Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717223*x^8 - 46483976*x^7 + 257685716*x^6 - 618838472*x^5 + 2865167258*x^4 - 4749147848*x^3 + 18507674124*x^2 - 16229095208*x + 53156046238)
gp: K = bnfinit(y^16 - 8*y^15 + 172*y^14 - 1064*y^13 + 12790*y^12 - 63272*y^11 + 545756*y^10 - 2173192*y^9 + 14717223*y^8 - 46483976*y^7 + 257685716*y^6 - 618838472*y^5 + 2865167258*y^4 - 4749147848*y^3 + 18507674124*y^2 - 16229095208*y + 53156046238, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717223*x^8 - 46483976*x^7 + 257685716*x^6 - 618838472*x^5 + 2865167258*x^4 - 4749147848*x^3 + 18507674124*x^2 - 16229095208*x + 53156046238);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717223*x^8 - 46483976*x^7 + 257685716*x^6 - 618838472*x^5 + 2865167258*x^4 - 4749147848*x^3 + 18507674124*x^2 - 16229095208*x + 53156046238)
\( x^{16} - 8 x^{15} + 172 x^{14} - 1064 x^{13} + 12790 x^{12} - 63272 x^{11} + 545756 x^{10} + \cdots + 53156046238 \)
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: |
\(2801731803266966736034164664958976\)
\(\medspace = 2^{48}\cdot 3^{8}\cdot 79^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(123.16\) | 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: | $2^{3}3^{1/2}79^{1/2}\approx 123.15843454672522$ | ||
| Ramified primes: |
\(2\), \(3\), \(79\)
| 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: | \(3792=2^{4}\cdot 3\cdot 79\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3792}(1,·)$, $\chi_{3792}(2053,·)$, $\chi_{3792}(1739,·)$, $\chi_{3792}(3791,·)$, $\chi_{3792}(1105,·)$, $\chi_{3792}(791,·)$, $\chi_{3792}(2843,·)$, $\chi_{3792}(157,·)$, $\chi_{3792}(1895,·)$, $\chi_{3792}(1897,·)$, $\chi_{3792}(2845,·)$, $\chi_{3792}(3635,·)$, $\chi_{3792}(947,·)$, $\chi_{3792}(949,·)$, $\chi_{3792}(3001,·)$, $\chi_{3792}(2687,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{10\!\cdots\!29}a^{14}-\frac{1}{15\!\cdots\!47}a^{13}+\frac{42\!\cdots\!80}{10\!\cdots\!29}a^{12}-\frac{43\!\cdots\!31}{10\!\cdots\!29}a^{11}+\frac{67\!\cdots\!57}{10\!\cdots\!29}a^{10}-\frac{18\!\cdots\!24}{10\!\cdots\!29}a^{9}-\frac{32\!\cdots\!24}{10\!\cdots\!29}a^{8}-\frac{49\!\cdots\!51}{15\!\cdots\!47}a^{7}+\frac{35\!\cdots\!23}{10\!\cdots\!29}a^{6}+\frac{43\!\cdots\!66}{10\!\cdots\!29}a^{5}+\frac{14\!\cdots\!26}{10\!\cdots\!29}a^{4}+\frac{32\!\cdots\!14}{10\!\cdots\!29}a^{3}+\frac{48\!\cdots\!41}{10\!\cdots\!29}a^{2}+\frac{39\!\cdots\!64}{10\!\cdots\!29}a-\frac{26\!\cdots\!04}{10\!\cdots\!29}$, $\frac{1}{59\!\cdots\!29}a^{15}+\frac{28402793}{59\!\cdots\!29}a^{14}+\frac{10\!\cdots\!47}{59\!\cdots\!29}a^{13}-\frac{41\!\cdots\!80}{85\!\cdots\!47}a^{12}+\frac{89\!\cdots\!07}{59\!\cdots\!29}a^{11}+\frac{96\!\cdots\!70}{59\!\cdots\!29}a^{10}+\frac{14\!\cdots\!80}{91\!\cdots\!31}a^{9}+\frac{22\!\cdots\!50}{59\!\cdots\!29}a^{8}-\frac{10\!\cdots\!49}{59\!\cdots\!29}a^{7}+\frac{17\!\cdots\!72}{85\!\cdots\!47}a^{6}-\frac{11\!\cdots\!60}{59\!\cdots\!29}a^{5}-\frac{16\!\cdots\!03}{59\!\cdots\!29}a^{4}-\frac{28\!\cdots\!14}{59\!\cdots\!29}a^{3}-\frac{14\!\cdots\!99}{59\!\cdots\!29}a^{2}+\frac{71\!\cdots\!98}{59\!\cdots\!29}a+\frac{23\!\cdots\!10}{59\!\cdots\!29}$
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
$C_{2}\times C_{4}\times C_{80}\times C_{4080}$, which has order $2611200$ (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: |
$\frac{37811933135700}{10\!\cdots\!29}a^{14}-\frac{37811933135700}{15\!\cdots\!47}a^{13}+\frac{64\!\cdots\!06}{10\!\cdots\!29}a^{12}-\frac{35\!\cdots\!36}{10\!\cdots\!29}a^{11}+\frac{48\!\cdots\!04}{10\!\cdots\!29}a^{10}-\frac{21\!\cdots\!90}{10\!\cdots\!29}a^{9}+\frac{21\!\cdots\!25}{10\!\cdots\!29}a^{8}-\frac{10\!\cdots\!08}{15\!\cdots\!47}a^{7}+\frac{64\!\cdots\!96}{10\!\cdots\!29}a^{6}-\frac{16\!\cdots\!44}{10\!\cdots\!29}a^{5}+\frac{13\!\cdots\!14}{10\!\cdots\!29}a^{4}-\frac{24\!\cdots\!88}{10\!\cdots\!29}a^{3}+\frac{18\!\cdots\!07}{10\!\cdots\!29}a^{2}-\frac{17\!\cdots\!38}{10\!\cdots\!29}a+\frac{13\!\cdots\!79}{10\!\cdots\!29}$, $\frac{56\!\cdots\!68}{59\!\cdots\!29}a^{15}-\frac{42\!\cdots\!60}{59\!\cdots\!29}a^{14}+\frac{96\!\cdots\!16}{59\!\cdots\!29}a^{13}-\frac{56\!\cdots\!94}{59\!\cdots\!29}a^{12}+\frac{71\!\cdots\!76}{59\!\cdots\!29}a^{11}-\frac{33\!\cdots\!76}{59\!\cdots\!29}a^{10}+\frac{30\!\cdots\!72}{59\!\cdots\!29}a^{9}-\frac{11\!\cdots\!65}{59\!\cdots\!29}a^{8}+\frac{84\!\cdots\!44}{59\!\cdots\!29}a^{7}-\frac{24\!\cdots\!24}{59\!\cdots\!29}a^{6}+\frac{15\!\cdots\!24}{59\!\cdots\!29}a^{5}-\frac{46\!\cdots\!58}{85\!\cdots\!47}a^{4}+\frac{18\!\cdots\!48}{59\!\cdots\!29}a^{3}-\frac{24\!\cdots\!84}{59\!\cdots\!29}a^{2}+\frac{11\!\cdots\!79}{59\!\cdots\!29}a-\frac{54\!\cdots\!09}{59\!\cdots\!29}$, $\frac{18\!\cdots\!34}{59\!\cdots\!29}a^{15}-\frac{13\!\cdots\!55}{59\!\cdots\!29}a^{14}+\frac{25\!\cdots\!67}{59\!\cdots\!29}a^{13}-\frac{14\!\cdots\!18}{59\!\cdots\!29}a^{12}+\frac{15\!\cdots\!59}{59\!\cdots\!29}a^{11}-\frac{68\!\cdots\!35}{59\!\cdots\!29}a^{10}+\frac{51\!\cdots\!18}{59\!\cdots\!29}a^{9}-\frac{18\!\cdots\!83}{59\!\cdots\!29}a^{8}+\frac{10\!\cdots\!39}{59\!\cdots\!29}a^{7}-\frac{29\!\cdots\!07}{59\!\cdots\!29}a^{6}+\frac{13\!\cdots\!40}{59\!\cdots\!29}a^{5}-\frac{38\!\cdots\!42}{85\!\cdots\!47}a^{4}+\frac{99\!\cdots\!97}{59\!\cdots\!29}a^{3}-\frac{12\!\cdots\!19}{59\!\cdots\!29}a^{2}+\frac{31\!\cdots\!07}{59\!\cdots\!29}a-\frac{13\!\cdots\!75}{59\!\cdots\!29}$, $\frac{25\!\cdots\!86}{83\!\cdots\!47}a^{15}-\frac{18\!\cdots\!45}{83\!\cdots\!47}a^{14}+\frac{35\!\cdots\!69}{83\!\cdots\!47}a^{13}-\frac{20\!\cdots\!16}{83\!\cdots\!47}a^{12}+\frac{21\!\cdots\!05}{83\!\cdots\!47}a^{11}-\frac{95\!\cdots\!73}{83\!\cdots\!47}a^{10}+\frac{72\!\cdots\!70}{83\!\cdots\!47}a^{9}-\frac{25\!\cdots\!64}{83\!\cdots\!47}a^{8}+\frac{15\!\cdots\!69}{83\!\cdots\!47}a^{7}-\frac{41\!\cdots\!33}{83\!\cdots\!47}a^{6}+\frac{19\!\cdots\!52}{83\!\cdots\!47}a^{5}-\frac{54\!\cdots\!00}{11\!\cdots\!21}a^{4}+\frac{14\!\cdots\!35}{83\!\cdots\!47}a^{3}-\frac{17\!\cdots\!29}{83\!\cdots\!47}a^{2}+\frac{45\!\cdots\!98}{83\!\cdots\!47}a-\frac{19\!\cdots\!65}{83\!\cdots\!47}$, $\frac{13\!\cdots\!96}{83\!\cdots\!47}a^{15}-\frac{10\!\cdots\!70}{83\!\cdots\!47}a^{14}+\frac{22\!\cdots\!90}{83\!\cdots\!47}a^{13}-\frac{13\!\cdots\!40}{83\!\cdots\!47}a^{12}+\frac{16\!\cdots\!28}{83\!\cdots\!47}a^{11}-\frac{77\!\cdots\!63}{83\!\cdots\!47}a^{10}+\frac{70\!\cdots\!19}{83\!\cdots\!47}a^{9}-\frac{25\!\cdots\!18}{83\!\cdots\!47}a^{8}+\frac{18\!\cdots\!20}{83\!\cdots\!47}a^{7}-\frac{53\!\cdots\!94}{83\!\cdots\!47}a^{6}+\frac{31\!\cdots\!06}{83\!\cdots\!47}a^{5}-\frac{92\!\cdots\!56}{11\!\cdots\!21}a^{4}+\frac{30\!\cdots\!43}{83\!\cdots\!47}a^{3}-\frac{39\!\cdots\!11}{83\!\cdots\!47}a^{2}+\frac{13\!\cdots\!70}{83\!\cdots\!47}a-\frac{54\!\cdots\!95}{83\!\cdots\!47}$, $\frac{18\!\cdots\!34}{59\!\cdots\!29}a^{15}-\frac{68\!\cdots\!53}{59\!\cdots\!29}a^{14}+\frac{20\!\cdots\!53}{59\!\cdots\!29}a^{13}-\frac{39\!\cdots\!47}{59\!\cdots\!29}a^{12}+\frac{96\!\cdots\!15}{59\!\cdots\!29}a^{11}+\frac{83\!\cdots\!92}{59\!\cdots\!29}a^{10}+\frac{22\!\cdots\!86}{59\!\cdots\!29}a^{9}+\frac{72\!\cdots\!81}{59\!\cdots\!29}a^{8}+\frac{22\!\cdots\!63}{59\!\cdots\!29}a^{7}+\frac{28\!\cdots\!34}{59\!\cdots\!29}a^{6}-\frac{73\!\cdots\!58}{59\!\cdots\!29}a^{5}+\frac{74\!\cdots\!57}{85\!\cdots\!47}a^{4}-\frac{36\!\cdots\!39}{59\!\cdots\!29}a^{3}+\frac{49\!\cdots\!03}{59\!\cdots\!29}a^{2}-\frac{23\!\cdots\!13}{59\!\cdots\!29}a+\frac{19\!\cdots\!43}{59\!\cdots\!29}$, $\frac{56\!\cdots\!68}{59\!\cdots\!29}a^{15}-\frac{21\!\cdots\!60}{59\!\cdots\!29}a^{14}+\frac{81\!\cdots\!16}{59\!\cdots\!29}a^{13}-\frac{19\!\cdots\!88}{59\!\cdots\!29}a^{12}+\frac{51\!\cdots\!40}{59\!\cdots\!29}a^{11}-\frac{56\!\cdots\!72}{59\!\cdots\!29}a^{10}+\frac{18\!\cdots\!82}{59\!\cdots\!29}a^{9}+\frac{95\!\cdots\!60}{59\!\cdots\!29}a^{8}+\frac{41\!\cdots\!88}{59\!\cdots\!29}a^{7}+\frac{12\!\cdots\!72}{59\!\cdots\!29}a^{6}+\frac{82\!\cdots\!40}{85\!\cdots\!47}a^{5}+\frac{45\!\cdots\!08}{59\!\cdots\!29}a^{4}+\frac{45\!\cdots\!60}{59\!\cdots\!29}a^{3}+\frac{82\!\cdots\!23}{59\!\cdots\!29}a^{2}+\frac{22\!\cdots\!63}{85\!\cdots\!47}a+\frac{65\!\cdots\!41}{59\!\cdots\!29}$
| 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: | \( 11964.310642723332 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 11964.310642723332 \cdot 2611200}{2\cdot\sqrt{2801731803266966736034164664958976}}\cr\approx \mathstrut & 0.716842075968216 \end{aligned}\] (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 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717223*x^8 - 46483976*x^7 + 257685716*x^6 - 618838472*x^5 + 2865167258*x^4 - 4749147848*x^3 + 18507674124*x^2 - 16229095208*x + 53156046238)
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 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717223*x^8 - 46483976*x^7 + 257685716*x^6 - 618838472*x^5 + 2865167258*x^4 - 4749147848*x^3 + 18507674124*x^2 - 16229095208*x + 53156046238, 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 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717223*x^8 - 46483976*x^7 + 257685716*x^6 - 618838472*x^5 + 2865167258*x^4 - 4749147848*x^3 + 18507674124*x^2 - 16229095208*x + 53156046238);
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 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717223*x^8 - 46483976*x^7 + 257685716*x^6 - 618838472*x^5 + 2865167258*x^4 - 4749147848*x^3 + 18507674124*x^2 - 16229095208*x + 53156046238);
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
$C_2^2\times C_4$ (as 16T2):
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 $C_4\times C_2^2$ |
| Character table for $C_4\times C_2^2$ |
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 | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.24.9 | $x^{8} + 8 x^{6} + 14 x^{4} + 4 x^{2} + 8 x + 30$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $[2, 3, 4]$ |
| 2.8.24.9 | $x^{8} + 8 x^{6} + 14 x^{4} + 4 x^{2} + 8 x + 30$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $[2, 3, 4]$ | |
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(79\)
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |