Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 118*x^13 - 454*x^12 - 450*x^11 + 9075*x^10 - 36523*x^9 + 80287*x^8 - 79156*x^7 - 58067*x^6 + 350155*x^5 - 455538*x^4 + 197266*x^3 + 619715*x^2 - 823177*x + 636931)
gp: K = bnfinit(y^16 - 6*y^15 + 118*y^13 - 454*y^12 - 450*y^11 + 9075*y^10 - 36523*y^9 + 80287*y^8 - 79156*y^7 - 58067*y^6 + 350155*y^5 - 455538*y^4 + 197266*y^3 + 619715*y^2 - 823177*y + 636931, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 118*x^13 - 454*x^12 - 450*x^11 + 9075*x^10 - 36523*x^9 + 80287*x^8 - 79156*x^7 - 58067*x^6 + 350155*x^5 - 455538*x^4 + 197266*x^3 + 619715*x^2 - 823177*x + 636931);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 118*x^13 - 454*x^12 - 450*x^11 + 9075*x^10 - 36523*x^9 + 80287*x^8 - 79156*x^7 - 58067*x^6 + 350155*x^5 - 455538*x^4 + 197266*x^3 + 619715*x^2 - 823177*x + 636931)
\( x^{16} - 6 x^{15} + 118 x^{13} - 454 x^{12} - 450 x^{11} + 9075 x^{10} - 36523 x^{9} + 80287 x^{8} + \cdots + 636931 \)
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: |
\(129672479863204507348386828961\)
\(\medspace = 41^{14}\cdot 43^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(66.00\) | 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: | $41^{7/8}43^{1/2}\approx 169.01299681773574$ | ||
| Ramified primes: |
\(41\), \(43\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{2}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{8}a^{5}-\frac{1}{8}a^{4}-\frac{3}{8}a^{3}+\frac{1}{4}a^{2}+\frac{1}{8}a+\frac{1}{8}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{3}{8}a^{6}+\frac{3}{8}a^{5}-\frac{3}{8}a^{4}+\frac{1}{4}a^{3}-\frac{3}{8}a^{2}+\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{612208}a^{14}-\frac{17785}{612208}a^{13}+\frac{2909}{306104}a^{12}-\frac{72051}{612208}a^{11}-\frac{37927}{612208}a^{10}-\frac{127859}{306104}a^{9}-\frac{35361}{153052}a^{8}-\frac{61225}{612208}a^{7}+\frac{125195}{306104}a^{6}-\frac{141781}{612208}a^{5}-\frac{109403}{306104}a^{4}+\frac{127275}{306104}a^{3}+\frac{9375}{306104}a^{2}-\frac{121259}{306104}a-\frac{256493}{612208}$, $\frac{1}{40\!\cdots\!68}a^{15}+\frac{96\!\cdots\!99}{40\!\cdots\!68}a^{14}+\frac{27\!\cdots\!17}{20\!\cdots\!84}a^{13}+\frac{16\!\cdots\!27}{40\!\cdots\!68}a^{12}+\frac{28\!\cdots\!13}{40\!\cdots\!68}a^{11}+\frac{12\!\cdots\!75}{50\!\cdots\!46}a^{10}+\frac{60\!\cdots\!75}{10\!\cdots\!92}a^{9}+\frac{13\!\cdots\!55}{40\!\cdots\!68}a^{8}+\frac{73\!\cdots\!35}{20\!\cdots\!84}a^{7}-\frac{68\!\cdots\!21}{40\!\cdots\!68}a^{6}-\frac{23\!\cdots\!13}{10\!\cdots\!92}a^{5}+\frac{31\!\cdots\!35}{10\!\cdots\!92}a^{4}-\frac{64\!\cdots\!81}{10\!\cdots\!92}a^{3}-\frac{41\!\cdots\!47}{20\!\cdots\!84}a^{2}-\frac{13\!\cdots\!87}{40\!\cdots\!68}a+\frac{81\!\cdots\!49}{20\!\cdots\!84}$
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
Trivial group, which has order $1$ (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: |
$\frac{29\!\cdots\!63}{94\!\cdots\!13}a^{15}-\frac{11\!\cdots\!87}{18\!\cdots\!76}a^{14}+\frac{18\!\cdots\!61}{15\!\cdots\!08}a^{13}+\frac{18\!\cdots\!69}{18\!\cdots\!26}a^{12}-\frac{79\!\cdots\!29}{15\!\cdots\!08}a^{11}+\frac{63\!\cdots\!01}{15\!\cdots\!08}a^{10}+\frac{61\!\cdots\!67}{75\!\cdots\!04}a^{9}-\frac{14\!\cdots\!65}{37\!\cdots\!52}a^{8}+\frac{12\!\cdots\!97}{15\!\cdots\!08}a^{7}-\frac{63\!\cdots\!15}{75\!\cdots\!04}a^{6}-\frac{24\!\cdots\!69}{15\!\cdots\!08}a^{5}+\frac{75\!\cdots\!47}{18\!\cdots\!26}a^{4}-\frac{24\!\cdots\!07}{37\!\cdots\!52}a^{3}-\frac{45\!\cdots\!43}{75\!\cdots\!04}a^{2}+\frac{40\!\cdots\!49}{37\!\cdots\!52}a-\frac{22\!\cdots\!85}{15\!\cdots\!08}$, $\frac{61\!\cdots\!84}{94\!\cdots\!13}a^{15}-\frac{22\!\cdots\!91}{90\!\cdots\!88}a^{14}-\frac{71\!\cdots\!11}{75\!\cdots\!04}a^{13}+\frac{71\!\cdots\!77}{94\!\cdots\!13}a^{12}-\frac{71\!\cdots\!21}{75\!\cdots\!04}a^{11}-\frac{74\!\cdots\!07}{75\!\cdots\!04}a^{10}+\frac{18\!\cdots\!71}{37\!\cdots\!52}a^{9}-\frac{16\!\cdots\!03}{18\!\cdots\!26}a^{8}-\frac{23\!\cdots\!31}{75\!\cdots\!04}a^{7}+\frac{14\!\cdots\!37}{37\!\cdots\!52}a^{6}-\frac{44\!\cdots\!69}{75\!\cdots\!04}a^{5}-\frac{24\!\cdots\!64}{94\!\cdots\!13}a^{4}+\frac{30\!\cdots\!69}{18\!\cdots\!26}a^{3}-\frac{21\!\cdots\!31}{37\!\cdots\!52}a^{2}-\frac{10\!\cdots\!97}{18\!\cdots\!26}a+\frac{88\!\cdots\!07}{75\!\cdots\!04}$, $\frac{17\!\cdots\!59}{94\!\cdots\!13}a^{15}+\frac{22\!\cdots\!63}{18\!\cdots\!76}a^{14}-\frac{12\!\cdots\!93}{15\!\cdots\!08}a^{13}+\frac{13\!\cdots\!13}{18\!\cdots\!26}a^{12}+\frac{23\!\cdots\!17}{15\!\cdots\!08}a^{11}-\frac{91\!\cdots\!73}{15\!\cdots\!08}a^{10}-\frac{48\!\cdots\!63}{75\!\cdots\!04}a^{9}+\frac{45\!\cdots\!77}{37\!\cdots\!52}a^{8}-\frac{69\!\cdots\!85}{15\!\cdots\!08}a^{7}+\frac{69\!\cdots\!39}{75\!\cdots\!04}a^{6}-\frac{10\!\cdots\!59}{15\!\cdots\!08}a^{5}-\frac{22\!\cdots\!21}{18\!\cdots\!26}a^{4}+\frac{16\!\cdots\!31}{37\!\cdots\!52}a^{3}-\frac{40\!\cdots\!45}{75\!\cdots\!04}a^{2}+\frac{13\!\cdots\!71}{37\!\cdots\!52}a-\frac{10\!\cdots\!83}{15\!\cdots\!08}$, $\frac{15\!\cdots\!65}{50\!\cdots\!46}a^{15}-\frac{84\!\cdots\!75}{40\!\cdots\!68}a^{14}+\frac{82\!\cdots\!39}{40\!\cdots\!68}a^{13}+\frac{84\!\cdots\!41}{25\!\cdots\!23}a^{12}-\frac{68\!\cdots\!19}{40\!\cdots\!68}a^{11}+\frac{13\!\cdots\!15}{40\!\cdots\!68}a^{10}+\frac{54\!\cdots\!85}{20\!\cdots\!84}a^{9}-\frac{13\!\cdots\!41}{10\!\cdots\!92}a^{8}+\frac{15\!\cdots\!39}{40\!\cdots\!68}a^{7}-\frac{11\!\cdots\!17}{20\!\cdots\!84}a^{6}+\frac{12\!\cdots\!45}{40\!\cdots\!68}a^{5}+\frac{49\!\cdots\!21}{50\!\cdots\!46}a^{4}-\frac{28\!\cdots\!57}{10\!\cdots\!92}a^{3}+\frac{84\!\cdots\!03}{20\!\cdots\!84}a^{2}-\frac{32\!\cdots\!63}{10\!\cdots\!92}a+\frac{59\!\cdots\!17}{40\!\cdots\!68}$, $\frac{56\!\cdots\!23}{40\!\cdots\!68}a^{15}-\frac{82\!\cdots\!55}{10\!\cdots\!92}a^{14}-\frac{92\!\cdots\!63}{40\!\cdots\!68}a^{13}+\frac{71\!\cdots\!43}{40\!\cdots\!68}a^{12}-\frac{15\!\cdots\!72}{25\!\cdots\!23}a^{11}-\frac{42\!\cdots\!05}{40\!\cdots\!68}a^{10}+\frac{27\!\cdots\!57}{20\!\cdots\!84}a^{9}-\frac{22\!\cdots\!05}{49\!\cdots\!96}a^{8}+\frac{34\!\cdots\!35}{40\!\cdots\!68}a^{7}-\frac{16\!\cdots\!71}{40\!\cdots\!68}a^{6}-\frac{61\!\cdots\!57}{40\!\cdots\!68}a^{5}+\frac{10\!\cdots\!91}{25\!\cdots\!23}a^{4}-\frac{58\!\cdots\!35}{24\!\cdots\!48}a^{3}-\frac{36\!\cdots\!63}{20\!\cdots\!84}a^{2}+\frac{39\!\cdots\!67}{40\!\cdots\!68}a-\frac{23\!\cdots\!69}{40\!\cdots\!68}$, $\frac{13\!\cdots\!81}{40\!\cdots\!68}a^{15}+\frac{21\!\cdots\!35}{40\!\cdots\!68}a^{14}-\frac{13\!\cdots\!05}{10\!\cdots\!92}a^{13}+\frac{72\!\cdots\!37}{40\!\cdots\!68}a^{12}+\frac{78\!\cdots\!91}{40\!\cdots\!68}a^{11}-\frac{18\!\cdots\!97}{20\!\cdots\!84}a^{10}-\frac{24\!\cdots\!67}{10\!\cdots\!92}a^{9}+\frac{46\!\cdots\!83}{40\!\cdots\!68}a^{8}-\frac{65\!\cdots\!91}{20\!\cdots\!84}a^{7}+\frac{13\!\cdots\!93}{40\!\cdots\!68}a^{6}+\frac{39\!\cdots\!95}{10\!\cdots\!92}a^{5}-\frac{13\!\cdots\!19}{10\!\cdots\!92}a^{4}+\frac{36\!\cdots\!59}{20\!\cdots\!84}a^{3}+\frac{84\!\cdots\!37}{10\!\cdots\!92}a^{2}-\frac{85\!\cdots\!19}{40\!\cdots\!68}a+\frac{24\!\cdots\!79}{10\!\cdots\!92}$, $\frac{10\!\cdots\!83}{40\!\cdots\!68}a^{15}-\frac{87\!\cdots\!37}{40\!\cdots\!68}a^{14}-\frac{24\!\cdots\!33}{20\!\cdots\!84}a^{13}+\frac{16\!\cdots\!79}{40\!\cdots\!68}a^{12}-\frac{60\!\cdots\!25}{40\!\cdots\!68}a^{11}-\frac{12\!\cdots\!86}{25\!\cdots\!23}a^{10}+\frac{10\!\cdots\!59}{50\!\cdots\!46}a^{9}-\frac{21\!\cdots\!43}{40\!\cdots\!68}a^{8}+\frac{87\!\cdots\!31}{10\!\cdots\!92}a^{7}+\frac{35\!\cdots\!07}{40\!\cdots\!68}a^{6}-\frac{53\!\cdots\!11}{20\!\cdots\!84}a^{5}+\frac{27\!\cdots\!01}{50\!\cdots\!46}a^{4}+\frac{80\!\cdots\!01}{20\!\cdots\!84}a^{3}-\frac{36\!\cdots\!75}{10\!\cdots\!92}a^{2}+\frac{60\!\cdots\!07}{40\!\cdots\!68}a+\frac{25\!\cdots\!39}{20\!\cdots\!84}$
| 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: | \( 53320614.8559 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 53320614.8559 \cdot 1}{2\cdot\sqrt{129672479863204507348386828961}}\cr\approx \mathstrut & 0.179837473142 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 118*x^13 - 454*x^12 - 450*x^11 + 9075*x^10 - 36523*x^9 + 80287*x^8 - 79156*x^7 - 58067*x^6 + 350155*x^5 - 455538*x^4 + 197266*x^3 + 619715*x^2 - 823177*x + 636931)
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 - 6*x^15 + 118*x^13 - 454*x^12 - 450*x^11 + 9075*x^10 - 36523*x^9 + 80287*x^8 - 79156*x^7 - 58067*x^6 + 350155*x^5 - 455538*x^4 + 197266*x^3 + 619715*x^2 - 823177*x + 636931, 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 - 6*x^15 + 118*x^13 - 454*x^12 - 450*x^11 + 9075*x^10 - 36523*x^9 + 80287*x^8 - 79156*x^7 - 58067*x^6 + 350155*x^5 - 455538*x^4 + 197266*x^3 + 619715*x^2 - 823177*x + 636931);
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 - 6*x^15 + 118*x^13 - 454*x^12 - 450*x^11 + 9075*x^10 - 36523*x^9 + 80287*x^8 - 79156*x^7 - 58067*x^6 + 350155*x^5 - 455538*x^4 + 197266*x^3 + 619715*x^2 - 823177*x + 636931);
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
| \(\Q(\sqrt{41}) \), 4.2.2963603.1, 4.4.68921.1, 4.2.72283.1, 8.4.360100652405969.1, 8.0.194754273881.1, 8.4.8782942741609.1 |
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: | 16.8.443324403828803432927172239238695761.1 |
| 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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ | R | R | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(41\)
| 41.16.14.1 | $x^{16} + 304 x^{15} + 40480 x^{14} + 3085600 x^{13} + 147416080 x^{12} + 4529584192 x^{11} + 87831092608 x^{10} + 996302227840 x^{9} + 5391168776882 x^{8} + 5977813379504 x^{7} + 3161920977824 x^{6} + 978514601120 x^{5} + 196936323920 x^{4} + 202153692608 x^{3} + 3372805705856 x^{2} + 36445904670848 x + 172395305267889$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |
|
\(43\)
| 43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 43.8.4.1 | $x^{8} + 182 x^{6} + 84 x^{5} + 11555 x^{4} - 6804 x^{3} + 301934 x^{2} - 447636 x + 2755621$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |