Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 227*x^14 - 394*x^13 + 23377*x^12 - 34694*x^11 + 1422906*x^10 - 1763952*x^9 + 55896239*x^8 - 55806478*x^7 + 1449580514*x^6 - 1097221888*x^5 + 24220970150*x^4 - 12405931900*x^3 + 238353759596*x^2 - 62226612592*x + 1057852414081)
gp: K = bnfinit(y^16 - 2*y^15 + 227*y^14 - 394*y^13 + 23377*y^12 - 34694*y^11 + 1422906*y^10 - 1763952*y^9 + 55896239*y^8 - 55806478*y^7 + 1449580514*y^6 - 1097221888*y^5 + 24220970150*y^4 - 12405931900*y^3 + 238353759596*y^2 - 62226612592*y + 1057852414081, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 227*x^14 - 394*x^13 + 23377*x^12 - 34694*x^11 + 1422906*x^10 - 1763952*x^9 + 55896239*x^8 - 55806478*x^7 + 1449580514*x^6 - 1097221888*x^5 + 24220970150*x^4 - 12405931900*x^3 + 238353759596*x^2 - 62226612592*x + 1057852414081);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 227*x^14 - 394*x^13 + 23377*x^12 - 34694*x^11 + 1422906*x^10 - 1763952*x^9 + 55896239*x^8 - 55806478*x^7 + 1449580514*x^6 - 1097221888*x^5 + 24220970150*x^4 - 12405931900*x^3 + 238353759596*x^2 - 62226612592*x + 1057852414081)
\( x^{16} - 2 x^{15} + 227 x^{14} - 394 x^{13} + 23377 x^{12} - 34694 x^{11} + 1422906 x^{10} + \cdots + 1057852414081 \)
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: |
\(7239938343280505619539558400000000\)
\(\medspace = 2^{24}\cdot 3^{8}\cdot 5^{8}\cdot 17^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(130.69\) | 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: | $2^{3/2}3^{1/2}5^{1/2}17^{7/8}\approx 130.68731314985814$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(17\)
| 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: | \(2040=2^{3}\cdot 3\cdot 5\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2040}(1,·)$, $\chi_{2040}(389,·)$, $\chi_{2040}(961,·)$, $\chi_{2040}(1801,·)$, $\chi_{2040}(749,·)$, $\chi_{2040}(1681,·)$, $\chi_{2040}(149,·)$, $\chi_{2040}(1441,·)$, $\chi_{2040}(869,·)$, $\chi_{2040}(361,·)$, $\chi_{2040}(1709,·)$, $\chi_{2040}(1589,·)$, $\chi_{2040}(841,·)$, $\chi_{2040}(121,·)$, $\chi_{2040}(509,·)$, $\chi_{2040}(1109,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{101}a^{14}+\frac{33}{101}a^{13}+\frac{15}{101}a^{12}-\frac{35}{101}a^{11}+\frac{31}{101}a^{10}-\frac{5}{101}a^{9}-\frac{13}{101}a^{8}+\frac{30}{101}a^{7}+\frac{46}{101}a^{6}-\frac{49}{101}a^{5}-\frac{26}{101}a^{4}+\frac{14}{101}a^{3}+\frac{40}{101}a^{2}+\frac{47}{101}a-\frac{28}{101}$, $\frac{1}{42\!\cdots\!93}a^{15}+\frac{12\!\cdots\!46}{42\!\cdots\!93}a^{14}-\frac{35\!\cdots\!23}{42\!\cdots\!93}a^{13}+\frac{18\!\cdots\!71}{42\!\cdots\!93}a^{12}-\frac{72\!\cdots\!25}{42\!\cdots\!93}a^{11}-\frac{93\!\cdots\!36}{42\!\cdots\!93}a^{10}-\frac{87\!\cdots\!65}{42\!\cdots\!93}a^{9}-\frac{27\!\cdots\!08}{42\!\cdots\!93}a^{8}-\frac{84\!\cdots\!70}{42\!\cdots\!93}a^{7}+\frac{19\!\cdots\!45}{42\!\cdots\!93}a^{6}-\frac{17\!\cdots\!98}{42\!\cdots\!93}a^{5}+\frac{16\!\cdots\!70}{42\!\cdots\!93}a^{4}-\frac{55\!\cdots\!41}{42\!\cdots\!93}a^{3}-\frac{16\!\cdots\!95}{42\!\cdots\!93}a^{2}+\frac{11\!\cdots\!33}{42\!\cdots\!93}a-\frac{49\!\cdots\!21}{42\!\cdots\!93}$
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
$C_{8}\times C_{24}\times C_{24672}$, which has order $4737024$ (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: |
$\frac{92\!\cdots\!00}{42\!\cdots\!93}a^{15}-\frac{21\!\cdots\!00}{42\!\cdots\!93}a^{14}+\frac{20\!\cdots\!00}{42\!\cdots\!93}a^{13}-\frac{42\!\cdots\!55}{42\!\cdots\!93}a^{12}+\frac{20\!\cdots\!00}{42\!\cdots\!93}a^{11}-\frac{37\!\cdots\!40}{42\!\cdots\!93}a^{10}+\frac{11\!\cdots\!00}{42\!\cdots\!93}a^{9}-\frac{18\!\cdots\!70}{42\!\cdots\!93}a^{8}+\frac{39\!\cdots\!00}{42\!\cdots\!93}a^{7}-\frac{59\!\cdots\!40}{42\!\cdots\!93}a^{6}+\frac{87\!\cdots\!52}{42\!\cdots\!93}a^{5}-\frac{11\!\cdots\!75}{42\!\cdots\!93}a^{4}+\frac{11\!\cdots\!40}{42\!\cdots\!93}a^{3}-\frac{13\!\cdots\!20}{42\!\cdots\!93}a^{2}+\frac{61\!\cdots\!60}{42\!\cdots\!93}a-\frac{66\!\cdots\!10}{42\!\cdots\!93}$, $\frac{30\!\cdots\!00}{42\!\cdots\!93}a^{15}-\frac{14\!\cdots\!00}{42\!\cdots\!93}a^{14}+\frac{60\!\cdots\!00}{42\!\cdots\!93}a^{13}-\frac{28\!\cdots\!00}{42\!\cdots\!93}a^{12}+\frac{53\!\cdots\!00}{42\!\cdots\!93}a^{11}-\frac{24\!\cdots\!03}{42\!\cdots\!93}a^{10}+\frac{27\!\cdots\!00}{42\!\cdots\!93}a^{9}-\frac{12\!\cdots\!20}{42\!\cdots\!93}a^{8}+\frac{87\!\cdots\!90}{42\!\cdots\!93}a^{7}-\frac{39\!\cdots\!05}{42\!\cdots\!93}a^{6}+\frac{17\!\cdots\!70}{42\!\cdots\!93}a^{5}-\frac{76\!\cdots\!00}{42\!\cdots\!93}a^{4}+\frac{20\!\cdots\!60}{42\!\cdots\!93}a^{3}-\frac{85\!\cdots\!75}{42\!\cdots\!93}a^{2}+\frac{10\!\cdots\!70}{42\!\cdots\!93}a-\frac{42\!\cdots\!44}{42\!\cdots\!93}$, $\frac{68\!\cdots\!20}{42\!\cdots\!93}a^{15}+\frac{89\!\cdots\!20}{42\!\cdots\!93}a^{14}+\frac{13\!\cdots\!00}{42\!\cdots\!93}a^{13}+\frac{19\!\cdots\!20}{42\!\cdots\!93}a^{12}+\frac{11\!\cdots\!74}{42\!\cdots\!93}a^{11}+\frac{18\!\cdots\!20}{42\!\cdots\!93}a^{10}+\frac{57\!\cdots\!86}{42\!\cdots\!93}a^{9}+\frac{10\!\cdots\!00}{42\!\cdots\!93}a^{8}+\frac{17\!\cdots\!16}{42\!\cdots\!93}a^{7}+\frac{35\!\cdots\!61}{42\!\cdots\!93}a^{6}+\frac{34\!\cdots\!62}{42\!\cdots\!93}a^{5}+\frac{74\!\cdots\!44}{42\!\cdots\!93}a^{4}+\frac{38\!\cdots\!10}{42\!\cdots\!93}a^{3}+\frac{90\!\cdots\!49}{42\!\cdots\!93}a^{2}+\frac{18\!\cdots\!66}{42\!\cdots\!93}a+\frac{48\!\cdots\!68}{42\!\cdots\!93}$, $\frac{30\!\cdots\!00}{42\!\cdots\!93}a^{15}-\frac{14\!\cdots\!00}{42\!\cdots\!93}a^{14}+\frac{60\!\cdots\!00}{42\!\cdots\!93}a^{13}-\frac{28\!\cdots\!00}{42\!\cdots\!93}a^{12}+\frac{53\!\cdots\!00}{42\!\cdots\!93}a^{11}-\frac{24\!\cdots\!03}{42\!\cdots\!93}a^{10}+\frac{27\!\cdots\!00}{42\!\cdots\!93}a^{9}-\frac{12\!\cdots\!20}{42\!\cdots\!93}a^{8}+\frac{87\!\cdots\!90}{42\!\cdots\!93}a^{7}-\frac{39\!\cdots\!05}{42\!\cdots\!93}a^{6}+\frac{17\!\cdots\!70}{42\!\cdots\!93}a^{5}-\frac{76\!\cdots\!00}{42\!\cdots\!93}a^{4}+\frac{20\!\cdots\!60}{42\!\cdots\!93}a^{3}-\frac{85\!\cdots\!75}{42\!\cdots\!93}a^{2}+\frac{10\!\cdots\!70}{42\!\cdots\!93}a-\frac{42\!\cdots\!37}{42\!\cdots\!93}$, $\frac{38\!\cdots\!42}{42\!\cdots\!93}a^{15}-\frac{63\!\cdots\!40}{42\!\cdots\!93}a^{14}+\frac{74\!\cdots\!50}{42\!\cdots\!93}a^{13}-\frac{10\!\cdots\!95}{42\!\cdots\!93}a^{12}+\frac{65\!\cdots\!60}{42\!\cdots\!93}a^{11}-\frac{76\!\cdots\!80}{42\!\cdots\!93}a^{10}+\frac{33\!\cdots\!98}{42\!\cdots\!93}a^{9}-\frac{32\!\cdots\!09}{42\!\cdots\!93}a^{8}+\frac{10\!\cdots\!68}{42\!\cdots\!93}a^{7}-\frac{80\!\cdots\!28}{42\!\cdots\!93}a^{6}+\frac{21\!\cdots\!92}{42\!\cdots\!93}a^{5}-\frac{12\!\cdots\!55}{42\!\cdots\!93}a^{4}+\frac{25\!\cdots\!20}{42\!\cdots\!93}a^{3}-\frac{10\!\cdots\!61}{42\!\cdots\!93}a^{2}+\frac{12\!\cdots\!02}{42\!\cdots\!93}a-\frac{35\!\cdots\!68}{42\!\cdots\!93}$, $\frac{27\!\cdots\!38}{42\!\cdots\!93}a^{15}-\frac{51\!\cdots\!47}{42\!\cdots\!93}a^{14}+\frac{62\!\cdots\!26}{42\!\cdots\!93}a^{13}-\frac{10\!\cdots\!57}{42\!\cdots\!93}a^{12}+\frac{65\!\cdots\!26}{42\!\cdots\!93}a^{11}-\frac{89\!\cdots\!62}{42\!\cdots\!93}a^{10}+\frac{40\!\cdots\!28}{42\!\cdots\!93}a^{9}-\frac{45\!\cdots\!51}{42\!\cdots\!93}a^{8}+\frac{16\!\cdots\!82}{42\!\cdots\!93}a^{7}-\frac{14\!\cdots\!14}{42\!\cdots\!93}a^{6}+\frac{44\!\cdots\!32}{42\!\cdots\!93}a^{5}-\frac{28\!\cdots\!06}{42\!\cdots\!93}a^{4}+\frac{83\!\cdots\!00}{42\!\cdots\!93}a^{3}-\frac{32\!\cdots\!40}{42\!\cdots\!93}a^{2}+\frac{80\!\cdots\!84}{42\!\cdots\!93}a+\frac{26\!\cdots\!62}{42\!\cdots\!93}$, $\frac{33\!\cdots\!62}{42\!\cdots\!93}a^{15}+\frac{58\!\cdots\!60}{42\!\cdots\!93}a^{14}+\frac{64\!\cdots\!74}{42\!\cdots\!93}a^{13}+\frac{13\!\cdots\!60}{42\!\cdots\!93}a^{12}+\frac{56\!\cdots\!48}{42\!\cdots\!93}a^{11}+\frac{14\!\cdots\!60}{42\!\cdots\!93}a^{10}+\frac{28\!\cdots\!10}{42\!\cdots\!93}a^{9}+\frac{94\!\cdots\!00}{42\!\cdots\!93}a^{8}+\frac{87\!\cdots\!76}{42\!\cdots\!93}a^{7}+\frac{37\!\cdots\!40}{42\!\cdots\!93}a^{6}+\frac{16\!\cdots\!32}{42\!\cdots\!93}a^{5}+\frac{94\!\cdots\!77}{42\!\cdots\!93}a^{4}+\frac{18\!\cdots\!36}{42\!\cdots\!93}a^{3}+\frac{13\!\cdots\!47}{42\!\cdots\!93}a^{2}+\frac{93\!\cdots\!42}{42\!\cdots\!93}a+\frac{87\!\cdots\!27}{42\!\cdots\!93}$
| 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
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.012213375973 \cdot 4737024}{2\cdot\sqrt{7239938343280505619539558400000000}}\cr\approx \mathstrut & 0.246121721035075 \end{aligned}\] (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 + 227*x^14 - 394*x^13 + 23377*x^12 - 34694*x^11 + 1422906*x^10 - 1763952*x^9 + 55896239*x^8 - 55806478*x^7 + 1449580514*x^6 - 1097221888*x^5 + 24220970150*x^4 - 12405931900*x^3 + 238353759596*x^2 - 62226612592*x + 1057852414081)
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 + 227*x^14 - 394*x^13 + 23377*x^12 - 34694*x^11 + 1422906*x^10 - 1763952*x^9 + 55896239*x^8 - 55806478*x^7 + 1449580514*x^6 - 1097221888*x^5 + 24220970150*x^4 - 12405931900*x^3 + 238353759596*x^2 - 62226612592*x + 1057852414081, 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 + 227*x^14 - 394*x^13 + 23377*x^12 - 34694*x^11 + 1422906*x^10 - 1763952*x^9 + 55896239*x^8 - 55806478*x^7 + 1449580514*x^6 - 1097221888*x^5 + 24220970150*x^4 - 12405931900*x^3 + 238353759596*x^2 - 62226612592*x + 1057852414081);
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 + 227*x^14 - 394*x^13 + 23377*x^12 - 34694*x^11 + 1422906*x^10 - 1763952*x^9 + 55896239*x^8 - 55806478*x^7 + 1449580514*x^6 - 1097221888*x^5 + 24220970150*x^4 - 12405931900*x^3 + 238353759596*x^2 - 62226612592*x + 1057852414081);
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
| \(\Q(\sqrt{-510}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{17}, \sqrt{-30})\), 4.4.4913.1, 4.0.70747200.2, 8.0.5005166307840000.133, 8.0.85087827233280000.1, \(\Q(\zeta_{17})^+\) |
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 | R | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | 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 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
|
\(3\)
| 3.16.8.1 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(17\)
| 17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |