\!\cdots\!07}{21\!\cdots\!50}a^{10}+\frac{19\!\cdots\!13}{10\!\cdots\!75}a^{9}+\frac{43\!\cdots\!47}{21\!\cdots\!50}a^{8}-\frac{39\!\cdots\!59}{85\!\cdots\!10}a^{7}-\frac{13\!\cdots\!96}{10\!\cdots\!75}a^{6}+\frac{11\!\cdots\!77}{21\!\cdots\!50}a^{5}-\frac{11\!\cdots\!19}{21\!\cdots\!50}a^{4}-\frac{24\!\cdots\!47}{10\!\cdots\!75}a^{3}-\frac{11\!\cdots\!59}{85\!\cdots\!10}a^{2}+\frac{22\!\cdots\!79}{85\!\cdots\!10}a-\frac{31\!\cdots\!17}{85\!\cdots\!71}$, $\frac{19\!\cdots\!09}{16\!\cdots\!50}a^{15}-\frac{14\!\cdots\!79}{16\!\cdots\!75}a^{14}+\frac{24\!\cdots\!47}{16\!\cdots\!75}a^{13}+\frac{38\!\cdots\!26}{84\!\cdots\!75}a^{12}-\frac{91\!\cdots\!09}{16\!\cdots\!75}a^{11}-\frac{10\!\cdots\!12}{84\!\cdots\!75}a^{10}+\frac{22\!\cdots\!91}{84\!\cdots\!75}a^{9}+\frac{85\!\cdots\!27}{84\!\cdots\!75}a^{8}-\frac{12\!\cdots\!27}{91\!\cdots\!15}a^{7}-\frac{56\!\cdots\!72}{84\!\cdots\!75}a^{6}+\frac{17\!\cdots\!82}{84\!\cdots\!75}a^{5}-\frac{24\!\cdots\!56}{14\!\cdots\!25}a^{4}-\frac{92\!\cdots\!54}{84\!\cdots\!75}a^{3}-\frac{55\!\cdots\!19}{33\!\cdots\!55}a^{2}+\frac{37\!\cdots\!64}{33\!\cdots\!55}a-\frac{49\!\cdots\!61}{36\!\cdots\!86}$, $\frac{34\!\cdots\!17}{21\!\cdots\!50}a^{15}-\frac{28\!\cdots\!29}{42\!\cdots\!50}a^{14}-\frac{41\!\cdots\!53}{42\!\cdots\!50}a^{13}+\frac{53\!\cdots\!88}{10\!\cdots\!75}a^{12}+\frac{46\!\cdots\!91}{42\!\cdots\!50}a^{11}-\frac{20\!\cdots\!87}{21\!\cdots\!50}a^{10}-\frac{48\!\cdots\!92}{10\!\cdots\!75}a^{9}+\frac{20\!\cdots\!77}{21\!\cdots\!50}a^{8}-\frac{43\!\cdots\!99}{85\!\cdots\!10}a^{7}-\frac{70\!\cdots\!86}{10\!\cdots\!75}a^{6}+\frac{35\!\cdots\!57}{21\!\cdots\!50}a^{5}-\frac{23\!\cdots\!29}{21\!\cdots\!50}a^{4}-\frac{50\!\cdots\!27}{10\!\cdots\!75}a^{3}-\frac{18\!\cdots\!69}{85\!\cdots\!10}a^{2}+\frac{69\!\cdots\!89}{85\!\cdots\!10}a-\frac{11\!\cdots\!15}{17\!\cdots\!42}$, $\frac{42\!\cdots\!81}{16\!\cdots\!50}a^{15}-\frac{45\!\cdots\!47}{33\!\cdots\!50}a^{14}+\frac{64\!\cdots\!21}{33\!\cdots\!50}a^{13}+\frac{50\!\cdots\!84}{84\!\cdots\!75}a^{12}+\frac{99\!\cdots\!13}{33\!\cdots\!50}a^{11}-\frac{24\!\cdots\!41}{16\!\cdots\!50}a^{10}+\frac{15\!\cdots\!94}{84\!\cdots\!75}a^{9}+\frac{20\!\cdots\!11}{16\!\cdots\!50}a^{8}-\frac{60\!\cdots\!81}{18\!\cdots\!30}a^{7}-\frac{41\!\cdots\!98}{84\!\cdots\!75}a^{6}+\frac{67\!\cdots\!51}{16\!\cdots\!50}a^{5}-\frac{14\!\cdots\!47}{16\!\cdots\!50}a^{4}+\frac{65\!\cdots\!39}{84\!\cdots\!75}a^{3}+\frac{35\!\cdots\!33}{67\!\cdots\!10}a^{2}-\frac{45\!\cdots\!73}{67\!\cdots\!10}a+\frac{13\!\cdots\!69}{36\!\cdots\!86}$, $\frac{57\!\cdots\!73}{16\!\cdots\!75}a^{15}-\frac{12\!\cdots\!58}{16\!\cdots\!75}a^{14}-\frac{34\!\cdots\!11}{33\!\cdots\!55}a^{13}+\frac{11\!\cdots\!79}{16\!\cdots\!75}a^{12}+\frac{56\!\cdots\!61}{16\!\cdots\!75}a^{11}-\frac{20\!\cdots\!73}{16\!\cdots\!75}a^{10}-\frac{45\!\cdots\!63}{16\!\cdots\!75}a^{9}+\frac{20\!\cdots\!19}{16\!\cdots\!75}a^{8}+\frac{30\!\cdots\!86}{45\!\cdots\!75}a^{7}-\frac{16\!\cdots\!43}{16\!\cdots\!75}a^{6}+\frac{31\!\cdots\!56}{16\!\cdots\!75}a^{5}-\frac{19\!\cdots\!14}{16\!\cdots\!75}a^{4}+\frac{19\!\cdots\!92}{33\!\cdots\!55}a^{3}-\frac{54\!\cdots\!64}{16\!\cdots\!75}a^{2}+\frac{67\!\cdots\!04}{67\!\cdots\!91}a-\frac{18\!\cdots\!51}{18\!\cdots\!43}$, $\frac{16\!\cdots\!59}{84\!\cdots\!75}a^{15}-\frac{49\!\cdots\!73}{16\!\cdots\!75}a^{14}+\frac{73\!\cdots\!89}{16\!\cdots\!75}a^{13}+\frac{79\!\cdots\!02}{84\!\cdots\!75}a^{12}-\frac{76\!\cdots\!38}{16\!\cdots\!75}a^{11}-\frac{31\!\cdots\!49}{84\!\cdots\!75}a^{10}+\frac{66\!\cdots\!32}{84\!\cdots\!75}a^{9}+\frac{27\!\cdots\!79}{84\!\cdots\!75}a^{8}-\frac{10\!\cdots\!42}{91\!\cdots\!15}a^{7}-\frac{12\!\cdots\!19}{84\!\cdots\!75}a^{6}+\frac{84\!\cdots\!64}{84\!\cdots\!75}a^{5}-\frac{10\!\cdots\!33}{84\!\cdots\!75}a^{4}-\frac{49\!\cdots\!58}{84\!\cdots\!75}a^{3}+\frac{37\!\cdots\!17}{67\!\cdots\!91}a^{2}+\frac{30\!\cdots\!28}{67\!\cdots\!91}a-\frac{15\!\cdots\!76}{18\!\cdots\!43}$
| 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: | \( 3814663.53647 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3814663.53647 \cdot 8}{2\cdot\sqrt{35028437828275611780530528881}}\cr\approx \mathstrut & 0.198036294488 \end{aligned}\] (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 + 5*x^14 + 28*x^13 + 30*x^12 - 611*x^11 + 468*x^10 + 5496*x^9 - 10545*x^8 - 31116*x^7 + 150416*x^6 - 219357*x^5 + 89578*x^4 - 68160*x^3 + 648725*x^2 - 1271875*x + 855625)
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 + 5*x^14 + 28*x^13 + 30*x^12 - 611*x^11 + 468*x^10 + 5496*x^9 - 10545*x^8 - 31116*x^7 + 150416*x^6 - 219357*x^5 + 89578*x^4 - 68160*x^3 + 648725*x^2 - 1271875*x + 855625, 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 + 5*x^14 + 28*x^13 + 30*x^12 - 611*x^11 + 468*x^10 + 5496*x^9 - 10545*x^8 - 31116*x^7 + 150416*x^6 - 219357*x^5 + 89578*x^4 - 68160*x^3 + 648725*x^2 - 1271875*x + 855625);
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 + 5*x^14 + 28*x^13 + 30*x^12 - 611*x^11 + 468*x^10 + 5496*x^9 - 10545*x^8 - 31116*x^7 + 150416*x^6 - 219357*x^5 + 89578*x^4 - 68160*x^3 + 648725*x^2 - 1271875*x + 855625);
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
| \(\Q(\sqrt{41}) \), 4.2.2136551.1, 4.4.68921.1, 4.2.52111.1, 8.0.194754273881.1, 8.4.187158857199641.1, 8.4.4564850175601.2 |
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: | 16.8.32349497931606921267167334562710001.2 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(31\)
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(41\)
| 41.8.7.3 | $x^{8} + 41$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 41.8.7.3 | $x^{8} + 41$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |